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Theorem ovnovollem1 43225
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 2825 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → {⟨𝐴, (𝐹𝑗)⟩} = {⟨𝐴, (𝐹𝑗)⟩})
2 ovnovollem1.a . . . . . . . . 9 (𝜑𝐴𝑉)
32adantr 484 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝐴𝑉)
4 ovnovollem1.f . . . . . . . . . 10 (𝜑𝐹 ∈ ((ℝ × ℝ) ↑m ℕ))
5 elmapi 8424 . . . . . . . . . 10 (𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
64, 5syl 17 . . . . . . . . 9 (𝜑𝐹:ℕ⟶(ℝ × ℝ))
76ffvelrnda 6842 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∈ (ℝ × ℝ))
8 fsng 6890 . . . . . . . 8 ((𝐴𝑉 ∧ (𝐹𝑗) ∈ (ℝ × ℝ)) → ({⟨𝐴, (𝐹𝑗)⟩}:{𝐴}⟶{(𝐹𝑗)} ↔ {⟨𝐴, (𝐹𝑗)⟩} = {⟨𝐴, (𝐹𝑗)⟩}))
93, 7, 8syl2anc 587 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ({⟨𝐴, (𝐹𝑗)⟩}:{𝐴}⟶{(𝐹𝑗)} ↔ {⟨𝐴, (𝐹𝑗)⟩} = {⟨𝐴, (𝐹𝑗)⟩}))
101, 9mpbird 260 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → {⟨𝐴, (𝐹𝑗)⟩}:{𝐴}⟶{(𝐹𝑗)})
117snssd 4726 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → {(𝐹𝑗)} ⊆ (ℝ × ℝ))
1210, 11fssd 6518 . . . . 5 ((𝜑𝑗 ∈ ℕ) → {⟨𝐴, (𝐹𝑗)⟩}:{𝐴}⟶(ℝ × ℝ))
13 reex 10626 . . . . . . . 8 ℝ ∈ V
1413, 13xpex 7470 . . . . . . 7 (ℝ × ℝ) ∈ V
1514a1i 11 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (ℝ × ℝ) ∈ V)
16 snex 5319 . . . . . . 7 {𝐴} ∈ V
1716a1i 11 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → {𝐴} ∈ V)
1815, 17elmapd 8416 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ({⟨𝐴, (𝐹𝑗)⟩} ∈ ((ℝ × ℝ) ↑m {𝐴}) ↔ {⟨𝐴, (𝐹𝑗)⟩}:{𝐴}⟶(ℝ × ℝ)))
1912, 18mpbird 260 . . . 4 ((𝜑𝑗 ∈ ℕ) → {⟨𝐴, (𝐹𝑗)⟩} ∈ ((ℝ × ℝ) ↑m {𝐴}))
20 ovnovollem1.i . . . 4 𝐼 = (𝑗 ∈ ℕ ↦ {⟨𝐴, (𝐹𝑗)⟩})
2119, 20fmptd 6869 . . 3 (𝜑𝐼:ℕ⟶((ℝ × ℝ) ↑m {𝐴}))
22 ovexd 7184 . . . 4 (𝜑 → ((ℝ × ℝ) ↑m {𝐴}) ∈ V)
23 nnex 11640 . . . . 5 ℕ ∈ V
2423a1i 11 . . . 4 (𝜑 → ℕ ∈ V)
2522, 24elmapd 8416 . . 3 (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m {𝐴})))
2621, 25mpbird 260 . 2 (𝜑𝐼 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ))
27 ovnovollem1.s . . . . . 6 (𝜑𝐵 ran ([,) ∘ 𝐹))
28 icof 41777 . . . . . . . . . . 11 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
2928a1i 11 . . . . . . . . . 10 (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
30 rexpssxrxp 10684 . . . . . . . . . . 11 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
3130a1i 11 . . . . . . . . . 10 (𝜑 → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
3229, 31, 6fcoss 41768 . . . . . . . . 9 (𝜑 → ([,) ∘ 𝐹):ℕ⟶𝒫 ℝ*)
3332ffnd 6504 . . . . . . . 8 (𝜑 → ([,) ∘ 𝐹) Fn ℕ)
34 fniunfv 6998 . . . . . . . 8 (([,) ∘ 𝐹) Fn ℕ → 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
3533, 34syl 17 . . . . . . 7 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ran ([,) ∘ 𝐹))
3635eqcomd 2830 . . . . . 6 (𝜑 ran ([,) ∘ 𝐹) = 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗))
3727, 36sseqtrd 3993 . . . . 5 (𝜑𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗))
38 ovnovollem1.b . . . . . 6 (𝜑𝐵𝑊)
39 fvex 6674 . . . . . . . 8 (([,) ∘ 𝐹)‘𝑗) ∈ V
4023, 39iunex 7664 . . . . . . 7 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V
4140a1i 11 . . . . . 6 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V)
4216a1i 11 . . . . . 6 (𝜑 → {𝐴} ∈ V)
432snn0d 41645 . . . . . 6 (𝜑 → {𝐴} ≠ ∅)
