Metamath Proof Explorer

Theorem cgrcoml

Description: Congruence commutes on the left. Biconditional version of Theorem 2.4 of Schwabhauser p. 27. (Contributed by Scott Fenton, 12-Jun-2013)

Ref Expression
Assertion 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 >. ) )

Proof

Step Hyp Ref Expression
1 simp1 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> N e. NN )
2 simp2l 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
3 simp2r 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
4 1 2 3 cgrrflx2d 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> <. A , B >. Cgr <. B , A >. )
5 simp3l 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
6 simp3r 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
7 axcgrtr 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( <. A , B >. Cgr <. B , A >. /\ <. A , B >. Cgr <. C , D >. ) -> <. B , A >. Cgr <. C , D >. ) )
8 1 2 3 3 2 5 6 7 syl133anc 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( <. A , B >. Cgr <. B , A >. /\ <. A , B >. Cgr <. C , D >. ) -> <. B , A >. Cgr <. C , D >. ) )
9 4 8 mpand 
 |-  ( ( 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 >. ) )
10 1 3 2 cgrrflx2d 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> <. B , A >. Cgr <. A , B >. )
11 axcgrtr 
 |-  ( ( N e. NN /\ ( B e. ( EE ` N ) /\ A e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( <. B , A >. Cgr <. A , B >. /\ <. B , A >. Cgr <. C , D >. ) -> <. A , B >. Cgr <. C , D >. ) )
12 1 3 2 2 3 5 6 11 syl133anc 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( <. B , A >. Cgr <. A , B >. /\ <. B , A >. Cgr <. C , D >. ) -> <. A , B >. Cgr <. C , D >. ) )
13 10 12 mpand 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. B , A >. Cgr <. C , D >. -> <. A , B >. Cgr <. C , D >. ) )
14 9 13 impbid 
 |-  ( ( 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 >. ) )