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Theorem ovnovollem2 44888
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 8787 . . . . . . . . 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 485 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
64, 5ffvelcdmd 7036 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) ∈ ((ℝ × ℝ) ↑m {𝐴}))
7 elmapi 8787 . . . . . 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 4620 . . . . . . 7 (𝐴𝑉𝐴 ∈ {𝐴})
119, 10syl 17 . . . . . 6 (𝜑𝐴 ∈ {𝐴})
1211adantr 481 . . . . 5 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ {𝐴})
138, 12ffvelcdmd 7036 . . . 4 ((𝜑𝑗 ∈ ℕ) → ((𝐼𝑗)‘𝐴) ∈ (ℝ × ℝ))
14 ovnovollem2.f . . . 4 𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼𝑗)‘𝐴))
1513, 14fmptd 7062 . . 3 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
16 reex 11142 . . . . . 6 ℝ ∈ V
1716, 16xpex 7687 . . . . 5 (ℝ × ℝ) ∈ V
18 nnex 12159 . . . . 5 ℕ ∈ V
1917, 18elmap 8809 . . . 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 256 . 2 (𝜑𝐹 ∈ ((ℝ × ℝ) ↑m ℕ))
22 ovnovollem2.s . . . . . 6 (𝜑 → (𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
23 elsni 4603 . . . . . . . . . . . . 13 (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
2423fveq2d 6846 . . . . . . . . . . . 12 (𝑘 ∈ {𝐴} → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝐴))
2524adantl 482 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝐴))
26 elmapfun 8804 . . . . . . . . . . . . . 14 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑m {𝐴}) → Fun (𝐼𝑗))
276, 26syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → Fun (𝐼𝑗))
288fdmd 6679 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → dom (𝐼𝑗) = {𝐴})
2928eqcomd 2742 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → {𝐴} = dom (𝐼𝑗))
3012, 29eleqtrd 2840 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ dom (𝐼𝑗))
31 fvco 6939 . . . . . . . . . . . . 13 ((Fun (𝐼𝑗) ∧ 𝐴 ∈ dom (𝐼𝑗)) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
3227, 30, 31syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
3332adantr 481 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝐴) = ([,)‘((𝐼𝑗)‘𝐴)))
34 id 22 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
35 fvexd 6857 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ → ((𝐼𝑗)‘𝐴) ∈ V)
3614fvmpt2 6959 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ ∧ ((𝐼𝑗)‘𝐴) ∈ V) → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
3734, 35, 36syl2anc 584 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
3837eqcomd 2742 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → ((𝐼𝑗)‘𝐴) = (𝐹𝑗))
3938fveq2d 6846 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → ([,)‘((𝐼𝑗)‘𝐴)) = ([,)‘(𝐹𝑗)))
4039adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ([,)‘((𝐼𝑗)‘𝐴)) = ([,)‘(𝐹𝑗)))
4115ffund 6672 . . . . . . . . . . . . . . . 16 (𝜑 → Fun 𝐹)
4241adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → Fun 𝐹)
4314, 13dmmptd 6646 . . . . . . . . . . . . . . . . . 18 (𝜑 → dom 𝐹 = ℕ)
4443eqcomd 2742 . . . . . . . . . . . . . . . . 17 (𝜑 → ℕ = dom 𝐹)
4544adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ℕ = dom 𝐹)
465, 45eleqtrd 2840 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ dom 𝐹)
47 fvco 6939 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑗 ∈ dom 𝐹) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
4842, 46, 47syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
4948eqcomd 2742 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ([,)‘(𝐹𝑗)) = (([,) ∘ 𝐹)‘𝑗))
5040, 49eqtrd 2776 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ([,)‘((𝐼𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗))
5150adantr 481 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ([,)‘((𝐼𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗))
5225, 33, 513eqtrd 2780 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = (([,) ∘ 𝐹)‘𝑗))
5352ixpeq2dva 8850 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗))
54 snex 5388 . . . . . . . . . . 11 {𝐴} ∈ V
55 fvex 6855 . . . . . . . . . . 11 (([,) ∘ 𝐹)‘𝑗) ∈ V
5654, 55ixpconst 8845 . . . . . . . . . 10 X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗) = ((([,) ∘ 𝐹)‘𝑗) ↑m {𝐴})
5756a1i 11 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗) = ((([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
5853, 57eqtrd 2776 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = ((([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
5958iuneq2dv 4978 . . . . . . 7 (𝜑 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
60 nfv 1917 . . . . . . . 8 𝑗𝜑
6118a1i 11 . . . . . . . 8 (𝜑 → ℕ ∈ V)
62 fvexd 6857 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) ∈ V)
6360, 61, 62, 9iunmapsn 43428 . . . . . . 7 (𝜑 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}) = ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
6459, 63eqtrd 2776 . . . . . 6 (𝜑 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
6522, 64sseqtrd 3984 . . . . 5 (𝜑 → (𝐵m {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
66 ovnovollem2.b . . . . . 6 (𝜑𝐵𝑊)
6718, 55iunex 7901 . . . . . . 7 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V
6867a1i 11 . . . . . 6 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V)
6954a1i 11 . . . . . 6 (𝜑 → {𝐴} ∈ V)
7011ne0d 4295 . . . . . 6 (𝜑 → {𝐴} ≠ ∅)
