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Theorem ovnovollem2 40208
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 7839 . . . . . . . . 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 481 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 {𝐴}))
5 simpr 477 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
64, 5ffvelrnd 6326 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 {𝐴}))
7 elmapi 7839 . . . . . 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 4184 . . . . . . 7 (𝐴𝑉𝐴 ∈ {𝐴})
119, 10syl 17 . . . . . 6 (𝜑𝐴 ∈ {𝐴})
1211adantr 481 . . . . 5 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ {𝐴})
138, 12ffvelrnd 6326 . . . 4 ((𝜑𝑗 ∈ ℕ) → ((𝐼𝑗)‘𝐴) ∈ (ℝ × ℝ))
14 ovnovollem2.f . . . 4 𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼𝑗)‘𝐴))
1513, 14fmptd 6351 . . 3 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
16 reex 9987 . . . . . 6 ℝ ∈ V
1716, 16xpex 6927 . . . . 5 (ℝ × ℝ) ∈ V
18 nnex 10986 . . . . 5 ℕ ∈ V
1917, 18elmap 7846 . . . 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 247 . 2 (𝜑𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))
22 ovnovollem2.s . . . . . 6 (𝜑 → (𝐵𝑚 {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
23 elsni 4172 . . . . . . . . . . . . 13 (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
2423fveq2d 6162 . . . . . . . . . . . 12 (𝑘 ∈ {𝐴} → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝐴))
2524adantl 482 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝐴))
26 elmapfun 7841 . . . . . . . . . . . . . 14 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 {𝐴}) → Fun (𝐼𝑗))
276, 26syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → Fun (𝐼𝑗))
28 fdm 6018 . . . . . . . . . . . . . . . 16 ((𝐼𝑗):{𝐴}⟶(ℝ × ℝ) → dom (𝐼𝑗) = {𝐴})
298, 28syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → dom (𝐼𝑗) = {𝐴})
3029eqcomd 2627 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → {𝐴} = dom (𝐼𝑗))
3112, 30eleqtrd 2700 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ dom (𝐼𝑗))
32 fvco 6241 . . . . . . . . . . . . 13 ((Fun (𝐼𝑗) ∧ 𝐴 ∈ dom (𝐼𝑗)) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
3327, 31, 32syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
3433adantr 481 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
35 id 22 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
36 fvexd 6170 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ → ((𝐼𝑗)‘𝐴) ∈ V)
3714fvmpt2 6258 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ ∧ ((𝐼𝑗)‘𝐴) ∈ V) → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
3835, 36, 37syl2anc 692 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
3938eqcomd 2627 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → ((𝐼𝑗)‘𝐴) = (𝐹𝑗))
4039fveq2d 6162 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → ([,)‘((𝐼𝑗)‘𝐴)) = ([,)‘(𝐹𝑗)))
4140adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ([,)‘((𝐼𝑗)‘𝐴)) = ([,)‘(𝐹𝑗)))
4215ffund 6016 . . . . . . . . . . . . . . . 16 (𝜑 → Fun 𝐹)
4342adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → Fun 𝐹)
4414, 13dmmptd 5991 . . . . . . . . . . . . . . . . . 18 (𝜑 → dom 𝐹 = ℕ)
4544eqcomd 2627 . . . . . . . . . . . . . . . . 17 (𝜑 → ℕ = dom 𝐹)
4645adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ℕ = dom 𝐹)
475, 46eleqtrd 2700 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ dom 𝐹)
48 fvco 6241 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑗 ∈ dom 𝐹) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
4943, 47, 48syl2anc 692 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
5049eqcomd 2627 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ([,)‘(𝐹𝑗)) = (([,) ∘ 𝐹)‘𝑗))
5141, 50eqtrd 2655 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ([,)‘((𝐼𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗))
5251adantr 481 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ([,)‘((𝐼𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗))
5325, 34, 523eqtrd 2659 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ 𝐹)‘𝑗))
5453ixpeq2dva 7883 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗))
55 snex 4879 . . . . . . . . . . 11 {𝐴} ∈ V
56 fvex 6168 . . . . . . . . . . 11 (([,) ∘ 𝐹)‘𝑗) ∈ V
5755, 56ixpconst 7878 . . . . . . . . . 10 X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴})
5857a1i 11 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
5954, 58eqtrd 2655 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
6059iuneq2dv 4515 . . . . . . 7 (𝜑 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
61 nfv 1840 . . . . . . . 8 𝑗𝜑
6218a1i 11 . . . . . . . 8 (𝜑 → ℕ ∈ V)
63 fvexd 6170 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) ∈ V)
6461, 62, 63, 9iunmapsn 38918 . . . . . . 7 (𝜑 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}) = ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
6560, 64eqtrd 2655 . . . . . 6 (𝜑 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
6622, 65sseqtrd 3626 . . . . 5 (𝜑 → (𝐵𝑚 {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴}))
67 ovnovollem2.b . . . . . 6 (𝜑𝐵𝑊)
6818, 56iunex 7108 . . . . . . 7 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V
6968a1i 11 . . . . . 6 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V)
7055a1i 11 . . . . . 6 (𝜑 → {𝐴} ∈ V)
71 ne0i 3903 . . . . . . 7 (𝐴 ∈ {𝐴} → {𝐴} ≠ ∅)
7211, 71syl 17 . . . . . 6 (𝜑 → {𝐴} ≠ ∅)
7367, 69, 70, 72mapss2 38906 . . . . 5 (𝜑 → (𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↔ (𝐵𝑚 {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑𝑚 {𝐴})))
7466, 73mpbird 247 . . . 4 (𝜑𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗))
75 icof 38920 . . . . . . . 8 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
