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Theorem ovnovollem2 41514
Description: if 𝐼 is a cover of (𝐵𝑚 {𝐴}) in ℝ^1, then 𝐹 is the corresponding cover in the reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
ovnovollem2.a (𝜑𝐴𝑉)
ovnovollem2.b (𝜑𝐵𝑊)
ovnovollem2.i (𝜑𝐼 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ))
ovnovollem2.s (𝜑 → (𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
ovnovollem2.z (𝜑𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
ovnovollem2.f 𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼𝑗)‘𝐴))
Assertion
Ref Expression
ovnovollem2 (𝜑 → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
Distinct variable groups:   𝐴,𝑗,𝑘   𝐵,𝑓   𝑓,𝐹   𝑗,𝐹,𝑘   𝑘,𝐼   𝑘,𝑉   𝑓,𝑍   𝜑,𝑗,𝑘
Allowed substitution hints:   𝜑(𝑓)   𝐴(𝑓)   𝐵(𝑗,𝑘)   𝐼(𝑓,𝑗)   𝑉(𝑓,𝑗)   𝑊(𝑓,𝑗,𝑘)   𝑍(𝑗,𝑘)

Proof of Theorem ovnovollem2
StepHypRef Expression
1 ovnovollem2.i . . . . . . . . 9 (𝜑𝐼 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ))
2 elmapi 8084 . . . . . . . . 9 (𝐼 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 {𝐴}))
31, 2syl 17 . . . . . . . 8 (𝜑𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 {𝐴}))
43adantr 472 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 {𝐴}))
5 simpr 477 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
64, 5ffvelrnd 6552 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 {𝐴}))
7 elmapi 8084 . . . . . 6 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 {𝐴}) → (𝐼𝑗):{𝐴}⟶(ℝ × ℝ))
86, 7syl 17 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗):{𝐴}⟶(ℝ × ℝ))
9 ovnovollem2.a . . . . . . 7 (𝜑𝐴𝑉)
10 snidg 4366 . . . . . . 7 (𝐴𝑉𝐴 ∈ {𝐴})
119, 10syl 17 . . . . . 6 (𝜑𝐴 ∈ {𝐴})
1211adantr 472 . . . . 5 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ {𝐴})
138, 12ffvelrnd 6552 . . . 4 ((𝜑𝑗 ∈ ℕ) → ((𝐼𝑗)‘𝐴) ∈ (ℝ × ℝ))
14 ovnovollem2.f . . . 4 𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼𝑗)‘𝐴))
1513, 14fmptd 6576 . . 3 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
16 reex 10282 . . . . . 6 ℝ ∈ V
1716, 16xpex 7162 . . . . 5 (ℝ × ℝ) ∈ V
18 nnex 11283 . . . . 5 ℕ ∈ V
1917, 18elmap 8091 . . . 4 (𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ↔ 𝐹:ℕ⟶(ℝ × ℝ))
2019a1i 11 . . 3 (𝜑 → (𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ↔ 𝐹:ℕ⟶(ℝ × ℝ)))
2115, 20mpbird 248 . 2 (𝜑𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
22 ovnovollem2.s . . . . . 6 (𝜑 → (𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
23 elsni 4353 . . . . . . . . . . . . 13 (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
2423fveq2d 6381 . . . . . . . . . . . 12 (𝑘 ∈ {𝐴} → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝐴))
2524adantl 473 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝐴))
26 elmapfun 8086 . . . . . . . . . . . . . 14 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 {𝐴}) → Fun (𝐼𝑗))
276, 26syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → Fun (𝐼𝑗))
288fdmd 6234 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → dom (𝐼𝑗) = {𝐴})
2928eqcomd 2771 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → {𝐴} = dom (𝐼𝑗))
3012, 29eleqtrd 2846 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ dom (𝐼𝑗))
31 fvco 6465 . . . . . . . . . . . . 13 ((Fun (𝐼𝑗) ∧ 𝐴 ∈ dom (𝐼𝑗)) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
3227, 30, 31syl2anc 579 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
3332adantr 472 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
34 id 22 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
35 fvexd 6392 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ → ((𝐼𝑗)‘𝐴) ∈ V)
3614fvmpt2 6482 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ ∧ ((𝐼𝑗)‘𝐴) ∈ V) → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
3734, 35, 36syl2anc 579 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
3837eqcomd 2771 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → ((𝐼𝑗)‘𝐴) = (𝐹𝑗))
3938fveq2d 6381 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → ([,)‘((𝐼𝑗)‘𝐴)) = ([,)‘(𝐹𝑗)))
4039adantl 473 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ([,)‘((𝐼𝑗)‘𝐴)) = ([,)‘(𝐹𝑗)))
4115ffund 6229 . . . . . . . . . . . . . . . 16 (𝜑 → Fun 𝐹)
4241adantr 472 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → Fun 𝐹)
4314, 13dmmptd 6204 . . . . . . . . . . . . . . . . . 18 (𝜑 → dom 𝐹 = ℕ)
4443eqcomd 2771 . . . . . . . . . . . . . . . . 17 (𝜑 → ℕ = dom 𝐹)
