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Theorem ovoliunlem3 25558

Description: Lemma for ovoliun 25559. (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 2908	. . . 4 ⊢ Ⅎ𝑚𝐴
2		nfcsb1v 3946	. . . 4 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
3		csbeq1a 3935	. . . 4 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
4	1, 2, 3	cbviun 5059	. . 3 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴
5	4	fveq2i 6923	. 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 12366	. . . . . . . . 9 ⊢ 2 ∈ ℕ
10		nnnn0 12560	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
11		nnexpcl 14125	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
12	9, 10, 11	sylancr 586	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
13	12	nnrpd 13097	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
14		rpdivcl 13082	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ⁺ ∧ (2↑𝑛) ∈ ℝ⁺) → (𝐵 / (2↑𝑛)) ∈ ℝ⁺)
15	8, 13, 14	syl2an 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ⁺)
16		eqid 2740	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
17	16	ovolgelb 25534	. . . . . 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 1371	. . . . 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 3152	. . . 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 7481	. . . . 5 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∈ V
21		nnenom 14031	. . . . 5 ⊢ ℕ ≈ ω
22		coeq2 5883	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑛) → ((,) ∘ 𝑓) = ((,) ∘ (𝑔‘𝑛)))
23	22	rneqd 5963	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑛) → ran ((,) ∘ 𝑓) = ran ((,) ∘ (𝑔‘𝑛)))
24	23	unieqd 4944	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑛) → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ (𝑔‘𝑛)))
25	24	sseq2d 4041	. . . . . 6 ⊢ (𝑓 = (𝑔‘𝑛) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛))))
26		coeq2 5883	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔‘𝑛) → ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ (𝑔‘𝑛)))
27	26	seqeq3d 14060	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑛) → seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))))
28	27	rneqd 5963	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑛) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))))
29	28	supeq1d 9515	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑛) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ))
30	29	breq1d 5176	. . . . . 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 631	. . . . 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 10508	. . . 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 17	. . 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 16259	. . . . . . 7 ⊢ (ℕ × ℕ) ≈ ℕ
35	34	ensymi 9064	. . . . . 6 ⊢ ℕ ≈ (ℕ × ℕ)
36		bren 9013	. . . . . 6 ⊢ (ℕ ≈ (ℕ × ℕ) ↔ ∃𝑗 𝑗:ℕ–1-1-onto→(ℕ × ℕ))
37	35, 36	mpbi 230	. . . . 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 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑚(vol*‘𝐴)
41		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol*
42	41, 2	nffv 6930	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
43	3	fveq2d 6924	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (vol‘𝐴) = (vol‘⦋𝑚 / 𝑛⦌𝐴))
44	40, 42, 43	cbvmpt 5277	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol‘⦋𝑚 / 𝑛⦌𝐴))
45	39, 44	eqtri 2768	. . . . . . . 8 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
46	6	ralrimiva 3152	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
47		nfv 1913	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
48		nfcv 2908	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℝ
49	2, 48	nfss 4001	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
50	3	sseq1d 4040	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
51	47, 49, 50	cbvralw 3312	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
52	46, 51	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
53	52	r19.21bi 3257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
54	53	ad4ant14 751	. . . . . . . 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 3152	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
56	40	nfel1 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
57	42	nfel1 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
58	43	eleq1d 2829	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol‘𝐴) ∈ ℝ ↔ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
59	56, 57, 58	cbvralw 3312	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
60	55, 59	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
61	60	r19.21bi 3257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
62	61	ad4ant14 751	. . . . . . . 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 725	. . . . . . . 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 725	. . . . . . . 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 2740	. . . . . . . 8 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚)))
67		eqid 2740	. . . . . . . 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 2740	. . . . . . . 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 768	. . . . . . . 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 770	. . . . . . . 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 772	. . . . . . . . . . 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 1913	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))
73		nfcv 2908	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛∪ ran ((,) ∘ (𝑔‘𝑚))
74	2, 73	nfss 4001	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚))
75		nfcv 2908	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^*, < )
76		nfcv 2908	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 ≤
77		nfcv 2908	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛 +
78		nfcv 2908	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝐵 / (2↑𝑚))
79	42, 77, 78	nfov 7478	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((vol*‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚)))
80	75, 76, 79	nfbr 5213	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚)))
81	74, 80	nfan 1898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚))))
82		fveq2 6920	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝑔‘𝑛) = (𝑔‘𝑚))
83	82	coeq2d 5887	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → ((,) ∘ (𝑔‘𝑛)) = ((,) ∘ (𝑔‘𝑚)))
84	83	rneqd 5963	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ran ((,) ∘ (𝑔‘𝑛)) = ran ((,) ∘ (𝑔‘𝑚)))
85	84	unieqd 4944	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ∪ ran ((,) ∘ (𝑔‘𝑛)) = ∪ ran ((,) ∘ (𝑔‘𝑚)))
86	3, 85	sseq12d 4042	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚))))
87	82	coeq2d 5887	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → ((abs ∘ − ) ∘ (𝑔‘𝑛)) = ((abs ∘ − ) ∘ (𝑔‘𝑚)))
88	87	seqeq3d 14060	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))))
89	88	rneqd 5963	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))))
90	89	supeq1d 9515	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ))
91		oveq2 7456	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
92	91	oveq2d 7464	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐵 / (2↑𝑛)) = (𝐵 / (2↑𝑚)))
93	43, 92	oveq12d 7466	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((vol‘𝐴) + (𝐵 / (2↑𝑛))) = ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚))))
94	90, 93	breq12d 5179	. . . . . . . . . . . . 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 631	. . . . . . . . . . . 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 3312	. . . . . . . . . . 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 218	. . . . . . . . . 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 3257	. . . . . . . . 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 494	. . . . . . . 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 495	. . . . . . . 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 25557	. . . . . . 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 419	. . . . . 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 1932	. . . . 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 20	. . . 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 1932	. . 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 15	. 2 ⊢ (𝜑 → (vol‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵))
107	5, 106	eqbrtrid 5201	1 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∃wex 1777 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 ⦋csb 3921 ∩ cin 3975 ⊆ wss 3976 ∪ cuni 4931 ∪ ciun 5015 class class class wbr 5166 ↦ cmpt 5249 × cxp 5698 ran crn 5701 ∘ ccom 5704 ⟶wf 6569 –1-1-onto→wf1o 6572 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 ↑_m cmap 8884 ≈ cen 9000 supcsup 9509 ℝcr 11183 1c1 11185 + caddc 11187 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℝ⁺crp 13057 (,)cioo 13407 seqcseq 14052 ↑cexp 14112 abscabs 15283 vol*covol 25516

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-ioo 13411 df-ico 13413 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-rlim 15535 df-sum 15735 df-ovol 25518

This theorem is referenced by: ovoliun 25559

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