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Theorem ovoliunlem3 25825

Description: Lemma for ovoliun 25826. (Contributed by Mario Carneiro, 12-Jun-2014.)

**Hypotheses**
Ref	Expression
ovoliun.t	⊢ 𝑇 = seq1( + , 𝐺)
ovoliun.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
ovoliun.a	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
ovoliun.v	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
ovoliun.r	⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) ∈ ℝ)
ovoliun.b	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)

**Assertion**
Ref	Expression
ovoliunlem3	⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵))

Distinct variable groups: 𝐵,𝑛 𝜑,𝑛 𝑛,𝐺 𝑇,𝑛 Allowed substitution hint: 𝐴(𝑛)

Proof of Theorem ovoliunlem3 Dummy variables 𝑓 𝑔 𝑗 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2922	. . . 4 ⊢ Ⅎ𝑚𝐴
2		nfcsb1v 3870	. . . 4 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
3		csbeq1a 3860	. . . 4 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
4	1, 2, 3	cbviun 4992	. . 3 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴
5	4	fveq2i 6888	. 2 ⊢ (vol‘∪ 𝑛 ∈ ℕ 𝐴) = (vol‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴)
6		ovoliun.a	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
7		ovoliun.v	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
8		ovoliun.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
9		2nn 12416	. . . . . . . . 9 ⊢ 2 ∈ ℕ
10		nnnn0 12613	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
11		nnexpcl 14217	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
12	9, 10, 11	sylancr 599	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
13	12	nnrpd 13162	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
14		rpdivcl 13147	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ⁺ ∧ (2↑𝑛) ∈ ℝ⁺) → (𝐵 / (2↑𝑛)) ∈ ℝ⁺)
15	8, 13, 14	syl2an 608	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ⁺)
16		eqid 2760	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
17	16	ovolgelb 25801	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ (𝐵 / (2↑𝑛)) ∈ ℝ⁺) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛)))))
18	6, 7, 15, 17	syl3anc 1398	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))
19	18	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))
20		ovex 7453	. . . . 5 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∈ V
21		nnenom 14123	. . . . 5 ⊢ ℕ ≈ ω
22		coeq2 5835	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑛) → ((,) ∘ 𝑓) = ((,) ∘ (𝑔‘𝑛)))
23	22	rneqd 5919	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑛) → ran ((,) ∘ 𝑓) = ran ((,) ∘ (𝑔‘𝑛)))
24	23	unieqd 4879	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑛) → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ (𝑔‘𝑛)))
25	24	sseq2d 3940	. . . . . 6 ⊢ (𝑓 = (𝑔‘𝑛) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛))))
26		coeq2 5835	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔‘𝑛) → ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ (𝑔‘𝑛)))
27	26	seqeq3d 14152	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑛) → seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))))
28	27	rneqd 5919	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑛) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))))
29	28	supeq1d 9438	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑛) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ))
30	29	breq1d 5112	. . . . . 6 ⊢ (𝑓 = (𝑔‘𝑛) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))) ↔ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))
31	25, 30	anbi12d 644	. . . . 5 ⊢ (𝑓 = (𝑔‘𝑛) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))))
32	20, 21, 31	axcc4 10517	. . . 4 ⊢ (∀𝑛 ∈ ℕ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))) → ∃𝑔(𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))))
33	19, 32	syl 18	. . 3 ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))))
34		xpnnen 16379	. . . . . . 7 ⊢ (ℕ × ℕ) ≈ ℕ
35	34	ensymi 9031	. . . . . 6 ⊢ ℕ ≈ (ℕ × ℕ)
36		bren 8983	. . . . . 6 ⊢ (ℕ ≈ (ℕ × ℕ) ↔ ∃𝑗 𝑗:ℕ–1-1-onto→(ℕ × ℕ))
37	35, 36	mpbi 233	. . . . 5 ⊢ ∃𝑗 𝑗:ℕ–1-1-onto→(ℕ × ℕ)
38		ovoliun.t	. . . . . . . 8 ⊢ 𝑇 = seq1( + , 𝐺)
39		ovoliun.g	. . . . . . . . 9 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
40		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑚(vol*‘𝐴)
41		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol*
42	41, 2	nffv 6895	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
43	3	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (vol‘𝐴) = (vol‘⦋𝑚 / 𝑛⦌𝐴))
44	40, 42, 43	cbvmpt 5206	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol‘⦋𝑚 / 𝑛⦌𝐴))
45	39, 44	eqtri 2783	. . . . . . . 8 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
46	6	ralrimiva 3154	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
47		nfv 1946	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
48		nfcv 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℝ
49	2, 48	nfss 3901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
50	3	sseq1d 3939	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
51	47, 49, 50	cbvralw 3304	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
52	46, 51	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
53	52	r19.21bi 3254	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
54	53	ad4ant14 765	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))) ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
55	7	ralrimiva 3154	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
56	40	nfel1 2938	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
57	42	nfel1 2938	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
58	43	eleq1d 2845	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol‘𝐴) ∈ ℝ ↔ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
59	56, 57, 58	cbvralw 3304	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
60	55, 59	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
61	60	r19.21bi 3254	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
62	61	ad4ant14 765	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))) ∧ 𝑚 ∈ ℕ) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
