Theorem addccncf

Description: Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009) (Proof shortened by Mario Carneiro, 12-Sep-2015)

	Ref	Expression
Hypothesis	addccncf.1	⊢ 𝐹 = ( 𝑥 ∈ ℂ ↦ ( 𝑥 + 𝐴 ) )
Assertion	addccncf	⊢ ( 𝐴 ∈ ℂ → 𝐹 ∈ ( ℂ –cn→ ℂ ) )

Proof

Step	Hyp	Ref	Expression
1		addccncf.1	⊢ 𝐹 = ( 𝑥 ∈ ℂ ↦ ( 𝑥 + 𝐴 ) )
2		eqid	⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld )
3	2	addcn	⊢ + ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) )
4	3	a1i	⊢ ( 𝐴 ∈ ℂ → + ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) ) )
5		ssid	⊢ ℂ ⊆ ℂ
6		cncfmptid	⊢ ( ( ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( 𝑥 ∈ ℂ ↦ 𝑥 ) ∈ ( ℂ –cn→ ℂ ) )
7	5 5 6	mp2an	⊢ ( 𝑥 ∈ ℂ ↦ 𝑥 ) ∈ ( ℂ –cn→ ℂ )
8	7	a1i	⊢ ( 𝐴 ∈ ℂ → ( 𝑥 ∈ ℂ ↦ 𝑥 ) ∈ ( ℂ –cn→ ℂ ) )
9		cncfmptc	⊢ ( ( 𝐴 ∈ ℂ ∧ ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( 𝑥 ∈ ℂ ↦ 𝐴 ) ∈ ( ℂ –cn→ ℂ ) )
10	5 5 9	mp3an23	⊢ ( 𝐴 ∈ ℂ → ( 𝑥 ∈ ℂ ↦ 𝐴 ) ∈ ( ℂ –cn→ ℂ ) )
11	2 4 8 10	cncfmpt2f	⊢ ( 𝐴 ∈ ℂ → ( 𝑥 ∈ ℂ ↦ ( 𝑥 + 𝐴 ) ) ∈ ( ℂ –cn→ ℂ ) )
12	1 11	eqeltrid	⊢ ( 𝐴 ∈ ℂ → 𝐹 ∈ ( ℂ –cn→ ℂ ) )