Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cncfuni Structured version   Visualization version   GIF version

Theorem cncfuni 44022
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 3942 . . . . . 6 ((𝜑𝑥𝐴) → 𝑥 𝐵)
4 eluni2 4867 . . . . . 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 3924 . . . . . . . . . 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 6676 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = 𝐴)
1413ineq2d 4170 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑏 ∩ dom 𝐹) = (𝑏𝐴))
15 incom 4159 . . . . . . . . . . . . . . . . . . 19 (𝑏𝐴) = (𝐴𝑏)
1614, 15eqtr2di 2793 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴𝑏) = (𝑏 ∩ dom 𝐹))
1716reseq2d 5935 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐹 ↾ (𝐴𝑏)) = (𝐹 ↾ (𝑏 ∩ dom 𝐹)))
18 frel 6670 . . . . . . . . . . . . . . . . . . 19 (𝐹:𝐴⟶ℂ → Rel 𝐹)
191, 18syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → Rel 𝐹)
20 resindm 5984 . . . . . . . . . . . . . . . . . 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 4186 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑏) ⊆ 𝐴
2423a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐴𝑏) ⊆ 𝐴)
25 cncfuni.acn . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ⊆ ℂ)
2624, 25sstrd 3952 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴𝑏) ⊆ ℂ)
27 ssidd 3965 . . . . . . . . . . . . . . . . . 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 24093 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘ℂfld) ∈ Top
31 unicntop 24095 . . . . . . . . . . . . . . . . . . . . . 22 ℂ = (TopOpen‘ℂfld)
3231restid 17269 . . . . . . . . . . . . . . . . . . . . 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 24219 . . . . . . . . . . . . . . . . . 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 2832 . . . . . . . . . . . . . . 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 24092 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
4342a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
44 resttopon 22458 . . . . . . . . . . . . . . 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 22577 . . . . . . . . . . . . 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 3229 . . . . . . . . . 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 11090 . . . . . . . . . . . . . . . 16 ℂ ∈ V
5756ssex 5276 . . . . . . . . . . . . . . 15 (𝐴 ⊆ ℂ → 𝐴 ∈ V)
5825, 57syl 17 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ V)
59 restabs 22462 . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . 12 (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴𝑏)) = (((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)))
6261oveq1d 7366 . . . . . . . . . . 11 (𝜑 → (((TopOpen‘ℂfld) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld)) = ((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld)))
6362fveq1d 6841 . . . . . . . . . 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 2840 . . . . . . . 8 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐹 ↾ (𝐴𝑏)) ∈ (((((TopOpen‘ℂfld) ↾t 𝐴) ↾t (𝐴𝑏)) CnP (TopOpen‘ℂfld))‘𝑥))
66 resttop 22457 . . . . . . . . . . 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 22459 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ∈ Top ∧ 𝐴 ⊆ ℂ) → 𝐴 = ((TopOpen‘ℂfld) ↾t 𝐴))
7055, 25, 69syl2anc 584 . . . . . . . . . . 11 (𝜑𝐴 = ((TopOpen‘ℂfld) ↾t 𝐴))
7124, 70sseqtrd 3982 . . . . . . . . . 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 22363 . . . . . . . . . . . . 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 2742 . . . . . . . . . 10 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → (𝐴𝑏) = ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)))
8052, 79eleqtrd 2840 . . . . . . . . 9 ((𝜑𝑏𝐵𝑥 ∈ (𝐴𝑏)) → 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝐴))‘(𝐴𝑏)))
8170feq2d 6651 . . . . . . . . . . 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 22586 . . . . . . . . 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 3154 . . . . 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 3141 . . 3 (𝜑 → ∀𝑥𝐴 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝐴) CnP (TopOpen‘ℂfld))‘𝑥))
91 resttopon 22458 . . . . 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 22577 . . . 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 2736 . . . . 5 ((TopOpen‘ℂfld) ↾t 𝐴) = ((TopOpen‘ℂfld) ↾t 𝐴)
9728, 96, 34cncfcn 24219 . . . 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 2840 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 3062  wrex 3071  Vcvv 3443  cin 3907  wss 3908   cuni 4863  dom cdm 5631  cres 5633  Rel wrel 5636  wf 6489  cfv 6493  (class class class)co 7351  cc 11007  t crest 17256  TopOpenctopn 17257  fldccnfld 20743  Topctop 22188  TopOnctopon 22205  intcnt 22314   Cn ccn 22521   CnP ccnp 22522  cnccncf 24185
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 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-1st 7913  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-1o 8404  df-er 8606  df-map 8725  df-en 8842  df-dom 8843  df-sdom 8844  df-fin 8845  df-fi 9305  df-sup 9336  df-inf 9337  df-pnf 11149  df-mnf 11150  df-xr 11151  df-ltxr 11152  df-le 11153  df-sub 11345  df-neg 11346  df-div 11771  df-nn 12112  df-2 12174  df-3 12175  df-4 12176  df-5 12177  df-6 12178  df-7 12179  df-8 12180  df-9 12181  df-n0 12372  df-z 12458  df-dec 12577  df-uz 12722  df-q 12828  df-rp 12870  df-xneg 12987  df-xadd 12988  df-xmul 12989  df-fz 13379  df-seq 13861  df-exp 13922  df-cj 14938  df-re 14939  df-im 14940  df-sqrt 15074  df-abs 15075  df-struct 16973  df-slot 17008  df-ndx 17020  df-base 17038  df-plusg 17100  df-mulr 17101  df-starv 17102  df-tset 17106  df-ple 17107  df-ds 17109  df-unif 17110  df-rest 17258  df-topn 17259  df-topgen 17279  df-psmet 20735  df-xmet 20736  df-met 20737  df-bl 20738  df-mopn 20739  df-cnfld 20744  df-top 22189  df-topon 22206  df-topsp 22228  df-bases 22242  df-ntr 22317  df-cn 22524  df-cnp 22525  df-xms 23619  df-ms 23620  df-cncf 24187
This theorem is referenced by:  fouriersw  44367
  Copyright terms: Public domain W3C validator