Theorem ovnovollem3 47668

Description: The 1-dimensional Lebesgue outer measure agrees with the Lebesgue outer measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
ovnovollem3.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ovnovollem3.b	⊢ (𝜑 → 𝐵 ⊆ ℝ)
ovnovollem3.m	⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ)((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}
ovnovollem3.n	⊢ 𝑁 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}

Assertion

Ref	Expression
ovnovollem3	⊢ (𝜑 → ((voln*‘{𝐴})‘(𝐵 ↑m {𝐴})) = (vol*‘𝐵))

Distinct variable groups:   𝐴,𝑓,𝑖,𝑗,𝑘,𝑧   𝐵,𝑓,𝑖,𝑗,𝑘,𝑧   𝑧,𝑁   𝑘,𝑉   𝜑,𝑓,𝑖,𝑗,𝑘,𝑧

Allowed substitution hints:   𝑀(𝑧, 𝑓, 𝑖, 𝑗, 𝑘)   𝑁(𝑓, 𝑖, 𝑗, 𝑘)   𝑉(𝑧, 𝑓, 𝑖, 𝑗)

Proof of Theorem ovnovollem3

Dummy variables 𝑛 𝑙 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovnovollem3.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2	1	snn0d 4735	. . . 4 ⊢ (𝜑 → {𝐴} ≠ ∅)
3	2	neneqd 2960	. . 3 ⊢ (𝜑 → ¬ {𝐴} = ∅)
4	3	iffalsed 4492	. 2 ⊢ (𝜑 → if({𝐴} = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
5		snfi 9071	. . . 4 ⊢ {𝐴} ∈ Fin
6	5	a1i 11	. . 3 ⊢ (𝜑 → {𝐴} ∈ Fin)
7		reex 11291	. . . . 5 ⊢ ℝ ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ V)
9		ovnovollem3.b	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
10		mapss 8917	. . . 4 ⊢ ((ℝ ∈ V ∧ 𝐵 ⊆ ℝ) → (𝐵 ↑m {𝐴}) ⊆ (ℝ ↑m {𝐴}))
11	8, 9, 10	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐵 ↑m {𝐴}) ⊆ (ℝ ↑m {𝐴}))
12		ovnovollem3.m	. . 3 ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ)((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}
13	6, 11, 12	ovnval2 47555	. 2 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝐵 ↑m {𝐴})) = if({𝐴} = ∅, 0, inf(𝑀, ℝ*, < )))
14		ovnovollem3.n	. . . 4 ⊢ 𝑁 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}
15	9, 14	ovolval5 47665	. . 3 ⊢ (𝜑 → (vol*‘𝐵) = inf(𝑁, ℝ*, < ))
16	1	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐴 ∈ 𝑉)
17		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ))
18		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → (𝑓‘𝑛) = (𝑓‘𝑗))
19	18	opeq2d 4839	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ⟨𝐴, (𝑓‘𝑛)⟩ = ⟨𝐴, (𝑓‘𝑗)⟩)
20	19	sneqd 4595	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → {⟨𝐴, (𝑓‘𝑛)⟩} = {⟨𝐴, (𝑓‘𝑗)⟩})
21	20	cbvmptv 5208	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ {⟨𝐴, (𝑓‘𝑛)⟩}) = (𝑗 ∈ ℕ ↦ {⟨𝐴, (𝑓‘𝑗)⟩})
22		simprl 783	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐵 ⊆ ∪ ran ([,) ∘ 𝑓))
23	8, 9	ssexd 5285	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ V)
24	23	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) → 𝐵 ∈ V)
25	24	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐵 ∈ V)
26		simprr 785	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
27	16, 17, 21, 22, 25, 26	ovnovollem1 47666	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ)((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
28	27	rexlimdva2 3165	. . . . . . 7 ⊢ (𝜑 → (∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ)((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
29	1	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ) ∧ ((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → 𝐴 ∈ 𝑉)
30	23	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ) ∧ ((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → 𝐵 ∈ V)
31		simp2 1155	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ) ∧ ((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → 𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ))
32		simp3l 1220	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ) ∧ ((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → (𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘))
33		fveq2 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑛 → (𝑖‘𝑗) = (𝑖‘𝑛))
34	33	coeq2d 5839	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑛 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝑖‘𝑛)))
35	34	fveq1d 6887	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑛 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑛))‘𝑘))
36	35	ixpeq2dv 8941	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑛 → X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑘))
37		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → (([,) ∘ (𝑖‘𝑛))‘𝑘) = (([,) ∘ (𝑖‘𝑛))‘𝑙))
38	37	cbvixpv 8943	. . . . . . . . . . . . . . . 16 ⊢ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑘) = X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙)
39	38	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑛 → X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑘) = X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙))
40	36, 39	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑛 → X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙))
41	40	cbviunv 4996	. . . . . . . . . . . . 13 ⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙)