4544adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ℕ = dom 𝐹)
465, 45eleqtrd 2846 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ dom 𝐹)
47 fvco 6465 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑗 ∈ dom 𝐹) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
4842, 46, 47syl2anc 579 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
4948eqcomd 2771 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ([,)‘(𝐹𝑗)) = (([,) ∘ 𝐹)‘𝑗))
5040, 49eqtrd 2799 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ([,)‘((𝐼𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗))
5150adantr 472 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ([,)‘((𝐼𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗))
5225, 33, 513eqtrd 2803 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ 𝐹)‘𝑗))
5352ixpeq2dva 8130 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗))
54 snex 5066 . . . . . . . . . . 11 {𝐴} ∈ V
55 fvex 6390 . . . . . . . . . . 11 (([,) ∘ 𝐹)‘𝑗) ∈ V
5654, 55ixpconst 8125 . . . . . . . . . 10 X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴})
5756a1i 11 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
5853, 57eqtrd 2799 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
5958iuneq2dv 4700 . . . . . . 7 (𝜑 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
60 nfv 2009 . . . . . . . 8 𝑗𝜑
6118a1i 11 . . . . . . . 8 (𝜑 → ℕ ∈ V)
62 fvexd 6392 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) ∈ V)
6360, 61, 62, 9iunmapsn 40057 . . . . . . 7 (𝜑 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}) = ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
6459, 63eqtrd 2799 . . . . . 6 (𝜑 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
6522, 64sseqtrd 3803 . . . . 5 (𝜑 → (𝐵𝑚 {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
66 ovnovollem2.b . . . . . 6 (𝜑𝐵𝑊)
6718, 55iunex 7347 . . . . . . 7 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V
6867a1i 11 . . . . . 6 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V)
6954a1i 11 . . . . . 6 (𝜑 → {𝐴} ∈ V)
7011ne0d 4088 . . . . . 6 (𝜑 → {𝐴} ≠ ∅)
7166, 68, 69, 70mapss2 40045 . . . . 5 (𝜑 → (𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↔ (𝐵𝑚 {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴})))
7265, 71mpbird 248 . . . 4 (𝜑𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗))
73 icof 40059 . . . . . . . 8 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
7473a1i 11 . . . . . . 7 (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
75 rexpssxrxp 10340 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
7675a1i 11 . . . . . . 7 (𝜑 → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
7774, 76, 15fcoss 40050 . . . . . 6 (𝜑 → ([,) ∘ 𝐹):ℕ⟶𝒫 ℝ*)
7877ffnd 6226 . . . . 5 (𝜑 → ([,) ∘ 𝐹) Fn ℕ)
79 fniunfv 6699 . . . . 5 (([,) ∘ 𝐹) Fn ℕ → 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
8078, 79syl 17 . . . 4 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
8172, 80sseqtrd 3803 . . 3 (𝜑𝐵 ran ([,) ∘ 𝐹))
82 ovnovollem2.z . . . 4 (𝜑𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
83 nfcv 2907 . . . . . . 7 𝑗𝐹
84 ressxr 10339 . . . . . . . . . 10 ℝ ⊆ ℝ*
85 xpss2 5299 . . . . . . . . . 10 (ℝ ⊆ ℝ* → (ℝ × ℝ) ⊆ (ℝ × ℝ*))
8684, 85ax-mp 5 . . . . . . . . 9 (ℝ × ℝ) ⊆ (ℝ × ℝ*)
8786a1i 11 . . . . . . . 8 (𝜑 → (ℝ × ℝ) ⊆ (ℝ × ℝ*))
8815, 87fssd 6239 . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ × ℝ*))
8983, 88volicofmpt 40854 . . . . . 6 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))))
909adantr 472 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 𝐴𝑉)
91 fvexd 6392 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → ((𝐼𝑗)‘𝐴) ∈ V)
925, 91, 36syl2anc 579 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
9392, 13eqeltrd 2844 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∈ (ℝ × ℝ))
94 1st2nd2 7407 . . . . . . . . . . . . . . . 16 ((𝐹𝑗) ∈ (ℝ × ℝ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
9593, 94syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
9695fveq2d 6381 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ([,)‘(𝐹𝑗)) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩))
97 df-ov 6847 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
9897eqcomi 2774 . . . . . . . . . . . . . . 15 ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))
