Theorem ovolscalem1 25397

Description: Lemma for ovolsca 25399. (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 4072	. . . 4 ⊢ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ⊆ ℝ
3	1, 2	eqsstrdi 4031	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
4		ovolcl 25362	. . 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 25350	. . . . . . . . . . . 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 579	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
9	8	simp3d 1141	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))
10	8	simp1d 1139	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
11	8	simp2d 1140	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
12		ovolsca.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ+)
13	12	rpregt0d 13028	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
14	13	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
15		lediv1 12083	. . . . . . . . . . 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 1368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)) ↔ ((1st ‘(𝐹‘𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) / 𝐶)))
17	9, 16	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) / 𝐶))
18		df-br 5142	. . . . . . . . 9 ⊢ (((1st ‘(𝐹‘𝑛)) / 𝐶) ≤ ((2nd ‘(𝐹‘𝑛)) / 𝐶) ↔ ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ ≤ )
19	17, 18	sylib 217	. . . . . . . 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 13053	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛)) / 𝐶) ∈ ℝ)
22	11, 20	rerpdivcld 13053	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) / 𝐶) ∈ ℝ)
23	21, 22	opelxpd 5708	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ (ℝ × ℝ))
24	19, 23	elind 4189	. . . . . . 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 7109	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
27		eqid 2726	. . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
28		eqid 2726	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝐺)) = seq1( + , ((abs ∘ − ) ∘ 𝐺))
29	27, 28	ovolsf 25356	. . . . . 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 6719	. . . 4 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ (0[,)+∞))
32		icossxr 13415	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ*
33	31, 32	sstrdi 3989	. . 3 ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐺)) ⊆ ℝ*)
34		supxrcl 13300	. . 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 13053	. . . 4 ⊢ (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℝ)
38		ovolsca.9	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
39	38	rpred 13022	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ)
40	37, 39	readdcld 11247	. . 3 ⊢ (𝜑 → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ)
41	40	rexrd 11268	. 2 ⊢ (𝜑 → (((vol*‘𝐴) / 𝐶) + 𝑅) ∈ ℝ*)
42	1	eleq2d 2813	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}))
43		oveq2 7413	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
44	43	eleq1d 2812	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐶 · 𝑥) ∈ 𝐴 ↔ (𝐶 · 𝑦) ∈ 𝐴))
45	44	elrab 3678	. . . . . . 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 5145	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐶 · 𝑦) → ((1st ‘(𝐹‘𝑛)) < 𝑥 ↔ (1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦)))
48		breq1 5144	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐶 · 𝑦) → (𝑥 < (2nd ‘(𝐹‘𝑛)) ↔ (𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛))))
49	47, 48	anbi12d 630	. . . . . . . . . 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 3172	. . . . . . . . 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 25351	. . . . . . . . . . . 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 583	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ⊆ ∪ ran ((,) ∘ 𝐹) ↔ ∀𝑥 ∈ 𝐴 ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛)))))
55	51, 54	mpbid 231	. . . . . . . . . 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 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → (𝐶 · 𝑦) ∈ 𝐴)
58	50, 56, 57	rspcdva 3607	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) → ∃𝑛 ∈ ℕ ((1st ‘(𝐹‘𝑛)) < (𝐶 · 𝑦) ∧ (𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛))))
59		opex 5457	. . . . . . . . . . . . . . . 16 ⊢ ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩ ∈ V
60	25	fvmpt2 7003	. . . . . . . . . . . . . . . 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 688	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = ⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩)
62	61	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (1st ‘(𝐺‘𝑛)) = (1st ‘⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩))
63		ovex 7438	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘(𝐹‘𝑛)) / 𝐶) ∈ V
64		ovex 7438	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘(𝐹‘𝑛)) / 𝐶) ∈ V
65	63, 64	op1st 7982	. . . . . . . . . . . . . 14 ⊢ (1st ‘⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩) = ((1st ‘(𝐹‘𝑛)) / 𝐶)
