Theorem ovolscalem1 25480

Description: Lemma for ovolsca 25482. (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 4020	. . . 4 ⊢ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ⊆ ℝ
3	1, 2	eqsstrdi 3966	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
4		ovolcl 25445	. . 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 25433	. . . . . . . . . . . 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 581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
9	8	simp3d 1145	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))
10	8	simp1d 1143	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
11	8	simp2d 1144	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
12		ovolsca.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ+)
13	12	rpregt0d 12992	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
14	13	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
15		lediv1 12021	. . . . . . . . . . 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 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)) ↔ ((1st ‘(𝐹‘𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) / 𝐶)))
17	9, 16	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) / 𝐶))
18		df-br 5086	. . . . . . . . 9 ⊢ (((1st ‘(𝐹‘𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) / 𝐶) ↔ ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ ≤ )
19	17, 18	sylib 218	. . . . . . . 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 13017	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) / 𝐶) ∈ ℝ)
22	11, 20	rerpdivcld 13017	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) / 𝐶) ∈ ℝ)
23	21, 22	opelxpd 5670	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ (ℝ × ℝ))
24	19, 23	elind 4140	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ ( ≤ ∩ (ℝ × ℝ)))
25		ovolsca.6	. . . . . . 7 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩)
26	24, 25	fmptd 7066	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
27		eqid 2736	. . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
28		eqid 2736	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
29	27, 28	ovolsf 25439	. . . . . 6 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
30	26, 29	syl 17	. . . . 5 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)):ℕ⟶(0[,)+∞))
31	30	frnd 6676	. . . 4 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
32		icossxr 13385	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ*
33	31, 32	sstrdi 3934	. . 3 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ*)
34		supxrcl 13267	. . 3 ⊢ (ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ ℝ*)
35	33, 34	syl 17	. 2 ⊢ (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ∈ ℝ*)
36		ovolsca.4	. . . . 5 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)
37	36, 12	rerpdivcld 13017	. . . 4 ⊢ (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℝ)
38		ovolsca.9	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
39	38	rpred 12986	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ)
40	37, 39	readdcld 11174	. . 3 ⊢ (𝜑 → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ)
41	40	rexrd 11195	. 2 ⊢ (𝜑 → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ*)
42	1	eleq2d 2822	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}))
43		oveq2 7375	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
44	43	eleq1d 2821	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐶 · 𝑥) ∈ 𝐴 ↔ (𝐶 · 𝑦) ∈ 𝐴))
45	44	elrab 3634	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ↔ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴))
46	42, 45	bitrdi 287	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)))
47		breq2 5089	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐶 · 𝑦) → ((1st ‘(𝐹‘𝑛)) < 𝑥 ↔ (1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦)))
48		breq1 5088	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐶 · 𝑦) → (𝑥 < (2nd ‘(𝐹‘𝑛)) ↔ (𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛))))
49	47, 48	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑥 = (𝐶 · 𝑦) → (((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) ↔ ((1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛)))))
50	49	rexbidv 3161	. . . . . . . . 9 ⊢ (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)))))
51		ovolsca.8	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ((,) ∘ 𝐹))
52		ovolsca.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
53		ovolfioo 25434	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛)))))
54	52, 6, 53	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛)))))
55	51, 54	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))
56	55	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))
57		simprr 773	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → (𝐶 · 𝑦) ∈ 𝐴)
58	50, 56, 57	rspcdva 3565	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛))))
59		opex 5416	. . . . . . . . . . . . . . . 16 ⊢ ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ V
60	25	fvmpt2 6959	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ V) → (𝐺‘𝑛) = ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩)
61	59, 60	mpan2 692	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩)
62	61	fveq2d 6844	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = (1st ‘⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩))
63		ovex 7400	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘(𝐹‘𝑛)) / 𝐶) ∈ V
64		ovex 7400	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘(𝐹‘𝑛)) / 𝐶) ∈ V
65	63, 64	op1st 7950	. . . . . . . . . . . . . 14 ⊢ (1st ‘⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩) = ((1st ‘(𝐹‘𝑛)) / 𝐶)
66	62, 65	eqtrdi 2787	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) / 𝐶))
