ovolval4lem1 - Mathbox for Glauco Siliprandi

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Theorem ovolval4lem1 39539

Description: |- ( ( ph /\ n e. A ) -> ( ( (,) o. G ) 𝑛) = (((,) ∘ 𝐹) n ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
ovolval4lem1.f	⊢ (𝜑 → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
ovolval4lem1.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩)
ovolval4lem1.a	⊢ 𝐴 = {𝑛 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))}

**Assertion**
Ref	Expression
ovolval4lem1	⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) = ∪ ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺))))

Distinct variable groups: 𝐴,𝑛 𝑛,𝐹 𝑛,𝐺 𝜑,𝑛

**Proof of Theorem ovolval4lem1**
Step	Hyp	Ref	Expression
1		ioof 12142	. . . . . . . 8 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2	1	a1i 11	. . . . . . 7 ⊢ (𝜑 → (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ)
3		ovolval4lem1.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
4		fco 5971	. . . . . . 7 ⊢ (((,):(ℝ^* × ℝ^)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
5	2, 3, 4	syl2anc 691	. . . . . 6 ⊢ (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
6		ffn 5958	. . . . . 6 ⊢ (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
8		fniunfv 6409	. . . . 5 ⊢ (((,) ∘ 𝐹) Fn ℕ → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ∪ ran ((,) ∘ 𝐹))
9	7, 8	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ∪ ran ((,) ∘ 𝐹))
10	9	eqcomd 2616	. . 3 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) = ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛))
11		ovolval4lem1.a	. . . . . . . . 9 ⊢ 𝐴 = {𝑛 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))}
12		ssrab2 3650	. . . . . . . . 9 ⊢ {𝑛 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))} ⊆ ℕ
13	11, 12	eqsstri 3598	. . . . . . . 8 ⊢ 𝐴 ⊆ ℕ
14		undif 4001	. . . . . . . 8 ⊢ (𝐴 ⊆ ℕ ↔ (𝐴 ∪ (ℕ ∖ 𝐴)) = ℕ)
15	13, 14	mpbi 219	. . . . . . 7 ⊢ (𝐴 ∪ (ℕ ∖ 𝐴)) = ℕ
16	15	eqcomi 2619	. . . . . 6 ⊢ ℕ = (𝐴 ∪ (ℕ ∖ 𝐴))
17	16	iuneq1i 38286	. . . . 5 ⊢ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ∪ 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐹)‘𝑛)
18		iunxun 4541	. . . . 5 ⊢ ∪ 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐹)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
19	17, 18	eqtri 2632	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
20	19	a1i 11	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)))
21	3	ffvelrnda 6267	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ^* × ℝ^*))
22		xp1st 7089	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑛) ∈ (ℝ^* × ℝ^) → (1^st ‘(𝐹‘𝑛)) ∈ ℝ^)
23	21, 22	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐹‘𝑛)) ∈ ℝ^*)
24		xp2nd 7090	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∈ (ℝ^* × ℝ^) → (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^)
25	21, 24	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^*)
26	25, 23	ifcld 4081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))) ∈ ℝ^*)
27	23, 26	opelxpd 5073	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩ ∈ (ℝ^* × ℝ^*))
28		ovolval4lem1.g	. . . . . . . . 9 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩)
29	27, 28	fmptd 6292	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ⟶(ℝ^* × ℝ^*))
30		fco 5971	. . . . . . . 8 ⊢ (((,):(ℝ^* × ℝ^)⟶𝒫 ℝ ∧ 𝐺:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
31	2, 29, 30	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
32		ffn 5958	. . . . . . 7 ⊢ (((,) ∘ 𝐺):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐺) Fn ℕ)
33	31, 32	syl 17	. . . . . 6 ⊢ (𝜑 → ((,) ∘ 𝐺) Fn ℕ)
34		fniunfv 6409	. . . . . 6 ⊢ (((,) ∘ 𝐺) Fn ℕ → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ∪ ran ((,) ∘ 𝐺))
35	33, 34	syl 17	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ∪ ran ((,) ∘ 𝐺))
36	35	eqcomd 2616	. . . 4 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐺) = ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛))
37	16	iuneq1i 38286	. . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ∪ 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐺)‘𝑛)
38		iunxun 4541	. . . . . 6 ⊢ ∪ 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐺)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛))
39	37, 38	eqtri 2632	. . . . 5 ⊢ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛))
40	39	a1i 11	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛)))
41	29	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐺:ℕ⟶(ℝ^* × ℝ^*))
42	13	sseli 3564	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐴 → 𝑛 ∈ ℕ)
43	42	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ ℕ)
44		fvco3 6185	. . . . . . . 8 ⊢ ((𝐺:ℕ⟶(ℝ^* × ℝ^*) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺‘𝑛)))
45	41, 43, 44	syl2anc 691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺‘𝑛)))
46	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
47		fvco3 6185	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(ℝ^* × ℝ^*) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
