Theorem ovolscalem1 23088

Description: Lemma for ovolsca 23090. (Contributed by Mario Carneiro, 6-Apr-2015.)

Hypotheses

Ref	Expression
ovolsca.1	⊢ (𝜑 → 𝐴 ⊆ ℝ)
ovolsca.2	⊢ (𝜑 → 𝐶 ∈ ℝ+)
ovolsca.3	⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})
ovolsca.4	⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)
ovolsca.5	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovolsca.6	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩)
ovolsca.7	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ovolsca.8	⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹))
ovolsca.9	⊢ (𝜑 → 𝑅 ∈ ℝ+)
ovolsca.10	⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))

Assertion

Ref	Expression
ovolscalem1	⊢ (𝜑 → (vol*‘𝐵) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))

Distinct variable groups:   𝑥,𝑛,𝐴   𝐵,𝑛   𝑛,𝐹,𝑥   𝑛,𝐺   𝑥,𝑅   𝐶,𝑛,𝑥   𝜑,𝑛   𝑥,𝑆

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑅(𝑛)   𝑆(𝑛)   𝐺(𝑥)

Proof of Theorem ovolscalem1

Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolsca.3	. . . 4 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴})
2		ssrab2 3650	. . . 4 ⊢ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ⊆ ℝ
3	1, 2	syl6eqss 3618	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
4		ovolcl 23053	. . 3 ⊢ (𝐵 ⊆ ℝ → (vol*‘𝐵) ∈ ℝ*)
5	3, 4	syl 17	. 2 ⊢ (𝜑 → (vol*‘𝐵) ∈ ℝ*)
6		ovolsca.7	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
7		ovolfcl 23042	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
8	6, 7	sylan 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
9	8	simp3d 1068	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))
10	8	simp1d 1066	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
11	8	simp2d 1067	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
12		ovolsca.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ+)
13	12	rpregt0d 11754	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
14	13	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
15		lediv1 10767	. . . . . . . . . . 11 ⊢ (((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)) ↔ ((1st ‘(𝐹‘𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) / 𝐶)))
16	10, 11, 14, 15	syl3anc 1318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)) ↔ ((1st ‘(𝐹‘𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) / 𝐶)))
17	9, 16	mpbid 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) / 𝐶))
18		df-br 4584	. . . . . . . . 9 ⊢ (((1st ‘(𝐹‘𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) / 𝐶) ↔ ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ ≤ )
19	17, 18	sylib 207	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ ≤ )
20	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ+)
21	10, 20	rerpdivcld 11779	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) / 𝐶) ∈ ℝ)
22	11, 20	rerpdivcld 11779	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) / 𝐶) ∈ ℝ)
23		opelxpi 5072	. . . . . . . . 9 ⊢ ((((1st ‘(𝐹‘𝑛)) / 𝐶) ∈ ℝ ∧ ((2nd ‘(𝐹‘𝑛)) / 𝐶) ∈ ℝ) → ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ (ℝ × ℝ))
24	21, 22, 23	syl2anc 691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ (ℝ × ℝ))
25	19, 24	elind 3760	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
26		ovolsca.6	. . . . . . 7 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩)
27	25, 26	fmptd 6292	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
28		eqid 2610	. . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
29		eqid 2610	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
30	28, 29	ovolsf 23048	. . . . . 6 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
31	27, 30	syl 17	. . . . 5 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
32		frn 5966	. . . . 5 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
33	31, 32	syl 17	. . . 4 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
34		icossxr 12129	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ*
35	33, 34	syl6ss 3580	. . 3 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ*)
36		supxrcl 12017	. . 3 ⊢ (ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ ℝ*)
37	35, 36	syl 17	. 2 ⊢ (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ ℝ*)
38		ovolsca.4	. . . . 5 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)
39	38, 12	rerpdivcld 11779	. . . 4 ⊢ (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℝ)
40		ovolsca.9	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
41	40	rpred 11748	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ)
42	39, 41	readdcld 9948	. . 3 ⊢ (𝜑 → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ)
43	42	rexrd 9968	. 2 ⊢ (𝜑 → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ*)
44	1	eleq2d 2673	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}))
45		oveq2 6557	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
46	45	eleq1d 2672	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐶 · 𝑥) ∈ 𝐴 ↔ (𝐶 · 𝑦) ∈ 𝐴))
47	46	elrab 3331	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ↔ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴))
48	44, 47	syl6bb 275	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)))
49		simprr 792	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → (𝐶 · 𝑦) ∈ 𝐴)
50		ovolsca.8	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹))
51		ovolsca.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
52		ovolfioo 23043	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛)))))
53	51, 6, 52	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛)))))
54	50, 53	mpbid 221	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))
55	54	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))
56		breq2 4587	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐶 · 𝑦) → ((1st ‘(𝐹‘𝑛)) < 𝑥 ↔ (1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦)))
57		breq1 4586	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐶 · 𝑦) → (𝑥 < (2nd ‘(𝐹‘𝑛)) ↔ (𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛))))
58	56, 57	anbi12d 743	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐶 · 𝑦) → (((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) ↔ ((1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛)))))
59	58	rexbidv 3034	. . . . . . . . . 10 ⊢ (𝑥 = (𝐶 · 𝑦) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛)))))
60	59	rspcv 3278	. . . . . . . . 9 ⊢ ((𝐶 · 𝑦) ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛)))))
61	49, 55, 60	sylc 63	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛))))