67	66	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐺‘𝑛)) = ((1st ‘(𝐹‘𝑛)) / 𝐶))
68	67	breq1d 5095	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺‘𝑛)) < 𝑦 ↔ ((1st ‘(𝐹‘𝑛)) / 𝐶) < 𝑦))
69	10	adantlr 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
70		simplrl 777	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ ℝ)
71	14	adantlr 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
72		ltdivmul 12031	. . . . . . . . . . . 12 ⊢ (((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (((1st ‘(𝐹‘𝑛)) / 𝐶) < 𝑦 ↔ (1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦)))
73	69, 70, 71, 72	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (((1st ‘(𝐹‘𝑛)) / 𝐶) < 𝑦 ↔ (1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦)))
74	68, 73	bitr2d 280	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦) ↔ (1st ‘(𝐺‘𝑛)) < 𝑦))
75	11	adantlr 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
76		ltmuldiv2 12030	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹‘𝑛)) / 𝐶)))
77	70, 75, 71, 76	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹‘𝑛)) / 𝐶)))
78	61	fveq2d 6844	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = (2nd ‘⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩))
79	63, 64	op2nd 7951	. . . . . . . . . . . . . 14 ⊢ (2nd ‘⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩) = ((2nd ‘(𝐹‘𝑛)) / 𝐶)
80	78, 79	eqtrdi 2787	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = ((2nd ‘(𝐹‘𝑛)) / 𝐶))
81	80	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐺‘𝑛)) = ((2nd ‘(𝐹‘𝑛)) / 𝐶))
82	81	breq2d 5097	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑦 < (2nd ‘(𝐺‘𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹‘𝑛)) / 𝐶)))
83	77, 82	bitr4d 282	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛)) ↔ 𝑦 < (2nd ‘(𝐺‘𝑛))))
84	74, 83	anbi12d 633	. . . . . . . . 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)))))
85	84	rexbidva 3159	. . . . . . . 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)))))
86	58, 85	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
87	86	ex 412	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴) → ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
88	46, 87	sylbid 240	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
89	88	ralrimiv 3128	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
90		ovolfioo 25434	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
91	3, 26, 90	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝐺) ↔ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
92	89, 91	mpbird 257	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺))
93	28	ovollb 25446	. . 3 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺)) → (vol*‘𝐵) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
94	26, 92, 93	syl2anc 585	. 2 ⊢ (𝜑 → (vol*‘𝐵) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
95		fzfid 13935	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
96	12	rpcnd 12988	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
97	96	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐶 ∈ ℂ)
98		simpl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝜑)
99		elfznn 13507	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
100	11, 10	resubcld 11578	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ∈ ℝ)
101	98, 99, 100	syl2an 597	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ∈ ℝ)
102	101	recnd 11173	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ∈ ℂ)
103	12	rpne0d 12991	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≠ 0)
104	103	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐶 ≠ 0)
105	95, 97, 102, 104	fsumdivc 15748	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) = Σ𝑛 ∈ (1...𝑘)(((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶))
106	80, 66	oveq12d 7385	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = (((2nd ‘(𝐹‘𝑛)) / 𝐶) − ((1st ‘(𝐹‘𝑛)) / 𝐶)))
107	106	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))) = (((2nd ‘(𝐹‘𝑛)) / 𝐶) − ((1st ‘(𝐹‘𝑛)) / 𝐶)))
108	27	ovolfsval 25437	. . . . . . . . . . 11 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
109	26, 108	sylan 581	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
110	11	recnd 11173	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℂ)
111	10	recnd 11173	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℂ)
112	12	rpcnne0d 12995	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
113	112	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
114		divsubdir 11848	. . . . . . . . . . 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)))
115	110, 111, 113, 114	syl3anc 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) = (((2nd ‘(𝐹‘𝑛)) / 𝐶) − ((1st ‘(𝐹‘𝑛)) / 𝐶)))
116	107, 109, 115	3eqtr4d 2781	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶))
117	98, 99, 116	syl2an 597	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶))
118		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
119		nnuz 12827	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
120	118, 119	eleqtrdi 2846	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
121	100, 20	rerpdivcld 13017	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) ∈ ℝ)
122	121	recnd 11173	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) ∈ ℂ)
123	98, 99, 122	syl2an 597	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) ∈ ℂ)
124	117, 120, 123	fsumser 15692	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)(((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
125	105, 124	eqtrd 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
126		ovolsca.10	. . . . . . . . . . 11 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
127		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
