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Theorem cncfuni 45906
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 4910 . . . . . 6 (𝑥 𝐵 ↔ ∃𝑏𝐵 𝑥𝑏)
53, 4sylib 218 . . . . 5 ((𝜑𝑥𝐴) → ∃𝑏𝐵 𝑥𝑏)
6 simp1l 1197 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑏𝐵𝑥𝑏) → 𝜑)
7 simp2 1137 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑏𝐵𝑥𝑏) → 𝑏𝐵)
8 elin 3966 . . . . . . . . . 10 (𝑥 ∈ (𝐴𝑏) ↔ (𝑥𝐴𝑥𝑏))
98biimpri 228 . . . . . . . . 9 ((𝑥𝐴𝑥𝑏) → 𝑥 ∈ (𝐴𝑏))
109adantll 714 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑥𝑏) → 𝑥 ∈ (𝐴𝑏))
11103adant2 1131 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑏𝐵𝑥𝑏) → 𝑥 ∈ (𝐴𝑏))
12 cncfuni.fcn . . . . . . . . . . . . . 14 ((𝜑𝑏𝐵) → (𝐹𝑏) ∈ ((𝐴𝑏)–cn→ℂ))
131fdmd 6745 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = 𝐴)
1413ineq2d 4219 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑏 ∩ dom 𝐹) = (𝑏𝐴))
15 incom 4208 . . . . . . . . . . . . . . . . . . 19 (𝑏𝐴) = (𝐴𝑏)
1614, 15eqtr2di 2793 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴𝑏) = (𝑏 ∩ dom 𝐹))
1716reseq2d 5996 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 ↾ (𝐴𝑏)) = (𝐹 ↾ (𝑏 ∩ dom 𝐹)))
18 frel 6740 . . . . . . . . . . . . . . . . . . 19 (𝐹:𝐴⟶ℂ → Rel 𝐹)
191, 18syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → Rel 𝐹)
20 resindm 6047 . . . . . . . . . . . . . . . . . 18 (Rel 𝐹 → (𝐹 ↾ (𝑏 ∩ dom 𝐹)) = (𝐹𝑏))
2119, 20syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 ↾ (𝑏 ∩ dom 𝐹)) = (𝐹𝑏))
2217, 21eqtrd 2776 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹 ↾ (𝐴𝑏)) = (𝐹𝑏))
23 inss1 4236 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑏) ⊆ 𝐴
2423a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐴𝑏) ⊆ 𝐴)
25 cncfuni.acn . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ⊆ ℂ)
2624, 25sstrd 3993 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴𝑏) ⊆ ℂ)
27 ssidd 4006 . . . . . . . . . . . . . . . . . 18 (𝜑 → ℂ ⊆ ℂ)
28 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
29 eqid 2736 . . . . . . . . . . . . . . . . . . 19 ((TopOpen‘ℂfld) ↾t (𝐴𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴𝑏))
3028cnfldtop 24805 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘ℂfld) ∈ Top
31 unicntop 24807 . . . . . . . . . . . . . . . . . . . . . 22 ℂ = (TopOpen‘ℂfld)
3231restid 17479 . . . . . . . . . . . . . . . . . . . . 21 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
3330, 32ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
3433eqcomi 2745 . . . . . . . . . . . . . . . . . . 19 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
3528, 29, 34cncfcn 24937 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑏) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴𝑏)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)))
3626, 27, 35syl2anc 584 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐴𝑏)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)))
3736eqcomd 2742 . . . . . . . . . . . . . . . 16 (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)) = ((𝐴𝑏)–cn→ℂ))
3822, 37eleq12d 2834 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐹 ↾ (𝐴𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)) ↔ (𝐹𝑏) ∈ ((𝐴𝑏)–cn→ℂ)))
3938adantr 480 . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . 12 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐹 ↾ (𝐴𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)))
4228cnfldtopon 24804 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
4342a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
44 resttopon 23170 . . . . . . . . . . . . . . 15 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝐴𝑏) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝐴𝑏)) ∈ (TopOn‘(𝐴𝑏)))
4543, 26, 44syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴𝑏)) ∈ (TopOn‘(𝐴𝑏)))
46453ad2ant1 1133 . . . . . . . . . . . . 13 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((TopOpen‘ℂfld) ↾t (𝐴𝑏)) ∈ (TopOn‘(𝐴𝑏)))
4742a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
48 cncnp 23289 . . . . . . . . . . . . 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 584 . . . . . . . . . . . 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 232 . . . . . . . . . . 11 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((𝐹 ↾ (𝐴𝑏)):(𝐴𝑏)⟶ℂ ∧ ∀𝑥 ∈ (𝐴𝑏)(𝐹 ↾ (𝐴𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥)))
5150simprd 495 . . . . . . . . . 10 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ∀𝑥 ∈ (𝐴𝑏)(𝐹 ↾ (𝐴𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
52 simp3 1138 . . . . . . . . . 10 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → 𝑥 ∈ (𝐴𝑏))
53 rspa 3247 . . . . . . . . . 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 584 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐹 ↾ (𝐴𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
5530a1i 11 . . . . . . . . . . . . . 14 (𝜑 → (TopOpen‘ℂfld) ∈ Top)
56 cnex 11237 . . . . . . . . . . . . . . . 16 ℂ ∈ V
5756ssex 5320 . . . . . . . . . . . . . . 15 (𝐴 ⊆ ℂ → 𝐴 ∈ V)
5825, 57syl 17 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ V)