66	62, 65	eqtrdi 2782	. . . . . . . . . . . . 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 5151	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐺‘𝑛)) < 𝑦 ↔ ((1st ‘(𝐹‘𝑛)) / 𝐶) < 𝑦))
69	10	adantlr 712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ)
70		simplrl 774	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑦 ∈ ℝ)
71	14	adantlr 712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (𝐶 ∈ ℝ ∧ 0 < 𝐶))
72		ltdivmul 12093	. . . . . . . . . . . 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 1368	. . . . . . . . . . 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 712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
76		ltmuldiv2 12092	. . . . . . . . . . . 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 1368	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ (𝐶 · 𝑦) ∈ 𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝐶 · 𝑦) < (2nd ‘(𝐹‘𝑛)) ↔ 𝑦 < ((2nd ‘(𝐹‘𝑛)) / 𝐶)))
78	61	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (2nd ‘(𝐺‘𝑛)) = (2nd ‘⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩))
79	63, 64	op2nd 7983	. . . . . . . . . . . . . 14 ⊢ (2nd ‘⟨((1st ‘(𝐹‘𝑛)) / 𝐶), ((2nd ‘(𝐹‘𝑛)) / 𝐶)⟩) = ((2nd ‘(𝐹‘𝑛)) / 𝐶)
80	78, 79	eqtrdi 2782	. . . . . . . . . . . . 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 5153	. . . . . . . . . . 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 630	. . . . . . . . 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 3170	. . . . . . . 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 231	. . . . . . 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 239	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛)))))
89	88	ralrimiv 3139	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℕ ((1st ‘(𝐺‘𝑛)) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐺‘𝑛))))
90		ovolfioo 25351	. . . . 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 583	. . . 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 25363	. . 3 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐵 ⊆ ∪ ran ((,) ∘ 𝐺)) → (vol*‘𝐵) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
94	26, 92, 93	syl2anc 583	. 2 ⊢ (𝜑 → (vol*‘𝐵) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝐺)), ℝ*, < ))
95		fzfid 13944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin)
96	12	rpcnd 13024	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
97	96	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐶 ∈ ℂ)
98		simpl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝜑)
99		elfznn 13536	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
100	11, 10	resubcld 11646	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ∈ ℝ)
101	98, 99, 100	syl2an 595	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ∈ ℝ)
102	101	recnd 11246	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ∈ ℂ)
103	12	rpne0d 13027	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≠ 0)
104	103	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐶 ≠ 0)
105	95, 97, 102, 104	fsumdivc 15738	. . . . . . 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 7423	. . . . . . . . . . 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 25354	. . . . . . . . . . 11 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
109	26, 108	sylan 579	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛))))
110	11	recnd 11246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℂ)
111	10	recnd 11246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℂ)
112	12	rpcnne0d 13031	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
113	112	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))
114		divsubdir 11912	. . . . . . . . . . 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 1368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) = (((2nd ‘(𝐹‘𝑛)) / 𝐶) − ((1st ‘(𝐹‘𝑛)) / 𝐶)))
116	107, 109, 115	3eqtr4d 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐺)‘𝑛) = (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶))
117	98, 99, 116	syl2an 595	. . . . . . . 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 12869	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
120	118, 119	eleqtrdi 2837	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
121	100, 20	rerpdivcld 13053	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) ∈ ℝ)
122	121	recnd 11246	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) ∈ ℂ)
123	98, 99, 122	syl2an 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) ∈ ℂ)
124	117, 120, 123	fsumser 15682	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)(((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) / 𝐶) = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘))
125	105, 124	eqtrd 2766	. . . . . 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 2726	. . . . . . . . . . . . . . . 16 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
