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Metamath Proof Explorer

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 ) )