Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovnovollem1 Structured version   Visualization version   GIF version

Theorem ovnovollem1 44887
Description: if 𝐹 is a cover of 𝐵 in , then 𝐼 is the corresponding cover in the space of 1-dimensional reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
ovnovollem1.a (𝜑𝐴𝑉)
ovnovollem1.f (𝜑𝐹 ∈ ((ℝ × ℝ) ↑m ℕ))
ovnovollem1.i 𝐼 = (𝑗 ∈ ℕ ↦ {⟨𝐴, (𝐹𝑗)⟩})
ovnovollem1.s (𝜑𝐵 ran ([,) ∘ 𝐹))
ovnovollem1.b (𝜑𝐵𝑊)
ovnovollem1.z (𝜑𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
Assertion
Ref Expression
ovnovollem1 (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ)((𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
Distinct variable groups:   𝐴,𝑖,𝑗,𝑘   𝐵,𝑖   𝑗,𝐹,𝑘   𝑖,𝐼,𝑗,𝑘   𝑘,𝑉   𝑖,𝑍   𝜑,𝑗,𝑘
Allowed substitution hints:   𝜑(𝑖)   𝐵(𝑗,𝑘)   𝐹(𝑖)   𝑉(𝑖,𝑗)   𝑊(𝑖,𝑗,𝑘)   𝑍(𝑗,𝑘)

Proof of Theorem ovnovollem1
StepHypRef Expression
1 eqidd 2737 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → {⟨𝐴, (𝐹𝑗)⟩} = {⟨𝐴, (𝐹𝑗)⟩})
2 ovnovollem1.a . . . . . . . . 9 (𝜑𝐴𝑉)
32adantr 481 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝐴𝑉)
4 ovnovollem1.f . . . . . . . . . 10 (𝜑𝐹 ∈ ((ℝ × ℝ) ↑m ℕ))
5 elmapi 8787 . . . . . . . . . 10 (𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
64, 5syl 17 . . . . . . . . 9 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
76ffvelcdmda 7035 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∈ (ℝ × ℝ))
8 fsng 7083 . . . . . . . 8 ((𝐴𝑉 ∧ (𝐹𝑗) ∈ (ℝ × ℝ)) → ({⟨𝐴, (𝐹𝑗)⟩}:{𝐴}⟶{(𝐹𝑗)} ↔ {⟨𝐴, (𝐹𝑗)⟩} = {⟨𝐴, (𝐹𝑗)⟩}))
93, 7, 8syl2anc 584 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ({⟨𝐴, (𝐹𝑗)⟩}:{𝐴}⟶{(𝐹𝑗)} ↔ {⟨𝐴, (𝐹𝑗)⟩} = {⟨𝐴, (𝐹𝑗)⟩}))
101, 9mpbird 256 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → {⟨𝐴, (𝐹𝑗)⟩}:{𝐴}⟶{(𝐹𝑗)})
117snssd 4769 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → {(𝐹𝑗)} ⊆ (ℝ × ℝ))
1210, 11fssd 6686 . . . . 5 ((𝜑𝑗 ∈ ℕ) → {⟨𝐴, (𝐹𝑗)⟩}:{𝐴}⟶(ℝ × ℝ))
13 reex 11142 . . . . . . . 8 ℝ ∈ V
1413, 13xpex 7687 . . . . . . 7 (ℝ × ℝ) ∈ V
1514a1i 11 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (ℝ × ℝ) ∈ V)
16 snex 5388 . . . . . . 7 {𝐴} ∈ V
1716a1i 11 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → {𝐴} ∈ V)
1815, 17elmapd 8779 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ({⟨𝐴, (𝐹𝑗)⟩} ∈ ((ℝ × ℝ) ↑m {𝐴}) ↔ {⟨𝐴, (𝐹𝑗)⟩}:{𝐴}⟶(ℝ × ℝ)))
1912, 18mpbird 256 . . . 4 ((𝜑𝑗 ∈ ℕ) → {⟨𝐴, (𝐹𝑗)⟩} ∈ ((ℝ × ℝ) ↑m {𝐴}))
20 ovnovollem1.i . . . 4 𝐼 = (𝑗 ∈ ℕ ↦ {⟨𝐴, (𝐹𝑗)⟩})
2119, 20fmptd 7062 . . 3 (𝜑𝐼:ℕ⟶((ℝ × ℝ) ↑m {𝐴}))
22 ovexd 7392 . . . 4 (𝜑 → ((ℝ × ℝ) ↑m {𝐴}) ∈ V)
23 nnex 12159 . . . . 5 ℕ ∈ V
2423a1i 11 . . . 4 (𝜑 → ℕ ∈ V)
2522, 24elmapd 8779 . . 3 (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m {𝐴})))
2621, 25mpbird 256 . 2 (𝜑𝐼 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ))
27 ovnovollem1.s . . . . . 6 (𝜑𝐵 ran ([,) ∘ 𝐹))
28 icof 43430 . . . . . . . . . . 11 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
2928a1i 11 . . . . . . . . . 10 (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
30 rexpssxrxp 11200 . . . . . . . . . . 11 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
3130a1i 11 . . . . . . . . . 10 (𝜑 → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
3229, 31, 6fcoss 43421 . . . . . . . . 9 (𝜑 → ([,) ∘ 𝐹):ℕ⟶𝒫 ℝ*)
3332ffnd 6669 . . . . . . . 8 (𝜑 → ([,) ∘ 𝐹) Fn ℕ)
34 fniunfv 7194 . . . . . . . 8 (([,) ∘ 𝐹) Fn ℕ → 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
3533, 34syl 17 . . . . . . 7 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
3635eqcomd 2742 . . . . . 6 (𝜑 ran ([,) ∘ 𝐹) = 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗))
3727, 36sseqtrd 3984 . . . . 5 (𝜑𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗))
38 ovnovollem1.b . . . . . 6 (𝜑𝐵𝑊)
39 fvex 6855 . . . . . . . 8 (([,) ∘ 𝐹)‘𝑗) ∈ V
4023, 39iunex 7901 . . . . . . 7 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V
4140a1i 11 . . . . . 6 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V)
