Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovnovollem2 Structured version   Visualization version   GIF version

Theorem ovnovollem2 39347
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 7738 . . . . . . . . 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 479 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 {𝐴}))
5 simpr 475 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
64, 5ffvelrnd 6249 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 {𝐴}))
7 elmapi 7738 . . . . . 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 4148 . . . . . . 7 (𝐴𝑉𝐴 ∈ {𝐴})
119, 10syl 17 . . . . . 6 (𝜑𝐴 ∈ {𝐴})
1211adantr 479 . . . . 5 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ {𝐴})
138, 12ffvelrnd 6249 . . . 4 ((𝜑𝑗 ∈ ℕ) → ((𝐼𝑗)‘𝐴) ∈ (ℝ × ℝ))
14 ovnovollem2.f . . . 4 𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼𝑗)‘𝐴))
1513, 14fmptd 6273 . . 3 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
16 reex 9879 . . . . . 6 ℝ ∈ V
1716, 16xpex 6833 . . . . 5 (ℝ × ℝ) ∈ V
18 nnex 10869 . . . . 5 ℕ ∈ V
1917, 18elmap 7745 . . . 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 245 . 2 (𝜑𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
22 ovnovollem2.s . . . . . 6 (𝜑 → (𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
23 elsni 4137 . . . . . . . . . . . . 13 (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
2423fveq2d 6088 . . . . . . . . . . . 12 (𝑘 ∈ {𝐴} → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝐴))
2524adantl 480 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝐴))
26 elmapfun 7740 . . . . . . . . . . . . . 14 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 {𝐴}) → Fun (𝐼𝑗))
276, 26syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → Fun (𝐼𝑗))
28 fdm 5946 . . . . . . . . . . . . . . . 16 ((𝐼𝑗):{𝐴}⟶(ℝ × ℝ) → dom (𝐼𝑗) = {𝐴})
298, 28syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → dom (𝐼𝑗) = {𝐴})
3029eqcomd 2611 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → {𝐴} = dom (𝐼𝑗))
3112, 30eleqtrd 2685 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ dom (𝐼𝑗))
32 fvco 6165 . . . . . . . . . . . . 13 ((Fun (𝐼𝑗) ∧ 𝐴 ∈ dom (𝐼𝑗)) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
3327, 31, 32syl2anc 690 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
3433adantr 479 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
35 id 22 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
36 fvex 6094 . . . . . . . . . . . . . . . . . 18 ((𝐼𝑗)‘𝐴) ∈ V
3736a1i 11 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ → ((𝐼𝑗)‘𝐴) ∈ V)
3814fvmpt2 6181 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ ∧ ((𝐼𝑗)‘𝐴) ∈ V) → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
3935, 37, 38syl2anc 690 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
4039eqcomd 2611 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → ((𝐼𝑗)‘𝐴) = (𝐹𝑗))
4140fveq2d 6088 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → ([,)‘((𝐼𝑗)‘𝐴)) = ([,)‘(𝐹𝑗)))
4241adantl 480 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ([,)‘((𝐼𝑗)‘𝐴)) = ([,)‘(𝐹𝑗)))
4315ffund 5944 . . . . . . . . . . . . . . . 16 (𝜑 → Fun 𝐹)
4443adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → Fun 𝐹)
4514, 13dmmptd 5919 . . . . . . . . . . . . . . . . . 18 (𝜑 → dom 𝐹 = ℕ)
4645eqcomd 2611 . . . . . . . . . . . . . . . . 17 (𝜑 → ℕ = dom 𝐹)
4746adantr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ℕ = dom 𝐹)
485, 47eleqtrd 2685 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ dom 𝐹)
49 fvco 6165 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑗 ∈ dom 𝐹) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
5044, 48, 49syl2anc 690 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
5150eqcomd 2611 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ([,)‘(𝐹𝑗)) = (([,) ∘ 𝐹)‘𝑗))
5242, 51eqtrd 2639 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ([,)‘((𝐼𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗))
5352adantr 479 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ([,)‘((𝐼𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗))