48	46, 43, 47	syl2anc 691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
49		simpl 472	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝜑)
50		1st2nd2 7096	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∈ (ℝ^* × ℝ^*) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
51	21, 50	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
52	49, 43, 51	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
53	28	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩))
54	27	elexd 3187	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩ ∈ V)
55	53, 54	fvmpt2d 6202	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩)
56	49, 43, 55	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩)
57	11	eleq2i 2680	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝐴 ↔ 𝑛 ∈ {𝑛 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))})
58	57	biimpi 205	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝐴 → 𝑛 ∈ {𝑛 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))})
59		rabid 3095	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ {𝑛 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))} ↔ (𝑛 ∈ ℕ ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))))
60	58, 59	sylib 207	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝐴 → (𝑛 ∈ ℕ ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))))
61	60	simprd 478	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝐴 → (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)))
62	61	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)))
63	62	iftrued 4044	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))) = (2^nd ‘(𝐹‘𝑛)))
64	63	opeq2d 4347	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩ = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
65		eqidd 2611	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩ = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
66	56, 64, 65	3eqtrd 2648	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
67	52, 66	eqtr4d 2647	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐹‘𝑛) = (𝐺‘𝑛))
68	67	fveq2d 6107	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((,)‘(𝐹‘𝑛)) = ((,)‘(𝐺‘𝑛)))
69	48, 68	eqtrd 2644	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐺‘𝑛)))
70	45, 69	eqtr4d 2647	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) = (((,) ∘ 𝐹)‘𝑛))
71	70	iuneq2dv 4478	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) = ∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛))
72	29	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → 𝐺:ℕ⟶(ℝ^* × ℝ^*))
73		eldifi 3694	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℕ ∖ 𝐴) → 𝑛 ∈ ℕ)
74	73	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → 𝑛 ∈ ℕ)
75	72, 74, 44	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺‘𝑛)))
76		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → 𝜑)
77	76, 74, 55	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩)
78	73	anim1i 590	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))) → (𝑛 ∈ ℕ ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))))
79	78, 59	sylibr 223	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))) → 𝑛 ∈ {𝑛 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))})
80	79, 57	sylibr 223	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))) → 𝑛 ∈ 𝐴)
81	80	adantll 746	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))) → 𝑛 ∈ 𝐴)
82		eldifn 3695	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℕ ∖ 𝐴) → ¬ 𝑛 ∈ 𝐴)
83	82	ad2antlr 759	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))) → ¬ 𝑛 ∈ 𝐴)
84	81, 83	pm2.65da 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ¬ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)))
85	84	iffalsed 4047	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))) = (1^st ‘(𝐹‘𝑛)))
86	85	opeq2d 4347	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩ = ⟨(1^st ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))⟩)
87	77, 86	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))⟩)
88	87	fveq2d 6107	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘(𝐺‘𝑛)) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))⟩))
89		iooid 12074	. . . . . . . . . . . 12 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(1^st ‘(𝐹‘𝑛))) = ∅
90	89	eqcomi 2619	. . . . . . . . . . 11 ⊢ ∅ = ((1^st ‘(𝐹‘𝑛))(,)(1^st ‘(𝐹‘𝑛)))
91		df-ov 6552	. . . . . . . . . . 11 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(1^st ‘(𝐹‘𝑛))) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))⟩)
92	90, 91	eqtr2i 2633	. . . . . . . . . 10 ⊢ ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))⟩) = ∅
93	92	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))⟩) = ∅)
94	75, 88, 93	3eqtrd 2648	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) = ∅)
95	94	iuneq2dv 4478	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖ 𝐴)∅)
96		iun0 4512	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)∅ = ∅
