Theorem itg2addnclem 32631

Description: An alternate expression for the ∫₂ integral that includes an arbitrarily small but strictly positive "buffer zone" wherever the simple function is nonzero. (Contributed by Brendan Leahy, 10-Oct-2017.) (Revised by Brendan Leahy, 10-Mar-2018.)

Hypothesis

Ref	Expression
itg2addnclem.1	⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}

Assertion

Ref	Expression
itg2addnclem	⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < ))

Distinct variable group:   𝑥,𝑦,𝑧,𝑔,𝐹

Allowed substitution hints:   𝐿(𝑥,𝑦,𝑧,𝑔)

Proof of Theorem itg2addnclem

Dummy variables 𝑠 𝑢 𝑓 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. . 3 ⊢ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} = {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}
2	1	itg2val 23301	. 2 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
3		itg2addnclem.1	. . . 4 ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}
4	3	supeq1i 8236	. . 3 ⊢ sup(𝐿, ℝ*, < ) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )
5		xrltso 11850	. . . . 5 ⊢ < Or ℝ*
6	5	a1i 11	. . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → < Or ℝ*)
7		simprr 792	. . . . . . . 8 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 = (∫1‘𝑓))
8		itg1cl 23258	. . . . . . . . . 10 ⊢ (𝑓 ∈ dom ∫1 → (∫1‘𝑓) ∈ ℝ)
9	8	rexrd 9968	. . . . . . . . 9 ⊢ (𝑓 ∈ dom ∫1 → (∫1‘𝑓) ∈ ℝ*)
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → (∫1‘𝑓) ∈ ℝ*)
11	7, 10	eqeltrd 2688	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 ∈ ℝ*)
12	11	rexlimiva 3010	. . . . . 6 ⊢ (∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) → 𝑥 ∈ ℝ*)
13	12	abssi 3640	. . . . 5 ⊢ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ*
14		supxrcl 12017	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ* → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ∈ ℝ*)
15	13, 14	mp1i 13	. . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ∈ ℝ*)
16		fveq1 6102	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑧) = (𝑓‘𝑧))
17	16	eqeq1d 2612	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((𝑔‘𝑧) = 0 ↔ (𝑓‘𝑧) = 0))
18	16	oveq1d 6564	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((𝑔‘𝑧) + 𝑦) = ((𝑓‘𝑧) + 𝑦))
19	17, 18	ifbieq2d 4061	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦)) = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)))
20	19	mpteq2dv 4673	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))))
21	20	breq1d 4593	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹))
22	21	rexbidv 3034	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹))
23		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (∫1‘𝑔) = (∫1‘𝑓))
24	23	eqeq2d 2620	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (𝑥 = (∫1‘𝑔) ↔ 𝑥 = (∫1‘𝑓)))
25	22, 24	anbi12d 743	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))))
26	25	cbvrexv 3148	. . . . . . . . 9 ⊢ (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑓 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)))
27		breq2 4587	. . . . . . . . . . . . . . . . 17 ⊢ (0 = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) → ((𝑓‘𝑧) ≤ 0 ↔ (𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))))
28		breq2 4587	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓‘𝑧) + 𝑦) = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) → ((𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑦) ↔ (𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))))
29		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑧) = 0 → (𝑓‘𝑧) = 0)
30		0le0 10987	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ≤ 0
31	29, 30	syl6eqbr 4622	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑧) = 0 → (𝑓‘𝑧) ≤ 0)
32	31	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) → (𝑓‘𝑧) ≤ 0)
33		rpge0 11721	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
34	33	ad2antlr 759	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 0 ≤ 𝑦)
35		i1ff 23249	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∈ dom ∫1 → 𝑓:ℝ⟶ℝ)
36	35	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
37	36	adantlr 747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
38		rpre 11715	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
39	38	ad2antlr 759	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
40	37, 39	addge01d 10494	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (0 ≤ 𝑦 ↔ (𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑦)))
41	34, 40	mpbid 221	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑦))
42	41	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓‘𝑧) = 0) → (𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑦))
43	27, 28, 32, 42	ifbothda 4073	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)))
44	43	adantlll 750	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)))
45	35	ad2antlr 759	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝑓:ℝ⟶ℝ)
46	45	ffvelrnda 6267	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
47	46	rexrd 9968	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ*)
48		0re 9919	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
49	38	ad2antlr 759	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
50	46, 49	readdcld 9948	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + 𝑦) ∈ ℝ)
51		ifcl 4080	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ ((𝑓‘𝑧) + 𝑦) ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∈ ℝ)
52	48, 50, 51	sylancr 694	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∈ ℝ)
53	52	rexrd 9968	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∈ ℝ*)
54		iccssxr 12127	. . . . . . . . . . . . . . . . . . 19 ⊢ (0[,]+∞) ⊆ ℝ*
55		fss 5969	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐹:ℝ⟶ℝ*)
56	54, 55	mpan2 703	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹:ℝ⟶ℝ*)
57	56	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝐹:ℝ⟶ℝ*)
58	57	ffvelrnda 6267	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ ℝ*)
59		xrletr 11865	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑧) ∈ ℝ* ∧ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∈ ℝ* ∧ (𝐹‘𝑧) ∈ ℝ*) → (((𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∧ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧)) → (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
60	47, 53, 58, 59	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∧ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧)) → (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
61	44, 60	mpand 707	. . . . . . . . . . . . . 14 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧) → (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
62	61	ralimdva 2945	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → (∀𝑧 ∈ ℝ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧) → ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
63		reex 9906	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
64	63	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → ℝ ∈ V)
65		eqidd 2611	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))))
66		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹:ℝ⟶(0[,]+∞))
67	66	feqmptd 6159	. . . . . . . . . . . . . . 15 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹‘𝑧)))
68	67	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹‘𝑧)))
69	64, 52, 58, 65, 68	ofrfval2 6813	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧)))
70	35	feqmptd 6159	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 = (𝑧 ∈ ℝ ↦ (𝑓‘𝑧)))
71	70	ad2antlr 759	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝑓 = (𝑧 ∈ ℝ ↦ (𝑓‘𝑧)))
72	64, 46, 58, 71, 68	ofrfval2 6813	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → (𝑓 ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
73	62, 69, 72	3imtr4d 282	. . . . . . . . . . . 12 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 → 𝑓 ∘𝑟 ≤ 𝐹))
74	73	rexlimdva 3013	. . . . . . . . . . 11 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 → 𝑓 ∘𝑟 ≤ 𝐹))
75	74	anim1d 586	. . . . . . . . . 10 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) → (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))))
76	75	reximdva 3000	. . . . . . . . 9 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∃𝑓 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) → ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))))
77	26, 76	syl5bi 231	. . . . . . . 8 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) → ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))))
