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Theorem ovnovollem3 41370
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 4443 . . . . 5 (𝐴𝑉 → {𝐴} ≠ ∅)
31, 2syl 17 . . . 4 (𝜑 → {𝐴} ≠ ∅)
43neneqd 2929 . . 3 (𝜑 → ¬ {𝐴} = ∅)
54iffalsed 4233 . 2 (𝜑 → if({𝐴} = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
6 snfi 8195 . . . 4 {𝐴} ∈ Fin
76a1i 11 . . 3 (𝜑 → {𝐴} ∈ Fin)
8 reex 10211 . . . . 5 ℝ ∈ V
98a1i 11 . . . 4 (𝜑 → ℝ ∈ V)
10 ovnovollem3.b . . . 4 (𝜑𝐵 ⊆ ℝ)
11 mapss 8058 . . . 4 ((ℝ ∈ V ∧ 𝐵 ⊆ ℝ) → (𝐵𝑚 {𝐴}) ⊆ (ℝ ↑𝑚 {𝐴}))
129, 10, 11syl2anc 696 . . 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 41257 . 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 41367 . . 3 (𝜑 → (vol*‘𝐵) = inf(𝑁, ℝ*, < ))
171ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)) ∧ (𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐴𝑉)
18 simplr 809 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)) ∧ (𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
19 fveq2 6344 . . . . . . . . . . . . . 14 (𝑛 = 𝑗 → (𝑓𝑛) = (𝑓𝑗))
2019opeq2d 4552 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → ⟨𝐴, (𝑓𝑛)⟩ = ⟨𝐴, (𝑓𝑗)⟩)
2120sneqd 4325 . . . . . . . . . . . 12 (𝑛 = 𝑗 → {⟨𝐴, (𝑓𝑛)⟩} = {⟨𝐴, (𝑓𝑗)⟩})
2221cbvmptv 4894 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ {⟨𝐴, (𝑓𝑛)⟩}) = (𝑗 ∈ ℕ ↦ {⟨𝐴, (𝑓𝑗)⟩})
23 simprl 811 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)) ∧ (𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐵 ran ([,) ∘ 𝑓))
249, 10ssexd 4949 . . . . . . . . . . . . 13 (𝜑𝐵 ∈ V)
2524adantr 472 . . . . . . . . . . . 12 ((𝜑𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)) → 𝐵 ∈ V)
2625adantr 472 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)) ∧ (𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝐵 ∈ V)
27 simprr 813 . . . . . . . . . . 11 (((𝜑𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)) ∧ (𝐵 ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))) → 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))
2817, 18, 22, 23, 26, 27ovnovollem1 41368 . . . . . . . . . 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 1100 . . . . . . . . 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 1112 . . . . . . . 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 3160 . . . . . . 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 1127 . . . . . . . . . 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 1127 . . . . . . . . . 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 1131 . . . . . . . . . 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 1241 . . . . . . . . . . 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 6344 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑛 → (𝑖𝑗) = (𝑖𝑛))
3736coeq2d 5432 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑛 → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝑖𝑛)))
3837fveq1d 6346 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑛 → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝑖𝑛))‘𝑘))
3938ixpeq2dv 8082 . . . . . . . . . . . . . . 15 (𝑗 = 𝑛X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑘))
40 fveq2 6344 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑙 → (([,) ∘ (𝑖𝑛))‘𝑘) = (([,) ∘ (𝑖𝑛))‘𝑙))
4140cbvixpv 8084 . . . . . . . . . . . . . . . 16 X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑘) = X𝑙 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑙)
4241a1i 11 . . . . . . . . . . . . . . 15 (𝑗 = 𝑛X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑘) = X𝑙 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑙))
