Theorem add2cncf

Description: Addition to a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021)

	Ref	Expression
Hypotheses	add2cncf.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	add2cncf.f	⊢ 𝐹 = ( 𝑥 ∈ ℂ ↦ ( 𝐴 + 𝑥 ) )
Assertion	add2cncf	⊢ ( 𝜑 → 𝐹 ∈ ( ℂ –cn→ ℂ ) )

Proof

Step	Hyp	Ref	Expression
1		add2cncf.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		add2cncf.f	⊢ 𝐹 = ( 𝑥 ∈ ℂ ↦ ( 𝐴 + 𝑥 ) )
3		ssid	⊢ ℂ ⊆ ℂ
4	3	a1i	⊢ ( 𝐴 ∈ ℂ → ℂ ⊆ ℂ )
5		cncfmptc	⊢ ( ( 𝐴 ∈ ℂ ∧ ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( 𝑥 ∈ ℂ ↦ 𝐴 ) ∈ ( ℂ –cn→ ℂ ) )
6	4 4 5	mpd3an23	⊢ ( 𝐴 ∈ ℂ → ( 𝑥 ∈ ℂ ↦ 𝐴 ) ∈ ( ℂ –cn→ ℂ ) )
7	1 6	syl	⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ 𝐴 ) ∈ ( ℂ –cn→ ℂ ) )
8		cncfmptid	⊢ ( ( ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( 𝑥 ∈ ℂ ↦ 𝑥 ) ∈ ( ℂ –cn→ ℂ ) )
9	3 3 8	mp2an	⊢ ( 𝑥 ∈ ℂ ↦ 𝑥 ) ∈ ( ℂ –cn→ ℂ )
10	9	a1i	⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ 𝑥 ) ∈ ( ℂ –cn→ ℂ ) )
11	7 10	addcncf	⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ ( 𝐴 + 𝑥 ) ) ∈ ( ℂ –cn→ ℂ ) )
12	2 11	eqeltrid	⊢ ( 𝜑 → 𝐹 ∈ ( ℂ –cn→ ℂ ) )