Theorem subcncf

Description: The subtraction of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	subcncf.a	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ∈ ( 𝑋 –cn→ ℂ ) )
	subcncf.b	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) ∈ ( 𝑋 –cn→ ℂ ) )
Assertion	subcncf	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 − 𝐵 ) ) ∈ ( 𝑋 –cn→ ℂ ) )

Proof

Step	Hyp	Ref	Expression
1		subcncf.a	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ∈ ( 𝑋 –cn→ ℂ ) )
2		subcncf.b	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) ∈ ( 𝑋 –cn→ ℂ ) )
3		eqid	⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld )
4	3	subcn	⊢ − ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) )
5	4	a1i	⊢ ( 𝜑 → − ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) ) )
6	3 5 1 2	cncfmpt2f	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 − 𝐵 ) ) ∈ ( 𝑋 –cn→ ℂ ) )