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Theorem ovnovollem1 46612
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 2736 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → {⟨𝐴, (𝐹𝑗)⟩} = {⟨𝐴, (𝐹𝑗)⟩})
2 ovnovollem1.a . . . . . . . . 9 (𝜑𝐴𝑉)
32adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝐴𝑉)
4 ovnovollem1.f . . . . . . . . . 10 (𝜑𝐹 ∈ ((ℝ × ℝ) ↑m ℕ))
5 elmapi 8888 . . . . . . . . . 10 (𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
64, 5syl 17 . . . . . . . . 9 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
76ffvelcdmda 7104 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∈ (ℝ × ℝ))
8 fsng 7157 . . . . . . . 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 257 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → {⟨𝐴, (𝐹𝑗)⟩}:{𝐴}⟶{(𝐹𝑗)})
117snssd 4814 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → {(𝐹𝑗)} ⊆ (ℝ × ℝ))
1210, 11fssd 6754 . . . . 5 ((𝜑𝑗 ∈ ℕ) → {⟨𝐴, (𝐹𝑗)⟩}:{𝐴}⟶(ℝ × ℝ))
13 reex 11244 . . . . . . . 8 ℝ ∈ V
1413, 13xpex 7772 . . . . . . 7 (ℝ × ℝ) ∈ V
1514a1i 11 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (ℝ × ℝ) ∈ V)
16 snex 5442 . . . . . . 7 {𝐴} ∈ V
1716a1i 11 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → {𝐴} ∈ V)
1815, 17elmapd 8879 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ({⟨𝐴, (𝐹𝑗)⟩} ∈ ((ℝ × ℝ) ↑m {𝐴}) ↔ {⟨𝐴, (𝐹𝑗)⟩}:{𝐴}⟶(ℝ × ℝ)))
1912, 18mpbird 257 . . . 4 ((𝜑𝑗 ∈ ℕ) → {⟨𝐴, (𝐹𝑗)⟩} ∈ ((ℝ × ℝ) ↑m {𝐴}))
20 ovnovollem1.i . . . 4 𝐼 = (𝑗 ∈ ℕ ↦ {⟨𝐴, (𝐹𝑗)⟩})
2119, 20fmptd 7134 . . 3 (𝜑𝐼:ℕ⟶((ℝ × ℝ) ↑m {𝐴}))
22 ovexd 7466 . . . 4 (𝜑 → ((ℝ × ℝ) ↑m {𝐴}) ∈ V)
23 nnex 12270 . . . . 5 ℕ ∈ V
2423a1i 11 . . . 4 (𝜑 → ℕ ∈ V)
2522, 24elmapd 8879 . . 3 (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m {𝐴})))
2621, 25mpbird 257 . 2 (𝜑𝐼 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ))
27 ovnovollem1.s . . . . . 6 (𝜑𝐵 ran ([,) ∘ 𝐹))
28 icof 45162 . . . . . . . . . . 11 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
2928a1i 11 . . . . . . . . . 10 (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
30 rexpssxrxp 11304 . . . . . . . . . . 11 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
3130a1i 11 . . . . . . . . . 10 (𝜑 → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
3229, 31, 6fcoss 45153 . . . . . . . . 9 (𝜑 → ([,) ∘ 𝐹):ℕ⟶𝒫 ℝ*)
3332ffnd 6738 . . . . . . . 8 (𝜑 → ([,) ∘ 𝐹) Fn ℕ)
34 fniunfv 7267 . . . . . . . 8 (([,) ∘ 𝐹) Fn ℕ → 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
3533, 34syl 17 . . . . . . 7 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
3635eqcomd 2741 . . . . . 6 (𝜑 ran ([,) ∘ 𝐹) = 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗))
3727, 36sseqtrd 4036 . . . . 5 (𝜑𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗))
38 ovnovollem1.b . . . . . 6 (𝜑𝐵𝑊)
39 fvex 6920 . . . . . . . 8 (([,) ∘ 𝐹)‘𝑗) ∈ V
4023, 39iunex 7992 . . . . . . 7 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V
4140a1i 11 . . . . . 6 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V)
4216a1i 11 . . . . . 6 (𝜑 → {𝐴} ∈ V)
432snn0d 4780 . . . . . 6 (𝜑 → {𝐴} ≠ ∅)
4438, 41, 42, 43mapss2 45148 . . . . 5 (𝜑 → (𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↔ (𝐵m {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴})))
4537, 44mpbid 232 . . . 4 (𝜑 → (𝐵m {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
46 nfv 1912 . . . . . . 7 𝑗𝜑
47 fvexd 6922 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ([,)‘(𝐹𝑗)) ∈ V)
4846, 24, 47, 2iunmapsn 45160 . . . . . 6 (𝜑 𝑗 ∈ ℕ (([,)‘(𝐹𝑗)) ↑m {𝐴}) = ( 𝑗 ∈ ℕ ([,)‘(𝐹𝑗)) ↑m {𝐴}))
4948eqcomd 2741 . . . . 5 (𝜑 → ( 𝑗 ∈ ℕ ([,)‘(𝐹𝑗)) ↑m {𝐴}) = 𝑗 ∈ ℕ (([,)‘(𝐹𝑗)) ↑m {𝐴}))