62		opex 4859	. . . . . . . . . . . . . . . 16 ⊢ ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ V
63	26	fvmpt2 6200	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ V) → (𝐺‘𝑛) = ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩)
64	62, 63	mpan2 703	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩)
65	64	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = (1st ‘⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩))
66		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘(𝐹‘𝑛)) / 𝐶) ∈ V
67		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘(𝐹‘𝑛)) / 𝐶) ∈ V
68	66, 67	op1st 7067	. . . . . . . . . . . . . 14 ⊢ (1st ‘⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩) = ((1st ‘(𝐹‘𝑛)) / 𝐶)
69	65, 68	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) / 𝐶))
70	69	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) / 𝐶))
71	70	breq1d 4593	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺‘𝑛)) < 𝑦 ↔ ((1st ‘(𝐹‘𝑛)) / 𝐶) < 𝑦))
72	10	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
73		simplrl 796	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ ℝ)
74	14	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
75		ltdivmul 10777	. . . . . . . . . . . 12 ⊢ (((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (((1st ‘(𝐹‘𝑛)) / 𝐶) < 𝑦 ↔ (1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦)))
76	72, 73, 74, 75	syl3anc 1318	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹‘𝑛)) / 𝐶) < 𝑦 ↔ (1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦)))
77	71, 76	bitr2d 268	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦) ↔ (1st ‘(𝐺‘𝑛)) < 𝑦))
78	11	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
79		ltmuldiv2 10776	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹‘𝑛)) / 𝐶)))
80	73, 78, 74, 79	syl3anc 1318	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹‘𝑛)) / 𝐶)))
81	64	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = (2nd ‘⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩))
82	66, 67	op2nd 7068	. . . . . . . . . . . . . 14 ⊢ (2nd ‘⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩) = ((2nd ‘(𝐹‘𝑛)) / 𝐶)
83	81, 82	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = ((2nd ‘(𝐹‘𝑛)) / 𝐶))
84	83	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) = ((2nd ‘(𝐹‘𝑛)) / 𝐶))
85	84	breq2d 4595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺‘𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹‘𝑛)) / 𝐶)))
86	80, 85	bitr4d 270	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛)) ↔ 𝑦 < (2nd ‘(𝐺‘𝑛))))
87	77, 86	anbi12d 743	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛))) ↔ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
88	87	rexbidva 3031	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → (∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛))) ↔ ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
89	61, 88	mpbid 221	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
90	89	ex 449	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
91	48, 90	sylbid 229	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
92	91	ralrimiv 2948	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
93		ovolfioo 23043	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
94	3, 27, 93	syl2anc 691	. . . 4 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
95	92, 94	mpbird 246	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺))
96	29	ovollb 23054	. . 3 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺)) → (vol*‘𝐵) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
97	27, 95, 96	syl2anc 691	. 2 ⊢ (𝜑 → (vol*‘𝐵) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
98		fzfid 12634	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
99	12	rpcnd 11750	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
100	99	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐶 ∈ ℂ)
101		simpl 472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝜑)
102		elfznn 12241	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
103	11, 10	resubcld 10337	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ∈ ℝ)
104	101, 102, 103	syl2an 493	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ∈ ℝ)
105	104	recnd 9947	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ∈ ℂ)
106	12	rpne0d 11753	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≠ 0)
107	106	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐶 ≠ 0)
108	98, 100, 105, 107	fsumdivc 14360	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) = Σ𝑛 ∈ (1...𝑘)(((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶))
109	83, 69	oveq12d 6567	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = (((2nd ‘(𝐹‘𝑛)) / 𝐶) − ((1st ‘(𝐹‘𝑛)) / 𝐶)))
110	109	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = (((2nd ‘(𝐹‘𝑛)) / 𝐶) − ((1st ‘(𝐹‘𝑛)) / 𝐶)))
111	28	ovolfsval 23046	. . . . . . . . . . 11 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
112	27, 111	sylan 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
113	11	recnd 9947	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℂ)
114	10	recnd 9947	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℂ)
115	12	rpcnne0d 11757	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
116	115	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
117		divsubdir 10600	. . . . . . . . . . 11 ⊢ (((2nd ‘(𝐹‘𝑛)) ∈ ℂ ∧ (1st ‘(𝐹‘𝑛)) ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) = (((2nd ‘(𝐹‘𝑛)) / 𝐶) − ((1st ‘(𝐹‘𝑛)) / 𝐶)))
118	113, 114, 116, 117	syl3anc 1318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) = (((2nd ‘(𝐹‘𝑛)) / 𝐶) − ((1st ‘(𝐹‘𝑛)) / 𝐶)))
119	110, 112, 118	3eqtr4d 2654	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶))
120	101, 102, 119	syl2an 493	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶))
121		simpr 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
122		nnuz 11599	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
123	121, 122	syl6eleq 2698	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
124	103, 20	rerpdivcld 11779	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) ∈ ℝ)
125	124	recnd 9947	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) ∈ ℂ)
126	101, 102, 125	syl2an 493	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) ∈ ℂ)
127	120, 123, 126	fsumser 14308	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)(((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
128	108, 127	eqtrd 2644	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
129		ovolsca.10	. . . . . . . . . . 11 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