7166, 68, 69, 70mapss2 43416 . . . . 5 (𝜑 → (𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↔ (𝐵m {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴})))
7265, 71mpbird 256 . . . 4 (𝜑𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗))
73 icof 43430 . . . . . . . 8 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
7473a1i 11 . . . . . . 7 (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
75 rexpssxrxp 11200 . . . . . . . 8 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
7675a1i 11 . . . . . . 7 (𝜑 → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
7774, 76, 15fcoss 43421 . . . . . 6 (𝜑 → ([,) ∘ 𝐹):ℕ⟶𝒫 ℝ*)
7877ffnd 6669 . . . . 5 (𝜑 → ([,) ∘ 𝐹) Fn ℕ)
79 fniunfv 7194 . . . . 5 (([,) ∘ 𝐹) Fn ℕ → 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
8078, 79syl 17 . . . 4 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
8172, 80sseqtrd 3984 . . 3 (𝜑𝐵 ran ([,) ∘ 𝐹))
82 ovnovollem2.z . . . 4 (𝜑𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
83 nfcv 2907 . . . . . . 7 𝑗𝐹
84 ressxr 11199 . . . . . . . . . 10 ℝ ⊆ ℝ*
85 xpss2 5653 . . . . . . . . . 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 6686 . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ × ℝ*))
8983, 88volicofmpt 44228 . . . . . 6 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))))
909adantr 481 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 𝐴𝑉)
91 fvexd 6857 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → ((𝐼𝑗)‘𝐴) ∈ V)
925, 91, 36syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) = ((𝐼𝑗)‘𝐴))
9392, 13eqeltrd 2838 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∈ (ℝ × ℝ))
94 1st2nd2 7960 . . . . . . . . . . . . . . . 16 ((𝐹𝑗) ∈ (ℝ × ℝ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
9593, 94syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
9695fveq2d 6846 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ([,)‘(𝐹𝑗)) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩))
97 df-ov 7360 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
9897eqcomi 2745 . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
10132, 50, 1003eqtrd 2780 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
102101fveq2d 6846 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) = (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))))
103 xp1st 7953 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
10493, 103syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
105 xp2nd 7954 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
10693, 105syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
107 volicore 44812 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑗)) ∈ ℝ ∧ (2nd ‘(𝐹𝑗)) ∈ ℝ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) ∈ ℝ)
108104, 106, 107syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) ∈ ℝ)
109102, 108eqeltrd 2838 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℝ)
110109recnd 11183 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ)
111 2fveq3 6847 . . . . . . . . . 10 (𝑘 = 𝐴 → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
112111prodsn 15845 . . . . . . . . 9 ((𝐴𝑉 ∧ (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
11390, 110, 112syl2anc 584 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
114113, 102eqtr2d 2777 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
115114mpteq2dva 5205 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
11689, 115eqtrd 2776 . . . . 5 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
117116fveq2d 6846 . . . 4 (𝜑 → (Σ^‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
11882, 117eqtr4d 2779 . . 3 (𝜑𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
11981, 118jca 512 . 2 (𝜑 → (𝐵 ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹))))
120 coeq2 5814 . . . . . . 7 (𝑓 = 𝐹 → ([,) ∘ 𝑓) = ([,) ∘ 𝐹))
121120rneqd 5893 . . . . . 6 (𝑓 = 𝐹 → ran ([,) ∘ 𝑓) = ran ([,) ∘ 𝐹))
122121unieqd 4879 . . . . 5 (𝑓 = 𝐹 ran ([,) ∘ 𝑓) = ran ([,) ∘ 𝐹))
123122sseq2d 3976 . . . 4 (𝑓 = 𝐹 → (𝐵 ran ([,) ∘ 𝑓) ↔ 𝐵 ran ([,) ∘ 𝐹)))
124 coeq2 5814 . . . . . 6 (𝑓 = 𝐹 → ((vol ∘ [,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝐹))
125124fveq2d 6846 . . . . 5 (𝑓 = 𝐹 → (Σ^‘((vol ∘ [,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
126125eqeq2d 2747 . . . 4 (𝑓 = 𝐹 → (𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹))))
127123, 126anbi12d 631 . . 3 (𝑓 = 𝐹 → ((𝐵 ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) ↔ (𝐵 ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))))
128127rspcev 3581 . 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 584 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 205  wa 396   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  wss 3910  𝒫 cpw 4560  {csn 4586  cop 4592   cuni 4865   ciun 4954  cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634  ccom 5637  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  m cmap 8765  Xcixp 8835  cc 11049  cr 11050  *cxr 11188  cn 12153  [,)cico 13266  cprod 15788  volcvol 24827  Σ^csumge0 44593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-prod 15789  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828  df-vol 24829
This theorem is referenced by:  ovnovollem3  44889
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