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Theorem cncfuni 44602
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 3983 . . . . . 6 ((𝜑𝑥𝐴) → 𝑥 𝐵)
4 eluni2 4913 . . . . . 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 3965 . . . . . . . . . 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 6729 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = 𝐴)
1413ineq2d 4213 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑏 ∩ dom 𝐹) = (𝑏𝐴))
15 incom 4202 . . . . . . . . . . . . . . . . . . 19 (𝑏𝐴) = (𝐴𝑏)
1614, 15eqtr2di 2790 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴𝑏) = (𝑏 ∩ dom 𝐹))
1716reseq2d 5982 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 ↾ (𝐴𝑏)) = (𝐹 ↾ (𝑏 ∩ dom 𝐹)))
18 frel 6723 . . . . . . . . . . . . . . . . . . 19 (𝐹:𝐴⟶ℂ → Rel 𝐹)
191, 18syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → Rel 𝐹)
20 resindm 6031 . . . . . . . . . . . . . . . . . 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 4229 . . . . . . . . . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . . . . . 19 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
29 eqid 2733 . . . . . . . . . . . . . . . . . . 19 ((TopOpen‘ℂfld) ↾t (𝐴𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴𝑏))
3028cnfldtop 24300 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘ℂfld) ∈ Top
31 unicntop 24302 . . . . . . . . . . . . . . . . . . . . . 22 ℂ = (TopOpen‘ℂfld)
3231restid 17379 . . . . . . . . . . . . . . . . . . . . 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 24426 . . . . . . . . . . . . . . . . . 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 24299 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
4342a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
44 resttopon 22665 . . . . . . . . . . . . . . 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 22784 . . . . . . . . . . . . 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 11191 . . . . . . . . . . . . . . . 16 ℂ ∈ V
5756ssex 5322 . . . . . . . . . . . . . . 15 (𝐴 ⊆ ℂ → 𝐴 ∈ V)
5825, 57syl 17 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ V)
59 restabs 22669 . . . . . . . . . . . . . 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 7424 . . . . . . . . . . 11 (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld)) = ((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld)))
6362fveq1d 6894 . . . . . . . . . 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 22664 . . . . . . . . . . 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 22666 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ∈ Top ∧ 𝐴 ⊆ ℂ) → 𝐴 = ((TopOpen‘ℂfld) ↾t 𝐴))
7055, 25, 69syl2anc 585 . . . . . . . . . . 11 (𝜑𝐴 = ((TopOpen‘ℂfld) ↾t 𝐴))
7124, 70sseqtrd 4023 . . . . . . . . . 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 22570 . . . . . . . . . . . . 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 6704 . . . . . . . . . . 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 22793 . . . . . . . . 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 22665 . . . . 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 22784 . . . 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 24426 . . . 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 3948  wss 3949   cuni 4909  dom cdm 5677  cres 5679  Rel wrel 5682  wf 6540  cfv 6544  (class class class)co 7409  cc 11108  t crest 17366  TopOpenctopn 17367  fldccnfld 20944  Topctop 22395  TopOnctopon 22412  intcnt 22521   Cn ccn 22728   CnP ccnp 22729  cnccncf 24392
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fi 9406  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-fz 13485  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-struct 17080  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-mulr 17211  df-starv 17212  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-rest 17368  df-topn 17369  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-cnfld 20945  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-ntr 22524  df-cn 22731  df-cnp 22732  df-xms 23826  df-ms 23827  df-cncf 24394
This theorem is referenced by:  fouriersw  44947
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