78	77	ss2abdv 3638	. . . . . . 7 ⊢ (𝐹:ℝ⟶(0[,]+∞) → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⊆ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))})
79	78	sseld 3567	. . . . . 6 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} → 𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}))
80		simp3r 1083	. . . . . . . . . . 11 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 = (∫1‘𝑓))
81	9	3ad2ant2 1076	. . . . . . . . . . 11 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → (∫1‘𝑓) ∈ ℝ*)
82	80, 81	eqeltrd 2688	. . . . . . . . . 10 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 ∈ ℝ*)
83	82	rexlimdv3a 3015	. . . . . . . . 9 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) → 𝑥 ∈ ℝ*))
84	83	abssdv 3639	. . . . . . . 8 ⊢ (𝐹:ℝ⟶(0[,]+∞) → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ*)
85		xrsupss 12011	. . . . . . . 8 ⊢ ({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ* → ∃𝑎 ∈ ℝ* (∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ* (𝑏 < 𝑎 → ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠)))
86	84, 85	syl 17	. . . . . . 7 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∃𝑎 ∈ ℝ* (∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ* (𝑏 < 𝑎 → ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠)))
87	6, 86	supub 8248	. . . . . 6 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} → ¬ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) < 𝑏))
88	79, 87	syld 46	. . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} → ¬ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) < 𝑏))
89	88	imp 444	. . . 4 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}) → ¬ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) < 𝑏)
90		supxrlub 12027	. . . . . . . 8 ⊢ (({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ* ∧ 𝑏 ∈ ℝ*) → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ↔ ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠))
91	13, 90	mpan 702	. . . . . . 7 ⊢ (𝑏 ∈ ℝ* → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ↔ ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠))
92	91	adantl 481	. . . . . 6 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ↔ ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠))
93		simprrr 801	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → 𝑠 = (∫1‘𝑓))
94	93	breq2d 4595	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑏 < 𝑠 ↔ 𝑏 < (∫1‘𝑓)))
95		simplll 794	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → 𝐹:ℝ⟶(0[,]+∞))
96		i1f0 23260	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × {0}) ∈ dom ∫1
97		2rp 11713	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ+
98	97	ne0ii 3882	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ+ ≠ ∅
99		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ (0[,]+∞))
100		elxrge0 12152	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑧) ∈ (0[,]+∞) ↔ ((𝐹‘𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑧)))
101	99, 100	sylib 207	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑧)))
102	101	simprd 478	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐹‘𝑧))
103	102	ralrimiva 2949	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∀𝑧 ∈ ℝ 0 ≤ (𝐹‘𝑧))
104	63	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ℝ ∈ V)
105		c0ex 9913	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ V
106	105	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → 0 ∈ V)
107		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦ 0))
108	104, 106, 99, 107, 67	ofrfval2 6813	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ((𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐹‘𝑧)))
109	103, 108	mpbird 246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹)
110	109	ralrimivw 2950	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∀𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹)
111		r19.2z 4012	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℝ+ ≠ ∅ ∧ ∀𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹)
112	98, 110, 111	sylancr 694	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹)
113		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 = (ℝ × {0}) → (∫1‘𝑔) = (∫1‘(ℝ × {0})))
114		itg10 23261	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∫1‘(ℝ × {0})) = 0
115	113, 114	syl6req 2661	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 = (ℝ × {0}) → 0 = (∫1‘𝑔))
116	115	biantrud 527	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = (ℝ × {0}) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔))))
117		fveq1 6102	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔 = (ℝ × {0}) → (𝑔‘𝑧) = ((ℝ × {0})‘𝑧))
118	105	fvconst2 6374	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ ℝ → ((ℝ × {0})‘𝑧) = 0)
119	117, 118	sylan9eq 2664	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) = 0)
120		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔‘𝑧) = 0 → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦)) = 0)
121	119, 120	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦)) = 0)
122	121	mpteq2dva 4672	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ 0))
123	122	breq1d 4593	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹))
124	123	rexbidv 3034	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = (ℝ × {0}) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹))
125	116, 124	bitr3d 269	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = (ℝ × {0}) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔)) ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹))
126	125	rspcev 3282	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ℝ × {0}) ∈ dom ∫1 ∧ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔)))
127	96, 112, 126	sylancr 694	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔)))
128		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = -∞ → 𝑏 = -∞)
129		mnflt 11833	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ ℝ → -∞ < 0)
130	48, 129	mp1i 13	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = -∞ → -∞ < 0)
131	128, 130	eqbrtrd 4605	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = -∞ → 𝑏 < 0)
132		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 0 → (𝑎 = (∫1‘𝑔) ↔ 0 = (∫1‘𝑔)))
133	132	anbi2d 736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 0 → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔))))
134	133	rexbidv 3034	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 0 → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔))))
135		breq2 4587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 0 → (𝑏 < 𝑎 ↔ 𝑏 < 0))
136	134, 135	anbi12d 743	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 0 → ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎) ↔ (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔)) ∧ 𝑏 < 0)))
137	105, 136	spcev 3273	. . . . . . . . . . . . . . . . . 18 ⊢ ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔)) ∧ 𝑏 < 0) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
138	127, 131, 137	syl2an 493	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 = -∞) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
139	95, 138	sylan 487	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 = -∞) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
140		simp-4r 803	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ∈ ℝ*)
141	8	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → (∫1‘𝑓) ∈ ℝ)
142	141	ad3antlr 763	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → (∫1‘𝑓) ∈ ℝ)
143		simpllr 795	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → 𝑏 ∈ ℝ*)
144		ngtmnft 11872	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 ∈ ℝ* → (𝑏 = -∞ ↔ ¬ -∞ < 𝑏))
145	144	biimprd 237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 ∈ ℝ* → (¬ -∞ < 𝑏 → 𝑏 = -∞))
146	145	necon1ad 2799	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ℝ* → (𝑏 ≠ -∞ → -∞ < 𝑏))
147	146	imp 444	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℝ* ∧ 𝑏 ≠ -∞) → -∞ < 𝑏)
148	143, 147	sylan 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → -∞ < 𝑏)
149		simpr 476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → 𝑏 ∈ ℝ*)
150	9	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → (∫1‘𝑓) ∈ ℝ*)
151	149, 150	anim12i 588	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑏 ∈ ℝ* ∧ (∫1‘𝑓) ∈ ℝ*))
152		xrltle 11858	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℝ* ∧ (∫1‘𝑓) ∈ ℝ*) → (𝑏 < (∫1‘𝑓) → 𝑏 ≤ (∫1‘𝑓)))
153	152	imp 444	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑏 ∈ ℝ* ∧ (∫1‘𝑓) ∈ ℝ*) ∧ 𝑏 < (∫1‘𝑓)) → 𝑏 ≤ (∫1‘𝑓))
154	151, 153	sylan 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → 𝑏 ≤ (∫1‘𝑓))
