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Theorem ovnovollem3 41193
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 (𝜑𝐴𝑉)
ovnovollem3.b (𝜑𝐵 ⊆ ℝ)
ovnovollem3.m 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘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})‘(𝐵𝑚 {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.
StepHypRef Expression
1 ovnovollem3.a . . . . 5 (𝜑𝐴𝑉)
2 snnzg 4339 . . . . 5 (𝐴𝑉 → {𝐴} ≠ ∅)
31, 2syl 17 . . . 4 (𝜑 → {𝐴} ≠ ∅)
43neneqd 2828 . . 3 (𝜑 → ¬ {𝐴} = ∅)
54iffalsed 4130 . 2 (𝜑 → if({𝐴} = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
6 snfi 8079 . . . 4 {𝐴} ∈ Fin
76a1i 11 . . 3 (𝜑 → {𝐴} ∈ Fin)
8 reex 10065 . . . . 5 ℝ ∈ V
98a1i 11 . . . 4 (𝜑 → ℝ ∈ V)
10 ovnovollem3.b . . . 4 (𝜑𝐵 ⊆ ℝ)
11 mapss 7942 . . . 4 ((ℝ ∈ V ∧ 𝐵 ⊆ ℝ) → (𝐵𝑚 {𝐴}) ⊆ (ℝ ↑𝑚 {𝐴}))
129, 10, 11syl2anc 694 . . 3 (𝜑 → (𝐵𝑚 {𝐴}) ⊆ (ℝ ↑𝑚 {𝐴}))
13 ovnovollem3.m . . 3 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}
147, 12, 13ovnval2 41080 . 2 (𝜑 → ((voln*‘{𝐴})‘(𝐵𝑚 {𝐴})) = if({𝐴} = ∅, 0, inf(𝑀, ℝ*, < )))
15 ovnovollem3.n . . . 4 𝑁 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}
1610, 15ovolval5 41190 . . 3 (𝜑 → (vol*‘𝐵) = inf(𝑁, ℝ*, < ))
171ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)) ∧ (𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐴𝑉)
18 simplr 807 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)) ∧ (𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
19 fveq2 6229 . . . . . . . . . . . . . 14 (𝑛 = 𝑗 → (𝑓𝑛) = (𝑓𝑗))
2019opeq2d 4440 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → ⟨𝐴, (𝑓𝑛)⟩ = ⟨𝐴, (𝑓𝑗)⟩)
2120sneqd 4222 . . . . . . . . . . . 12 (𝑛 = 𝑗 → {⟨𝐴, (𝑓𝑛)⟩} = {⟨𝐴, (𝑓𝑗)⟩})
2221cbvmptv 4783 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ {⟨𝐴, (𝑓𝑛)⟩}) = (𝑗 ∈ ℕ ↦ {⟨𝐴, (𝑓𝑗)⟩})
23 simprl 809 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)) ∧ (𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐵 ran ([,) ∘ 𝑓))
249, 10ssexd 4838 . . . . . . . . . . . . 13 (𝜑𝐵 ∈ V)
2524adantr 480 . . . . . . . . . . . 12 ((𝜑𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)) → 𝐵 ∈ V)
2625adantr 480 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)) ∧ (𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐵 ∈ V)
27 simprr 811 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)) ∧ (𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
2817, 18, 22, 23, 26, 27ovnovollem1 41191 . . . . . . . . . 10 (((𝜑𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)) ∧ (𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
29283impa 1278 . . . . . . . . 9 ((𝜑𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
30293exp 1283 . . . . . . . 8 (𝜑 → (𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) → ((𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))))
3130rexlimdv 3059 . . . . . . 7 (𝜑 → (∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
3213ad2ant1 1102 . . . . . . . . . 10 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ) ∧ ((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → 𝐴𝑉)
33243ad2ant1 1102 . . . . . . . . . 10 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ) ∧ ((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → 𝐵 ∈ V)
34 simp2 1082 . . . . . . . . . 10 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ) ∧ ((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → 𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ))
35 simp3l 1109 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ) ∧ ((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → (𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘))
36 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑛 → (𝑖𝑗) = (𝑖𝑛))
3736coeq2d 5317 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑛 → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝑖𝑛)))
3837fveq1d 6231 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑛 → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝑖𝑛))‘𝑘))
3938ixpeq2dv 7966 . . . . . . . . . . . . . . 15 (𝑗 = 𝑛X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑘))
40 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑙 → (([,) ∘ (𝑖𝑛))‘𝑘) = (([,) ∘ (𝑖𝑛))‘𝑙))
4140cbvixpv 7968 . . . . . . . . . . . . . . . 16 X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑘) = X𝑙 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑙)
