Theorem subcncf

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

	Ref	Expression
Hypotheses	subcncf.a	|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )
	subcncf.b	|- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )
Assertion	subcncf	|- ( ph -> ( x e. X |-> ( A - B ) ) e. ( X -cn-> CC ) )

Proof

Step	Hyp	Ref	Expression
1		subcncf.a	|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )
2		subcncf.b	|- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )
3		eqid	|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
4	3	subcn	|- - e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) )
5	4	a1i	|- ( ph -> - e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) )
6	3 5 1 2	cncfmpt2f	|- ( ph -> ( x e. X |-> ( A - B ) ) e. ( X -cn-> CC ) )