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Theorem ovoliunlem3 24104

Description: Lemma for ovoliun 24105. (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 2977	. . . 4 ⊢ Ⅎ𝑚𝐴
2		nfcsb1v 3906	. . . 4 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
3		csbeq1a 3896	. . . 4 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
4	1, 2, 3	cbviun 4960	. . 3 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴
5	4	fveq2i 6672	. 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 11709	. . . . . . . . 9 ⊢ 2 ∈ ℕ
10		nnnn0 11903	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
11		nnexpcl 13441	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
12	9, 10, 11	sylancr 589	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
13	12	nnrpd 12428	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
14		rpdivcl 12413	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ⁺ ∧ (2↑𝑛) ∈ ℝ⁺) → (𝐵 / (2↑𝑛)) ∈ ℝ⁺)
15	8, 13, 14	syl2an 597	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ⁺)
16		eqid 2821	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
17	16	ovolgelb 24080	. . . . . 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 1367	. . . . 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 3182	. . . 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 7188	. . . . 5 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∈ V
21		nnenom 13347	. . . . 5 ⊢ ℕ ≈ ω
22		coeq2 5728	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑛) → ((,) ∘ 𝑓) = ((,) ∘ (𝑔‘𝑛)))
23	22	rneqd 5807	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑛) → ran ((,) ∘ 𝑓) = ran ((,) ∘ (𝑔‘𝑛)))
24	23	unieqd 4851	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑛) → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ (𝑔‘𝑛)))
25	24	sseq2d 3998	. . . . . 6 ⊢ (𝑓 = (𝑔‘𝑛) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛))))
26		coeq2 5728	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔‘𝑛) → ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ (𝑔‘𝑛)))
27	26	seqeq3d 13376	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑛) → seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))))
28	27	rneqd 5807	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑛) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))))
29	28	supeq1d 8909	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑛) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ))
30	29	breq1d 5075	. . . . . 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 632	. . . . 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 9860	. . . 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 15563	. . . . . . 7 ⊢ (ℕ × ℕ) ≈ ℕ
35	34	ensymi 8558	. . . . . 6 ⊢ ℕ ≈ (ℕ × ℕ)
36		bren 8517	. . . . . 6 ⊢ (ℕ ≈ (ℕ × ℕ) ↔ ∃𝑗 𝑗:ℕ–1-1-onto→(ℕ × ℕ))
37	35, 36	mpbi 232	. . . . 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 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑚(vol*‘𝐴)
41		nfcv 2977	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol*
42	41, 2	nffv 6679	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
43	3	fveq2d 6673	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (vol‘𝐴) = (vol‘⦋𝑚 / 𝑛⦌𝐴))
44	40, 42, 43	cbvmpt 5166	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol‘⦋𝑚 / 𝑛⦌𝐴))
45	39, 44	eqtri 2844	. . . . . . . 8 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
46	6	ralrimiva 3182	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
47		nfv 1911	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
48		nfcv 2977	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℝ
49	2, 48	nfss 3959	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
50	3	sseq1d 3997	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
51	47, 49, 50	cbvralw 3441	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
52	46, 51	sylib 220	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
53	52	r19.21bi 3208	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
54	53	ad4ant14 750	. . . . . . . 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 3182	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
56	40	nfel1 2994	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
57	42	nfel1 2994	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
58	43	eleq1d 2897	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol‘𝐴) ∈ ℝ ↔ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
59	56, 57, 58	cbvralw 3441	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
60	55, 59	sylib 220	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
61	60	r19.21bi 3208	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
62	61	ad4ant14 750	. . . . . . . 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 724	. . . . . . . 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 724	. . . . . . . 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 2821	. . . . . . . 8 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚)))
67		eqid 2821	. . . . . . . 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 2821	. . . . . . . 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 767	. . . . . . . 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 769	. . . . . . . 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 771	. . . . . . . . . . 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 1911	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))
73		nfcv 2977	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛∪ ran ((,) ∘ (𝑔‘𝑚))
74	2, 73	nfss 3959	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚))
75		nfcv 2977	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^*, < )
76		nfcv 2977	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 ≤
77		nfcv 2977	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛 +
78		nfcv 2977	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝐵 / (2↑𝑚))
79	42, 77, 78	nfov 7185	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((vol*‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚)))
80	75, 76, 79	nfbr 5112	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚)))
81	74, 80	nfan 1896	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚))))
82		fveq2 6669	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝑔‘𝑛) = (𝑔‘𝑚))
83	82	coeq2d 5732	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → ((,) ∘ (𝑔‘𝑛)) = ((,) ∘ (𝑔‘𝑚)))
84	83	rneqd 5807	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ran ((,) ∘ (𝑔‘𝑛)) = ran ((,) ∘ (𝑔‘𝑚)))
85	84	unieqd 4851	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ∪ ran ((,) ∘ (𝑔‘𝑛)) = ∪ ran ((,) ∘ (𝑔‘𝑚)))
86	3, 85	sseq12d 3999	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚))))
87	82	coeq2d 5732	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → ((abs ∘ − ) ∘ (𝑔‘𝑛)) = ((abs ∘ − ) ∘ (𝑔‘𝑚)))
88	87	seqeq3d 13376	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))))
89	88	rneqd 5807	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))))
90	89	supeq1d 8909	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ))
91		oveq2 7163	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
92	91	oveq2d 7171	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐵 / (2↑𝑛)) = (𝐵 / (2↑𝑚)))
93	43, 92	oveq12d 7173	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((vol‘𝐴) + (𝐵 / (2↑𝑛))) = ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚))))
94	90, 93	breq12d 5078	. . . . . . . . . . . . 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 632	. . . . . . . . . . . 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 3441	. . . . . . . . . . 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 220	. . . . . . . . . 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 3208	. . . . . . . . 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 497	. . . . . . . 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 498	. . . . . . . 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 24103	. . . . . . 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 422	. . . . . 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 1930	. . . . 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 1930	. . 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 5100	1 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ⦋csb 3882 ∩ cin 3934 ⊆ wss 3935 ∪ cuni 4837 ∪ ciun 4918 class class class wbr 5065 ↦ cmpt 5145 × cxp 5552 ran crn 5555 ∘ ccom 5558 ⟶wf 6350 –1-1-onto→wf1o 6353 ‘cfv 6354 (class class class)co 7155 1^st c1st 7686 2^nd c2nd 7687 ↑_m cmap 8405 ≈ cen 8505 supcsup 8903 ℝcr 10535 1c1 10537 + caddc 10539 ℝ^*cxr 10673 < clt 10674 ≤ cle 10675 − cmin 10869 / cdiv 11296 ℕcn 11637 2c2 11691 ℕ₀cn0 11896 ℝ⁺crp 12388 (,)cioo 12737 seqcseq 13368 ↑cexp 13428 abscabs 14592 vol*covol 24062

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cc 9856 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-ioo 12741 df-ico 12743 df-fz 12892 df-fzo 13033 df-fl 13161 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-rlim 14845 df-sum 15042 df-ovol 24064

This theorem is referenced by: ovoliun 24105

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