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Theorem cncfuni 44681
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 1197 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑏𝐵𝑥𝑏) → 𝜑)
7 simp2 1137 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑏𝐵𝑥𝑏) → 𝑏𝐵)
8 elin 3964 . . . . . . . . . 10 (𝑥 ∈ (𝐴𝑏) ↔ (𝑥𝐴𝑥𝑏))
98biimpri 227 . . . . . . . . 9 ((𝑥𝐴𝑥𝑏) → 𝑥 ∈ (𝐴𝑏))
109adantll 712 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑥𝑏) → 𝑥 ∈ (𝐴𝑏))
11103adant2 1131 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑏𝐵𝑥𝑏) → 𝑥 ∈ (𝐴𝑏))
12 cncfuni.fcn . . . . . . . . . . . . . 14 ((𝜑𝑏𝐵) → (𝐹𝑏) ∈ ((𝐴𝑏)–cn→ℂ))
131fdmd 6728 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = 𝐴)
1413ineq2d 4212 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑏 ∩ dom 𝐹) = (𝑏𝐴))
15 incom 4201 . . . . . . . . . . . . . . . . . . 19 (𝑏𝐴) = (𝐴𝑏)
1614, 15eqtr2di 2789 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴𝑏) = (𝑏 ∩ dom 𝐹))
1716reseq2d 5981 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 ↾ (𝐴𝑏)) = (𝐹 ↾ (𝑏 ∩ dom 𝐹)))
18 frel 6722 . . . . . . . . . . . . . . . . . . 19 (𝐹:𝐴⟶ℂ → Rel 𝐹)
191, 18syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → Rel 𝐹)
20 resindm 6030 . . . . . . . . . . . . . . . . . 18 (Rel 𝐹 → (𝐹 ↾ (𝑏 ∩ dom 𝐹)) = (𝐹𝑏))
2119, 20syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 ↾ (𝑏 ∩ dom 𝐹)) = (𝐹𝑏))
2217, 21eqtrd 2772 . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . 19 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
29 eqid 2732 . . . . . . . . . . . . . . . . . . 19 ((TopOpen‘ℂfld) ↾t (𝐴𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴𝑏))
3028cnfldtop 24307 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘ℂfld) ∈ Top
31 unicntop 24309 . . . . . . . . . . . . . . . . . . . . . 22 ℂ = (TopOpen‘ℂfld)
3231restid 17381 . . . . . . . . . . . . . . . . . . . . 21 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
3330, 32ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
3433eqcomi 2741 . . . . . . . . . . . . . . . . . . 19 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
3528, 29, 34cncfcn 24433 . . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . 16 (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)) = ((𝐴𝑏)–cn→ℂ))
3822, 37eleq12d 2827 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐹 ↾ (𝐴𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)) ↔ (𝐹𝑏) ∈ ((𝐴𝑏)–cn→ℂ)))
3938adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑏𝐵) → ((𝐹 ↾ (𝐴𝑏)) ∈ (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) Cn (TopOpen‘ℂfld)) ↔ (𝐹𝑏) ∈ ((𝐴𝑏)–cn→ℂ)))
4012, 39mpbird 256 . . . . . . . . . . . . 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 24306 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
4342a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
44 resttopon 22672 . . . . . . . . . . . . . . 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 22791 . . . . . . . . . . . . 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 231 . . . . . . . . . . 11 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((𝐹 ↾ (𝐴𝑏)):(𝐴𝑏)⟶ℂ ∧ ∀𝑥 ∈ (𝐴𝑏)(𝐹 ↾ (𝐴𝑏)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥)))
5150simprd 496 . . . . . . . . . 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 3245 . . . . . . . . . 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 11193 . . . . . . . . . . . . . . . 16 ℂ ∈ V
5756ssex 5321 . . . . . . . . . . . . . . 15 (𝐴 ⊆ ℂ → 𝐴 ∈ V)
5825, 57syl 17 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ V)
59 restabs 22676 . . . . . . . . . . . . . 14 (((TopOpen‘ℂfld) ∈ Top ∧ (𝐴𝑏) ⊆ 𝐴𝐴 ∈ V) → (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴𝑏)))
6055, 24, 58, 59syl3anc 1371 . . . . . . . . . . . . 13 (𝜑 → (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) = ((TopOpen‘ℂfld) ↾t (𝐴𝑏)))
6160eqcomd 2738 . . . . . . . . . . . 12 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴𝑏)) = (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)))