7675a1i 11 . . . . . . 7 (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
77 rexpssxrxp 10044 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
7877a1i 11 . . . . . . 7 (𝜑 → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
7976, 78, 15fcoss 38911 . . . . . 6 (𝜑 → ([,) ∘ 𝐹):ℕ⟶𝒫 ℝ*)
8079ffnd 6013 . . . . 5 (𝜑 → ([,) ∘ 𝐹) Fn ℕ)
81 fniunfv 6470 . . . . 5 (([,) ∘ 𝐹) Fn ℕ → 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
8280, 81syl 17 . . . 4 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
8374, 82sseqtrd 3626 . . 3 (𝜑𝐵 ran ([,) ∘ 𝐹))
84 ovnovollem2.z . . . 4 (𝜑𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
85 nfcv 2761 . . . . . . 7 𝑗𝐹
86 ressxr 10043 . . . . . . . . . 10 ℝ ⊆ ℝ*
87 xpss2 5200 . . . . . . . . . 10 (ℝ ⊆ ℝ* → (ℝ × ℝ) ⊆ (ℝ × ℝ*))
8886, 87ax-mp 5 . . . . . . . . 9 (ℝ × ℝ) ⊆ (ℝ × ℝ*)
8988a1i 11 . . . . . . . 8 (𝜑 → (ℝ × ℝ) ⊆ (ℝ × ℝ*))
9015, 89fssd 6024 . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ × ℝ*))
9185, 90volicofmpt 39551 . . . . . 6 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))))
929adantr 481 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 𝐴𝑉)
93 fvexd 6170 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → ((𝐼𝑗)‘𝐴) ∈ V)
945, 93, 37syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
9594, 13eqeltrd 2698 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∈ (ℝ × ℝ))
96 1st2nd2 7165 . . . . . . . . . . . . . . . 16 ((𝐹𝑗) ∈ (ℝ × ℝ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
9795, 96syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
9897fveq2d 6162 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ([,)‘(𝐹𝑗)) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩))
99 df-ov 6618 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
10099eqcomi 2630 . . . . . . . . . . . . . . 15 ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))
101100a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
10249, 98, 1013eqtrd 2659 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
10333, 51, 1023eqtrd 2659 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
104103fveq2d 6162 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) = (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))))
105 xp1st 7158 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
10695, 105syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
107 xp2nd 7159 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
10895, 107syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
109 volicore 40132 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑗)) ∈ ℝ ∧ (2nd ‘(𝐹𝑗)) ∈ ℝ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) ∈ ℝ)
110106, 108, 109syl2anc 692 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) ∈ ℝ)
111104, 110eqeltrd 2698 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℝ)
112111recnd 10028 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ)
113 fveq2 6158 . . . . . . . . . . 11 (𝑘 = 𝐴 → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝐴))
114113fveq2d 6162 . . . . . . . . . 10 (𝑘 = 𝐴 → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
115114prodsn 14636 . . . . . . . . 9 ((𝐴𝑉 ∧ (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
11692, 112, 115syl2anc 692 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
117116, 104eqtr2d 2656 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
118117mpteq2dva 4714 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
11991, 118eqtrd 2655 . . . . 5 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
120119fveq2d 6162 . . . 4 (𝜑 → (Σ^‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
12184, 120eqtr4d 2658 . . 3 (𝜑𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
12283, 121jca 554 . 2 (𝜑 → (𝐵 ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹))))
123 coeq2 5250 . . . . . . 7 (𝑓 = 𝐹 → ([,) ∘ 𝑓) = ([,) ∘ 𝐹))
124123rneqd 5323 . . . . . 6 (𝑓 = 𝐹 → ran ([,) ∘ 𝑓) = ran ([,) ∘ 𝐹))
125124unieqd 4419 . . . . 5 (𝑓 = 𝐹 ran ([,) ∘ 𝑓) = ran ([,) ∘ 𝐹))
126125sseq2d 3618 . . . 4 (𝑓 = 𝐹 → (𝐵 ran ([,) ∘ 𝑓) ↔ 𝐵 ran ([,) ∘ 𝐹)))
127 coeq2 5250 . . . . . 6 (𝑓 = 𝐹 → ((vol ∘ [,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝐹))
128127fveq2d 6162 . . . . 5 (𝑓 = 𝐹 → (Σ^‘((vol ∘ [,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
129128eqeq2d 2631 . . . 4 (𝑓 = 𝐹 → (𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹))))
130126, 129anbi12d 746 . . 3 (𝑓 = 𝐹 → ((𝐵 ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) ↔ (𝐵 ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))))
131130rspcev 3299 . 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))))
13221, 122, 131syl2anc 692 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 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wrex 2909  Vcvv 3190  wss 3560  c0 3897  𝒫 cpw 4136  {csn 4155  cop 4161   cuni 4409   ciun 4492  cmpt 4683   × cxp 5082  dom cdm 5084  ran crn 5085  ccom 5088  Fun wfun 5851   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  𝑚 cmap 7817  Xcixp 7868  cc 9894  cr 9895  *cxr 10033  cn 10980  [,)cico 12135  cprod 14579  volcvol 23172  Σ^csumge0 39916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fi 8277  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-z 11338  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-ioo 12137  df-ico 12139  df-icc 12140  df-fz 12285  df-fzo 12423  df-fl 12549  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-rlim 14170  df-sum 14367  df-prod 14580  df-rest 16023  df-topgen 16044  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-top 20639  df-topon 20656  df-bases 20690  df-cmp 21130  df-ovol 23173  df-vol 23174
This theorem is referenced by:  ovnovollem3  40209
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