Theorem add2cncf

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

	Ref	Expression
Hypotheses	add2cncf.a	|- ( ph -> A e. CC )
	add2cncf.f	|- F = ( x e. CC |-> ( A + x ) )
Assertion	add2cncf	|- ( ph -> F e. ( CC -cn-> CC ) )

Proof

Step	Hyp	Ref	Expression
1		add2cncf.a	|- ( ph -> A e. CC )
2		add2cncf.f	|- F = ( x e. CC |-> ( A + x ) )
3		ssid	|- CC C_ CC
4	3	a1i	|- ( A e. CC -> CC C_ CC )
5		cncfmptc	|- ( ( A e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( x e. CC |-> A ) e. ( CC -cn-> CC ) )
6	4 4 5	mpd3an23	|- ( A e. CC -> ( x e. CC |-> A ) e. ( CC -cn-> CC ) )
7	1 6	syl	|- ( ph -> ( x e. CC |-> A ) e. ( CC -cn-> CC ) )
8		cncfmptid	|- ( ( CC C_ CC /\ CC C_ CC ) -> ( x e. CC |-> x ) e. ( CC -cn-> CC ) )
9	3 3 8	mp2an	|- ( x e. CC |-> x ) e. ( CC -cn-> CC )
10	9	a1i	|- ( ph -> ( x e. CC |-> x ) e. ( CC -cn-> CC ) )
11	7 10	addcncf	|- ( ph -> ( x e. CC |-> ( A + x ) ) e. ( CC -cn-> CC ) )
12	2 11	eqeltrid	|- ( ph -> F e. ( CC -cn-> CC ) )