Theorem ovnovollem3 44196

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 4711	. . . 4 ⊢ (𝜑 → {𝐴} ≠ ∅)
3	2	neneqd 2948	. . 3 ⊢ (𝜑 → ¬ {𝐴} = ∅)
4	3	iffalsed 4470	. 2 ⊢ (𝜑 → if({𝐴} = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
5		snfi 8834	. . . 4 ⊢ {𝐴} ∈ Fin
6	5	a1i 11	. . 3 ⊢ (𝜑 → {𝐴} ∈ Fin)
7		reex 10962	. . . . 5 ⊢ ℝ ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ V)
9		ovnovollem3.b	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
10		mapss 8677	. . . 4 ⊢ ((ℝ ∈ V ∧ 𝐵 ⊆ ℝ) → (𝐵 ↑m {𝐴}) ⊆ (ℝ ↑m {𝐴}))
11	8, 9, 10	syl2anc 584	. . 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 44083	. 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 44193	. . 3 ⊢ (𝜑 → (vol*‘𝐵) = inf(𝑁, ℝ*, < ))
16	1	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐴 ∈ 𝑉)
17		simplr 766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ))
18		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → (𝑓‘𝑛) = (𝑓‘𝑗))
19	18	opeq2d 4811	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ⟨𝐴, (𝑓‘𝑛)⟩ = ⟨𝐴, (𝑓‘𝑗)⟩)
20	19	sneqd 4573	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → {⟨𝐴, (𝑓‘𝑛)⟩} = {⟨𝐴, (𝑓‘𝑗)⟩})
21	20	cbvmptv 5187	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ {⟨𝐴, (𝑓‘𝑛)⟩}) = (𝑗 ∈ ℕ ↦ {⟨𝐴, (𝑓‘𝑗)⟩})
22		simprl 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐵 ⊆ ∪ ran ([,) ∘ 𝑓))
23	8, 9	ssexd 5248	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ V)
24	23	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) → 𝐵 ∈ V)
25	24	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐵 ∈ V)
26		simprr 770	. . . . . . . . 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 44194	. . . . . . . 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 3216	. . . . . . 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 1132	. . . . . . . . . 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 1132	. . . . . . . . . 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 1136	. . . . . . . . . 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 1200	. . . . . . . . . . 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 6774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑛 → (𝑖‘𝑗) = (𝑖‘𝑛))
34	33	coeq2d 5771	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑛 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝑖‘𝑛)))
35	34	fveq1d 6776	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑛 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑛))‘𝑘))
36	35	ixpeq2dv 8701	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑛 → X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑘))
37		fveq2 6774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → (([,) ∘ (𝑖‘𝑛))‘𝑘) = (([,) ∘ (𝑖‘𝑛))‘𝑙))
38	37	cbvixpv 8703	. . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑛 → X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙))
41	40	cbviunv 4970	. . . . . . . . . . . . 13 ⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙)
42	41	sseq2i 3950	. . . . . . . . . . . 12 ⊢ ((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ (𝐵 ↑m {𝐴}) ⊆ ∪ 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙))
43	42	biimpi 215	. . . . . . . . . . 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 17	. . . . . . . . . 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 1201	. . . . . . . . . . 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 6778	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑛 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)))
47	46	prodeq2ad 43133	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)))
48	37	fveq2d 6778	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
49	48	cbvprodv 15626	. . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
52	51	cbvmptv 5187	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
53	52	fveq2i 6777	. . . . . . . . . . . . 13 ⊢ (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙))))
54	53	eqeq2i 2751	. . . . . . . . . . . 12 ⊢ (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))))
55	54	biimpi 215	. . . . . . . . . . 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 17	. . . . . . . . . 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 6774	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑖‘𝑚) = (𝑖‘𝑛))
58	57	fveq1d 6776	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑖‘𝑚)‘𝐴) = ((𝑖‘𝑛)‘𝐴))
59	58	cbvmptv 5187	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ ((𝑖‘𝑚)‘𝐴)) = (𝑛 ∈ ℕ ↦ ((𝑖‘𝑛)‘𝐴))
60	29, 30, 31, 44, 56, 59	ovnovollem2 44195	. . . . . . . . 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 1118	. . . . . . . 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 3212	. . . . . . 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 211	. . . . . 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 3414	. . . . 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 2788	. . . 4 ⊢ (𝜑 → 𝑁 = 𝑀)
68	67	infeq1d 9236	. . 3 ⊢ (𝜑 → inf(𝑁, ℝ*, < ) = inf(𝑀, ℝ*, < ))
69	15, 68	eqtrd 2778	. 2 ⊢ (𝜑 → (vol*‘𝐵) = inf(𝑀, ℝ*, < ))
70	4, 13, 69	3eqtr4d 2788	1 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝐵 ↑m {𝐴})) = (vol*‘𝐵))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  {crab 3068  Vcvv 3432   ⊆ wss 3887  ∅c0 4256  ifcif 4459  {csn 4561  ⟨cop 4567  ∪ cuni 4839  ∪ ciun 4924   ↦ cmpt 5157   × cxp 5587  ran crn 5590   ∘ ccom 5593  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  Xcixp 8685  Fincfn 8733  infcinf 9200  ℝcr 10870  0cc0 10871  ℝ^*cxr 11008   < clt 11009  ℕcn 11973  [,)cico 13081  ∏cprod 15615  vol*covol 24626  volcvol 24627  Σ^{^}csumge0 43900  voln*covoln 44074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-sum 15398  df-prod 15616  df-rest 17133  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-top 22043  df-topon 22060  df-bases 22096  df-cmp 22538  df-ovol 24628  df-vol 24629  df-sumge0 43901  df-ovoln 44075

This theorem is referenced by:  ovnovol  44197