Theorem sub1cncfd

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

	Ref	Expression
Hypotheses	sub1cncfd.1	|- ( ph -> A e. CC )
	sub1cncfd.2	|- F = ( x e. CC |-> ( x - A ) )
Assertion	sub1cncfd	|- ( ph -> F e. ( CC -cn-> CC ) )

Proof

Step	Hyp	Ref	Expression
1		sub1cncfd.1	|- ( ph -> A e. CC )
2		sub1cncfd.2	|- F = ( x e. CC |-> ( x - A ) )
3		ssid	|- CC C_ CC
4		cncfmptid	|- ( ( CC C_ CC /\ CC C_ CC ) -> ( x e. CC |-> x ) e. ( CC -cn-> CC ) )
5	3 3 4	mp2an	|- ( x e. CC |-> x ) e. ( CC -cn-> CC )
6	5	a1i	|- ( ph -> ( x e. CC |-> x ) e. ( CC -cn-> CC ) )
7	3	a1i	|- ( ph -> CC C_ CC )
8		cncfmptc	|- ( ( A e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( x e. CC |-> A ) e. ( CC -cn-> CC ) )
9	1 7 7 8	syl3anc	|- ( ph -> ( x e. CC |-> A ) e. ( CC -cn-> CC ) )
10	6 9	subcncf	|- ( ph -> ( x e. CC |-> ( x - A ) ) e. ( CC -cn-> CC ) )
11	2 10	eqeltrid	|- ( ph -> F e. ( CC -cn-> CC ) )