50 elmapfun 8905 . . . . . . . . . 10 (𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) → Fun 𝐹)
514, 50syl 17 . . . . . . . . 9 (𝜑 → Fun 𝐹)
5251adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → Fun 𝐹)
53 simpr 484 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
546fdmd 6747 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = ℕ)
5554eqcomd 2741 . . . . . . . . . 10 (𝜑 → ℕ = dom 𝐹)
5655adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ℕ = dom 𝐹)
5753, 56eleqtrd 2841 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ dom 𝐹)
58 fvco 7007 . . . . . . . 8 ((Fun 𝐹𝑗 ∈ dom 𝐹) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
5952, 57, 58syl2anc 584 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
6059iuneq2dv 5021 . . . . . 6 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = 𝑗 ∈ ℕ ([,)‘(𝐹𝑗)))
6160oveq1d 7446 . . . . 5 (𝜑 → ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}) = ( 𝑗 ∈ ℕ ([,)‘(𝐹𝑗)) ↑m {𝐴}))
6210ffund 6741 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → Fun {⟨𝐴, (𝐹𝑗)⟩})
63 id 22 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
64 snex 5442 . . . . . . . . . . . . . . . 16 {⟨𝐴, (𝐹𝑗)⟩} ∈ V
6564a1i 11 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → {⟨𝐴, (𝐹𝑗)⟩} ∈ V)
6620fvmpt2 7027 . . . . . . . . . . . . . . 15 ((𝑗 ∈ ℕ ∧ {⟨𝐴, (𝐹𝑗)⟩} ∈ V) → (𝐼𝑗) = {⟨𝐴, (𝐹𝑗)⟩})
6763, 65, 66syl2anc 584 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → (𝐼𝑗) = {⟨𝐴, (𝐹𝑗)⟩})
6867adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) = {⟨𝐴, (𝐹𝑗)⟩})
6968funeqd 6590 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (Fun (𝐼𝑗) ↔ Fun {⟨𝐴, (𝐹𝑗)⟩}))
7062, 69mpbird 257 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → Fun (𝐼𝑗))
7170adantr 480 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → Fun (𝐼𝑗))
72 simpr 484 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → 𝑘 ∈ {𝐴})
7368dmeqd 5919 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → dom (𝐼𝑗) = dom {⟨𝐴, (𝐹𝑗)⟩})
7410fdmd 6747 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → dom {⟨𝐴, (𝐹𝑗)⟩} = {𝐴})
7573, 74eqtrd 2775 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → dom (𝐼𝑗) = {𝐴})
7675eleq2d 2825 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (𝑘 ∈ dom (𝐼𝑗) ↔ 𝑘 ∈ {𝐴}))
7776adantr 480 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (𝑘 ∈ dom (𝐼𝑗) ↔ 𝑘 ∈ {𝐴}))
7872, 77mpbird 257 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → 𝑘 ∈ dom (𝐼𝑗))
79 fvco 7007 . . . . . . . . . 10 ((Fun (𝐼𝑗) ∧ 𝑘 ∈ dom (𝐼𝑗)) → (([,) ∘ (𝐼𝑗))‘𝑘) = ([,)‘((𝐼𝑗)‘𝑘)))
8071, 78, 79syl2anc 584 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = ([,)‘((𝐼𝑗)‘𝑘)))
8167fveq1d 6909 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → ((𝐼𝑗)‘𝑘) = ({⟨𝐴, (𝐹𝑗)⟩}‘𝑘))
8281ad2antlr 727 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ((𝐼𝑗)‘𝑘) = ({⟨𝐴, (𝐹𝑗)⟩}‘𝑘))
83 elsni 4648 . . . . . . . . . . . . 13 (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
8483fveq2d 6911 . . . . . . . . . . . 12 (𝑘 ∈ {𝐴} → ({⟨𝐴, (𝐹𝑗)⟩}‘𝑘) = ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴))
8584adantl 481 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝑘) = ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴))
86 fvexd 6922 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑗) ∈ V)
87 fvsng 7200 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝐹𝑗) ∈ V) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = (𝐹𝑗))
882, 86, 87syl2anc 584 . . . . . . . . . . . 12 (𝜑 → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = (𝐹𝑗))
8988ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = (𝐹𝑗))
