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Theorem cncfuni 44589
Description: A complex function on a subset of the complex numbers is continuous if its domain is the union of relatively open subsets over which the function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
cncfuni.acn (𝜑𝐴 ⊆ ℂ)
cncfuni.f (𝜑𝐹:𝐴⟶ℂ)
cncfuni.auni (𝜑𝐴 𝐵)
cncfuni.opn ((𝜑𝑏𝐵) → (𝐴𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴))
cncfuni.fcn ((𝜑𝑏𝐵) → (𝐹𝑏) ∈ ((𝐴𝑏)–cn→ℂ))
Assertion
Ref Expression
cncfuni (𝜑𝐹 ∈ (𝐴cn→ℂ))
Distinct variable groups:   𝐴,𝑏   𝐵,𝑏   𝐹,𝑏   𝜑,𝑏

Proof of Theorem cncfuni
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cncfuni.f . . 3 (𝜑𝐹:𝐴⟶ℂ)
2 cncfuni.auni . . . . . . 7 (𝜑𝐴 𝐵)
32sselda 3982 . . . . . 6 ((𝜑𝑥𝐴) → 𝑥 𝐵)
4 eluni2 4912 . . . . . 6 (𝑥 𝐵 ↔ ∃𝑏𝐵 𝑥𝑏)
53, 4sylib 217 . . . . 5 ((𝜑𝑥𝐴) → ∃𝑏𝐵 𝑥𝑏)
6 simp1l 1198 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑏𝐵𝑥𝑏) → 𝜑)
7 simp2 1138 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑏𝐵𝑥𝑏) → 𝑏𝐵)
8 elin 3964 . . . . . . . . . 10 (𝑥 ∈ (𝐴𝑏) ↔ (𝑥𝐴𝑥𝑏))
98biimpri 227 . . . . . . . . 9 ((𝑥𝐴𝑥𝑏) → 𝑥 ∈ (𝐴𝑏))
109adantll 713 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑥𝑏) → 𝑥 ∈ (𝐴𝑏))
11103adant2 1132 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑏𝐵𝑥𝑏) → 𝑥 ∈ (𝐴𝑏))
12 cncfuni.fcn . . . . . . . . . . . . . 14 ((𝜑𝑏𝐵) → (𝐹𝑏) ∈ ((𝐴𝑏)–cn→ℂ))
131fdmd 6726 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = 𝐴)
1413ineq2d 4212 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑏 ∩ dom 𝐹) = (𝑏𝐴))
15 incom 4201 . . . . . . . . . . . . . . . . . . 19 (𝑏𝐴) = (𝐴𝑏)
1614, 15eqtr2di 2790 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴𝑏) = (𝑏 ∩ dom 𝐹))
1716reseq2d 5980 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 ↾ (𝐴𝑏)) = (𝐹 ↾ (𝑏 ∩ dom 𝐹)))
18 frel 6720 . . . . . . . . . . . . . . . . . . 19 (𝐹:𝐴⟶ℂ → Rel 𝐹)
191, 18syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → Rel 𝐹)
20 resindm 6029 . . . . . . . . . . . . . . . . . 18 (Rel 𝐹 → (𝐹 ↾ (𝑏 ∩ dom 𝐹)) = (𝐹𝑏))
2119, 20syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 ↾ (𝑏 ∩ dom 𝐹)) = (𝐹𝑏))
2217, 21eqtrd 2773 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹 ↾ (𝐴𝑏)) = (𝐹𝑏))
23 inss1 4228 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑏) ⊆ 𝐴
2423a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐴𝑏) ⊆ 𝐴)
25 cncfuni.acn . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ⊆ ℂ)
2624, 25sstrd 3992 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴𝑏) ⊆ ℂ)
27 ssidd 4005 . . . . . . . . . . . . . . . . . 18 (𝜑 → ℂ ⊆ ℂ)
28 eqid 2733 . . . . . . . . . . . . . . . . . . 19 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
29 eqid 2733 . . . . . . . . . . . . . . . . . . 19 ((TopOpen‘ℂfld) ↾t (𝐴𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴𝑏))
3028cnfldtop 24292 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘ℂfld) ∈ Top
31 unicntop 24294 . . . . . . . . . . . . . . . . . . . . . 22 ℂ = (TopOpen‘ℂfld)
3231restid 17376 . . . . . . . . . . . . . . . . . . . . 21 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
3330, 32ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
3433eqcomi 2742 . . . . . . . . . . . . . . . . . . 19 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
3528, 29, 34cncfcn 24418 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑏) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴𝑏)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)))
3626, 27, 35syl2anc 585 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐴𝑏)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)))
3736eqcomd 2739 . . . . . . . . . . . . . . . 16 (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)) = ((𝐴𝑏)–cn→ℂ))
3822, 37eleq12d 2828 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐹 ↾ (𝐴𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)) ↔ (𝐹𝑏) ∈ ((𝐴𝑏)–cn→ℂ)))
3938adantr 482 . . . . . . . . . . . . . 14 ((𝜑𝑏𝐵) → ((𝐹 ↾ (𝐴𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)) ↔ (𝐹𝑏) ∈ ((𝐴𝑏)–cn→ℂ)))
4012, 39mpbird 257 . . . . . . . . . . . . 13 ((𝜑𝑏𝐵) → (𝐹 ↾ (𝐴𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)))
41403adant3 1133 . . . . . . . . . . . 12 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐹 ↾ (𝐴𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)))
4228cnfldtopon 24291 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