5425, 34, 533eqtrd 2643 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ 𝐹)‘𝑗))
5554ixpeq2dva 7782 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗))
56 snex 4826 . . . . . . . . . . 11 {𝐴} ∈ V
57 fvex 6094 . . . . . . . . . . 11 (([,) ∘ 𝐹)‘𝑗) ∈ V
5856, 57ixpconst 7777 . . . . . . . . . 10 X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴})
5958a1i 11 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
6055, 59eqtrd 2639 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
6160iuneq2dv 4468 . . . . . . 7 (𝜑 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
62 nfv 1828 . . . . . . . 8 𝑗𝜑
6318a1i 11 . . . . . . . 8 (𝜑 → ℕ ∈ V)
6457a1i 11 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) ∈ V)
6562, 63, 64, 9iunmapsn 38203 . . . . . . 7 (𝜑 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}) = ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
6661, 65eqtrd 2639 . . . . . 6 (𝜑 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
6722, 66sseqtrd 3599 . . . . 5 (𝜑 → (𝐵𝑚 {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
68 ovnovollem2.b . . . . . 6 (𝜑𝐵𝑊)
6918, 57iunex 7012 . . . . . . 7 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V
7069a1i 11 . . . . . 6 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V)
7156a1i 11 . . . . . 6 (𝜑 → {𝐴} ∈ V)
72 ne0i 3875 . . . . . . 7 (𝐴 ∈ {𝐴} → {𝐴} ≠ ∅)
7311, 72syl 17 . . . . . 6 (𝜑 → {𝐴} ≠ ∅)
7468, 70, 71, 73mapss2 38191 . . . . 5 (𝜑 → (𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↔ (𝐵𝑚 {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴})))
7567, 74mpbird 245 . . . 4 (𝜑𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗))
76 icof 38205 . . . . . . . 8 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
7776a1i 11 . . . . . . 7 (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
78 rexpssxrxp 9936 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
7978a1i 11 . . . . . . 7 (𝜑 → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
8077, 79, 15fcoss 38196 . . . . . 6 (𝜑 → ([,) ∘ 𝐹):ℕ⟶𝒫 ℝ*)
8180ffnd 5941 . . . . 5 (𝜑 → ([,) ∘ 𝐹) Fn ℕ)
82 fniunfv 6383 . . . . 5 (([,) ∘ 𝐹) Fn ℕ → 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
8381, 82syl 17 . . . 4 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
8475, 83sseqtrd 3599 . . 3 (𝜑𝐵 ran ([,) ∘ 𝐹))
85 ovnovollem2.z . . . 4 (𝜑𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
86 nfcv 2746 . . . . . . 7 𝑗𝐹
87 ressxr 9935 . . . . . . . . . 10 ℝ ⊆ ℝ*
88 xpss2 5137 . . . . . . . . . 10 (ℝ ⊆ ℝ* → (ℝ × ℝ) ⊆ (ℝ × ℝ*))
8987, 88ax-mp 5 . . . . . . . . 9 (ℝ × ℝ) ⊆ (ℝ × ℝ*)
9089a1i 11 . . . . . . . 8 (𝜑 → (ℝ × ℝ) ⊆ (ℝ × ℝ*))
9115, 90fssd 5952 . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ × ℝ*))
9286, 91volicofmpt 38690 . . . . . 6 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))))
939adantr 479 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 𝐴𝑉)
9413elexd 3182 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → ((𝐼𝑗)‘𝐴) ∈ V)
955, 94, 38syl2anc 690 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
9695, 13eqeltrd 2683 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∈ (ℝ × ℝ))
97 1st2nd2 7069 . . . . . . . . . . . . . . . 16 ((𝐹𝑗) ∈ (ℝ × ℝ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
9896, 97syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
9998fveq2d 6088 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ([,)‘(𝐹𝑗)) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩))
100 df-ov 6526 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
101100eqcomi 2614 . . . . . . . . . . . . . . 15 ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))
102101a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
10350, 99, 1023eqtrd 2643 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
10433, 52, 1033eqtrd 2643 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
105104fveq2d 6088 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) = (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))))