9082, 85, 893eqtrd 2779 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ((𝐼𝑗)‘𝑘) = (𝐹𝑗))
9190fveq2d 6911 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ([,)‘((𝐼𝑗)‘𝑘)) = ([,)‘(𝐹𝑗)))
92 eqidd 2736 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ([,)‘(𝐹𝑗)) = ([,)‘(𝐹𝑗)))
9380, 91, 923eqtrd 2779 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = ([,)‘(𝐹𝑗)))
9493ixpeq2dva 8951 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = X𝑘 ∈ {𝐴} ([,)‘(𝐹𝑗)))
95 fvex 6920 . . . . . . . . 9 ([,)‘(𝐹𝑗)) ∈ V
9616, 95ixpconst 8946 . . . . . . . 8 X𝑘 ∈ {𝐴} ([,)‘(𝐹𝑗)) = (([,)‘(𝐹𝑗)) ↑m {𝐴})
9796a1i 11 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} ([,)‘(𝐹𝑗)) = (([,)‘(𝐹𝑗)) ↑m {𝐴}))
9894, 97eqtrd 2775 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = (([,)‘(𝐹𝑗)) ↑m {𝐴}))
9998iuneq2dv 5021 . . . . 5 (𝜑 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = 𝑗 ∈ ℕ (([,)‘(𝐹𝑗)) ↑m {𝐴}))
10049, 61, 993eqtr4d 2785 . . . 4 (𝜑 → ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}) = 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
10145, 100sseqtrd 4036 . . 3 (𝜑 → (𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
102 ovnovollem1.z . . . 4 (𝜑𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
103 nfcv 2903 . . . . . . 7 𝑗𝐹
104 ressxr 11303 . . . . . . . . . 10 ℝ ⊆ ℝ*
105 xpss2 5709 . . . . . . . . . 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 6754 . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ × ℝ*))
109103, 108volicofmpt 45953 . . . . . 6 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))))
11067coeq2d 5876 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → ([,) ∘ (𝐼𝑗)) = ([,) ∘ {⟨𝐴, (𝐹𝑗)⟩}))
111110fveq1d 6909 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → (([,) ∘ (𝐼𝑗))‘𝐴) = (([,) ∘ {⟨𝐴, (𝐹𝑗)⟩})‘𝐴))
112111adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = (([,) ∘ {⟨𝐴, (𝐹𝑗)⟩})‘𝐴))
113 snidg 4665 . . . . . . . . . . . . . . . . 17 (𝐴𝑉𝐴 ∈ {𝐴})
1142, 113syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐴 ∈ {𝐴})
115 dmsnopg 6235 . . . . . . . . . . . . . . . . 17 ((𝐹𝑗) ∈ V → dom {⟨𝐴, (𝐹𝑗)⟩} = {𝐴})
11686, 115syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → dom {⟨𝐴, (𝐹𝑗)⟩} = {𝐴})
117114, 116eleqtrrd 2842 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ dom {⟨𝐴, (𝐹𝑗)⟩})
118117adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ dom {⟨𝐴, (𝐹𝑗)⟩})
119 fvco 7007 . . . . . . . . . . . . . 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 6922 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∈ V)
1223, 121, 87syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = (𝐹𝑗))
123 1st2nd2 8052 . . . . . . . . . . . . . . . . 17 ((𝐹𝑗) ∈ (ℝ × ℝ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
1247, 123syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
125122, 124eqtrd 2775 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
126125fveq2d 6911 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ([,)‘({⟨𝐴, (𝐹𝑗)⟩}‘𝐴)) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩))
127 df-ov 7434 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
128127eqcomi 2744 . . . . . . . . . . . . . . 15 ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))
129128a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
130126, 129eqtrd 2775 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ([,)‘({⟨𝐴, (𝐹𝑗)⟩}‘𝐴)) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
131112, 120, 1303eqtrd 2779 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
132131fveq2d 6911 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) = (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))))