128		ovolsca.5	. . . . . . . . . . . . . . . 16 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
129	127, 128	ovolsf 25439	. . . . . . . . . . . . . . 15 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
130	6, 129	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞))
131	130	frnd 6676	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
132	131, 32	sstrdi 3934	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*)
133	12, 38	rpmulcld 13002	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶 · 𝑅) ∈ ℝ+)
134	133	rpred 12986	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 · 𝑅) ∈ ℝ)
135	36, 134	readdcld 11174	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ)
136	135	rexrd 11195	. . . . . . . . . . . 12 ⊢ (𝜑 → ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ*)
137		supxrleub 13278	. . . . . . . . . . . 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))))
138	132, 136, 137	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → (sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
139	126, 138	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
140	130	ffnd 6669	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 Fn ℕ)
141		breq1 5088	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑆‘𝑘) → (𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ (𝑆‘𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
142	141	ralrn 7040	. . . . . . . . . . 11 ⊢ (𝑆 Fn ℕ → (∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
143	140, 142	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
144	139, 143	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
145	144	r19.21bi 3229	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
146	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
147	127	ovolfsval 25437	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
148	146, 99, 147	syl2an 597	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
149	148, 120, 102	fsumser 15692	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘))
150	128	fveq1i 6841	. . . . . . . . 9 ⊢ (𝑆‘𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘)
151	149, 150	eqtr4di 2789	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) = (𝑆‘𝑘))
152	37	recnd 11173	. . . . . . . . . . 11 ⊢ (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℂ)
153	38	rpcnd 12988	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℂ)
154	96, 152, 153	adddid 11169	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((𝐶 · ((vol*‘𝐴) / 𝐶)) + (𝐶 · 𝑅)))
155	36	recnd 11173	. . . . . . . . . . . 12 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℂ)
156	155, 96, 103	divcan2d 11933	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 · ((vol*‘𝐴) / 𝐶)) = (vol*‘𝐴))
157	156	oveq1d 7382	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶 · ((vol*‘𝐴) / 𝐶)) + (𝐶 · 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
158	154, 157	eqtrd 2771	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
159	158	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
160	145, 151, 159	3brtr4d 5117	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ≤ (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)))
161	95, 101	fsumrecl 15696	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ∈ ℝ)
162	40	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ)
163	13	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
164		ledivmul 12032	. . . . . . . 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))))
165	161, 162, 163, 164	syl3anc 1374	. . . . . . 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))))
166	160, 165	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
167	125, 166	eqbrtrrd 5109	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
168	167	ralrimiva 3129	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
169	30	ffnd 6669	. . . . 5 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
170		breq1 5088	. . . . . 6 ⊢ (𝑦 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) → (𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
171	170	ralrn 7040	. . . . 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)))
172	169, 171	syl 17	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ ∀𝑘 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
173	168, 172	mpbird 257	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
174		supxrleub 13278	. . . 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)))
175	33, 41, 174	syl2anc 585	. . 3 ⊢ (𝜑 → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ ∀𝑦 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝐺))𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
176	173, 175	mpbird 257	. 2 ⊢ (𝜑 → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
177	5, 35, 41, 94, 176	xrletrd 13113	1 ⊢ (𝜑 → (vol*‘𝐵) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  {crab 3389  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  ⟨cop 4573  ∪ cuni 4850   class class class wbr 5085   ↦ cmpt 5166   × cxp 5629  ran crn 5632   ∘ ccom 5635   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  supcsup 9353  ℂcc 11036  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   · cmul 11043  +∞cpnf 11176  ℝ^*cxr 11178   < clt 11179   ≤ cle 11180   − cmin 11377   / cdiv 11807  ℕcn 12174  ℤ_≥cuz 12788  ℝ⁺crp 12942  (,)cioo 13298  [,)cico 13300  ...cfz 13461  seqcseq 13963  abscabs 15196  Σcsu 15648  vol*covol 25429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-z 12525  df-uz 12789  df-rp 12943  df-ioo 13302  df-ico 13304  df-fz 13462  df-fzo 13609  df-seq 13964  df-exp 14024  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-clim 15450  df-sum 15649  df-ovol 25431

This theorem is referenced by:  ovolscalem2  25481