Metamath Proof Explorer


Theorem dvcnre

Description: From compex differentiation to real differentiation. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Assertion dvcnre
|- ( ( F : CC --> CC /\ RR C_ dom ( CC _D F ) ) -> ( RR _D ( F |` RR ) ) = ( ( CC _D F ) |` RR ) )

Proof

Step Hyp Ref Expression
1 reelprrecn
 |-  RR e. { RR , CC }
2 1 a1i
 |-  ( ( F : CC --> CC /\ RR C_ dom ( CC _D F ) ) -> RR e. { RR , CC } )
3 simpl
 |-  ( ( F : CC --> CC /\ RR C_ dom ( CC _D F ) ) -> F : CC --> CC )
4 ssidd
 |-  ( ( F : CC --> CC /\ RR C_ dom ( CC _D F ) ) -> CC C_ CC )
5 simpr
 |-  ( ( F : CC --> CC /\ RR C_ dom ( CC _D F ) ) -> RR C_ dom ( CC _D F ) )
6 dvres3
 |-  ( ( ( RR e. { RR , CC } /\ F : CC --> CC ) /\ ( CC C_ CC /\ RR C_ dom ( CC _D F ) ) ) -> ( RR _D ( F |` RR ) ) = ( ( CC _D F ) |` RR ) )
7 2 3 4 5 6 syl22anc
 |-  ( ( F : CC --> CC /\ RR C_ dom ( CC _D F ) ) -> ( RR _D ( F |` RR ) ) = ( ( CC _D F ) |` RR ) )