59 restabs 23174 . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . 12 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴𝑏)) = (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)))
6261oveq1d 7447 . . . . . . . . . . 11 (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld)) = ((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld)))
6362fveq1d 6907 . . . . . . . . . 10 (𝜑 → ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥) = (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
64633ad2ant1 1133 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥) = (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
6554, 64eleqtrd 2842 . . . . . . . 8 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐹 ↾ (𝐴𝑏)) ∈ (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
66 resttop 23169 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ∈ Top ∧ 𝐴 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
6755, 58, 66syl2anc 584 . . . . . . . . . 10 (𝜑 → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
68673ad2ant1 1133 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ Top)
6931restuni 23171 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ∈ Top ∧ 𝐴 ⊆ ℂ) → 𝐴 = ((TopOpen‘ℂfld) ↾t 𝐴))
7055, 25, 69syl2anc 584 . . . . . . . . . . 11 (𝜑𝐴 = ((TopOpen‘ℂfld) ↾t 𝐴))
7124, 70sseqtrd 4019 . . . . . . . . . 10 (𝜑 → (𝐴𝑏) ⊆ ((TopOpen‘ℂfld) ↾t 𝐴))
72713ad2ant1 1133 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐴𝑏) ⊆ ((TopOpen‘ℂfld) ↾t 𝐴))
73 cncfuni.opn . . . . . . . . . . . . 13 ((𝜑𝑏𝐵) → (𝐴𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴))
74733adant3 1132 . . . . . . . . . . . 12 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐴𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴))
75 eqid 2736 . . . . . . . . . . . . . 14 ((TopOpen‘ℂfld) ↾t 𝐴) = ((TopOpen‘ℂfld) ↾t 𝐴)
7675isopn3 23075 . . . . . . . . . . . . 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 584 . . . . . . . . . . . 12 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((𝐴𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)) = (𝐴𝑏)))
7874, 77mpbid 232 . . . . . . . . . . 11 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)) = (𝐴𝑏))
7978eqcomd 2742 . . . . . . . . . 10 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐴𝑏) = ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)))
8052, 79eleqtrd 2842 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)))
8170feq2d 6721 . . . . . . . . . . 11 (𝜑 → (𝐹:𝐴⟶ℂ ↔ 𝐹: ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ))
821, 81mpbid 232 . . . . . . . . . 10 (𝜑𝐹: ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ)
83823ad2ant1 1133 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → 𝐹: ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ)
8475, 31cnprest 23298 . . . . . . . . 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 3158 . . . . 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 3145 . . 3 (𝜑 → ∀𝑥𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
91 resttopon 23170 . . . . 5 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝐴 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴))
9243, 25, 91syl2anc 584 . . . 4 (𝜑 → ((TopOpen‘ℂfld) ↾t 𝐴) ∈ (TopOn‘𝐴))
93 cncnp 23289 . . . 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 584 . . 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 713 . 2 (𝜑𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
96 eqid 2736 . . . . 5 ((TopOpen‘ℂfld) ↾t 𝐴) = ((TopOpen‘ℂfld) ↾t 𝐴)
9728, 96, 34cncfcn 24937 . . . 4 ((𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐴cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
9825, 27, 97syl2anc 584 . . 3 (𝜑 → (𝐴cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
9998eqcomd 2742 . 2 (𝜑 → (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)) = (𝐴cn→ℂ))
10095, 99eleqtrd 2842 1 (𝜑𝐹 ∈ (𝐴cn→ℂ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wrex 3069  Vcvv 3479  cin 3949  wss 3950   cuni 4906  dom cdm 5684  cres 5686  Rel wrel 5689  wf 6556  cfv 6560  (class class class)co 7432  cc 11154  t crest 17466  TopOpenctopn 17467  fldccnfld 21365  Topctop 22900  TopOnctopon 22917  intcnt 23026   Cn ccn 23233   CnP ccnp 23234  cnccncf 24903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fi 9452  df-sup 9483  df-inf 9484  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-q 12992  df-rp 13036  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-fz 13549  df-seq 14044  df-exp 14104  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17249  df-plusg 17311  df-mulr 17312  df-starv 17313  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-rest 17468  df-topn 17469  df-topgen 17489  df-psmet 21357  df-xmet 21358  df-met 21359  df-bl 21360  df-mopn 21361  df-cnfld 21366  df-top 22901  df-topon 22918  df-topsp 22940  df-bases 22954  df-ntr 23029  df-cn 23236  df-cnp 23237  df-xms 24331  df-ms 24332  df-cncf 24905
This theorem is referenced by:  fouriersw  46251
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