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Theorem ovoliunlem3 25028

Description: Lemma for ovoliun 25029. (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 2903	. . . 4 ⊢ Ⅎ𝑚𝐴
2		nfcsb1v 3918	. . . 4 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
3		csbeq1a 3907	. . . 4 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
4	1, 2, 3	cbviun 5039	. . 3 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴
5	4	fveq2i 6894	. 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 12287	. . . . . . . . 9 ⊢ 2 ∈ ℕ
10		nnnn0 12481	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
11		nnexpcl 14042	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
12	9, 10, 11	sylancr 587	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
13	12	nnrpd 13016	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
14		rpdivcl 13001	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ⁺ ∧ (2↑𝑛) ∈ ℝ⁺) → (𝐵 / (2↑𝑛)) ∈ ℝ⁺)
15	8, 13, 14	syl2an 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 / (2↑𝑛)) ∈ ℝ⁺)
16		eqid 2732	. . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
17	16	ovolgelb 25004	. . . . . 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 3146	. . . 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 7444	. . . . 5 ⊢ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∈ V
21		nnenom 13947	. . . . 5 ⊢ ℕ ≈ ω
22		coeq2 5858	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑛) → ((,) ∘ 𝑓) = ((,) ∘ (𝑔‘𝑛)))
23	22	rneqd 5937	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑛) → ran ((,) ∘ 𝑓) = ran ((,) ∘ (𝑔‘𝑛)))
24	23	unieqd 4922	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑛) → ∪ ran ((,) ∘ 𝑓) = ∪ ran ((,) ∘ (𝑔‘𝑛)))
25	24	sseq2d 4014	. . . . . 6 ⊢ (𝑓 = (𝑔‘𝑛) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ 𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛))))
26		coeq2 5858	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔‘𝑛) → ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ (𝑔‘𝑛)))
27	26	seqeq3d 13976	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑛) → seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))))
28	27	rneqd 5937	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑛) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))))
29	28	supeq1d 9443	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑛) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ))
30	29	breq1d 5158	. . . . . 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 10436	. . . 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 16156	. . . . . . 7 ⊢ (ℕ × ℕ) ≈ ℕ
35	34	ensymi 9002	. . . . . 6 ⊢ ℕ ≈ (ℕ × ℕ)
36		bren 8951	. . . . . 6 ⊢ (ℕ ≈ (ℕ × ℕ) ↔ ∃𝑗 𝑗:ℕ–1-1-onto→(ℕ × ℕ))
37	35, 36	mpbi 229	. . . . 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 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑚(vol*‘𝐴)
41		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol*
42	41, 2	nffv 6901	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
43	3	fveq2d 6895	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (vol‘𝐴) = (vol‘⦋𝑚 / 𝑛⦌𝐴))
44	40, 42, 43	cbvmpt 5259	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol‘⦋𝑚 / 𝑛⦌𝐴))
45	39, 44	eqtri 2760	. . . . . . . 8 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
46	6	ralrimiva 3146	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
47		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
48		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℝ
49	2, 48	nfss 3974	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
50	3	sseq1d 4013	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
51	47, 49, 50	cbvralw 3303	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
52	46, 51	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
53	52	r19.21bi 3248	. . . . . . . . 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 3146	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
56	40	nfel1 2919	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
57	42	nfel1 2919	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
58	43	eleq1d 2818	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol‘𝐴) ∈ ℝ ↔ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
59	56, 57, 58	cbvralw 3303	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
60	55, 59	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
61	60	r19.21bi 3248	. . . . . . . . 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 2732	. . . . . . . 8 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚)))
67		eqid 2732	. . . . . . . 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 2732	. . . . . . . 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 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))
73		nfcv 2903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛∪ ran ((,) ∘ (𝑔‘𝑚))
74	2, 73	nfss 3974	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚))
75		nfcv 2903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^*, < )
76		nfcv 2903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 ≤
77		nfcv 2903	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛 +
78		nfcv 2903	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝐵 / (2↑𝑚))
79	42, 77, 78	nfov 7441	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((vol*‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚)))
80	75, 76, 79	nfbr 5195	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚)))
81	74, 80	nfan 1902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚)) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ) ≤ ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚))))
82		fveq2 6891	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝑔‘𝑛) = (𝑔‘𝑚))
83	82	coeq2d 5862	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → ((,) ∘ (𝑔‘𝑛)) = ((,) ∘ (𝑔‘𝑚)))
84	83	rneqd 5937	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ran ((,) ∘ (𝑔‘𝑛)) = ran ((,) ∘ (𝑔‘𝑚)))
85	84	unieqd 4922	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ∪ ran ((,) ∘ (𝑔‘𝑛)) = ∪ ran ((,) ∘ (𝑔‘𝑚)))
86	3, 85	sseq12d 4015	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑛)) ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚))))
87	82	coeq2d 5862	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → ((abs ∘ − ) ∘ (𝑔‘𝑛)) = ((abs ∘ − ) ∘ (𝑔‘𝑚)))
88	87	seqeq3d 13976	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))) = seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))))
89	88	rneqd 5937	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))) = ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))))
90	89	supeq1d 9443	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑛))), ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑔‘𝑚))), ℝ^, < ))
91		oveq2 7419	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚))
92	91	oveq2d 7427	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐵 / (2↑𝑛)) = (𝐵 / (2↑𝑚)))
93	43, 92	oveq12d 7429	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((vol‘𝐴) + (𝐵 / (2↑𝑛))) = ((vol‘⦋𝑚 / 𝑛⦌𝐴) + (𝐵 / (2↑𝑚))))
94	90, 93	breq12d 5161	. . . . . . . . . . . . 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 3303	. . . . . . . . . . 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 217	. . . . . . . . . 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 3248	. . . . . . . . 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 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) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ∪ ran ((,) ∘ (𝑔‘𝑚)))
100	98	simprd 496	. . . . . . . 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 25027	. . . . . . 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 420	. . . . . 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 1936	. . . . 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 1936	. . 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 5183	1 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ⦋csb 3893 ∩ cin 3947 ⊆ wss 3948 ∪ cuni 4908 ∪ ciun 4997 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ran crn 5677 ∘ ccom 5680 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 1^st c1st 7975 2^nd c2nd 7976 ↑_m cmap 8822 ≈ cen 8938 supcsup 9437 ℝcr 11111 1c1 11113 + caddc 11115 ℝ^*cxr 11249 < clt 11250 ≤ cle 11251 − cmin 11446 / cdiv 11873 ℕcn 12214 2c2 12269 ℕ₀cn0 12474 ℝ⁺crp 12976 (,)cioo 13326 seqcseq 13968 ↑cexp 14029 abscabs 15183 vol*covol 24986

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cc 10432 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 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 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 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-z 12561 df-uz 12825 df-rp 12977 df-ioo 13330 df-ico 13332 df-fz 13487 df-fzo 13630 df-fl 13759 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-rlim 15435 df-sum 15635 df-ovol 24988

This theorem is referenced by: ovoliun 25029

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