Theorem ovnovollem3 47551

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 4736	. . . 4 ⊢ (𝜑 → {𝐴} ≠ ∅)
3	2	neneqd 2960	. . 3 ⊢ (𝜑 → ¬ {𝐴} = ∅)
4	3	iffalsed 4493	. 2 ⊢ (𝜑 → if({𝐴} = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
5		snfi 9057	. . . 4 ⊢ {𝐴} ∈ Fin
6	5	a1i 11	. . 3 ⊢ (𝜑 → {𝐴} ∈ Fin)
7		reex 11240	. . . . 5 ⊢ ℝ ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ V)
9		ovnovollem3.b	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
10		mapss 8903	. . . 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 47438	. 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 47548	. . 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 6881	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → (𝑓‘𝑛) = (𝑓‘𝑗))
19	18	opeq2d 4840	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ⟨𝐴, (𝑓‘𝑛)⟩ = ⟨𝐴, (𝑓‘𝑗)⟩)
20	19	sneqd 4596	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → {⟨𝐴, (𝑓‘𝑛)⟩} = {⟨𝐴, (𝑓‘𝑗)⟩})
21	20	cbvmptv 5209	. . . . . . . . 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 5289	. . . . . . . . . . 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 47549	. . . . . . . 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 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑛 → (𝑖‘𝑗) = (𝑖‘𝑛))
34	33	coeq2d 5844	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑛 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝑖‘𝑛)))
35	34	fveq1d 6883	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑛 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑛))‘𝑘))
36	35	ixpeq2dv 8927	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑛 → X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑘))
37		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → (([,) ∘ (𝑖‘𝑛))‘𝑘) = (([,) ∘ (𝑖‘𝑛))‘𝑙))
38	37	cbvixpv 8929	. . . . . . . . . . . . . . . 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 4997	. . . . . . . . . . . . 13 ⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙)
42	41	sseq2i 3960	. . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑛 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)))
47	46	prodeq2ad 46487	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)))
48	37	fveq2d 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
49	48	cbvprodv 16028	. . . . . . . . . . . . . . . . 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 5209	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
53	52	fveq2i 6884	. . . . . . . . . . . . 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 6881	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑖‘𝑚) = (𝑖‘𝑛))
58	57	fveq1d 6883	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑖‘𝑚)‘𝐴) = ((𝑖‘𝑛)‘𝐴))
59	58	cbvmptv 5209	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ ((𝑖‘𝑚)‘𝐴)) = (𝑛 ∈ ℕ ↦ ((𝑖‘𝑛)‘𝐴))
60	29, 30, 31, 44, 56, 59	ovnovollem2 47550	. . . . . . . . 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 9455	. . 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 1570   ∈ wcel 2145  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3899  ∅c0 4279  ifcif 4482  {csn 4584  ⟨cop 4590  ∪ cuni 4867  ∪ ciun 4951   ↦ cmpt 5186   × cxp 5653  ran crn 5656   ∘ ccom 5659  ‘cfv 6535  (class class class)co 7416   ↑_m cmap 8833  Xcixp 8911  Fincfn 8959  infcinf 9418  ℝcr 11148  0cc0 11149  ℝ^*cxr 11291   < clt 11292  ℕcn 12282  [,)cico 13425  ∏cprod 16017  vol*covol 25722  volcvol 25723  Σ^{^}csumge0 47255  voln*covoln 47429

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-inf2 9627  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226  ax-pre-sup 11227

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7684  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-2o 8463  df-er 8703  df-map 8835  df-pm 8836  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fi 9388  df-sup 9419  df-inf 9420  df-oi 9489  df-dju 9931  df-card 9969  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11492  df-neg 11493  df-div 11921  df-nn 12283  df-2 12352  df-3 12353  df-n0 12554  df-z 12641  df-uz 12913  df-q 13023  df-rp 13068  df-xneg 13188  df-xadd 13189  df-xmul 13190  df-ioo 13427  df-ico 13429  df-icc 13430  df-fz 13587  df-fzo 13735  df-fl 13878  df-seq 14091  df-exp 14151  df-hash 14420  df-cj 15211  df-re 15212  df-im 15213  df-sqrt 15347  df-abs 15348  df-clim 15600  df-rlim 15601  df-sum 15799  df-prod 16018  df-rest 17532  df-topgen 17553  df-psmet 21609  df-xmet 21610  df-met 21611  df-bl 21612  df-mopn 21613  df-top 23151  df-topon 23168  df-bases 23203  df-cmp 23644  df-ovol 25724  df-vol 25725  df-sumge0 47256  df-ovoln 47430

This theorem is used by:  ovnovol  47552