9998a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
10048, 96, 993eqtrd 2803 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
10132, 50, 1003eqtrd 2803 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
102101fveq2d 6381 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) = (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))))
103 xp1st 7400 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
10493, 103syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
105 xp2nd 7401 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
10693, 105syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
107 volicore 41438 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑗)) ∈ ℝ ∧ (2nd ‘(𝐹𝑗)) ∈ ℝ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) ∈ ℝ)
108104, 106, 107syl2anc 579 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) ∈ ℝ)
109102, 108eqeltrd 2844 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℝ)
110109recnd 10324 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ)
111 2fveq3 6382 . . . . . . . . . 10 (𝑘 = 𝐴 → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
112111prodsn 14978 . . . . . . . . 9 ((𝐴𝑉 ∧ (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
11390, 110, 112syl2anc 579 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
114113, 102eqtr2d 2800 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
115114mpteq2dva 4905 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
11689, 115eqtrd 2799 . . . . 5 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
117116fveq2d 6381 . . . 4 (𝜑 → (Σ^‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
11882, 117eqtr4d 2802 . . 3 (𝜑𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
11981, 118jca 507 . 2 (𝜑 → (𝐵 ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹))))
120 coeq2 5451 . . . . . . 7 (𝑓 = 𝐹 → ([,) ∘ 𝑓) = ([,) ∘ 𝐹))
121120rneqd 5523 . . . . . 6 (𝑓 = 𝐹 → ran ([,) ∘ 𝑓) = ran ([,) ∘ 𝐹))
122121unieqd 4606 . . . . 5 (𝑓 = 𝐹 ran ([,) ∘ 𝑓) = ran ([,) ∘ 𝐹))
123122sseq2d 3795 . . . 4 (𝑓 = 𝐹 → (𝐵 ran ([,) ∘ 𝑓) ↔ 𝐵 ran ([,) ∘ 𝐹)))
124 coeq2 5451 . . . . . 6 (𝑓 = 𝐹 → ((vol ∘ [,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝐹))
125124fveq2d 6381 . . . . 5 (𝑓 = 𝐹 → (Σ^‘((vol ∘ [,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
126125eqeq2d 2775 . . . 4 (𝑓 = 𝐹 → (𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹))))
127123, 126anbi12d 624 . . 3 (𝑓 = 𝐹 → ((𝐵 ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) ↔ (𝐵 ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))))
128127rspcev 3462 . 2 ((𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ) ∧ (𝐵 ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
12921, 119, 128syl2anc 579 1 (𝜑 → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wrex 3056  Vcvv 3350  wss 3734  𝒫 cpw 4317  {csn 4336  cop 4342   cuni 4596   ciun 4678  cmpt 4890   × cxp 5277  dom cdm 5279  ran crn 5280  ccom 5283  Fun wfun 6064   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6844  1st c1st 7366  2nd c2nd 7367  𝑚 cmap 8062  Xcixp 8115  cc 10189  cr 10190  *cxr 10329  cn 11276  [,)cico 12382  cprod 14921  volcvol 23524  Σ^csumge0 41219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-inf2 8755  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268  ax-pre-sup 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-of 7097  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-2o 7767  df-oadd 7770  df-er 7949  df-map 8064  df-pm 8065  df-ixp 8116  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-fi 8526  df-sup 8557  df-inf 8558  df-oi 8624  df-card 9018  df-cda 9245  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-div 10941  df-nn 11277  df-2 11337  df-3 11338  df-n0 11541  df-z 11627  df-uz 11890  df-q 11993  df-rp 12032  df-xneg 12149  df-xadd 12150  df-xmul 12151  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12537  df-fzo 12677  df-fl 12804  df-seq 13012  df-exp 13071  df-hash 13325  df-cj 14127  df-re 14128  df-im 14129  df-sqrt 14263  df-abs 14264  df-clim 14507  df-rlim 14508  df-sum 14705  df-prod 14922  df-rest 16352  df-topgen 16373  df-psmet 20014  df-xmet 20015  df-met 20016  df-bl 20017  df-mopn 20018  df-top 20981  df-topon 20998  df-bases 21033  df-cmp 21473  df-ovol 23525  df-vol 23526
This theorem is referenced by:  ovnovollem3  41515
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