Theorem dvsubcncf

Description: A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	dvsubcncf.s	|- ( ph -> S e. { RR , CC } )
	dvsubcncf.f	|- ( ph -> F : X --> CC )
	dvsubcncf.g	|- ( ph -> G : X --> CC )
	dvsubcncf.fdv	|- ( ph -> ( S _D F ) e. ( X -cn-> CC ) )
	dvsubcncf.gdv	|- ( ph -> ( S _D G ) e. ( X -cn-> CC ) )
Assertion	dvsubcncf	|- ( ph -> ( S _D ( F oF - G ) ) e. ( X -cn-> CC ) )

Proof

Step	Hyp	Ref	Expression
1		dvsubcncf.s	|- ( ph -> S e. { RR , CC } )
2		dvsubcncf.f	|- ( ph -> F : X --> CC )
3		dvsubcncf.g	|- ( ph -> G : X --> CC )
4		dvsubcncf.fdv	|- ( ph -> ( S _D F ) e. ( X -cn-> CC ) )
5		dvsubcncf.gdv	|- ( ph -> ( S _D G ) e. ( X -cn-> CC ) )
6		cncff	|- ( ( S _D F ) e. ( X -cn-> CC ) -> ( S _D F ) : X --> CC )
7		fdm	|- ( ( S _D F ) : X --> CC -> dom ( S _D F ) = X )
8	4 6 7	3syl	|- ( ph -> dom ( S _D F ) = X )
9		cncff	|- ( ( S _D G ) e. ( X -cn-> CC ) -> ( S _D G ) : X --> CC )
10		fdm	|- ( ( S _D G ) : X --> CC -> dom ( S _D G ) = X )
11	5 9 10	3syl	|- ( ph -> dom ( S _D G ) = X )
12	1 2 3 8 11	dvsubf	|- ( ph -> ( S _D ( F oF - G ) ) = ( ( S _D F ) oF - ( S _D G ) ) )
13	4 5	subcncff	|- ( ph -> ( ( S _D F ) oF - ( S _D G ) ) e. ( X -cn-> CC ) )
14	12 13	eqeltrd	|- ( ph -> ( S _D ( F oF - G ) ) e. ( X -cn-> CC ) )