4342a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
44 resttopon 22657 . . . . . . . . . . . . . . 15 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝐴𝑏) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝐴𝑏)) ∈ (TopOn‘(𝐴𝑏)))
4543, 26, 44syl2anc 585 . . . . . . . . . . . . . 14 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴𝑏)) ∈ (TopOn‘(𝐴𝑏)))
46453ad2ant1 1134 . . . . . . . . . . . . 13 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((TopOpen‘ℂfld) ↾t (𝐴𝑏)) ∈ (TopOn‘(𝐴𝑏)))
4742a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
48 cncnp 22776 . . . . . . . . . . . . 13 ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) ∈ (TopOn‘(𝐴𝑏)) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) → ((𝐹 ↾ (𝐴𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)) ↔ ((𝐹 ↾ (𝐴𝑏)):(𝐴𝑏)⟶ℂ ∧ ∀𝑥 ∈ (𝐴𝑏)(𝐹 ↾ (𝐴𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))))
4946, 47, 48syl2anc 585 . . . . . . . . . . . 12 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((𝐹 ↾ (𝐴𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)) ↔ ((𝐹 ↾ (𝐴𝑏)):(𝐴𝑏)⟶ℂ ∧ ∀𝑥 ∈ (𝐴𝑏)(𝐹 ↾ (𝐴𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))))
5041, 49mpbid 231 . . . . . . . . . . 11 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((𝐹 ↾ (𝐴𝑏)):(𝐴𝑏)⟶ℂ ∧ ∀𝑥 ∈ (𝐴𝑏)(𝐹 ↾ (𝐴𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥)))
5150simprd 497 . . . . . . . . . 10 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ∀𝑥 ∈ (𝐴𝑏)(𝐹 ↾ (𝐴𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
52 simp3 1139 . . . . . . . . . 10 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → 𝑥 ∈ (𝐴𝑏))
53 rspa 3246 . . . . . . . . . 10 ((∀𝑥 ∈ (𝐴𝑏)(𝐹 ↾ (𝐴𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥) ∧ 𝑥 ∈ (𝐴𝑏)) → (𝐹 ↾ (𝐴𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
5451, 52, 53syl2anc 585 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐹 ↾ (𝐴𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
5530a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (TopOpen‘ℂfld) ∈ Top)
56 cnex 11188 . . . . . . . . . . . . . . . 16 ℂ ∈ V
5756ssex 5321 . . . . . . . . . . . . . . 15 (𝐴 ⊆ ℂ → 𝐴 ∈ V)
5825, 57syl 17 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ V)
59 restabs 22661 . . . . . . . . . . . . . 14 (((TopOpen‘ℂfld) ∈ Top ∧ (𝐴𝑏) ⊆ 𝐴𝐴 ∈ V) → (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴𝑏)))
6055, 24, 58, 59syl3anc 1372 . . . . . . . . . . . . 13 (𝜑 → (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴𝑏)))
6160eqcomd 2739 . . . . . . . . . . . 12 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴𝑏)) = (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)))
6261oveq1d 7421 . . . . . . . . . . 11 (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld)) = ((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld)))
6362fveq1d 6891 . . . . . . . . . 10 (𝜑 → ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥) = (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
64633ad2ant1 1134 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥) = (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
6554, 64eleqtrd 2836 . . . . . . . 8 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐹 ↾ (𝐴𝑏)) ∈ (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
66 resttop 22656 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ∈ Top ∧ 𝐴 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
6755, 58, 66syl2anc 585 . . . . . . . . . 10 (𝜑 → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
68673ad2ant1 1134 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
6931restuni 22658 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ∈ Top ∧ 𝐴 ⊆ ℂ) → 𝐴 = ((TopOpen‘ℂfld) ↾t 𝐴))
7055, 25, 69syl2anc 585 . . . . . . . . . . 11 (𝜑𝐴 = ((TopOpen‘ℂfld) ↾t 𝐴))
7124, 70sseqtrd 4022 . . . . . . . . . 10 (𝜑 → (𝐴𝑏) ⊆ ((TopOpen‘ℂfld) ↾t 𝐴))
72713ad2ant1 1134 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐴𝑏) ⊆ ((TopOpen‘ℂfld) ↾t 𝐴))
73 cncfuni.opn . . . . . . . . . . . . 13 ((𝜑𝑏𝐵) → (𝐴𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴))
74733adant3 1133 . . . . . . . . . . . 12 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐴𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴))
75 eqid 2733 . . . . . . . . . . . . . 14 ((TopOpen‘ℂfld) ↾t 𝐴) = ((TopOpen‘ℂfld) ↾t 𝐴)