4438, 41, 42, 43mapss2 41763 . . . . 5 (𝜑 → (𝐵 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↔ (𝐵m {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴})))
4537, 44mpbid 235 . . . 4 (𝜑 → (𝐵m {𝐴}) ⊆ ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
46 nfv 1916 . . . . . . 7 𝑗𝜑
47 fvexd 6676 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ([,)‘(𝐹𝑗)) ∈ V)
4846, 24, 47, 2iunmapsn 41775 . . . . . 6 (𝜑 𝑗 ∈ ℕ (([,)‘(𝐹𝑗)) ↑m {𝐴}) = ( 𝑗 ∈ ℕ ([,)‘(𝐹𝑗)) ↑m {𝐴}))
4948eqcomd 2830 . . . . 5 (𝜑 → ( 𝑗 ∈ ℕ ([,)‘(𝐹𝑗)) ↑m {𝐴}) = 𝑗 ∈ ℕ (([,)‘(𝐹𝑗)) ↑m {𝐴}))
50 elmapfun 8426 . . . . . . . . . 10 (𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) → Fun 𝐹)
514, 50syl 17 . . . . . . . . 9 (𝜑 → Fun 𝐹)
5251adantr 484 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → Fun 𝐹)
53 simpr 488 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
546fdmd 6513 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = ℕ)
5554eqcomd 2830 . . . . . . . . . 10 (𝜑 → ℕ = dom 𝐹)
5655adantr 484 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ℕ = dom 𝐹)
5753, 56eleqtrd 2918 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ dom 𝐹)
58 fvco 6750 . . . . . . . 8 ((Fun 𝐹𝑗 ∈ dom 𝐹) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
5952, 57, 58syl2anc 587 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹𝑗)))
6059iuneq2dv 4929 . . . . . 6 (𝜑 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = 𝑗 ∈ ℕ ([,)‘(𝐹𝑗)))
6160oveq1d 7164 . . . . 5 (𝜑 → ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}) = ( 𝑗 ∈ ℕ ([,)‘(𝐹𝑗)) ↑m {𝐴}))
6210ffund 6507 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → Fun {⟨𝐴, (𝐹𝑗)⟩})
63 id 22 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
64 snex 5319 . . . . . . . . . . . . . . . 16 {⟨𝐴, (𝐹𝑗)⟩} ∈ V
6564a1i 11 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → {⟨𝐴, (𝐹𝑗)⟩} ∈ V)
6620fvmpt2 6770 . . . . . . . . . . . . . . 15 ((𝑗 ∈ ℕ ∧ {⟨𝐴, (𝐹𝑗)⟩} ∈ V) → (𝐼𝑗) = {⟨𝐴, (𝐹𝑗)⟩})
6763, 65, 66syl2anc 587 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → (𝐼𝑗) = {⟨𝐴, (𝐹𝑗)⟩})
6867adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) = {⟨𝐴, (𝐹𝑗)⟩})
6968funeqd 6365 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (Fun (𝐼𝑗) ↔ Fun {⟨𝐴, (𝐹𝑗)⟩}))
7062, 69mpbird 260 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → Fun (𝐼𝑗))
7170adantr 484 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → Fun (𝐼𝑗))
72 simpr 488 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → 𝑘 ∈ {𝐴})
7368dmeqd 5761 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → dom (𝐼𝑗) = dom {⟨𝐴, (𝐹𝑗)⟩})
7410fdmd 6513 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → dom {⟨𝐴, (𝐹𝑗)⟩} = {𝐴})
7573, 74eqtrd 2859 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → dom (𝐼𝑗) = {𝐴})
7675eleq2d 2901 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (𝑘 ∈ dom (𝐼𝑗) ↔ 𝑘 ∈ {𝐴}))
7776adantr 484 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (𝑘 ∈ dom (𝐼𝑗) ↔ 𝑘 ∈ {𝐴}))
7872, 77mpbird 260 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → 𝑘 ∈ dom (𝐼𝑗))
79 fvco 6750 . . . . . . . . . 10 ((Fun (𝐼𝑗) ∧ 𝑘 ∈ dom (𝐼𝑗)) → (([,) ∘ (𝐼𝑗))‘𝑘) = ([,)‘((𝐼𝑗)‘𝑘)))
8071, 78, 79syl2anc 587 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = ([,)‘((𝐼𝑗)‘𝑘)))