155	154	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ≤ (∫1‘𝑓))
156		xrre 11874	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑏 ∈ ℝ* ∧ (∫1‘𝑓) ∈ ℝ) ∧ (-∞ < 𝑏 ∧ 𝑏 ≤ (∫1‘𝑓))) → 𝑏 ∈ ℝ)
157	140, 142, 148, 155, 156	syl22anc 1319	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ∈ ℝ)
158	127	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔)))
159		simplrl 796	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑏 < (∫1‘𝑓))
160		simplrl 796	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑓 ∈ dom ∫1)
161		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑓 ∈ dom ∫1)
162		cnvimass 5404	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ dom 𝑓
163		fdm 5964	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓:ℝ⟶ℝ → dom 𝑓 = ℝ)
164	35, 163	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 ∈ dom ∫1 → dom 𝑓 = ℝ)
165	162, 164	syl5sseq 3616	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 ∈ dom ∫1 → (◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
166	165	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
167		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0)
168	163	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓:ℝ⟶ℝ → ℝ = dom 𝑓)
169		ffun 5961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑓:ℝ⟶ℝ → Fun 𝑓)
170		difpreima 6251	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (Fun 𝑓 → (◡𝑓 “ (ran 𝑓 ∖ {0})) = ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})))
171	169, 170	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓:ℝ⟶ℝ → (◡𝑓 “ (ran 𝑓 ∖ {0})) = ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})))
172		cnvimarndm 5405	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (◡𝑓 “ ran 𝑓) = dom 𝑓
173	172	difeq1i 3686	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})) = (dom 𝑓 ∖ (◡𝑓 “ {0}))
174	171, 173	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓:ℝ⟶ℝ → (◡𝑓 “ (ran 𝑓 ∖ {0})) = (dom 𝑓 ∖ (◡𝑓 “ {0})))
175	168, 174	difeq12d 3691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓:ℝ⟶ℝ → (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) = (dom 𝑓 ∖ (dom 𝑓 ∖ (◡𝑓 “ {0}))))
176		cnvimass 5404	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (◡𝑓 “ {0}) ⊆ dom 𝑓
177		dfss4 3820	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((◡𝑓 “ {0}) ⊆ dom 𝑓 ↔ (dom 𝑓 ∖ (dom 𝑓 ∖ (◡𝑓 “ {0}))) = (◡𝑓 “ {0}))
178	176, 177	mpbi 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (dom 𝑓 ∖ (dom 𝑓 ∖ (◡𝑓 “ {0}))) = (◡𝑓 “ {0})
179	175, 178	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓:ℝ⟶ℝ → (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) = (◡𝑓 “ {0}))
180	179	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ 𝑧 ∈ (◡𝑓 “ {0})))
181		ffn 5958	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓:ℝ⟶ℝ → 𝑓 Fn ℝ)
182		fniniseg 6246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 Fn ℝ → (𝑧 ∈ (◡𝑓 “ {0}) ↔ (𝑧 ∈ ℝ ∧ (𝑓‘𝑧) = 0)))
183		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑧 ∈ ℝ ∧ (𝑓‘𝑧) = 0) → (𝑓‘𝑧) = 0)
184	182, 183	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 Fn ℝ → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
185	181, 184	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
186	180, 185	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓‘𝑧) = 0))
187	35, 186	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 ∈ dom ∫1 → (𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓‘𝑧) = 0))
188	187	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (𝑓‘𝑧) = 0)
189	188	adantlr 747	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) ∧ 𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (𝑓‘𝑧) = 0)
190	161, 166, 167, 189	itg10a 23283	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (∫1‘𝑓) = 0)
191	160, 190	sylan 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (∫1‘𝑓) = 0)
192	159, 191	breqtrd 4609	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑏 < 0)
193	158, 192, 137	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
194		simprl 790	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → 𝑓 ∈ dom ∫1)
195		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
196	194, 195	anim12i 588	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → (𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ))
197	63	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ℝ ∈ V)
198		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓‘𝑢) ∈ V
199	198	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑓‘𝑢) ∈ V)
200		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V
201	200, 105	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V
202	201	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V)
203	35	feqmptd 6159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 = (𝑢 ∈ ℝ ↦ (𝑓‘𝑢)))
204	203	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 = (𝑢 ∈ ℝ ↦ (𝑓‘𝑢)))
205		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
206	197, 199, 202, 204, 205	offval2 6812	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑓 − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑢 ∈ ℝ ↦ ((𝑓‘𝑢) − if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))))
207		ovif2 6636	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓‘𝑢) − if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑢) − 0))
208	172, 163	syl5eq 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓:ℝ⟶ℝ → (◡𝑓 “ ran 𝑓) = ℝ)
209	208	difeq1d 3689	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑓:ℝ⟶ℝ → ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})) = (ℝ ∖ (◡𝑓 “ {0})))
210	171, 209	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑓:ℝ⟶ℝ → (◡𝑓 “ (ran 𝑓 ∖ {0})) = (ℝ ∖ (◡𝑓 “ {0})))
211	210	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑓:ℝ⟶ℝ → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0}))))
212	35, 211	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑓 ∈ dom ∫1 → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0}))))
213	212	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0}))))
214		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
215	214	biantrurd 528	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (◡𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ ¬ 𝑢 ∈ (◡𝑓 “ {0}))))
216		eldif 3550	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0})) ↔ (𝑢 ∈ ℝ ∧ ¬ 𝑢 ∈ (◡𝑓 “ {0})))
217	215, 216	syl6bbr 277	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (◡𝑓 “ {0}) ↔ 𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0}))))
218	213, 217	bitr4d 270	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ 𝑢 ∈ (◡𝑓 “ {0})))
219	218	con2bid 343	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (◡𝑓 “ {0}) ↔ ¬ 𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))))
220		fniniseg 6246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑓 Fn ℝ → (𝑢 ∈ (◡𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0)))
221	35, 181, 220	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑓 ∈ dom ∫1 → (𝑢 ∈ (◡𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0)))
222	221	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (◡𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0)))
223	219, 222	bitr3d 269	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ (𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0)))
224		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑓‘𝑢) = 0 → ((𝑓‘𝑢) − 0) = (0 − 0))
225		0m0e0 11007	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (0 − 0) = 0
226	224, 225	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑓‘𝑢) = 0 → ((𝑓‘𝑢) − 0) = 0)
227	226	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0) → ((𝑓‘𝑢) − 0) = 0)
228	223, 227	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → ((𝑓‘𝑢) − 0) = 0))
229	228	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) ∧ ¬ 𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑢) − 0) = 0)
230	229	ifeq2da 4067	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑢) − 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
231	207, 230	syl5eq 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → ((𝑓‘𝑢) − if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
232	231	mpteq2dva 4672	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ ((𝑓‘𝑢) − if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
233	206, 232	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑓 − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
234		simpll 786	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 ∈ dom ∫1)
235	200	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V)
236		1ex 9914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 1 ∈ V