128		ovolsca.5	. . . . . . . . . . . . . . . 16 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
129	127, 128	ovolsf 25356	. . . . . . . . . . . . . . 15 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
130	6, 129	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞))
131	130	frnd 6719	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
132	131, 32	sstrdi 3989	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*)
133	12, 38	rpmulcld 13038	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶 · 𝑅) ∈ ℝ+)
134	133	rpred 13022	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 · 𝑅) ∈ ℝ)
135	36, 134	readdcld 11247	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ)
136	135	rexrd 11268	. . . . . . . . . . . 12 ⊢ (𝜑 → ((vol*‘𝐴) + (𝐶 · 𝑅)) ∈ ℝ*)
137		supxrleub 13311	. . . . . . . . . . . 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 583	. . . . . . . . . . 11 ⊢ (𝜑 → (sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ ∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
139	126, 138	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝑆 𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
140	130	ffnd 6712	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 Fn ℕ)
141		breq1 5144	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑆‘𝑘) → (𝑥 ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)) ↔ (𝑆‘𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅))))
142	141	ralrn 7083	. . . . . . . . . . 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 231	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ ((vol*‘𝐴) + (𝐶 · 𝑅)))
145	144	r19.21bi 3242	. . . . . . . 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 25354	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
148	146, 99, 147	syl2an 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (((abs ∘ − ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))))
149	148, 120, 102	fsumser 15682	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘))
150	128	fveq1i 6886	. . . . . . . . 9 ⊢ (𝑆‘𝑘) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑘)
151	149, 150	eqtr4di 2784	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) = (𝑆‘𝑘))
152	37	recnd 11246	. . . . . . . . . . 11 ⊢ (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℂ)
153	38	rpcnd 13024	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℂ)
154	96, 152, 153	adddid 11242	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)) = ((𝐶 · ((vol*‘𝐴) / 𝐶)) + (𝐶 · 𝑅)))
155	36	recnd 11246	. . . . . . . . . . . 12 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℂ)
156	155, 96, 103	divcan2d 11996	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 · ((vol*‘𝐴) / 𝐶)) = (vol*‘𝐴))
157	156	oveq1d 7420	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶 · ((vol*‘𝐴) / 𝐶)) + (𝐶 · 𝑅)) = ((vol*‘𝐴) + (𝐶 · 𝑅)))
158	154, 157	eqtrd 2766	. . . . . . . . 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 5173	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑛 ∈ (1...𝑘)((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) ≤ (𝐶 · (((vol*‘𝐴) / 𝐶) + 𝑅)))
161	95, 101	fsumrecl 15686	. . . . . . . 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 12094	. . . . . . . 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 1368	. . . . . . 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 5165	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
168	167	ralrimiva 3140	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))
169	30	ffnd 6712	. . . . 5 ⊢ (𝜑 → seq1( + , ((abs ∘ − ) ∘ 𝐺)) Fn ℕ)
170		breq1 5144	. . . . . 6 ⊢ (𝑦 = (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) → (𝑦 ≤ (((vol*‘𝐴) / 𝐶) + 𝑅) ↔ (seq1( + , ((abs ∘ − ) ∘ 𝐺))‘𝑘) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅)))
171	170	ralrn 7083	. . . . 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 13311	. . . 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 583	. . 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 13147	1 ⊢ (𝜑 → (vol*‘𝐵) ≤ (((vol*‘𝐴) / 𝐶) + 𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2934  ∀wral 3055  ∃wrex 3064  {crab 3426  Vcvv 3468   ∩ cin 3942   ⊆ wss 3943  ⟨cop 4629  ∪ cuni 4902   class class class wbr 5141   ↦ cmpt 5224   × cxp 5667  ran crn 5670   ∘ ccom 5673   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7405  1^st c1st 7972  2^nd c2nd 7973  supcsup 9437  ℂcc 11110  ℝcr 11111  0cc0 11112  1c1 11113   + caddc 11115   · cmul 11117  +∞cpnf 11249  ℝ^*cxr 11251   < clt 11252   ≤ cle 11253   − cmin 11448   / cdiv 11875  ℕcn 12216  ℤ_≥cuz 12826  ℝ⁺crp 12980  (,)cioo 13330  [,)cico 13332  ...cfz 13490  seqcseq 13972  abscabs 15187  Σcsu 15638  vol*covol 25346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12981  df-ioo 13334  df-ico 13336  df-fz 13491  df-fzo 13634  df-seq 13973  df-exp 14033  df-hash 14296  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15438  df-sum 15639  df-ovol 25348

This theorem is referenced by:  ovolscalem2  25398