130		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
131		ovolsca.5	. . . . . . . . . . . . . . . 16 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
132	130, 131	ovolsf 23048	. . . . . . . . . . . . . . 15 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
133	6, 132	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞))
134		frn 5966	. . . . . . . . . . . . . 14 ⊢ (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
135	133, 134	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
136	135, 34	syl6ss 3580	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*)
137	12, 40	rpmulcld 11764	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶 · 𝑅) ∈ ℝ+)
138	137	rpred 11748	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 · 𝑅) ∈ ℝ)
139	38, 138	readdcld 9948	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ)
140	139	rexrd 9968	. . . . . . . . . . . 12 ⊢ (𝜑 → ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ*)
141		supxrleub 12028	. . . . . . . . . . . 12 ⊢ ((ran 𝑆 ⊆ ℝ* ∧ ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ*) → (sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
142	136, 140, 141	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → (sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
143	129, 142	mpbid 221	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
144		ffn 5958	. . . . . . . . . . . 12 ⊢ (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
145	133, 144	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 Fn ℕ)
146		breq1 4586	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑆‘𝑘) → (𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ (𝑆‘𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
147	146	ralrn 6270	. . . . . . . . . . 11 ⊢ (𝑆 Fn ℕ → (∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
148	145, 147	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
149	143, 148	mpbid 221	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
150	149	r19.21bi 2916	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
151	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
152	130	ovolfsval 23046	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
153	151, 102, 152	syl2an 493	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
154	153, 123, 105	fsumser 14308	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘))
155	131	fveq1i 6104	. . . . . . . . 9 ⊢ (𝑆‘𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘)
156	154, 155	syl6eqr 2662	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) = (𝑆‘𝑘))
157	39	recnd 9947	. . . . . . . . . . 11 ⊢ (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℂ)
158	40	rpcnd 11750	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℂ)
159	99, 157, 158	adddid 9943	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((𝐶 · ((vol*‘𝐴) / 𝐶)) + (𝐶 · 𝑅)))
160	38	recnd 9947	. . . . . . . . . . . 12 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℂ)
161	160, 99, 106	divcan2d 10682	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 · ((vol*‘𝐴) / 𝐶)) = (vol*‘𝐴))
162	161	oveq1d 6564	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶 · ((vol*‘𝐴) / 𝐶)) + (𝐶 · 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
163	159, 162	eqtrd 2644	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
164	163	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
165	150, 156, 164	3brtr4d 4615	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ≤ (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)))
166	98, 104	fsumrecl 14312	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ∈ ℝ)
167	42	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ)
168	13	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
169		ledivmul 10778	. . . . . . . 8 ⊢ ((Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ∈ ℝ ∧ (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ≤ (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅))))
170	166, 167, 168, 169	syl3anc 1318	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ≤ (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅))))
171	165, 170	mpbird 246	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
172	128, 171	eqbrtrrd 4607	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
173	172	ralrimiva 2949	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
174		ffn 5958	. . . . . 6 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞) → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
175	31, 174	syl 17	. . . . 5 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
176		breq1 4586	. . . . . 6 ⊢ (𝑦 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) → (𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
177	176	ralrn 6270	. . . . 5 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ → (∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ ∀𝑘 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
178	175, 177	syl 17	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ ∀𝑘 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
179	173, 178	mpbird 246	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
180		supxrleub 12028	. . . 4 ⊢ ((ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ* ∧ (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ*) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ ∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
181	35, 43, 180	syl2anc 691	. . 3 ⊢ (𝜑 → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ ∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
182	179, 181	mpbird 246	. 2 ⊢ (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
183	5, 37, 43, 97, 182	xrletrd 11869	1 ⊢ (𝜑 → (vol*‘𝐵) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ran crn 5039   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  supcsup 8229  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  +∞cpnf 9950  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  ℕcn 10897  ℤ_≥cuz 11563  ℝ⁺crp 11708  (,)cioo 12046  [,)cico 12048  ...cfz 12197  seqcseq 12663  abscabs 13822  Σcsu 14264  vol*covol 23038

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-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  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-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-rp 11709  df-ioo 12050  df-ico 12052  df-fz 12198  df-fzo 12335  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-ovol 23040

This theorem is referenced by:  ovolscalem2  23089