237	236, 105	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V
238	237	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V)
239		fconstmpt 5085	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑢 ∈ ℝ ↦ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
240	239	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑢 ∈ ℝ ↦ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
241		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))
242	197, 235, 238, 240, 241	offval2 6812	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑢 ∈ ℝ ↦ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))))
243		ovif2 6636	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0))
244		resubcl 10224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((∫1‘𝑓) ∈ ℝ ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
245	8, 244	sylan 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
246	245	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
247		2re 10967	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ 2 ∈ ℝ
248		i1fima 23251	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑓 ∈ dom ∫1 → (◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol)
249		mblvol 23105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
250	248, 249	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑓 ∈ dom ∫1 → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
251		neldifsn 4262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ¬ 0 ∈ (ran 𝑓 ∖ {0})
252		i1fima2 23252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑓 ∈ dom ∫1 ∧ ¬ 0 ∈ (ran 𝑓 ∖ {0})) → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
253	251, 252	mpan2 703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑓 ∈ dom ∫1 → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
254	250, 253	eqeltrrd 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑓 ∈ dom ∫1 → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
255		remulcl 9900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((2 ∈ ℝ ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
256	247, 254, 255	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑓 ∈ dom ∫1 → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
257	256	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
258		2cnd 10970	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 2 ∈ ℂ)
259	254	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
260	259	recnd 9947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
261		2ne0 10990	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ 2 ≠ 0
262	261	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 2 ≠ 0)
263		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0)
264	258, 260, 262, 263	mulne0d 10558	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ≠ 0)
265	246, 257, 264	redivcld 10732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ)
266	265	recnd 9947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
267	266	mulid1d 9936	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1) = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
268	266	mul01d 10114	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0) = 0)
269	267, 268	ifeq12d 4056	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
270	243, 269	syl5eq 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
271	270	mpteq2dv 4673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
272	242, 271	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
273		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))
274	273	i1f1 23263	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol ∧ (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
275	248, 253, 274	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 ∈ dom ∫1 → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
276	275	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
277	276, 265	i1fmulc 23276	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) ∈ dom ∫1)
278	272, 277	eqeltrrd 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1)
279		i1fsub 23281	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1) → (𝑓 ∘𝑓 − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
280	234, 278, 279	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑓 − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
281	233, 280	eqeltrrd 2689	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
282		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑢 = 𝑧 → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))))
283		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
284	283	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑢 = 𝑧 → ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
285	282, 284	ifbieq1d 4059	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑢 = 𝑧 → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
286		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
287		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ V
288	287, 105	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V
289	285, 286, 288	fvmpt 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 ∈ ℝ → ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
290	289	breq2d 4595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ ℝ → (0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) ↔ 0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
291	290, 289	ifbieq1d 4059	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ ℝ → if(0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
292		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
293		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
294	293	breq2d 4595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
295	294, 293	ifbieq1d 4059	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
296		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
297	295, 296	sylan9eqr 2666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
298	292, 297	eqtr4d 2647	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
299		iffalse 4045	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
300		ianor 508	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∨ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))))
301	294	ifbid 4058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
302		iffalse 4045	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
303	301, 302	sylan9eqr 2666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
304	303	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0))
305		iffalse 4045	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
306		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 0 = 0
307		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) → (if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 ↔ if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0))
308		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (0 = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) → (0 = 0 ↔ if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0))
309	307, 308	ifboth 4074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 ∧ 0 = 0) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
310	305, 306, 309	sylancl 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
311	304, 310	pm2.61d1 170	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
312	311, 310	jaoi 393	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∨ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
313	300, 312	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
314	299, 313	eqtr4d 2647	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
315	298, 314	pm2.61i 175	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0)
316	291, 315	syl6reqr 2663	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ ℝ → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), 0))
317	316	mpteq2ia 4668	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑧 ∈ ℝ ↦ if(0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), 0))
318	317	i1fpos 23279	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
319	281, 318	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
320	196, 319	sylan 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
321	196, 265	sylan 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ)
322	8	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (∫1‘𝑓) ∈ ℝ)
323	322, 195, 244	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
324	323	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
325	256	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
