Metamath Proof Explorer

Theorem cgrcomlr

Description: Congruence commutes on both sides. (Contributed by Scott Fenton, 12-Jun-2013)

Ref Expression
Assertion cgrcomlr 
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. A , B >. Cgr <. C , D >. <-> <. B , A >. Cgr <. D , C >. ) )

Proof

Step Hyp Ref Expression
1 cgrcoml 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. A , B >. Cgr <. C , D >. <-> <. B , A >. Cgr <. C , D >. ) )
2 ancom 
 |-  ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) <-> ( B e. ( EE ` N ) /\ A e. ( EE ` N ) ) )
3 cgrcomr 
 |-  ( ( N e. NN /\ ( B e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. B , A >. Cgr <. C , D >. <-> <. B , A >. Cgr <. D , C >. ) )
4 2 3 syl3an2b 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. B , A >. Cgr <. C , D >. <-> <. B , A >. Cgr <. D , C >. ) )
5 1 4 bitrd 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. A , B >. Cgr <. C , D >. <-> <. B , A >. Cgr <. D , C >. ) )