Metamath Proof Explorer


Theorem resubd

Description: Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses recld.1
|- ( ph -> A e. CC )
readdd.2
|- ( ph -> B e. CC )
Assertion resubd
|- ( ph -> ( Re ` ( A - B ) ) = ( ( Re ` A ) - ( Re ` B ) ) )

Proof

Step Hyp Ref Expression
1 recld.1
 |-  ( ph -> A e. CC )
2 readdd.2
 |-  ( ph -> B e. CC )
3 resub
 |-  ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A - B ) ) = ( ( Re ` A ) - ( Re ` B ) ) )
4 1 2 3 syl2anc
 |-  ( ph -> ( Re ` ( A - B ) ) = ( ( Re ` A ) - ( Re ` B ) ) )