326	325	ad3antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
327		simprl 790	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑏 < (∫1‘𝑓))
328		simprr 792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑏 ∈ ℝ)
329	141	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → (∫1‘𝑓) ∈ ℝ)
330	328, 329	posdifd 10493	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → (𝑏 < (∫1‘𝑓) ↔ 0 < ((∫1‘𝑓) − 𝑏)))
331	327, 330	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → 0 < ((∫1‘𝑓) − 𝑏))
332	331	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < ((∫1‘𝑓) − 𝑏))
333	254	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
334	333	ad3antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
335		mblss 23106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol → (◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
336		ovolge0 23056	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ → 0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
337	248, 335, 336	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 ∈ dom ∫1 → 0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
338		ltlen 10017	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((0 ∈ ℝ ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ (0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0)))
339	48, 254, 338	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → (0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ (0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0)))
340	339	biimprd 237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 ∈ dom ∫1 → ((0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
341	337, 340	mpand 707	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 ∈ dom ∫1 → ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0 → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
342	341	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0 → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
343	342	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
344	343	adantlr 747	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
345		2pos 10989	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 < 2
346		mulgt0 9994	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((2 ∈ ℝ ∧ 0 < 2) ∧ ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ ∧ 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → 0 < (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
347	247, 345, 346	mpanl12 714	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ ∧ 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) → 0 < (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
348	334, 344, 347	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
349	324, 326, 332, 348	divgt0d 10838	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
350	321, 349	elrpd 11745	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ+)
351		simprl 790	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → 𝑓 ∘𝑟 ≤ 𝐹)
352	351	ad3antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 ∘𝑟 ≤ 𝐹)
353		ffn 5958	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹 Fn ℝ)
354	35, 181	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 Fn ℝ)
355	354	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → 𝑓 Fn ℝ)
356		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → 𝑓 Fn ℝ)
357		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → 𝐹 Fn ℝ)
358	63	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → ℝ ∈ V)
359		inidm 3784	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (ℝ ∩ ℝ) = ℝ
360		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) = (𝑓‘𝑧))
361		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧))
362	356, 357, 358, 358, 359, 360, 361	ofrfval 6803	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → (𝑓 ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
363	353, 355, 362	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑓 ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
364	363	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
365		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → 𝑓 ∈ dom ∫1)
366	365	anim2i 591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1))
367	366, 195	anim12i 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ))
368		breq1 4586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0 = if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → (0 ≤ (𝐹‘𝑧) ↔ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
369		breq1 4586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧) ↔ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
370		simplll 794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝐹:ℝ⟶(0[,]+∞))
371	370	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ (0[,]+∞))
372	371, 100	sylib 207	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑧)))
373	372	simprd 478	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐹‘𝑧))
374	373	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) ∧ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) → 0 ≤ (𝐹‘𝑧))
375		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
376	375	breq1d 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → ((((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧) ↔ (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧)))
377		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (0 = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → (0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
378	377	breq1d 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0 = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → ((0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧) ↔ (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧)))
379	35	ad3antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓:ℝ⟶ℝ)
380	379	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
381	380	recnd 9947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℂ)
382	245	recnd 9947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) − 𝑏) ∈ ℂ)
383	382	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − 𝑏) ∈ ℂ)
384	256	recnd 9947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓 ∈ dom ∫1 → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℂ)
385	384	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℂ)
386	383, 385, 264	divcld 10680	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
387	386	adantlll 750	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
388	387	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
389	381, 388	npcand 10275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (𝑓‘𝑧))
390	389	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (𝑓‘𝑧))
391		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) → (𝑓‘𝑧) ≤ (𝐹‘𝑧))
392	390, 391	eqbrtrd 4605	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
393	392	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) ∧ ¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) ∧ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
394	299	pm2.24d 146	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 → (0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧)))
395	394	impcom 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 ∧ ¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
396	395	adantll 746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) ∧ ¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) ∧ ¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
397	376, 378, 393, 396	ifbothda 4073	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) ∧ ¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) → (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
398	368, 369, 374, 397	ifbothda 4073	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧))
399	398	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) ≤ (𝐹‘𝑧) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
400	367, 399	sylanl1 680	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) ≤ (𝐹‘𝑧) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
401	400	ralimdva 2945	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧) → ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
402	364, 401	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑟 ≤ 𝐹 → ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
403	352, 402	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧))
404		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ V
405	105, 404	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ∈ V
406	405	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ∈ V)