4339, 42eqtrd 2786 . . . . . . . . . . . . . 14 (𝑗 = 𝑛X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑙 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑙))
4443cbviunv 4703 . . . . . . . . . . . . 13 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑛 ∈ ℕ X𝑙 ∈ {𝐴} (([,) ∘ (𝑖𝑛))‘𝑙)
4544sseq2i 3763 . . . . . . . . . . . 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 1242 . . . . . . . . . . 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 6348 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑛 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖𝑛))‘𝑘)))
5049prodeq2ad 40319 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑘)))
5140fveq2d 6348 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑙 → (vol‘(([,) ∘ (𝑖𝑛))‘𝑘)) = (vol‘(([,) ∘ (𝑖𝑛))‘𝑙)))
5251cbvprodv 14837 . . . . . . . . . . . . . . . . 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 2786 . . . . . . . . . . . . . . 15 (𝑗 = 𝑛 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑙)))
5554cbvmptv 4894 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑙)))
5655fveq2i 6347 . . . . . . . . . . . . 13 ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑙 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑛))‘𝑙))))
5756eqeq2i 2764 . . . . . . . . . . . 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 6344 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑖𝑚) = (𝑖𝑛))
6160fveq1d 6346 . . . . . . . . . . 11 (𝑚 = 𝑛 → ((𝑖𝑚)‘𝐴) = ((𝑖𝑛)‘𝐴))
6261cbvmptv 4894 . . . . . . . . . 10 (𝑚 ∈ ℕ ↦ ((𝑖𝑚)‘𝐴)) = (𝑛 ∈ ℕ ↦ ((𝑖𝑛)‘𝐴))
6332, 33, 34, 47, 59, 62ovnovollem2 41369 . . . . . . . . 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 1112 . . . . . . . 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 3160 . . . . . . 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 3321 . . . . 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 2796 . . . 4 (𝜑𝑁 = 𝑀)
7170infeq1d 8540 . . 3 (𝜑 → inf(𝑁, ℝ*, < ) = inf(𝑀, ℝ*, < ))
7216, 71eqtrd 2786 . 2 (𝜑 → (vol*‘𝐵) = inf(𝑀, ℝ*, < ))
735, 14, 723eqtr4d 2796 1 (𝜑 → ((voln*‘{𝐴})‘(𝐵𝑚 {𝐴})) = (vol*‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wrex 3043  {crab 3046  Vcvv 3332  wss 3707  c0 4050  ifcif 4222  {csn 4313  cop 4319   cuni 4580   ciun 4664  cmpt 4873   × cxp 5256  ran crn 5259  ccom 5262  cfv 6041  (class class class)co 6805  𝑚 cmap 8015  Xcixp 8066  Fincfn 8113  infcinf 8504  cr 10119  0cc0 10120  *cxr 10257   < clt 10258  cn 11204  [,)cico 12362  cprod 14826  vol*covol 23423  volcvol 23424  Σ^csumge0 41074  voln*covoln 41248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197  ax-pre-sup 10198
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-of 7054  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-er 7903  df-map 8017  df-pm 8018  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fi 8474  df-sup 8505  df-inf 8506  df-oi 8572  df-card 8947  df-cda 9174  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-div 10869  df-nn 11205  df-2 11263  df-3 11264  df-n0 11477  df-z 11562  df-uz 11872  df-q 11974  df-rp 12018  df-xneg 12131  df-xadd 12132  df-xmul 12133  df-ioo 12364  df-ico 12366  df-icc 12367  df-fz 12512  df-fzo 12652  df-fl 12779  df-seq 12988  df-exp 13047  df-hash 13304  df-cj 14030  df-re 14031  df-im 14032  df-sqrt 14166  df-abs 14167  df-clim 14410  df-rlim 14411  df-sum 14608  df-prod 14827  df-rest 16277  df-topgen 16298  df-psmet 19932  df-xmet 19933  df-met 19934  df-bl 19935  df-mopn 19936  df-top 20893  df-topon 20910  df-bases 20944  df-cmp 21384  df-ovol 23425  df-vol 23426  df-sumge0 41075  df-ovoln 41249
This theorem is referenced by:  ovnovol  41371
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