Theorem ovnovollem3 46755

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 4725	. . . 4 ⊢ (𝜑 → {𝐴} ≠ ∅)
3	2	neneqd 2933	. . 3 ⊢ (𝜑 → ¬ {𝐴} = ∅)
4	3	iffalsed 4483	. 2 ⊢ (𝜑 → if({𝐴} = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
5		snfi 8965	. . . 4 ⊢ {𝐴} ∈ Fin
6	5	a1i 11	. . 3 ⊢ (𝜑 → {𝐴} ∈ Fin)
7		reex 11097	. . . . 5 ⊢ ℝ ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ V)
9		ovnovollem3.b	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
10		mapss 8813	. . . 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 46642	. 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 46752	. . 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 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ))
18		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → (𝑓‘𝑛) = (𝑓‘𝑗))
19	18	opeq2d 4829	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ⟨𝐴, (𝑓‘𝑛)⟩ = ⟨𝐴, (𝑓‘𝑗)⟩)
20	19	sneqd 4585	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → {⟨𝐴, (𝑓‘𝑛)⟩} = {⟨𝐴, (𝑓‘𝑗)⟩})
21	20	cbvmptv 5193	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ {⟨𝐴, (𝑓‘𝑛)⟩}) = (𝑗 ∈ ℕ ↦ {⟨𝐴, (𝑓‘𝑗)⟩})
22		simprl 770	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐵 ⊆ ∪ ran ([,) ∘ 𝑓))
23	8, 9	ssexd 5260	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ V)
24	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) → 𝐵 ∈ V)
25	24	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐵 ∈ V)
26		simprr 772	. . . . . . . . 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 46753	. . . . . . . 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 3135	. . . . . . 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 1133	. . . . . . . . . 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 1133	. . . . . . . . . 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 1137	. . . . . . . . . 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 1202	. . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑛 → (𝑖‘𝑗) = (𝑖‘𝑛))
34	33	coeq2d 5801	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑛 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝑖‘𝑛)))
35	34	fveq1d 6824	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑛 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑛))‘𝑘))
36	35	ixpeq2dv 8837	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑛 → X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑘))
37		fveq2 6822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → (([,) ∘ (𝑖‘𝑛))‘𝑘) = (([,) ∘ (𝑖‘𝑛))‘𝑙))
38	37	cbvixpv 8839	. . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑛 → X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙))
41	40	cbviunv 4987	. . . . . . . . . . . . 13 ⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙)
42	41	sseq2i 3959	. . . . . . . . . . . 12 ⊢ ((𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ (𝐵 ↑m {𝐴}) ⊆ ∪ 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖‘𝑛))‘𝑙))
43	42	biimpi 216	. . . . . . . . . . 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 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)))))) → 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
46	35	fveq2d 6826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑛 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)))
47	46	prodeq2ad 45691	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)))
48	37	fveq2d 6826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
49	48	cbvprodv 15821	. . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
52	51	cbvmptv 5193	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))
53	52	fveq2i 6825	. . . . . . . . . . . . 13 ⊢ (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙))))
54	53	eqeq2i 2744	. . . . . . . . . . . 12 ⊢ (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑛))‘𝑙)))))
55	54	biimpi 216	. . . . . . . . . . 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 6822	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑖‘𝑚) = (𝑖‘𝑛))
58	57	fveq1d 6824	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑖‘𝑚)‘𝐴) = ((𝑖‘𝑛)‘𝐴))
59	58	cbvmptv 5193	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ ((𝑖‘𝑚)‘𝐴)) = (𝑛 ∈ ℕ ↦ ((𝑖‘𝑛)‘𝐴))
60	29, 30, 31, 44, 56, 59	ovnovollem2 46754	. . . . . . . . 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 1119	. . . . . . . 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 3131	. . . . . . 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 212	. . . . . 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 3402	. . . . 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 2776	. . . 4 ⊢ (𝜑 → 𝑁 = 𝑀)
68	67	infeq1d 9362	. . 3 ⊢ (𝜑 → inf(𝑁, ℝ*, < ) = inf(𝑀, ℝ*, < ))
69	15, 68	eqtrd 2766	. 2 ⊢ (𝜑 → (vol*‘𝐵) = inf(𝑀, ℝ*, < ))
70	4, 13, 69	3eqtr4d 2776	1 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝐵 ↑m {𝐴})) = (vol*‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  {crab 3395  Vcvv 3436   ⊆ wss 3897  ∅c0 4280  ifcif 4472  {csn 4573  ⟨cop 4579  ∪ cuni 4856  ∪ ciun 4939   ↦ cmpt 5170   × cxp 5612  ran crn 5615   ∘ ccom 5618  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750  Xcixp 8821  Fincfn 8869  infcinf 9325  ℝcr 11005  0cc0 11006  ℝ^*cxr 11145   < clt 11146  ℕcn 12125  [,)cico 13247  ∏cprod 15810  vol*covol 25390  volcvol 25391  Σ^{^}csumge0 46459  voln*covoln 46633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-map 8752  df-pm 8753  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fi 9295  df-sup 9326  df-inf 9327  df-oi 9396  df-dju 9794  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-z 12469  df-uz 12733  df-q 12847  df-rp 12891  df-xneg 13011  df-xadd 13012  df-xmul 13013  df-ioo 13249  df-ico 13251  df-icc 13252  df-fz 13408  df-fzo 13555  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-sum 15594  df-prod 15811  df-rest 17326  df-topgen 17347  df-psmet 21283  df-xmet 21284  df-met 21285  df-bl 21286  df-mopn 21287  df-top 22809  df-topon 22826  df-bases 22861  df-cmp 23302  df-ovol 25392  df-vol 25393  df-sumge0 46460  df-ovoln 46634

This theorem is referenced by:  ovnovol  46756