407		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) = (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))))
408	104, 406, 99, 407, 67	ofrfval2 6813	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ((𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
409	408	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
410	403, 409	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟 ≤ 𝐹)
411		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦) = (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
412	411	ifeq2d 4055	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦)) = if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
413	412	mpteq2dv 4673	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))))
414	413	breq1d 4593	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → ((𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟 ≤ 𝐹))
415	414	rspcev 3282	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ+ ∧ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟 ≤ 𝐹) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟 ≤ 𝐹)
416	350, 410, 415	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟 ≤ 𝐹)
417		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = 𝑔 → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))
418	417	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))
419	418	biantrud 527	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))))
420		nfmpt1 4675	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ Ⅎ𝑧(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
421	420	nfeq2 2766	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑧 𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
422		fveq1 6102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (𝑔‘𝑧) = ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧))
423	287, 105	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V
424		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
425	424	fvmpt2 6200	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑧 ∈ ℝ ∧ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V) → ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
426	423, 425	mpan2 703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 ∈ ℝ → ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
427	422, 426	sylan9eq 2664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
428	427	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) = 0 ↔ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0))
429	427	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) + 𝑦) = (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))
430	428, 429	ifbieq2d 4061	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦)) = if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦)))
431	421, 430	mpteq2da 4671	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))))
432	431	breq1d 4593	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟 ≤ 𝐹))
433	432	rexbidv 3034	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟 ≤ 𝐹))
434	419, 433	bitr3d 269	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)) ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟 ≤ 𝐹))
435	434	rspcev 3282	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 ∧ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟 ≤ 𝐹) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)))
436	320, 416, 435	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)))
437		simplrr 797	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 ∈ ℝ)
438	200	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V)
439	236, 105	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V
440	439	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V)
441		fconstmpt 5085	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑧 ∈ ℝ ↦ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
442	441	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑧 ∈ ℝ ↦ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
443		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))
444	197, 438, 440, 442, 443	offval2 6812	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑧 ∈ ℝ ↦ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))))
445		ovif2 6636	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0))
446	267, 268	ifeq12d 4056	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0)) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
447	445, 446	syl5eq 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
448	447	mpteq2dv 4673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
449	444, 448	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
450		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))
451	450	i1f1 23263	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol ∧ (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
452	248, 253, 451	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 ∈ dom ∫1 → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
453	452	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
454	453, 265	i1fmulc 23276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) ∈ dom ∫1)
455	449, 454	eqeltrrd 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1)
456		i1fsub 23281	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1) → (𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
457	234, 455, 456	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
458		itg1cl 23258	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1 → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ∈ ℝ)
459	457, 458	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ∈ ℝ)
460	459	adantlrl 752	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ∈ ℝ)
461	319	adantlrl 752	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
462		itg1cl 23258	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) ∈ ℝ)
463	461, 462	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) ∈ ℝ)
464		simplrl 796	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘𝑓))
465		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
466	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (∫1‘𝑓) ∈ ℝ)
467	97	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → 2 ∈ ℝ+)
468	465, 466, 467	ltdiv1d 11793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (𝑏 < (∫1‘𝑓) ↔ (𝑏 / 2) < ((∫1‘𝑓) / 2)))
469		recn 9905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑏 ∈ ℝ → 𝑏 ∈ ℂ)
470	469	2halvesd 11155	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑏 ∈ ℝ → ((𝑏 / 2) + (𝑏 / 2)) = 𝑏)
471	470	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑏 ∈ ℝ → (((𝑏 / 2) + (𝑏 / 2)) − (𝑏 / 2)) = (𝑏 − (𝑏 / 2)))
472	469	halfcld 11154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑏 ∈ ℝ → (𝑏 / 2) ∈ ℂ)
473	472, 472	pncand 10272	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑏 ∈ ℝ → (((𝑏 / 2) + (𝑏 / 2)) − (𝑏 / 2)) = (𝑏 / 2))
474	471, 473	eqtr3d 2646	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 ∈ ℝ → (𝑏 − (𝑏 / 2)) = (𝑏 / 2))
475	474	breq1d 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑏 ∈ ℝ → ((𝑏 − (𝑏 / 2)) < ((∫1‘𝑓) / 2) ↔ (𝑏 / 2) < ((∫1‘𝑓) / 2)))
476	475	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((𝑏 − (𝑏 / 2)) < ((∫1‘𝑓) / 2) ↔ (𝑏 / 2) < ((∫1‘𝑓) / 2)))
477		rehalfcl 11135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 ∈ ℝ → (𝑏 / 2) ∈ ℝ)
478	477	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (𝑏 / 2) ∈ ℝ)
479	8	rehalfcld 11156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → ((∫1‘𝑓) / 2) ∈ ℝ)
480	479	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) / 2) ∈ ℝ)
481	465, 478, 480	ltsubaddd 10502	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((𝑏 − (𝑏 / 2)) < ((∫1‘𝑓) / 2) ↔ 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2))))
482	468, 476, 481	3bitr2d 295	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (𝑏 < (∫1‘𝑓) ↔ 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2))))
483	482	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑏 < (∫1‘𝑓) ↔ 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2))))
484	483	adantlrl 752	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑏 < (∫1‘𝑓) ↔ 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2))))
485	464, 484	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2)))
486	453, 265	itg1mulc 23277	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))))
487	449	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))))
488	450	itg11 23264	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol ∧ (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