106 xp1st 7062 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
10796, 106syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
108 xp2nd 7063 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
10996, 108syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
110 volicore 39271 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑗)) ∈ ℝ ∧ (2nd ‘(𝐹𝑗)) ∈ ℝ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) ∈ ℝ)
111107, 109, 110syl2anc 690 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) ∈ ℝ)
112105, 111eqeltrd 2683 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℝ)
113112recnd 9920 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ)
114 fveq2 6084 . . . . . . . . . . 11 (𝑘 = 𝐴 → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝐴))
115114fveq2d 6088 . . . . . . . . . 10 (𝑘 = 𝐴 → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
116115prodsn 14473 . . . . . . . . 9 ((𝐴𝑉 ∧ (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
11793, 113, 116syl2anc 690 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
118117, 105eqtr2d 2640 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
119118mpteq2dva 4662 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
12092, 119eqtrd 2639 . . . . 5 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
121120fveq2d 6088 . . . 4 (𝜑 → (Σ^‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
12285, 121eqtr4d 2642 . . 3 (𝜑𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
12384, 122jca 552 . 2 (𝜑 → (𝐵 ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹))))
124 coeq2 5186 . . . . . . 7 (𝑓 = 𝐹 → ([,) ∘ 𝑓) = ([,) ∘ 𝐹))
125124rneqd 5257 . . . . . 6 (𝑓 = 𝐹 → ran ([,) ∘ 𝑓) = ran ([,) ∘ 𝐹))
126125unieqd 4372 . . . . 5 (𝑓 = 𝐹 ran ([,) ∘ 𝑓) = ran ([,) ∘ 𝐹))
127126sseq2d 3591 . . . 4 (𝑓 = 𝐹 → (𝐵 ran ([,) ∘ 𝑓) ↔ 𝐵 ran ([,) ∘ 𝐹)))
128 coeq2 5186 . . . . . 6 (𝑓 = 𝐹 → ((vol ∘ [,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝐹))
129128fveq2d 6088 . . . . 5 (𝑓 = 𝐹 → (Σ^‘((vol ∘ [,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
130129eqeq2d 2615 . . . 4 (𝑓 = 𝐹 → (𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹))))
131127, 130anbi12d 742 . . 3 (𝑓 = 𝐹 → ((𝐵 ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) ↔ (𝐵 ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))))
132131rspcev 3277 . 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))))
13321, 123, 132syl2anc 690 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 194  wa 382   = wceq 1474  wcel 1975  wne 2775  wrex 2892  Vcvv 3168  wss 3535  c0 3869  𝒫 cpw 4103  {csn 4120  cop 4126   cuni 4362   ciun 4445  cmpt 4633   × cxp 5022  dom cdm 5024  ran crn 5025  ccom 5028  Fun wfun 5780   Fn wfn 5781  wf 5782  cfv 5786  (class class class)co 6523  1st c1st 7030  2nd c2nd 7031  𝑚 cmap 7717  Xcixp 7767  cc 9786  cr 9787  *cxr 9925  cn 10863  [,)cico 12000  cprod 14416  volcvol 22952  Σ^csumge0 39055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-pre-sup 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-isom 5795  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-of 6768  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-map 7719  df-pm 7720  df-ixp 7768  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-fi 8173  df-sup 8204  df-inf 8205  df-oi 8271  df-card 8621  df-cda 8846  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-2 10922  df-3 10923  df-n0 11136  df-z 11207  df-uz 11516  df-q 11617  df-rp 11661  df-xneg 11774  df-xadd 11775  df-xmul 11776  df-ioo 12002  df-ico 12004  df-icc 12005  df-fz 12149  df-fzo 12286  df-fl 12406  df-seq 12615  df-exp 12674  df-hash 12931  df-cj 13629  df-re 13630  df-im 13631  df-sqrt 13765  df-abs 13766  df-clim 14009  df-rlim 14010  df-sum 14207  df-prod 14417  df-rest 15848  df-topgen 15869  df-psmet 19501  df-xmet 19502  df-met 19503  df-bl 19504  df-mopn 19505  df-top 20459  df-bases 20460  df-topon 20461  df-cmp 20938  df-ovol 22953  df-vol 22954
This theorem is referenced by:  ovnovollem3  39348
  Copyright terms: Public domain W3C validator