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

Theorem ovnovollem2 39547
 Description: if 𝐼 is a cover of (𝐵 ↑𝑚 {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 7765 . . . . . . . . 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 480 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 {𝐴}))
5 simpr 476 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
64, 5ffvelrnd 6268 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 {𝐴}))
7 elmapi 7765 . . . . . 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 4153 . . . . . . 7 (𝐴𝑉𝐴 ∈ {𝐴})
119, 10syl 17 . . . . . 6 (𝜑𝐴 ∈ {𝐴})
1211adantr 480 . . . . 5 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ {𝐴})
138, 12ffvelrnd 6268 . . . 4 ((𝜑𝑗 ∈ ℕ) → ((𝐼𝑗)‘𝐴) ∈ (ℝ × ℝ))
14 ovnovollem2.f . . . 4 𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼𝑗)‘𝐴))
1513, 14fmptd 6292 . . 3 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
16 reex 9906 . . . . . 6 ℝ ∈ V
1716, 16xpex 6860 . . . . 5 (ℝ × ℝ) ∈ V
18 nnex 10903 . . . . 5 ℕ ∈ V
1917, 18elmap 7772 . . . 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 246 . 2 (𝜑𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
22 ovnovollem2.s . . . . . 6 (𝜑 → (𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
23 elsni 4142 . . . . . . . . . . . . 13 (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
2423fveq2d 6107 . . . . . . . . . . . 12 (𝑘 ∈ {𝐴} → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝐴))
2524adantl 481 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝐴))
26 elmapfun 7767 . . . . . . . . . . . . . 14 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 {𝐴}) → Fun (𝐼𝑗))
276, 26syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → Fun (𝐼𝑗))
28 fdm 5964 . . . . . . . . . . . . . . . 16 ((𝐼𝑗):{𝐴}⟶(ℝ × ℝ) → dom (𝐼𝑗) = {𝐴})
298, 28syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → dom (𝐼𝑗) = {𝐴})
3029eqcomd 2616 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → {𝐴} = dom (𝐼𝑗))
3112, 30eleqtrd 2690 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ dom (𝐼𝑗))
32 fvco 6184 . . . . . . . . . . . . 13 ((Fun (𝐼𝑗) ∧ 𝐴 ∈ dom (𝐼𝑗)) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
3327, 31, 32syl2anc 691 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
3433adantr 480 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
35 id 22 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
36 fvex 6113 . . . . . . . . . . . . . . . . . 18 ((𝐼𝑗)‘𝐴) ∈ V
3736a1i 11 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ → ((𝐼𝑗)‘𝐴) ∈ V)
3814fvmpt2 6200 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ ∧ ((𝐼𝑗)‘𝐴) ∈ V) → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
3935, 37, 38syl2anc 691 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
4039eqcomd 2616 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → ((𝐼𝑗)‘𝐴) = (𝐹𝑗))
4140fveq2d 6107 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → ([,)‘((𝐼𝑗)‘𝐴)) = ([,)‘(𝐹𝑗)))
4241adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ([,)‘((𝐼𝑗)‘𝐴)) = ([,)‘(𝐹𝑗)))
4315ffund 5962 . . . . . . . . . . . . . . . 16 (𝜑 → Fun 𝐹)
4443adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → Fun 𝐹)
4514, 13dmmptd 5937 . . . . . . . . . . . . . . . . . 18 (𝜑 → dom 𝐹 = ℕ)
4645eqcomd 2616 . . . . . . . . . . . . . . . . 17 (𝜑 → ℕ = dom 𝐹)
4746adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ℕ = dom 𝐹)
485, 47eleqtrd 2690 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ dom 𝐹)
49 fvco 6184 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑗 ∈ dom 𝐹) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
5044, 48, 49syl2anc 691 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
5150eqcomd 2616 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ([,)‘(𝐹𝑗)) = (([,) ∘ 𝐹)‘𝑗))
5242, 51eqtrd 2644 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ([,)‘((𝐼𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗))
5352adantr 480 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ([,)‘((𝐼𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗))
5425, 34, 533eqtrd 2648 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ 𝐹)‘𝑗))
5554ixpeq2dva 7809 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗))
56 snex 4835 . . . . . . . . . . 11 {𝐴} ∈ V
57 fvex 6113 . . . . . . . . . . 11 (([,) ∘ 𝐹)‘𝑗) ∈ V
5856, 57ixpconst 7804 . . . . . . . . . 10 X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴})
5958a1i 11 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
6055, 59eqtrd 2644 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
6160iuneq2dv 4478 . . . . . . 7 (𝜑 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
62 nfv 1830 . . . . . . . 8 𝑗𝜑
6318a1i 11 . . . . . . . 8 (𝜑 → ℕ ∈ V)
6457a1i 11 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) ∈ V)
6562, 63, 64, 9iunmapsn 38404 . . . . . . 7 (𝜑 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}) = ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
6661, 65eqtrd 2644 . . . . . 6 (𝜑 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
6722, 66sseqtrd 3604 . . . . 5 (𝜑 → (𝐵𝑚 {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
68 ovnovollem2.b . . . . . 6 (𝜑𝐵𝑊)
6918, 57iunex 7039 . . . . . . 7 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V
7069a1i 11 . . . . . 6 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V)
