Theorem ovnovollem3 43871

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 4691	. . . 4 ⊢ (𝜑 → {𝐴} ≠ ∅)
3	2	neneqd 2945	. . 3 ⊢ (𝜑 → ¬ {𝐴} = ∅)
4	3	iffalsed 4450	. 2 ⊢ (𝜑 → if({𝐴} = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
5		snfi 8721	. . . 4 ⊢ {𝐴} ∈ Fin
6	5	a1i 11	. . 3 ⊢ (𝜑 → {𝐴} ∈ Fin)
7		reex 10820	. . . . 5 ⊢ ℝ ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ V)
9		ovnovollem3.b	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
10		mapss 8570	. . . 4 ⊢ ((ℝ ∈ V ∧ 𝐵 ⊆ ℝ) → (𝐵 ↑m {𝐴}) ⊆ (ℝ ↑m {𝐴}))
11	8, 9, 10	syl2anc 587	. . 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 43758	. 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 43868	. . 3 ⊢ (𝜑 → (vol*‘𝐵) = inf(𝑁, ℝ*, < ))
16	1	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐴 ∈ 𝑉)
17		simplr 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ))
18		fveq2 6717	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → (𝑓‘𝑛) = (𝑓‘𝑗))
19	18	opeq2d 4791	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ⟨𝐴, (𝑓‘𝑛)⟩ = ⟨𝐴, (𝑓‘𝑗)⟩)
20	19	sneqd 4553	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → {⟨𝐴, (𝑓‘𝑛)⟩} = {⟨𝐴, (𝑓‘𝑗)⟩})
21	20	cbvmptv 5158	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ {⟨𝐴, (𝑓‘𝑛)⟩}) = (𝑗 ∈ ℕ ↦ {⟨𝐴, (𝑓‘𝑗)⟩})
22		simprl 771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐵 ⊆ ∪ ran ([,) ∘ 𝑓))
23	8, 9	ssexd 5217	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ V)
24	23	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) → 𝐵 ∈ V)
25	24	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐵 ∈ V)
26		simprr 773	. . . . . . . . 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 43869	. . . . . . . 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 3206	. . . . . . 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 1135	. . . . . . . . . 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 1135	. . . . . . . . . 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 1139	. . . . . . . . . 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 1203	. . . . . . . . . . 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 6717	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑛 → (𝑖‘𝑗) = (𝑖‘𝑛))
34	33	coeq2d 5731	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑛 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝑖‘𝑛)))
35	34	fveq1d 6719	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑛 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑛))‘𝑘))
36	35	ixpeq2dv 8594	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑛 → X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑘))
37		fveq2 6717	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → (([,) ∘ (𝑖‘𝑛))‘𝑘) = (([,) ∘ (𝑖‘𝑛))‘𝑙))
38	37	cbvixpv 8596	. . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑛 → X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙))
41	40	cbviunv 4949	. . . . . . . . . . . . 13 ⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙)
42	41	sseq2i 3930	. . . . . . . . . . . 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 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 1204	. . . . . . . . . . 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 6721	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑛 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)))
47	46	prodeq2ad 42808	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)))
48	37	fveq2d 6721	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
49	48	cbvprodv 15478	. . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
52	51	cbvmptv 5158	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
53	52	fveq2i 6720	. . . . . . . . . . . . 13 ⊢ (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙))))
54	53	eqeq2i 2750	. . . . . . . . . . . 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 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 6717	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑖‘𝑚) = (𝑖‘𝑛))
58	57	fveq1d 6719	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑖‘𝑚)‘𝐴) = ((𝑖‘𝑛)‘𝐴))
59	58	cbvmptv 5158	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ ((𝑖‘𝑚)‘𝐴)) = (𝑛 ∈ ℕ ↦ ((𝑖‘𝑛)‘𝐴))
60	29, 30, 31, 44, 56, 59	ovnovollem2 43870	. . . . . . . . 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 1121	. . . . . . . 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 3202	. . . . . . 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 3390	. . . . 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 2787	. . . 4 ⊢ (𝜑 → 𝑁 = 𝑀)
68	67	infeq1d 9093	. . 3 ⊢ (𝜑 → inf(𝑁, ℝ*, < ) = inf(𝑀, ℝ*, < ))
69	15, 68	eqtrd 2777	. 2 ⊢ (𝜑 → (vol*‘𝐵) = inf(𝑀, ℝ*, < ))
70	4, 13, 69	3eqtr4d 2787	1 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝐵 ↑m {𝐴})) = (vol*‘𝐵))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∃wrex 3062  {crab 3065  Vcvv 3408   ⊆ wss 3866  ∅c0 4237  ifcif 4439  {csn 4541  ⟨cop 4547  ∪ cuni 4819  ∪ ciun 4904   ↦ cmpt 5135   × cxp 5549  ran crn 5552   ∘ ccom 5555  ‘cfv 6380  (class class class)co 7213   ↑_m cmap 8508  Xcixp 8578  Fincfn 8626  infcinf 9057  ℝcr 10728  0cc0 10729  ℝ^*cxr 10866   < clt 10867  ℕcn 11830  [,)cico 12937  ∏cprod 15467  vol*covol 24359  volcvol 24360  Σ^{^}csumge0 43575  voln*covoln 43749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-pm 8511  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fi 9027  df-sup 9058  df-inf 9059  df-oi 9126  df-dju 9517  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-z 12177  df-uz 12439  df-q 12545  df-rp 12587  df-xneg 12704  df-xadd 12705  df-xmul 12706  df-ioo 12939  df-ico 12941  df-icc 12942  df-fz 13096  df-fzo 13239  df-fl 13367  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-clim 15049  df-rlim 15050  df-sum 15250  df-prod 15468  df-rest 16927  df-topgen 16948  df-psmet 20355  df-xmet 20356  df-met 20357  df-bl 20358  df-mopn 20359  df-top 21791  df-topon 21808  df-bases 21843  df-cmp 22284  df-ovol 24361  df-vol 24362  df-sumge0 43576  df-ovoln 43750

This theorem is referenced by:  ovnovol  43872