97	96	a1i 11	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)∅ = ∅)
98	95, 97	eqtrd 2644	. . . . . 6 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = ∅)
99	76, 3	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
100	99, 74, 47	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
101	76, 74, 51	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
102	101	fveq2d 6107	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘(𝐹‘𝑛)) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
103		df-ov 6552	. . . . . . . . . . 11 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
104	103	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
105		simplr 788	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → 𝑛 ∈ (ℕ ∖ 𝐴))
106	74, 23	syldan 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (1^st ‘(𝐹‘𝑛)) ∈ ℝ^*)
107	106	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → (1^st ‘(𝐹‘𝑛)) ∈ ℝ^*)
108	74, 25	syldan 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^*)
109	108	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^*)
110		simpr 476	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛)))
111	107, 109	xrltnled 38520	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → ((1^st ‘(𝐹‘𝑛)) < (2^nd ‘(𝐹‘𝑛)) ↔ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))))
112	110, 111	mpbird 246	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → (1^st ‘(𝐹‘𝑛)) < (2^nd ‘(𝐹‘𝑛)))
113	107, 109, 112	xrltled 38427	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)))
114	105, 113, 80	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → 𝑛 ∈ 𝐴)
115	82	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → ¬ 𝑛 ∈ 𝐴)
116	114, 115	condan 831	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛)))
117		ioo0 12071	. . . . . . . . . . . 12 ⊢ (((1^st ‘(𝐹‘𝑛)) ∈ ℝ^* ∧ (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^*) → (((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ∅ ↔ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))))
118	106, 108, 117	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ∅ ↔ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))))
119	116, 118	mpbird 246	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ∅)
120	104, 119	eqtr3d 2646	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩) = ∅)
121	100, 102, 120	3eqtrd 2648	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = ∅)
122	121	iuneq2dv 4478	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖ 𝐴)∅)
123	122, 97	eqtrd 2644	. . . . . 6 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛) = ∅)
124	98, 123	eqtr4d 2647	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
125	71, 124	uneq12d 3730	. . . 4 ⊢ (𝜑 → (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛)) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)))
126	36, 40, 125	3eqtrrd 2649	. . 3 ⊢ (𝜑 → (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)) = ∪ ran ((,) ∘ 𝐺))
127	10, 20, 126	3eqtrd 2648	. 2 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) = ∪ ran ((,) ∘ 𝐺))
128		volf 23104	. . . . . 6 ⊢ vol:dom vol⟶(0[,]+∞)
129	128	a1i 11	. . . . 5 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞))
130	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
131		simpr 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
132	130, 131, 47	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
133	51	fveq2d 6107	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
134	103	eqcomi 2619	. . . . . . . . . . 11 ⊢ ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛)))
135	134	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
136	132, 133, 135	3eqtrd 2648	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
137		ioombl 23140	. . . . . . . . . 10 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) ∈ dom vol
138	137	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) ∈ dom vol)
139	136, 138	eqeltrd 2688	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
140	139	ralrimiva 2949	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
141	7, 140	jca 553	. . . . . 6 ⊢ (𝜑 → (((,) ∘ 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol))
142		ffnfv 6295	. . . . . 6 ⊢ (((,) ∘ 𝐹):ℕ⟶dom vol ↔ (((,) ∘ 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol))
143	141, 142	sylibr 223	. . . . 5 ⊢ (𝜑 → ((,) ∘ 𝐹):ℕ⟶dom vol)
144		fco 5971	. . . . 5 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ ((,) ∘ 𝐹):ℕ⟶dom vol) → (vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞))
145	129, 143, 144	syl2anc 691	. . . 4 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞))
146		ffn 5958	. . . 4 ⊢ ((vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞) → (vol ∘ ((,) ∘ 𝐹)) Fn ℕ)
147	145, 146	syl 17	. . 3 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐹)) Fn ℕ)
148	70	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) = (((,) ∘ 𝐹)‘𝑛))
149	139	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
150	148, 149	eqeltrd 2688	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