8167fveq1d 6663 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → ((𝐼𝑗)‘𝑘) = ({⟨𝐴, (𝐹𝑗)⟩}‘𝑘))
8281ad2antlr 726 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ((𝐼𝑗)‘𝑘) = ({⟨𝐴, (𝐹𝑗)⟩}‘𝑘))
83 elsni 4567 . . . . . . . . . . . . 13 (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
8483fveq2d 6665 . . . . . . . . . . . 12 (𝑘 ∈ {𝐴} → ({⟨𝐴, (𝐹𝑗)⟩}‘𝑘) = ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴))
8584adantl 485 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝑘) = ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴))
86 fvexd 6676 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑗) ∈ V)
87 fvsng 6933 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝐹𝑗) ∈ V) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = (𝐹𝑗))
882, 86, 87syl2anc 587 . . . . . . . . . . . 12 (𝜑 → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = (𝐹𝑗))
8988ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = (𝐹𝑗))
9082, 85, 893eqtrd 2863 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ((𝐼𝑗)‘𝑘) = (𝐹𝑗))
9190fveq2d 6665 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ([,)‘((𝐼𝑗)‘𝑘)) = ([,)‘(𝐹𝑗)))
92 eqidd 2825 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ([,)‘(𝐹𝑗)) = ([,)‘(𝐹𝑗)))
9380, 91, 923eqtrd 2863 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼𝑗))‘𝑘) = ([,)‘(𝐹𝑗)))
9493ixpeq2dva 8472 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = X𝑘 ∈ {𝐴} ([,)‘(𝐹𝑗)))
95 fvex 6674 . . . . . . . . 9 ([,)‘(𝐹𝑗)) ∈ V
9616, 95ixpconst 8467 . . . . . . . 8 X𝑘 ∈ {𝐴} ([,)‘(𝐹𝑗)) = (([,)‘(𝐹𝑗)) ↑m {𝐴})
9796a1i 11 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} ([,)‘(𝐹𝑗)) = (([,)‘(𝐹𝑗)) ↑m {𝐴}))
9894, 97eqtrd 2859 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = (([,)‘(𝐹𝑗)) ↑m {𝐴}))
9998iuneq2dv 4929 . . . . 5 (𝜑 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) = 𝑗 ∈ ℕ (([,)‘(𝐹𝑗)) ↑m {𝐴}))
10049, 61, 993eqtr4d 2869 . . . 4 (𝜑 → ( 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}) = 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
10145, 100sseqtrd 3993 . . 3 (𝜑 → (𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
102 ovnovollem1.z . . . 4 (𝜑𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
103 nfcv 2982 . . . . . . 7 𝑗𝐹
104 ressxr 10683 . . . . . . . . . 10 ℝ ⊆ ℝ*
105 xpss2 5562 . . . . . . . . . 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 6518 . . . . . . 7 (𝜑𝐹:ℕ⟶(ℝ × ℝ*))
109103, 108volicofmpt 42569 . . . . . 6 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))))
11067coeq2d 5720 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → ([,) ∘ (𝐼𝑗)) = ([,) ∘ {⟨𝐴, (𝐹𝑗)⟩}))
111110fveq1d 6663 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → (([,) ∘ (𝐼𝑗))‘𝐴) = (([,) ∘ {⟨𝐴, (𝐹𝑗)⟩})‘𝐴))
112111adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = (([,) ∘ {⟨𝐴, (𝐹𝑗)⟩})‘𝐴))
113 snidg 4584 . . . . . . . . . . . . . . . . 17 (𝐴𝑉𝐴 ∈ {𝐴})
1142, 113syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐴 ∈ {𝐴})
115 dmsnopg 6057 . . . . . . . . . . . . . . . . 17 ((𝐹𝑗) ∈ V → dom {⟨𝐴, (𝐹𝑗)⟩} = {𝐴})
11686, 115syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → dom {⟨𝐴, (𝐹𝑗)⟩} = {𝐴})
117114, 116eleqtrrd 2919 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ dom {⟨𝐴, (𝐹𝑗)⟩})
118117adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ dom {⟨𝐴, (𝐹𝑗)⟩})