7675isopn3 22562 . . . . . . . . . . . . 13 ((((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top ∧ (𝐴𝑏) ⊆ ((TopOpen‘ℂfld) ↾t 𝐴)) → ((𝐴𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)) = (𝐴𝑏)))
7768, 72, 76syl2anc 585 . . . . . . . . . . . 12 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((𝐴𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)) = (𝐴𝑏)))
7874, 77mpbid 231 . . . . . . . . . . 11 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)) = (𝐴𝑏))
7978eqcomd 2739 . . . . . . . . . 10 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐴𝑏) = ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)))
8052, 79eleqtrd 2836 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)))
8170feq2d 6701 . . . . . . . . . . 11 (𝜑 → (𝐹:𝐴⟶ℂ ↔ 𝐹: ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ))
821, 81mpbid 231 . . . . . . . . . 10 (𝜑𝐹: ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ)
83823ad2ant1 1134 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → 𝐹: ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ)
8475, 31cnprest 22785 . . . . . . . . 9 (((((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top ∧ (𝐴𝑏) ⊆ ((TopOpen‘ℂfld) ↾t 𝐴)) ∧ (𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)) ∧ 𝐹: ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ)) → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥) ↔ (𝐹 ↾ (𝐴𝑏)) ∈ (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥)))
8568, 72, 80, 83, 84syl22anc 838 . . . . . . . 8 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥) ↔ (𝐹 ↾ (𝐴𝑏)) ∈ (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥)))
8665, 85mpbird 257 . . . . . . 7 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
876, 7, 11, 86syl3anc 1372 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑏𝐵𝑥𝑏) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
8887rexlimdv3a 3160 . . . . 5 ((𝜑𝑥𝐴) → (∃𝑏𝐵 𝑥𝑏𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥)))
895, 88mpd 15 . . . 4 ((𝜑𝑥𝐴) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
9089ralrimiva 3147 . . 3 (𝜑 → ∀𝑥𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
91 resttopon 22657 . . . . 5 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝐴 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴))
9243, 25, 91syl2anc 585 . . . 4 (𝜑 → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴))
93 cncnp 22776 . . . 4 ((((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) → (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝐴⟶ℂ ∧ ∀𝑥𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))))
9492, 43, 93syl2anc 585 . . 3 (𝜑 → (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝐴⟶ℂ ∧ ∀𝑥𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))))
951, 90, 94mpbir2and 712 . 2 (𝜑𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
96 eqid 2733 . . . . 5 ((TopOpen‘ℂfld) ↾t 𝐴) = ((TopOpen‘ℂfld) ↾t 𝐴)
9728, 96, 34cncfcn 24418 . . . 4 ((𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐴cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
9825, 27, 97syl2anc 585 . . 3 (𝜑 → (𝐴cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
9998eqcomd 2739 . 2 (𝜑 → (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)) = (𝐴cn→ℂ))
10095, 99eleqtrd 2836 1 (𝜑𝐹 ∈ (𝐴cn→ℂ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wrex 3071  Vcvv 3475  cin 3947  wss 3948   cuni 4908  dom cdm 5676  cres 5678  Rel wrel 5681  wf 6537  cfv 6541  (class class class)co 7406  cc 11105  t crest 17363  TopOpenctopn 17364  fldccnfld 20937  Topctop 22387  TopOnctopon 22404  intcnt 22513   Cn ccn 22720   CnP ccnp 22721  cnccncf 24384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-fi 9403  df-sup 9434  df-inf 9435  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-q 12930  df-rp 12972  df-xneg 13089  df-xadd 13090  df-xmul 13091  df-fz 13482  df-seq 13964  df-exp 14025  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-struct 17077  df-slot 17112  df-ndx 17124  df-base 17142  df-plusg 17207  df-mulr 17208  df-starv 17209  df-tset 17213  df-ple 17214  df-ds 17216  df-unif 17217  df-rest 17365  df-topn 17366  df-topgen 17386  df-psmet 20929  df-xmet 20930  df-met 20931  df-bl 20932  df-mopn 20933  df-cnfld 20938  df-top 22388  df-topon 22405  df-topsp 22427  df-bases 22441  df-ntr 22516  df-cn 22723  df-cnp 22724  df-xms 23818  df-ms 23819  df-cncf 24386
This theorem is referenced by:  fouriersw  44934
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