7156a1i 11 . . . . . 6 (𝜑 → {𝐴} ∈ V)
72 ne0i 3880 . . . . . . 7 (𝐴 ∈ {𝐴} → {𝐴} ≠ ∅)
7311, 72syl 17 . . . . . 6 (𝜑 → {𝐴} ≠ ∅)
7468, 70, 71, 73mapss2 38392 . . . . 5 (𝜑 → (𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↔ (𝐵𝑚 {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴})))
7567, 74mpbird 246 . . . 4 (𝜑𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗))
76 icof 38406 . . . . . . . 8 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
7776a1i 11 . . . . . . 7 (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
78 rexpssxrxp 9963 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
7978a1i 11 . . . . . . 7 (𝜑 → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
8077, 79, 15fcoss 38397 . . . . . 6 (𝜑 → ([,) ∘ 𝐹):ℕ⟶𝒫 ℝ*)
8180ffnd 5959 . . . . 5 (𝜑 → ([,) ∘ 𝐹) Fn ℕ)
82 fniunfv 6409 . . . . 5 (([,) ∘ 𝐹) Fn ℕ → 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
8381, 82syl 17 . . . 4 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
8475, 83sseqtrd 3604 . . 3 (𝜑𝐵 ran ([,) ∘ 𝐹))
85 ovnovollem2.z . . . 4 (𝜑𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
86 nfcv 2751 . . . . . . 7 𝑗𝐹
87 ressxr 9962 . . . . . . . . . 10 ℝ ⊆ ℝ*
88 xpss2 5152 . . . . . . . . . 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 5970 . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ × ℝ*))
9286, 91volicofmpt 38890 . . . . . 6 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))))
939adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 𝐴𝑉)
9413elexd 3187 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → ((𝐼𝑗)‘𝐴) ∈ V)
955, 94, 38syl2anc 691 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
9695, 13eqeltrd 2688 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∈ (ℝ × ℝ))
97 1st2nd2 7096 . . . . . . . . . . . . . . . 16 ((𝐹𝑗) ∈ (ℝ × ℝ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
9896, 97syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
9998fveq2d 6107 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ([,)‘(𝐹𝑗)) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩))
100 df-ov 6552 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
101100eqcomi 2619 . . . . . . . . . . . . . . 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 2648 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
10433, 52, 1033eqtrd 2648 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
105104fveq2d 6107 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) = (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))))
106 xp1st 7089 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
10796, 106syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
108 xp2nd 7090 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
10996, 108syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
110 volicore 39471 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑗)) ∈ ℝ ∧ (2nd ‘(𝐹𝑗)) ∈ ℝ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) ∈ ℝ)
111107, 109, 110syl2anc 691 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) ∈ ℝ)
112105, 111eqeltrd 2688 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℝ)
113112recnd 9947 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ)
114 fveq2 6103 . . . . . . . . . . 11 (𝑘 = 𝐴 → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝐴))
115114fveq2d 6107 . . . . . . . . . 10 (𝑘 = 𝐴 → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
116115prodsn 14531 . . . . . . . . 9 ((𝐴𝑉 ∧ (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
11793, 113, 116syl2anc 691 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
118117, 105eqtr2d 2645 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
119118mpteq2dva 4672 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
12092, 119eqtrd 2644 . . . . 5 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
121120fveq2d 6107 . . . 4 (𝜑 → (Σ^‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
12285, 121eqtr4d 2647 . . 3 (𝜑𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
12384, 122jca 553 . 2 (𝜑 → (𝐵 ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹))))
124 coeq2 5202 . . . . . . 7 (𝑓 = 𝐹 → ([,) ∘ 𝑓) = ([,) ∘ 𝐹))
125124rneqd 5274 . . . . . 6 (𝑓 = 𝐹 → ran ([,) ∘ 𝑓) = ran ([,) ∘ 𝐹))
126125unieqd 4382 . . . . 5 (𝑓 = 𝐹 ran ([,) ∘ 𝑓) = ran ([,) ∘ 𝐹))
127126sseq2d 3596 . . . 4 (𝑓 = 𝐹 → (𝐵 ran ([,) ∘ 𝑓) ↔ 𝐵 ran ([,) ∘ 𝐹)))
128 coeq2 5202 . . . . . 6 (𝑓 = 𝐹 → ((vol ∘ [,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝐹))
129128fveq2d 6107 . . . . 5 (𝑓 = 𝐹 → (Σ^‘((vol ∘ [,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
130129eqeq2d 2620 . . . 4 (𝑓 = 𝐹 → (𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹))))
131127, 130anbi12d 743 . . 3 (𝑓 = 𝐹 → ((𝐵 ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) ↔ (𝐵 ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))))
132131rspcev 3282 . 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 691 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  Xcixp 7794  ℂcc 9813  ℝcr 9814  ℝ*cxr 9952  ℕcn 10897  [,)cico 12048  ∏cprod 14474  volcvol 23039  Σ^csumge0 39255 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-prod 14475  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041 This theorem is referenced by:  ovnovollem3  39548
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