489	248, 253, 488	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 ∈ dom ∫1 → (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
490	489	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
491	490	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
492	253	recnd 9947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 ∈ dom ∫1 → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
493	492	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
494	266, 493	mulcomd 9940	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) = ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
495	250	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
496	495	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) = ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)))
497	260, 383	mulcomd 9940	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) = (((∫1‘𝑓) − 𝑏) · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
498	496, 497	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) = (((∫1‘𝑓) − 𝑏) · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
499	498	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) = ((((∫1‘𝑓) − 𝑏) · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
500	493, 383, 385, 264	divassd 10715	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) = ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
501	383, 258, 260, 262, 263	divcan5rd 10707	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) = (((∫1‘𝑓) − 𝑏) / 2))
502	499, 500, 501	3eqtr3d 2652	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (((∫1‘𝑓) − 𝑏) / 2))
503	491, 494, 502	3eqtrd 2648	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = (((∫1‘𝑓) − 𝑏) / 2))
504	486, 487, 503	3eqtr3d 2652	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (((∫1‘𝑓) − 𝑏) / 2))
505	504	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = ((∫1‘𝑓) − (((∫1‘𝑓) − 𝑏) / 2)))
506		itg1sub 23282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = ((∫1‘𝑓) − (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))))
507	234, 455, 506	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = ((∫1‘𝑓) − (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))))
508	8	recnd 9947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → (∫1‘𝑓) ∈ ℂ)
509	508	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘𝑓) ∈ ℂ)
510	469	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 ∈ ℂ)
511	509, 510, 258, 262	divsubdird 10719	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / 2) = (((∫1‘𝑓) / 2) − (𝑏 / 2)))
512	511	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − (((∫1‘𝑓) − 𝑏) / 2)) = ((∫1‘𝑓) − (((∫1‘𝑓) / 2) − (𝑏 / 2))))
513	508	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (∫1‘𝑓) ∈ ℂ)
514	513	halfcld 11154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) / 2) ∈ ℂ)
515	472	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (𝑏 / 2) ∈ ℂ)
516	513, 514, 515	subsubd 10299	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) − (((∫1‘𝑓) / 2) − (𝑏 / 2))) = (((∫1‘𝑓) − ((∫1‘𝑓) / 2)) + (𝑏 / 2)))
517	516	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − (((∫1‘𝑓) / 2) − (𝑏 / 2))) = (((∫1‘𝑓) − ((∫1‘𝑓) / 2)) + (𝑏 / 2)))
518	508	2halvesd 11155	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 ∈ dom ∫1 → (((∫1‘𝑓) / 2) + ((∫1‘𝑓) / 2)) = (∫1‘𝑓))
519	518	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → ((((∫1‘𝑓) / 2) + ((∫1‘𝑓) / 2)) − ((∫1‘𝑓) / 2)) = ((∫1‘𝑓) − ((∫1‘𝑓) / 2)))
520	508	halfcld 11154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 ∈ dom ∫1 → ((∫1‘𝑓) / 2) ∈ ℂ)
521	520, 520	pncand 10272	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → ((((∫1‘𝑓) / 2) + ((∫1‘𝑓) / 2)) − ((∫1‘𝑓) / 2)) = ((∫1‘𝑓) / 2))
522	519, 521	eqtr3d 2646	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 ∈ dom ∫1 → ((∫1‘𝑓) − ((∫1‘𝑓) / 2)) = ((∫1‘𝑓) / 2))
523	522	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 ∈ dom ∫1 → (((∫1‘𝑓) − ((∫1‘𝑓) / 2)) + (𝑏 / 2)) = (((∫1‘𝑓) / 2) + (𝑏 / 2)))
524	523	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − ((∫1‘𝑓) / 2)) + (𝑏 / 2)) = (((∫1‘𝑓) / 2) + (𝑏 / 2)))
525	512, 517, 524	3eqtrrd 2649	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) / 2) + (𝑏 / 2)) = ((∫1‘𝑓) − (((∫1‘𝑓) − 𝑏) / 2)))
526	505, 507, 525	3eqtr4d 2654	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = (((∫1‘𝑓) / 2) + (𝑏 / 2)))
527	526	adantlrl 752	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = (((∫1‘𝑓) / 2) + (𝑏 / 2)))
528	485, 527	breqtrrd 4611	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))))
529	457	adantlrl 752	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
530		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0))
531	530	adantlrl 752	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0))
532	234, 36	sylan 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
533	265	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ)
534	532, 533	resubcld 10337	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ ℝ)
535	534	leidd 10473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
536	535	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
537	296	breq2d 4595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
538	537	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
539	536, 538	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
540	534	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ ℝ)
541	48	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → 0 ∈ ℝ)
542	48	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → 0 ∈ ℝ)
543	534, 542	ltnled 10063	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) < 0 ↔ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
544	543	biimpar 501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) < 0)
545	540, 541, 544	ltled 10064	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ 0)
546		iffalse 4045	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
547	546	breq2d 4595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ 0))
548	547	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ 0))
549	545, 548	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
550	539, 549	pm2.61dan 828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
551	531, 550	sylan 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
552	551	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
553		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
554	553	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
555		iba 523	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ↔ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})))))
556	555	bicomd 212	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → ((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
557	556	ifbid 4058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
558	554, 557	breq12d 4596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
559	558	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
560	552, 559	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
561	35	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓:ℝ⟶ℝ)
562	171	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑧 ∈ ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0}))))
563		eldif 3550	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑧 ∈ ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})) ↔ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0})))
564	562, 563	syl6bb 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0}))))
565	564	notbid 307	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑓:ℝ⟶ℝ → (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0}))))
566	565	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0}))))
567		pm4.53 512	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0})) ↔ (¬ 𝑧 ∈ (◡𝑓 “ ran 𝑓) ∨ 𝑧 ∈ (◡𝑓 “ {0})))
568	208	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ ran 𝑓) ↔ 𝑧 ∈ ℝ))
569	568	biimpar 501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ (◡𝑓 “ ran 𝑓))
570	569	pm2.24d 146	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (◡𝑓 “ ran 𝑓) → (𝑓‘𝑧) = 0))
571	182	simplbda 652	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑓 Fn ℝ ∧ 𝑧 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑧) = 0)
572	571	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓 Fn ℝ → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
573	181, 572	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
574	573	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
575	570, 574	jaod 394	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → ((¬ 𝑧 ∈ (◡𝑓 “ ran 𝑓) ∨ 𝑧 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑧) = 0))
