Metamath Proof Explorer

Theorem f1ovscpbl

Description: An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 15-Aug-2015)

Ref Expression
Hypothesis f1ocpbl.f 
|- ( ph -> F : V -1-1-onto-> X )
Assertion f1ovscpbl 
|- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> ( ( F ` B ) = ( F ` C ) -> ( F ` ( A .+ B ) ) = ( F ` ( A .+ C ) ) ) )

Proof

Step Hyp Ref Expression
1 f1ocpbl.f 
 |-  ( ph -> F : V -1-1-onto-> X )
2 f1of1 
 |-  ( F : V -1-1-onto-> X -> F : V -1-1-> X )
3 1 2 syl 
 |-  ( ph -> F : V -1-1-> X )
4 3 adantr 
 |-  ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> F : V -1-1-> X )
5 simpr2 
 |-  ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> B e. V )
6 simpr3 
 |-  ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> C e. V )
7 f1fveq 
 |-  ( ( F : V -1-1-> X /\ ( B e. V /\ C e. V ) ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
8 4 5 6 7 syl12anc 
 |-  ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
9 oveq2 
 |-  ( B = C -> ( A .+ B ) = ( A .+ C ) )
10 9 fveq2d 
 |-  ( B = C -> ( F ` ( A .+ B ) ) = ( F ` ( A .+ C ) ) )
11 8 10 biimtrdi 
 |-  ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> ( ( F ` B ) = ( F ` C ) -> ( F ` ( A .+ B ) ) = ( F ` ( A .+ C ) ) ) )