4216a1i 11 . . . . . 6 (𝜑 → {𝐴} ∈ V)
432snn0d 4736 . . . . . 6 (𝜑 → {𝐴} ≠ ∅)
4438, 41, 42, 43mapss2 43416 . . . . 5 (𝜑 → (𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↔ (𝐵m {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴})))
4537, 44mpbid 231 . . . 4 (𝜑 → (𝐵m {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
46 nfv 1917 . . . . . . 7 𝑗𝜑
47 fvexd 6857 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ([,)‘(𝐹𝑗)) ∈ V)
4846, 24, 47, 2iunmapsn 43428 . . . . . 6 (𝜑 𝑗 ∈ ℕ (([,)‘(𝐹𝑗)) ↑m {𝐴}) = ( 𝑗 ∈ ℕ ([,)‘(𝐹𝑗)) ↑m {𝐴}))
4948eqcomd 2742 . . . . 5 (𝜑 → ( 𝑗 ∈ ℕ ([,)‘(𝐹𝑗)) ↑m {𝐴}) = 𝑗 ∈ ℕ (([,)‘(𝐹𝑗)) ↑m {𝐴}))
50 elmapfun 8804 . . . . . . . . . 10 (𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) → Fun 𝐹)
514, 50syl 17 . . . . . . . . 9 (𝜑 → Fun 𝐹)
5251adantr 481 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → Fun 𝐹)
53 simpr 485 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
546fdmd 6679 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = ℕ)
5554eqcomd 2742 . . . . . . . . . 10 (𝜑 → ℕ = dom 𝐹)
5655adantr 481 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ℕ = dom 𝐹)
5753, 56eleqtrd 2840 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ dom 𝐹)
58 fvco 6939 . . . . . . . 8 ((Fun 𝐹𝑗 ∈ dom 𝐹) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
5952, 57, 58syl2anc 584 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
6059iuneq2dv 4978 . . . . . 6 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = 𝑗 ∈ ℕ ([,)‘(𝐹𝑗)))
6160oveq1d 7372 . . . . 5 (𝜑 → ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}) = ( 𝑗 ∈ ℕ ([,)‘(𝐹𝑗)) ↑m {𝐴}))
6210ffund 6672 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → Fun {⟨𝐴, (𝐹𝑗)⟩})
63 id 22 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
64 snex 5388 . . . . . . . . . . . . . . . 16 {⟨𝐴, (𝐹𝑗)⟩} ∈ V
6564a1i 11 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → {⟨𝐴, (𝐹𝑗)⟩} ∈ V)
6620fvmpt2 6959 . . . . . . . . . . . . . . 15 ((𝑗 ∈ ℕ ∧ {⟨𝐴, (𝐹𝑗)⟩} ∈ V) → (𝐼𝑗) = {⟨𝐴, (𝐹𝑗)⟩})
6763, 65, 66syl2anc 584 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → (𝐼𝑗) = {⟨𝐴, (𝐹𝑗)⟩})
6867adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) = {⟨𝐴, (𝐹𝑗)⟩})
6968funeqd 6523 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (Fun (𝐼𝑗) ↔ Fun {⟨𝐴, (𝐹𝑗)⟩}))
7062, 69mpbird 256 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → Fun (𝐼𝑗))
7170adantr 481 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → Fun (𝐼𝑗))
72 simpr 485 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → 𝑘 ∈ {𝐴})
7368dmeqd 5861 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → dom (𝐼𝑗) = dom {⟨𝐴, (𝐹𝑗)⟩})
7410fdmd 6679 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → dom {⟨𝐴, (𝐹𝑗)⟩} = {𝐴})
7573, 74eqtrd 2776 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → dom (𝐼𝑗) = {𝐴})
7675eleq2d 2823 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (𝑘 ∈ dom (𝐼𝑗) ↔ 𝑘 ∈ {𝐴}))
7776adantr 481 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (𝑘 ∈ dom (𝐼𝑗) ↔ 𝑘 ∈ {𝐴}))
7872, 77mpbird 256 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → 𝑘 ∈ dom (𝐼𝑗))
79 fvco 6939 . . . . . . . . . 10 ((Fun (𝐼𝑗) ∧ 𝑘 ∈ dom (𝐼𝑗)) → (([,) ∘ (𝐼𝑗))‘𝑘) = ([,)‘((𝐼𝑗)‘𝑘)))
8071, 78, 79syl2anc 584 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = ([,)‘((𝐼𝑗)‘𝑘)))
8167fveq1d 6844 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → ((𝐼𝑗)‘𝑘) = ({⟨𝐴, (𝐹𝑗)⟩}‘𝑘))
8281ad2antlr 725 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ((𝐼𝑗)‘𝑘) = ({⟨𝐴, (𝐹𝑗)⟩}‘𝑘))
83 elsni 4603 . . . . . . . . . . . . 13 (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
8483fveq2d 6846 . . . . . . . . . . . 12 (𝑘 ∈ {𝐴} → ({⟨𝐴, (𝐹𝑗)⟩}‘𝑘) = ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴))
8584adantl 482 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝑘) = ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴))
86 fvexd 6857 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑗) ∈ V)
87 fvsng 7126 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝐹𝑗) ∈ V) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = (𝐹𝑗))
882, 86, 87syl2anc 584 . . . . . . . . . . . 12 (𝜑 → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = (𝐹𝑗))
8988ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = (𝐹𝑗))
9082, 85, 893eqtrd 2780 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ((𝐼𝑗)‘𝑘) = (𝐹𝑗))
9190fveq2d 6846 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ([,)‘((𝐼𝑗)‘𝑘)) = ([,)‘(𝐹𝑗)))
92 eqidd 2737 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ([,)‘(𝐹𝑗)) = ([,)‘(𝐹𝑗)))
9380, 91, 923eqtrd 2780 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = ([,)‘(𝐹𝑗)))
9493ixpeq2dva 8850 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = X𝑘 ∈ {𝐴} ([,)‘(𝐹𝑗)))
95 fvex 6855 . . . . . . . . 9 ([,)‘(𝐹𝑗)) ∈ V
9616, 95ixpconst 8845 . . . . . . . 8 X𝑘 ∈ {𝐴} ([,)‘(𝐹𝑗)) = (([,)‘(𝐹𝑗)) ↑m {𝐴})
9796a1i 11 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} ([,)‘(𝐹𝑗)) = (([,)‘(𝐹𝑗)) ↑m {𝐴}))
9894, 97eqtrd 2776 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = (([,)‘(𝐹𝑗)) ↑m {𝐴}))
9998iuneq2dv 4978 . . . . 5 (𝜑 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = 𝑗 ∈ ℕ (([,)‘(𝐹𝑗)) ↑m {𝐴}))
10049, 61, 993eqtr4d 2786 . . . 4 (𝜑 → ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}) = 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
10145, 100sseqtrd 3984 . . 3 (𝜑 → (𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
102 ovnovollem1.z . . . 4 (𝜑𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
103 nfcv 2907 . . . . . . 7 𝑗𝐹
104 ressxr 11199 . . . . . . . . . 10 ℝ ⊆ ℝ*
105 xpss2 5653 . . . . . . . . . 10 (ℝ ⊆ ℝ* → (ℝ × ℝ) ⊆ (ℝ × ℝ*))
106104, 105ax-mp 5 . . . . . . . . 9 (ℝ × ℝ) ⊆ (ℝ × ℝ*)
107106a1i 11 . . . . . . . 8 (𝜑 → (ℝ × ℝ) ⊆ (ℝ × ℝ*))
1086, 107fssd 6686 . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ × ℝ*))
109103, 108volicofmpt 44228 . . . . . 6 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))))
11067coeq2d 5818 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → ([,) ∘ (𝐼𝑗)) = ([,) ∘ {⟨𝐴, (𝐹𝑗)⟩}))
111110fveq1d 6844 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → (([,) ∘ (𝐼𝑗))‘𝐴) = (([,) ∘ {⟨𝐴, (𝐹𝑗)⟩})‘𝐴))
112111adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = (([,) ∘ {⟨𝐴, (𝐹𝑗)⟩})‘𝐴))
113 snidg 4620 . . . . . . . . . . . . . . . . 17 (𝐴𝑉𝐴 ∈ {𝐴})
1142, 113syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐴 ∈ {𝐴})
115 dmsnopg 6165 . . . . . . . . . . . . . . . . 17 ((𝐹𝑗) ∈ V → dom {⟨𝐴, (𝐹𝑗)⟩} = {𝐴})
11686, 115syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → dom {⟨𝐴, (𝐹𝑗)⟩} = {𝐴})
117114, 116eleqtrrd 2841 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ dom {⟨𝐴, (𝐹𝑗)⟩})
118117adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ dom {⟨𝐴, (𝐹𝑗)⟩})
119 fvco 6939 . . . . . . . . . . . . . 14 ((Fun {⟨𝐴, (𝐹𝑗)⟩} ∧ 𝐴 ∈ dom {⟨𝐴, (𝐹𝑗)⟩}) → (([,) ∘ {⟨𝐴, (𝐹𝑗)⟩})‘𝐴) = ([,)‘({⟨𝐴, (𝐹𝑗)⟩}‘𝐴)))
12062, 118, 119syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ {⟨𝐴, (𝐹𝑗)⟩})‘𝐴) = ([,)‘({⟨𝐴, (𝐹𝑗)⟩}‘𝐴)))
121 fvexd 6857 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∈ V)
1223, 121, 87syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = (𝐹𝑗))
123 1st2nd2 7960 . . . . . . . . . . . . . . . . 17 ((𝐹𝑗) ∈ (ℝ × ℝ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
1247, 123syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
125122, 124eqtrd 2776 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
126125fveq2d 6846 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ([,)‘({⟨𝐴, (𝐹𝑗)⟩}‘𝐴)) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩))
127 df-ov 7360 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
128127eqcomi 2745 . . . . . . . . . . . . . . 15 ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))
129128a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
130126, 129eqtrd 2776 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ([,)‘({⟨𝐴, (𝐹𝑗)⟩}‘𝐴)) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