42	41	sseq2i 3937	. . . . . . . . . . . 12 ⊢ ((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ (𝐵 ↑m {𝐴}) ⊆ ∪ 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙))
43	42	biimpi 219	. . . . . . . . . . 11 ⊢ ((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) → (𝐵 ↑m {𝐴}) ⊆ ∪ 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙))
44	32, 43	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ) ∧ ((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → (𝐵 ↑m {𝐴}) ⊆ ∪ 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙))
45		simp3r 1221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ) ∧ ((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
46	35	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑛 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)))
47	46	prodeq2ad 46604	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)))
48	37	fveq2d 6889	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
49	48	cbvprodv 16083	. . . . . . . . . . . . . . . . 17 ⊢ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)) = ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙))
50	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)) = ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
51	47, 50	eqtrd 2795	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
52	51	cbvmptv 5208	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
53	52	fveq2i 6888	. . . . . . . . . . . . 13 ⊢ (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙))))
54	53	eqeq2i 2773	. . . . . . . . . . . 12 ⊢ (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))))
55	54	biimpi 219	. . . . . . . . . . 11 ⊢ (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) → 𝑧 = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))))
56	45, 55	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ) ∧ ((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → 𝑧 = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))))
57		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑖‘𝑚) = (𝑖‘𝑛))
58	57	fveq1d 6887	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑖‘𝑚)‘𝐴) = ((𝑖‘𝑛)‘𝐴))
59	58	cbvmptv 5208	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ ((𝑖‘𝑚)‘𝐴)) = (𝑛 ∈ ℕ ↦ ((𝑖‘𝑛)‘𝐴))
60	29, 30, 31, 44, 56, 59	ovnovollem2 47667	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ) ∧ ((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
61	60	3exp 1137	. . . . . . . 8 ⊢ (𝜑 → (𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ) → (((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))))
62	61	rexlimdv 3161	. . . . . . 7 ⊢ (𝜑 → (∃𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ)((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
63	28, 62	impbid 215	. . . . . 6 ⊢ (𝜑 → (∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ)((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
64	63	rabbidv 3419	. . . . 5 ⊢ (𝜑 → {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))} = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ)((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))})
65	14	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑁 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))})
66	12	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ)((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))})
67	64, 65, 66	3eqtr4d 2805	. . . 4 ⊢ (𝜑 → 𝑁 = 𝑀)
68	67	infeq1d 9470	. . 3 ⊢ (𝜑 → inf(𝑁, ℝ*, < ) = inf(𝑀, ℝ*, < ))
69	15, 68	eqtrd 2795	. 2 ⊢ (𝜑 → (vol*‘𝐵) = inf(𝑀, ℝ*, < ))
70	4, 13, 69	3eqtr4d 2805	1 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝐵 ↑m {𝐴})) = (vol*‘𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  ifcif 4481  {csn 4583  ⟨cop 4589  ∪ cuni 4866  ∪ ciun 4950   ↦ cmpt 5185   × cxp 5648  ran crn 5651   ∘ ccom 5654  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8847  Xcixp 8925  Fincfn 8973  infcinf 9433  ℝcr 11199  0cc0 11200  ℝ^*cxr 11342   < clt 11343  ℕcn 12335  [,)cico 13478  ∏cprod 16072  vol*covol 25783  volcvol 25784  Σ^{^}csumge0 47372  voln*covoln 47546

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-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-of 7693  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-2o 8477  df-er 8717  df-map 8849  df-pm 8850  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fi 9403  df-sup 9434  df-inf 9435  df-oi 9504  df-dju 9982  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-q 13076  df-rp 13121  df-xneg 13241  df-xadd 13242  df-xmul 13243  df-ioo 13480  df-ico 13482  df-icc 13483  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-prod 16073  df-rest 17593  df-topgen 17614  df-psmet 21670  df-xmet 21671  df-met 21672  df-bl 21673  df-mopn 21674  df-top 23212  df-topon 23229  df-bases 23264  df-cmp 23705  df-ovol 25785  df-vol 25786  df-sumge0 47373  df-ovoln 47547

This theorem is used by:  ovnovol  47669