133 xp1st 8045 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
1347, 133syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
135 xp2nd 8046 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
1367, 135syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
137 volicore 46537 . . . . . . . . . . . 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 2839 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℝ)
140139recnd 11287 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ)
141 2fveq3 6912 . . . . . . . . . 10 (𝑘 = 𝐴 → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
142141prodsn 15995 . . . . . . . . 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 2776 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
145144mpteq2dva 5248 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
146109, 145eqtrd 2775 . . . . 5 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
147146fveq2d 6911 . . . 4 (𝜑 → (Σ^‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
148102, 147eqtrd 2775 . . 3 (𝜑𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
149101, 148jca 511 . 2 (𝜑 → ((𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) ∧ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))))
150 fveq1 6906 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑖𝑗) = (𝐼𝑗))
151150coeq2d 5876 . . . . . . . 8 (𝑖 = 𝐼 → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐼𝑗)))
152151fveq1d 6909 . . . . . . 7 (𝑖 = 𝐼 → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
153152ixpeq2dv 8952 . . . . . 6 (𝑖 = 𝐼X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
154153iuneq2d 5027 . . . . 5 (𝑖 = 𝐼 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
155154sseq2d 4028 . . . 4 (𝑖 = 𝐼 → ((𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ↔ (𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘)))
156 simpl 482 . . . . . . . . . . . 12 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → 𝑖 = 𝐼)
157156fveq1d 6909 . . . . . . . . . . 11 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → (𝑖𝑗) = (𝐼𝑗))
158157coeq2d 5876 . . . . . . . . . 10 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐼𝑗)))
159158fveq1d 6909 . . . . . . . . 9 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
160159fveq2d 6911 . . . . . . . 8 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
161160prodeq2dv 15955 . . . . . . 7 (𝑖 = 𝐼 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
162161mpteq2dv 5250 . . . . . 6 (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
163162fveq2d 6911 . . . . 5 (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
164163eqeq2d 2746 . . . 4 (𝑖 = 𝐼 → (𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))))
165155, 164anbi12d 632 . . 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 3622 . 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 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  wss 3963  𝒫 cpw 4605  {csn 4631  cop 4637   cuni 4912   ciun 4996  cmpt 5231   × cxp 5687  dom cdm 5689  ran crn 5690  ccom 5693  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  m cmap 8865  Xcixp 8936  cc 11151  cr 11152  *cxr 11292  cn 12264  [,)cico 13386  cprod 15936  volcvol 25512  Σ^csumge0 46318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-rlim 15522  df-sum 15720  df-prod 15937  df-rest 17469  df-topgen 17490  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-top 22916  df-topon 22933  df-bases 22969  df-cmp 23411  df-ovol 25513  df-vol 25514
This theorem is referenced by:  ovnovollem3  46614
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