131112, 120, 1303eqtrd 2780 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
132131fveq2d 6846 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) = (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))))
133 xp1st 7953 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
1347, 133syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
135 xp2nd 7954 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
1367, 135syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
137 volicore 44812 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑗)) ∈ ℝ ∧ (2nd ‘(𝐹𝑗)) ∈ ℝ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) ∈ ℝ)
138134, 136, 137syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) ∈ ℝ)
139132, 138eqeltrd 2838 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℝ)
140139recnd 11183 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ)
141 2fveq3 6847 . . . . . . . . . 10 (𝑘 = 𝐴 → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
142141prodsn 15845 . . . . . . . . 9 ((𝐴𝑉 ∧ (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
1433, 140, 142syl2anc 584 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
144143, 132eqtr2d 2777 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
145144mpteq2dva 5205 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
146109, 145eqtrd 2776 . . . . 5 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
147146fveq2d 6846 . . . 4 (𝜑 → (Σ^‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
148102, 147eqtrd 2776 . . 3 (𝜑𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
149101, 148jca 512 . 2 (𝜑 → ((𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) ∧ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))))
150 fveq1 6841 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑖𝑗) = (𝐼𝑗))
151150coeq2d 5818 . . . . . . . 8 (𝑖 = 𝐼 → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐼𝑗)))
152151fveq1d 6844 . . . . . . 7 (𝑖 = 𝐼 → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
153152ixpeq2dv 8851 . . . . . 6 (𝑖 = 𝐼X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
154153iuneq2d 4983 . . . . 5 (𝑖 = 𝐼 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
155154sseq2d 3976 . . . 4 (𝑖 = 𝐼 → ((𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ↔ (𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘)))
156 simpl 483 . . . . . . . . . . . 12 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → 𝑖 = 𝐼)
157156fveq1d 6844 . . . . . . . . . . 11 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → (𝑖𝑗) = (𝐼𝑗))
158157coeq2d 5818 . . . . . . . . . 10 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐼𝑗)))
159158fveq1d 6844 . . . . . . . . 9 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
160159fveq2d 6846 . . . . . . . 8 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
161160prodeq2dv 15806 . . . . . . 7 (𝑖 = 𝐼 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
162161mpteq2dv 5207 . . . . . 6 (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
163162fveq2d 6846 . . . . 5 (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
164163eqeq2d 2747 . . . 4 (𝑖 = 𝐼 → (𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))))
165155, 164anbi12d 631 . . 3 (𝑖 = 𝐼 → (((𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ ((𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) ∧ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))))
166165rspcev 3581 . 2 ((𝐼 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ) ∧ ((𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) ∧ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ)((𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
16726, 149, 166syl2anc 584 1 (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ)((𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
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
  Copyright terms: Public domain W3C validator