63		ovoliun.r	. . . . . . . . 9 ⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) ∈ ℝ)
64	63	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))) → sup(ran 𝑇, ℝ^*, < ) ∈ ℝ)
65	8	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))) → 𝐵 ∈ ℝ⁺)
66		eqid 2760	. . . . . . . 8 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚)))
67		eqid 2760	. . . . . . . 8 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑘 ∈ ℕ ↦ ((𝑔‘(1^st ‘(𝑗‘𝑘)))‘(2^nd ‘(𝑗‘𝑘)))))) = seq1( + , ((abs ∘ − ) ∘ (𝑘 ∈ ℕ ↦ ((𝑔‘(1^st ‘(𝑗‘𝑘)))‘(2^nd ‘(𝑗‘𝑘))))))
68		eqid 2760	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ ↦ ((𝑔‘(1^st ‘(𝑗‘𝑘)))‘(2^nd ‘(𝑗‘𝑘)))) = (𝑘 ∈ ℕ ↦ ((𝑔‘(1^st ‘(𝑗‘𝑘)))‘(2^nd ‘(𝑗‘𝑘))))
69		simplr 781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))) → 𝑗:ℕ–1-1-onto→(ℕ × ℕ))
70		simprl 783	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))) → 𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
71		simprr 785	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))) → ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))
72		nfv 1946	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))
73		nfcv 2922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛∪ ran ((,) ∘ (𝑔‘𝑚))
74	2, 73	nfss 3901	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚))
75		nfcv 2922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^*, < )
76		nfcv 2922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 ≤
77		nfcv 2922	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛 +
78		nfcv 2922	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝐵 / (2↑𝑚))
79	42, 77, 78	nfov 7450	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((vol*‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚)))
80	75, 76, 79	nfbr 5151	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚)))
81	74, 80	nfan 1931	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚))))
82		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝑔‘𝑛) = (𝑔‘𝑚))
83	82	coeq2d 5839	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → ((,) ∘ (𝑔‘𝑛)) = ((,) ∘ (𝑔‘𝑚)))
84	83	rneqd 5919	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ran ((,) ∘ (𝑔‘𝑛)) = ran ((,) ∘ (𝑔‘𝑚)))
85	84	unieqd 4879	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ∪ ran ((,) ∘ (𝑔‘𝑛)) = ∪ ran ((,) ∘ (𝑔‘𝑚)))
86	3, 85	sseq12d 3941	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚))))
87	82	coeq2d 5839	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → ((abs ∘ − ) ∘ (𝑔‘𝑛)) = ((abs ∘ − ) ∘ (𝑔‘𝑚)))
88	87	seqeq3d 14152	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))))
89	88	rneqd 5919	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))))
90	89	supeq1d 9438	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ))
91		oveq2 7428	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
92	91	oveq2d 7436	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐵 / (2↑𝑛)) = (𝐵 / (2↑𝑚)))
93	43, 92	oveq12d 7438	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((vol‘𝐴) + (𝐵 / (2↑𝑛))) = ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚))))
94	90, 93	breq12d 5115	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))) ↔ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚)))))
95	86, 94	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))) ↔ (⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚))))))
96	72, 81, 95	cbvralw 3304	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))) ↔ ∀𝑚 ∈ ℕ (⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚)))))
97	71, 96	sylib 221	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))) → ∀𝑚 ∈ ℕ (⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚)))))
98	97	r19.21bi 3254	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))) ∧ 𝑚 ∈ ℕ) → (⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚)))))
99	98	simpld 500	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))) ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚)))
100	98	simprd 501	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))) ∧ 𝑚 ∈ ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚))))
101	38, 45, 54, 62, 64, 65, 66, 67, 68, 69, 70, 99, 100	ovoliunlem2 25824	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗:ℕ–1-1-onto→(ℕ × ℕ)) ∧ (𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛)))))) → (vol‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵))
102	101	exp31 425	. . . . . 6 ⊢ (𝜑 → (𝑗:ℕ–1-1-onto→(ℕ × ℕ) → ((𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))) → (vol‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵))))
103	102	exlimdv 1965	. . . . 5 ⊢ (𝜑 → (∃𝑗 𝑗:ℕ–1-1-onto→(ℕ × ℕ) → ((𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))) → (vol‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵))))
104	37, 103	mpi 21	. . . 4 ⊢ (𝜑 → ((𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))) → (vol‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵)))
105	104	exlimdv 1965	. . 3 ⊢ (𝜑 → (∃𝑔(𝑔:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))) → (vol‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵)))
106	33, 105	mpd 16	. 2 ⊢ (𝜑 → (vol‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵))
107	5, 106	eqbrtrid 5139	1 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∃wex 1811 ∈ wcel 2144 ∀wral 3076 ∃wrex 3086 ⦋csb 3846 ⊆ wss 3895 ∩ cin 4089 ∪ cuni 4866 ∪ ciun 4950 class class class wbr 5102 ↦ cmpt 5185 × cxp 5648 ran crn 5651 ∘ ccom 5654 ⟶wf 6534 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 2^nd c2nd 8000 ↑_m cmap 8847 ≈ cen 8970 supcsup 9432 ℝcr 11199 1c1 11201 + caddc 11203 ℝ^*cxr 11342 < clt 11343 ≤ cle 11344 − cmin 11541 / cdiv 11973 ℕcn 12335 2c2 12397 ℕ₀cn0 12606 ℝ⁺crp 13120 (,)cioo 13476 seqcseq 14144 ↑cexp 14204 abscabs 15401 vol*covol 25783

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cc 10513 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-map 8849 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-n0 12607 df-z 12694 df-uz 12966 df-rp 13121 df-ioo 13480 df-ico 13482 df-fz 13640 df-fzo 13789 df-fl 13932 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-rlim 15656 df-sum 15854 df-ovol 25785

This theorem is used by: ovoliun 25826

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