119 fvco 6750 . . . . . . . . . . . . . 14 ((Fun {⟨𝐴, (𝐹𝑗)⟩} ∧ 𝐴 ∈ dom {⟨𝐴, (𝐹𝑗)⟩}) → (([,) ∘ {⟨𝐴, (𝐹𝑗)⟩})‘𝐴) = ([,)‘({⟨𝐴, (𝐹𝑗)⟩}‘𝐴)))
12062, 118, 119syl2anc 587 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ {⟨𝐴, (𝐹𝑗)⟩})‘𝐴) = ([,)‘({⟨𝐴, (𝐹𝑗)⟩}‘𝐴)))
121 fvexd 6676 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) ∈ V)
1223, 121, 87syl2anc 587 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = (𝐹𝑗))
123 1st2nd2 7723 . . . . . . . . . . . . . . . . 17 ((𝐹𝑗) ∈ (ℝ × ℝ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
1247, 123syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (𝐹𝑗) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
125122, 124eqtrd 2859 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → ({⟨𝐴, (𝐹𝑗)⟩}‘𝐴) = ⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
126125fveq2d 6665 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ([,)‘({⟨𝐴, (𝐹𝑗)⟩}‘𝐴)) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩))
127 df-ov 7152 . . . . . . . . . . . . . . . 16 ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))) = ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩)
128127eqcomi 2833 . . . . . . . . . . . . . . 15 ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))
129128a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ([,)‘⟨(1st ‘(𝐹𝑗)), (2nd ‘(𝐹𝑗))⟩) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
130126, 129eqtrd 2859 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ([,)‘({⟨𝐴, (𝐹𝑗)⟩}‘𝐴)) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
131112, 120, 1303eqtrd 2863 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (([,) ∘ (𝐼𝑗))‘𝐴) = ((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))
132131fveq2d 6665 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) = (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))))
133 xp1st 7716 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
1347, 133syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (1st ‘(𝐹𝑗)) ∈ ℝ)
135 xp2nd 7717 . . . . . . . . . . . . 13 ((𝐹𝑗) ∈ (ℝ × ℝ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
1367, 135syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2nd ‘(𝐹𝑗)) ∈ ℝ)
137 volicore 43150 . . . . . . . . . . . 12 (((1st ‘(𝐹𝑗)) ∈ ℝ ∧ (2nd ‘(𝐹𝑗)) ∈ ℝ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) ∈ ℝ)
138134, 136, 137syl2anc 587 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) ∈ ℝ)
139132, 138eqeltrd 2916 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℝ)
140139recnd 10667 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ)
141 2fveq3 6666 . . . . . . . . . 10 (𝑘 = 𝐴 → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
142141prodsn 15316 . . . . . . . . 9 ((𝐴𝑉 ∧ (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)) ∈ ℂ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
1433, 140, 142syl2anc 587 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝐴)))
144143, 132eqtr2d 2860 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗)))) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
145144mpteq2dva 5147 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹𝑗))[,)(2nd ‘(𝐹𝑗))))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