6261oveq1d 7426 . . . . . . . . . . 11 (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld)) = ((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld)))
6362fveq1d 6893 . . . . . . . . . 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 2835 . . . . . . . 8 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐹 ↾ (𝐴𝑏)) ∈ (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
66 resttop 22671 . . . . . . . . . . 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 22673 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ∈ Top ∧ 𝐴 ⊆ ℂ) → 𝐴 = ((TopOpen‘ℂfld) ↾t 𝐴))
7055, 25, 69syl2anc 584 . . . . . . . . . . 11 (𝜑𝐴 = ((TopOpen‘ℂfld) ↾t 𝐴))
7124, 70sseqtrd 4022 . . . . . . . . . 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 2732 . . . . . . . . . . . . . 14 ((TopOpen‘ℂfld) ↾t 𝐴) = ((TopOpen‘ℂfld) ↾t 𝐴)
7675isopn3 22577 . . . . . . . . . . . . 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 231 . . . . . . . . . . 11 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)) = (𝐴𝑏))
7978eqcomd 2738 . . . . . . . . . 10 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐴𝑏) = ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)))
8052, 79eleqtrd 2835 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)))
8170feq2d 6703 . . . . . . . . . . 11 (𝜑 → (𝐹:𝐴⟶ℂ ↔ 𝐹: ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ))
821, 81mpbid 231 . . . . . . . . . 10 (𝜑𝐹: ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ)
83823ad2ant1 1133 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → 𝐹: ((TopOpen‘ℂfld) ↾t 𝐴)⟶ℂ)
8475, 31cnprest 22800 . . . . . . . . 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 837 . . . . . . . 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 256 . . . . . . 7 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
876, 7, 11, 86syl3anc 1371 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑏𝐵𝑥𝑏) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
8887rexlimdv3a 3159 . . . . 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 3146 . . 3 (𝜑 → ∀𝑥𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
91 resttopon 22672 . . . . 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 22791 . . . 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 711 . 2 (𝜑𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)))
96 eqid 2732 . . . . 5 ((TopOpen‘ℂfld) ↾t 𝐴) = ((TopOpen‘ℂfld) ↾t 𝐴)
9728, 96, 34cncfcn 24433 . . . 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 2738 . 2 (𝜑 → (((TopOpen‘ℂfld) ↾t 𝐴) Cn (TopOpen‘ℂfld)) = (𝐴cn→ℂ))
10095, 99eleqtrd 2835 1 (𝜑𝐹 ∈ (𝐴cn→ℂ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  wrex 3070  Vcvv 3474  cin 3947  wss 3948   cuni 4908  dom cdm 5676  cres 5678  Rel wrel 5681  wf 6539  cfv 6543  (class class class)co 7411  cc 11110  t crest 17368  TopOpenctopn 17369  fldccnfld 20950  Topctop 22402  TopOnctopon 22419  intcnt 22528   Cn ccn 22735   CnP ccnp 22736  cnccncf 24399
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fi 9408  df-sup 9439  df-inf 9440  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12475  df-z 12561  df-dec 12680  df-uz 12825  df-q 12935  df-rp 12977  df-xneg 13094  df-xadd 13095  df-xmul 13096  df-fz 13487  df-seq 13969  df-exp 14030  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-struct 17082  df-slot 17117  df-ndx 17129  df-base 17147  df-plusg 17212  df-mulr 17213  df-starv 17214  df-tset 17218  df-ple 17219  df-ds 17221  df-unif 17222  df-rest 17370  df-topn 17371  df-topgen 17391  df-psmet 20942  df-xmet 20943  df-met 20944  df-bl 20945  df-mopn 20946  df-cnfld 20951  df-top 22403  df-topon 22420  df-topsp 22442  df-bases 22456  df-ntr 22531  df-cn 22738  df-cnp 22739  df-xms 23833  df-ms 23834  df-cncf 24401
This theorem is referenced by:  fouriersw  45026
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