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Theorem ovoliunlem3 24108

Description: Lemma for ovoliun 24109. (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 2955	. . . 4 ⊢ Ⅎ𝑚𝐴
2		nfcsb1v 3852	. . . 4 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
3		csbeq1a 3842	. . . 4 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
4	1, 2, 3	cbviun 4923	. . 3 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴
5	4	fveq2i 6648	. 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 11698	. . . . . . . . 9 ⊢ 2 ∈ ℕ
10		nnnn0 11892	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
11		nnexpcl 13438	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
12	9, 10, 11	sylancr 590	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
13	12	nnrpd 12417	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
14		rpdivcl 12402	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ⁺ ∧ (2↑𝑛) ∈ ℝ⁺) → (𝐵 / (2↑𝑛)) ∈ ℝ⁺)
15	8, 13, 14	syl2an 598	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ⁺)
16		eqid 2798	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
17	16	ovolgelb 24084	. . . . . 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 1368	. . . . 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 3149	. . . 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 7168	. . . . 5 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∈ V
21		nnenom 13343	. . . . 5 ⊢ ℕ ≈ ω
22		coeq2 5693	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑛) → ((,) ∘ 𝑓) = ((,) ∘ (𝑔‘𝑛)))
23	22	rneqd 5772	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑛) → ran ((,) ∘ 𝑓) = ran ((,) ∘ (𝑔‘𝑛)))
24	23	unieqd 4814	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑛) → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ (𝑔‘𝑛)))
25	24	sseq2d 3947	. . . . . 6 ⊢ (𝑓 = (𝑔‘𝑛) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛))))
26		coeq2 5693	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔‘𝑛) → ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ (𝑔‘𝑛)))
27	26	seqeq3d 13372	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑛) → seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))))
28	27	rneqd 5772	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑛) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))))
29	28	supeq1d 8894	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑛) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ))
30	29	breq1d 5040	. . . . . 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 633	. . . . 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 9850	. . . 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 15556	. . . . . . 7 ⊢ (ℕ × ℕ) ≈ ℕ
35	34	ensymi 8542	. . . . . 6 ⊢ ℕ ≈ (ℕ × ℕ)
36		bren 8501	. . . . . 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 2955	. . . . . . . . . 10 ⊢ Ⅎ𝑚(vol*‘𝐴)
41		nfcv 2955	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol*
42	41, 2	nffv 6655	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
43	3	fveq2d 6649	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (vol‘𝐴) = (vol‘⦋𝑚 / 𝑛⦌𝐴))
44	40, 42, 43	cbvmpt 5131	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol‘⦋𝑚 / 𝑛⦌𝐴))
45	39, 44	eqtri 2821	. . . . . . . 8 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
46	6	ralrimiva 3149	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
47		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
48		nfcv 2955	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℝ
49	2, 48	nfss 3907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
50	3	sseq1d 3946	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
51	47, 49, 50	cbvralw 3387	. . . . . . . . . . 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 3173	. . . . . . . . 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 3149	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
56	40	nfel1 2971	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
57	42	nfel1 2971	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
58	43	eleq1d 2874	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol‘𝐴) ∈ ℝ ↔ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
59	56, 57, 58	cbvralw 3387	. . . . . . . . . . 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 3173	. . . . . . . . 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 2798	. . . . . . . 8 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚)))
67		eqid 2798	. . . . . . . 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 2798	. . . . . . . 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 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))
73		nfcv 2955	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛∪ ran ((,) ∘ (𝑔‘𝑚))
74	2, 73	nfss 3907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚))
75		nfcv 2955	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^*, < )
76		nfcv 2955	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 ≤
77		nfcv 2955	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛 +
78		nfcv 2955	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝐵 / (2↑𝑚))
79	42, 77, 78	nfov 7165	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((vol*‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚)))
80	75, 76, 79	nfbr 5077	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚)))
81	74, 80	nfan 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚))))
82		fveq2 6645	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝑔‘𝑛) = (𝑔‘𝑚))
83	82	coeq2d 5697	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → ((,) ∘ (𝑔‘𝑛)) = ((,) ∘ (𝑔‘𝑚)))
84	83	rneqd 5772	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ran ((,) ∘ (𝑔‘𝑛)) = ran ((,) ∘ (𝑔‘𝑚)))
85	84	unieqd 4814	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ∪ ran ((,) ∘ (𝑔‘𝑛)) = ∪ ran ((,) ∘ (𝑔‘𝑚)))
86	3, 85	sseq12d 3948	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚))))
87	82	coeq2d 5697	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → ((abs ∘ − ) ∘ (𝑔‘𝑛)) = ((abs ∘ − ) ∘ (𝑔‘𝑚)))
88	87	seqeq3d 13372	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))))
89	88	rneqd 5772	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))))
90	89	supeq1d 8894	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ))
91		oveq2 7143	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
92	91	oveq2d 7151	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐵 / (2↑𝑛)) = (𝐵 / (2↑𝑚)))
93	43, 92	oveq12d 7153	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((vol‘𝐴) + (𝐵 / (2↑𝑛))) = ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚))))
94	90, 93	breq12d 5043	. . . . . . . . . . . . 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 633	. . . . . . . . . . . 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 3387	. . . . . . . . . . 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 3173	. . . . . . . . 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 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) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚)))
100	98	simprd 499	. . . . . . . 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 24107	. . . . . . 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 423	. . . . . 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 1934	. . . . 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 1934	. . 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 5065	1 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ⦋csb 3828 ∩ cin 3880 ⊆ wss 3881 ∪ cuni 4800 ∪ ciun 4881 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 ran crn 5520 ∘ ccom 5523 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 1^st c1st 7669 2^nd c2nd 7670 ↑_m cmap 8389 ≈ cen 8489 supcsup 8888 ℝcr 10525 1c1 10527 + caddc 10529 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 ℝ⁺crp 12377 (,)cioo 12726 seqcseq 13364 ↑cexp 13425 abscabs 14585 vol*covol 24066

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ioo 12730 df-ico 12732 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-ovol 24068

This theorem is referenced by: ovoliun 24109

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