576	567, 575	syl5bi 231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑧) = 0))
577	566, 576	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (𝑓‘𝑧) = 0))
578	577	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓‘𝑧) = 0)
579	561, 578	sylanl1 680	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓‘𝑧) = 0)
580	579	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − 0) = (0 − 0))
581	580, 225	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − 0) = 0)
582	581, 30	syl6eqbr 4622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − 0) ≤ 0)
583		iffalse 4045	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) = 0)
584	583	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = ((𝑓‘𝑧) − 0))
585	300, 299	sylbir 224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∨ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
586	585	olcs 409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
587	584, 586	breq12d 4596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − 0) ≤ 0))
588	587	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − 0) ≤ 0))
589	582, 588	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
590	560, 589	pm2.61dan 828	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
591	590	ralrimiva 2949	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∀𝑧 ∈ ℝ ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
592	63	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ℝ ∈ V)
593		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ V
594	593	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ V)
595	423	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V)
596		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓‘𝑧) ∈ V
597	596	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ V)
598	200, 105	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V
599	598	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V)
600	70	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 = (𝑧 ∈ ℝ ↦ (𝑓‘𝑧)))
601		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
602	592, 597, 599, 600, 601	offval2 6812	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑧 ∈ ℝ ↦ ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))))
603		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
604	592, 594, 595, 602, 603	ofrfval2 6813	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∘𝑟 ≤ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ↔ ∀𝑧 ∈ ℝ ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
605	591, 604	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∘𝑟 ≤ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
606		itg1le 23286	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1 ∧ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 ∧ (𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∘𝑟 ≤ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ≤ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
607	529, 461, 605, 606	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ≤ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
608	437, 460, 463, 528, 607	ltletrd 10076	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
609	608	adantllr 751	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
610	609	adantlll 750	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
611		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) ∈ V
612		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (𝑎 = (∫1‘𝑔) ↔ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)))
613	612	anbi2d 736	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))))
614	613	rexbidv 3034	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))))
615		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (𝑏 < 𝑎 ↔ 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))))
616	614, 615	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎) ↔ (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)) ∧ 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))))
617	611, 616	spcev 3273	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)) ∧ 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
618	436, 610, 617	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
619	193, 618	pm2.61dane 2869	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
620	619	expr 641	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → (𝑏 ∈ ℝ → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
621	620	adantllr 751	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → (𝑏 ∈ ℝ → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
622	621	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → (𝑏 ∈ ℝ → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
623	157, 622	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
624	139, 623	pm2.61dane 2869	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
625	624	ex 449	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑏 < (∫1‘𝑓) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
626	94, 625	sylbid 229	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑏 < 𝑠 → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
627	626	imp 444	. . . . . . . . . . . 12 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < 𝑠) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
628	627	an32s 842	. . . . . . . . . . 11 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ 𝑏 < 𝑠) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
629	628	rexlimdvaa 3014	. . . . . . . . . 10 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ 𝑏 < 𝑠) → (∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
630	629	expimpd 627	. . . . . . . . 9 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → ((𝑏 < 𝑠 ∧ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
631	630	ancomsd 469	. . . . . . . 8 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → ((∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)) ∧ 𝑏 < 𝑠) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
632	631	exlimdv 1848	. . . . . . 7 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → (∃𝑠(∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)) ∧ 𝑏 < 𝑠) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
633		eqeq1 2614	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → (𝑥 = (∫1‘𝑓) ↔ 𝑠 = (∫1‘𝑓)))
634	633	anbi2d 736	. . . . . . . . 9 ⊢ (𝑥 = 𝑠 → ((𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) ↔ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))))
635	634	rexbidv 3034	. . . . . . . 8 ⊢ (𝑥 = 𝑠 → (∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) ↔ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))))
636	635	rexab 3336	. . . . . . 7 ⊢ (∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠 ↔ ∃𝑠(∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)) ∧ 𝑏 < 𝑠))
637		eqeq1 2614	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑥 = (∫1‘𝑔) ↔ 𝑎 = (∫1‘𝑔)))
638	637	anbi2d 736	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔))))
639	638	rexbidv 3034	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔))))
640	639	rexab 3336	. . . . . . 7 ⊢ (∃𝑎 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑏 < 𝑎 ↔ ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
641	632, 636, 640	3imtr4g 284	. . . . . 6 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → (∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠 → ∃𝑎 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑏 < 𝑎))
642	92, 641	sylbid 229	. . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) → ∃𝑎 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑏 < 𝑎))
643	642	impr 647	. . . 4 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑏 ∈ ℝ* ∧ 𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))) → ∃𝑎 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑏 < 𝑎)
644	6, 15, 89, 643	eqsupd 8246	. . 3 ⊢ (𝐹:ℝ⟶(0[,]+∞) → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
645	4, 644	syl5eq 2656	. 2 ⊢ (𝐹:ℝ⟶(0[,]+∞) → sup(𝐿, ℝ*, < ) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
646	2, 645	eqtr4d 2647	1 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   Or wor 4958   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793   ∘_𝑟 cofr 6794  supcsup 8229  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  +∞cpnf 9950  -∞cmnf 9951  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  2c2 10947  ℝ⁺crp 11708  [,]cicc 12049  vol*covol 23038  volcvol 23039  ∫₁citg1 23190  ∫₂citg2 23191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-mbf 23194  df-itg1 23195  df-itg2 23196

This theorem is referenced by:  itg2addnc  32634