4241a1i 11 . . . . . . . . . . . . . . 15 (𝑗 = 𝑛X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑘) = X𝑙 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑙))
4339, 42eqtrd 2685 . . . . . . . . . . . . . 14 (𝑗 = 𝑛X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑙 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑙))
4443cbviunv 4591 . . . . . . . . . . . . 13 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑙)
4544sseq2i 3663 . . . . . . . . . . . 12 ((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ↔ (𝐵𝑚 {𝐴}) ⊆ 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑙))
4645biimpi 206 . . . . . . . . . . 11 ((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) → (𝐵𝑚 {𝐴}) ⊆ 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑙))
4735, 46syl 17 . . . . . . . . . 10 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ) ∧ ((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → (𝐵𝑚 {𝐴}) ⊆ 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑙))
48 simp3r 1110 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ) ∧ ((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
4938fveq2d 6233 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑛 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖𝑛))‘𝑘)))
5049prodeq2ad 40142 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑘)))
5140fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑙 → (vol‘(([,) ∘ (𝑖𝑛))‘𝑘)) = (vol‘(([,) ∘ (𝑖𝑛))‘𝑙)))
5251cbvprodv 14690 . . . . . . . . . . . . . . . . 17 𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑘)) = ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑙))
5352a1i 11 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑘)) = ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑙)))
5450, 53eqtrd 2685 . . . . . . . . . . . . . . 15 (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑙)))
5554cbvmptv 4783 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑙)))
5655fveq2i 6232 . . . . . . . . . . . . 13 ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑙))))
5756eqeq2i 2663 . . . . . . . . . . . 12 (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑙)))))
5857biimpi 206 . . . . . . . . . . 11 (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) → 𝑧 = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑙)))))
5948, 58syl 17 . . . . . . . . . 10 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ) ∧ ((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → 𝑧 = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑙)))))
60 fveq2 6229 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑖𝑚) = (𝑖𝑛))
6160fveq1d 6231 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((𝑖𝑚)‘𝐴) = ((𝑖𝑛)‘𝐴))
6261cbvmptv 4783 . . . . . . . . . 10 (𝑚 ∈ ℕ ↦ ((𝑖𝑚)‘𝐴)) = (𝑛 ∈ ℕ ↦ ((𝑖𝑛)‘𝐴))
6332, 33, 34, 47, 59, 62ovnovollem2 41192 . . . . . . . . 9 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ) ∧ ((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
64633exp 1283 . . . . . . . 8 (𝜑 → (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ) → (((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))))
6564rexlimdv 3059 . . . . . . 7 (𝜑 → (∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))))
6631, 65impbid 202 . . . . . 6 (𝜑 → (∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
6766rabbidv 3220 . . . . 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)((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})
6815a1i 11 . . . . 5 (𝜑𝑁 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))})
6913a1i 11 . . . . 5 (𝜑𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})
7067, 68, 693eqtr4d 2695 . . . 4 (𝜑𝑁 = 𝑀)
7170infeq1d 8424 . . 3 (𝜑 → inf(𝑁, ℝ*, < ) = inf(𝑀, ℝ*, < ))
7216, 71eqtrd 2685 . 2 (𝜑 → (vol*‘𝐵) = inf(𝑀, ℝ*, < ))
735, 14, 723eqtr4d 2695 1 (𝜑 → ((voln*‘{𝐴})‘(𝐵𝑚 {𝐴})) = (vol*‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  c0 3948  ifcif 4119  {csn 4210  cop 4216   cuni 4468   ciun 4552  cmpt 4762   × cxp 5141  ran crn 5144  ccom 5147  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  Xcixp 7950  Fincfn 7997  infcinf 8388  cr 9973  0cc0 9974  *cxr 10111   < clt 10112  cn 11058  [,)cico 12215  cprod 14679  vol*covol 23277  volcvol 23278  Σ^csumge0 40897  voln*covoln 41071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-prod 14680  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280  df-sumge0 40898  df-ovoln 41072
This theorem is referenced by:  ovnovol  41194
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