146109, 145eqtrd 2859 . . . . 5 (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
147146fveq2d 6665 . . . 4 (𝜑 → (Σ^‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
148102, 147eqtrd 2859 . . 3 (𝜑𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
149101, 148jca 515 . 2 (𝜑 → ((𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘) ∧ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))))
150 fveq1 6660 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑖𝑗) = (𝐼𝑗))
151150coeq2d 5720 . . . . . . . 8 (𝑖 = 𝐼 → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐼𝑗)))
152151fveq1d 6663 . . . . . . 7 (𝑖 = 𝐼 → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
153152ixpeq2dv 8473 . . . . . 6 (𝑖 = 𝐼X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
154153iuneq2d 4934 . . . . 5 (𝑖 = 𝐼 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘))
155154sseq2d 3985 . . . 4 (𝑖 = 𝐼 → ((𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖𝑗))‘𝑘) ↔ (𝐵m {𝐴}) ⊆ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼𝑗))‘𝑘)))
156 simpl 486 . . . . . . . . . . . 12 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → 𝑖 = 𝐼)
157156fveq1d 6663 . . . . . . . . . . 11 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → (𝑖𝑗) = (𝐼𝑗))
158157coeq2d 5720 . . . . . . . . . 10 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐼𝑗)))
159158fveq1d 6663 . . . . . . . . 9 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
160159fveq2d 6665 . . . . . . . 8 ((𝑖 = 𝐼𝑘 ∈ {𝐴}) → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
161160prodeq2dv 15277 . . . . . . 7 (𝑖 = 𝐼 → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
162161mpteq2dv 5148 . . . . . 6 (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
163162fveq2d 6665 . . . . 5 (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
164163eqeq2d 2835 . . . 4 (𝑖 = 𝐼 → (𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))))
165155, 164anbi12d 633 . . 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 3609 . 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 587 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 209  wa 399   = wceq 1538  wcel 2115  wrex 3134  Vcvv 3480  wss 3919  𝒫 cpw 4522  {csn 4550  cop 4556   cuni 4824   ciun 4905  cmpt 5132   × cxp 5540  dom cdm 5542  ran crn 5543  ccom 5546  Fun wfun 6337   Fn wfn 6338  wf 6339  cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683  m cmap 8402  Xcixp 8457  cc 10533  cr 10534  *cxr 10672  cn 11634  [,)cico 12737  cprod 15259  volcvol 24073  Σ^csumge0 42931
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fi 8872  df-sup 8903  df-inf 8904  df-oi 8971  df-dju 9327  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-q 12346  df-rp 12387  df-xneg 12504  df-xadd 12505  df-xmul 12506  df-ioo 12739  df-ico 12741  df-icc 12742  df-fz 12895  df-fzo 13038  df-fl 13166  df-seq 13374  df-exp 13435  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-rlim 14846  df-sum 15043  df-prod 15260  df-rest 16696  df-topgen 16717  df-psmet 20139  df-xmet 20140  df-met 20141  df-bl 20142  df-mopn 20143  df-top 21505  df-topon 21522  df-bases 21557  df-cmp 21998  df-ovol 24074  df-vol 24075
This theorem is referenced by:  ovnovollem3  43227
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