151		simpll 786	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 ∈ 𝐴) → 𝜑)
152		eldif 3550	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℕ ∖ 𝐴) ↔ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐴))
153	152	bicomi 213	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐴) ↔ 𝑛 ∈ (ℕ ∖ 𝐴))
154	153	biimpi 205	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐴) → 𝑛 ∈ (ℕ ∖ 𝐴))
155	154	adantll 746	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 ∈ 𝐴) → 𝑛 ∈ (ℕ ∖ 𝐴))
156	119, 137	syl6eqelr 2697	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ∅ ∈ dom vol)
157	94, 156	eqeltrd 2688	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
158	151, 155, 157	syl2anc 691	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
159	150, 158	pm2.61dan 828	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
160	159	ralrimiva 2949	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
161	33, 160	jca 553	. . . . . 6 ⊢ (𝜑 → (((,) ∘ 𝐺) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol))
162		ffnfv 6295	. . . . . 6 ⊢ (((,) ∘ 𝐺):ℕ⟶dom vol ↔ (((,) ∘ 𝐺) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol))
163	161, 162	sylibr 223	. . . . 5 ⊢ (𝜑 → ((,) ∘ 𝐺):ℕ⟶dom vol)
164		fco 5971	. . . . 5 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ ((,) ∘ 𝐺):ℕ⟶dom vol) → (vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞))
165	129, 163, 164	syl2anc 691	. . . 4 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞))
166		ffn 5958	. . . 4 ⊢ ((vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞) → (vol ∘ ((,) ∘ 𝐺)) Fn ℕ)
167	165, 166	syl 17	. . 3 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐺)) Fn ℕ)
168	148	eqcomd 2616	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
169	121, 94	eqtr4d 2647	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
170	151, 155, 169	syl2anc 691	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
171	168, 170	pm2.61dan 828	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
172	171	fveq2d 6107	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(((,) ∘ 𝐹)‘𝑛)) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
173		fnfun 5902	. . . . . . 7 ⊢ (((,) ∘ 𝐹) Fn ℕ → Fun ((,) ∘ 𝐹))
174	7, 173	syl 17	. . . . . 6 ⊢ (𝜑 → Fun ((,) ∘ 𝐹))
175	174	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Fun ((,) ∘ 𝐹))
176		fdm 5964	. . . . . . . . 9 ⊢ (((,) ∘ 𝐹):ℕ⟶dom vol → dom ((,) ∘ 𝐹) = ℕ)
177	143, 176	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom ((,) ∘ 𝐹) = ℕ)
178	177	eqcomd 2616	. . . . . . 7 ⊢ (𝜑 → ℕ = dom ((,) ∘ 𝐹))
179	178	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℕ = dom ((,) ∘ 𝐹))
180	131, 179	eleqtrd 2690	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ dom ((,) ∘ 𝐹))
181		fvco 6184	. . . . 5 ⊢ ((Fun ((,) ∘ 𝐹) ∧ 𝑛 ∈ dom ((,) ∘ 𝐹)) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = (vol‘(((,) ∘ 𝐹)‘𝑛)))
182	175, 180, 181	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = (vol‘(((,) ∘ 𝐹)‘𝑛)))
183		fnfun 5902	. . . . . . 7 ⊢ (((,) ∘ 𝐺) Fn ℕ → Fun ((,) ∘ 𝐺))
184	33, 183	syl 17	. . . . . 6 ⊢ (𝜑 → Fun ((,) ∘ 𝐺))
185	184	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Fun ((,) ∘ 𝐺))
186		fdm 5964	. . . . . . . . 9 ⊢ (((,) ∘ 𝐺):ℕ⟶dom vol → dom ((,) ∘ 𝐺) = ℕ)
187	163, 186	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom ((,) ∘ 𝐺) = ℕ)
188	187	eqcomd 2616	. . . . . . 7 ⊢ (𝜑 → ℕ = dom ((,) ∘ 𝐺))
189	188	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℕ = dom ((,) ∘ 𝐺))
190	131, 189	eleqtrd 2690	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ dom ((,) ∘ 𝐺))
191		fvco 6184	. . . . 5 ⊢ ((Fun ((,) ∘ 𝐺) ∧ 𝑛 ∈ dom ((,) ∘ 𝐺)) → ((vol ∘ ((,) ∘ 𝐺))‘𝑛) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
192	185, 190, 191	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐺))‘𝑛) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
193	172, 182, 192	3eqtr4d 2654	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = ((vol ∘ ((,) ∘ 𝐺))‘𝑛))
194	147, 167, 193	eqfnfvd 6222	. 2 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺)))
195	127, 194	jca 553	1 ⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) = ∪ ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 ifcif 4036 𝒫 cpw 4108 ⟨cop 4131 ∪ cuni 4372 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 dom cdm 5038 ran crn 5039 ∘ ccom 5042 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 ℝcr 9814 0cc0 9815 +∞cpnf 9950 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 ℕcn 10897 (,)cioo 12046 [,]cicc 12049 volcvol 23039

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

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-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-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-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-xadd 11823 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-rlim 14068 df-sum 14265 df-xmet 19560 df-met 19561 df-ovol 23040 df-vol 23041

This theorem is referenced by: ovolval4lem2 39540

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