Theorem fourierdlem23

Description: If F is continuous and X is constant, then ( F( X + s ) ) is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	fourierdlem23.a	⊢ ( 𝜑 → 𝐴 ⊆ ℂ )
	fourierdlem23.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐴 –cn→ ℂ ) )
	fourierdlem23.b	⊢ ( 𝜑 → 𝐵 ⊆ ℂ )
	fourierdlem23.x	⊢ ( 𝜑 → 𝑋 ∈ ℂ )
	fourierdlem23.xps	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝐵 ) → ( 𝑋 + 𝑠 ) ∈ 𝐴 )
Assertion	fourierdlem23	⊢ ( 𝜑 → ( 𝑠 ∈ 𝐵 ↦ ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) ) ∈ ( 𝐵 –cn→ ℂ ) )

Proof

Step	Hyp	Ref	Expression
1		fourierdlem23.a	⊢ ( 𝜑 → 𝐴 ⊆ ℂ )
2		fourierdlem23.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐴 –cn→ ℂ ) )
3		fourierdlem23.b	⊢ ( 𝜑 → 𝐵 ⊆ ℂ )
4		fourierdlem23.x	⊢ ( 𝜑 → 𝑋 ∈ ℂ )
5		fourierdlem23.xps	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝐵 ) → ( 𝑋 + 𝑠 ) ∈ 𝐴 )
6		eqid	⊢ ( 𝑠 ∈ 𝐵 ↦ ( 𝑋 + 𝑠 ) ) = ( 𝑠 ∈ 𝐵 ↦ ( 𝑋 + 𝑠 ) )
7	6	addccncf2	⊢ ( ( 𝐵 ⊆ ℂ ∧ 𝑋 ∈ ℂ ) → ( 𝑠 ∈ 𝐵 ↦ ( 𝑋 + 𝑠 ) ) ∈ ( 𝐵 –cn→ ℂ ) )
8	3 4 7	syl2anc	⊢ ( 𝜑 → ( 𝑠 ∈ 𝐵 ↦ ( 𝑋 + 𝑠 ) ) ∈ ( 𝐵 –cn→ ℂ ) )
9		ssid	⊢ 𝐵 ⊆ 𝐵
10	9	a1i	⊢ ( 𝜑 → 𝐵 ⊆ 𝐵 )
11	6 8 10 1 5	cncfmptssg	⊢ ( 𝜑 → ( 𝑠 ∈ 𝐵 ↦ ( 𝑋 + 𝑠 ) ) ∈ ( 𝐵 –cn→ 𝐴 ) )
12	11 2	cncfcompt	⊢ ( 𝜑 → ( 𝑠 ∈ 𝐵 ↦ ( 𝐹 ‘ ( 𝑋 + 𝑠 ) ) ) ∈ ( 𝐵 –cn→ ℂ ) )