This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cnlimc

Description: F is a continuous function iff the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016)

Ref Expression
Assertion cnlimc ( 𝐴 ⊆ ℂ → ( 𝐹 ∈ ( 𝐴 –cn→ ℂ ) ↔ ( 𝐹 : 𝐴 ⟶ ℂ ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 limℂ 𝑥 ) ) ) )

Proof

Step Hyp Ref Expression
1 ssid ⊢ ℂ ⊆ ℂ
2 eqid ⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld )
3 eqid ⊢ ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) = ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 )
4 2 cnfldtopon ⊢ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ )
5 4 toponrestid ⊢ ( TopOpen ‘ ℂfld ) = ( ( TopOpen ‘ ℂfld ) ↾t ℂ )
6 2 3 5 cncfcn ⊢ ( ( 𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( 𝐴 –cn→ ℂ ) = ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) Cn ( TopOpen ‘ ℂfld ) ) )
7 1 6 mpan2 ⊢ ( 𝐴 ⊆ ℂ → ( 𝐴 –cn→ ℂ ) = ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) Cn ( TopOpen ‘ ℂfld ) ) )
8 7 eleq2d ⊢ ( 𝐴 ⊆ ℂ → ( 𝐹 ∈ ( 𝐴 –cn→ ℂ ) ↔ 𝐹 ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) Cn ( TopOpen ‘ ℂfld ) ) ) )
9 resttopon ⊢ ( ( ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ∧ 𝐴 ⊆ ℂ ) → ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) )
10 4 9 mpan ⊢ ( 𝐴 ⊆ ℂ → ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) )
11 cncnp ⊢ ( ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) ∧ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ) → ( 𝐹 ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) Cn ( TopOpen ‘ ℂfld ) ) ↔ ( 𝐹 : 𝐴 ⟶ ℂ ∧ ∀ 𝑥 ∈ 𝐴 𝐹 ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑥 ) ) ) )
12 10 4 11 sylancl ⊢ ( 𝐴 ⊆ ℂ → ( 𝐹 ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) Cn ( TopOpen ‘ ℂfld ) ) ↔ ( 𝐹 : 𝐴 ⟶ ℂ ∧ ∀ 𝑥 ∈ 𝐴 𝐹 ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑥 ) ) ) )
13 2 3 cnplimc ⊢ ( ( 𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑥 ) ↔ ( 𝐹 : 𝐴 ⟶ ℂ ∧ ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 limℂ 𝑥 ) ) ) )
14 13 baibd ⊢ ( ( ( 𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐹 : 𝐴 ⟶ ℂ ) → ( 𝐹 ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 limℂ 𝑥 ) ) )
15 14 an32s ⊢ ( ( ( 𝐴 ⊆ ℂ ∧ 𝐹 : 𝐴 ⟶ ℂ ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 limℂ 𝑥 ) ) )
16 15 ralbidva ⊢ ( ( 𝐴 ⊆ ℂ ∧ 𝐹 : 𝐴 ⟶ ℂ ) → ( ∀ 𝑥 ∈ 𝐴 𝐹 ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 limℂ 𝑥 ) ) )
17 16 pm5.32da ⊢ ( 𝐴 ⊆ ℂ → ( ( 𝐹 : 𝐴 ⟶ ℂ ∧ ∀ 𝑥 ∈ 𝐴 𝐹 ∈ ( ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐴 ) CnP ( TopOpen ‘ ℂfld ) ) ‘ 𝑥 ) ) ↔ ( 𝐹 : 𝐴 ⟶ ℂ ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 limℂ 𝑥 ) ) ) )
18 8 12 17 3bitrd ⊢ ( 𝐴 ⊆ ℂ → ( 𝐹 ∈ ( 𝐴 –cn→ ℂ ) ↔ ( 𝐹 : 𝐴 ⟶ ℂ ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 limℂ 𝑥 ) ) ) )