Metamath Proof Explorer

Theorem cgrcomim

Description: Congruence commutes on the two sides. Implication version. Theorem 2.2 of Schwabhauser p. 27. (Contributed by Scott Fenton, 12-Jun-2013)

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

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 simp3l 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
5 simp3r 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
6 simpr 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ <. A , B >. Cgr <. C , D >. ) -> <. A , B >. Cgr <. C , D >. )
7 simpl1 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ <. A , B >. Cgr <. C , D >. ) -> N e. NN )
8 simpl2l 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ <. A , B >. Cgr <. C , D >. ) -> A e. ( EE ` N ) )
9 simpl2r 
 |-  ( ( ( 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 e. ( EE ` N ) )
10 7 8 9 cgrrflxd 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ <. A , B >. Cgr <. C , D >. ) -> <. A , B >. Cgr <. A , B >. )
11 1 2 3 4 5 2 3 6 10 cgrtr4and 
 |-  ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ <. A , B >. Cgr <. C , D >. ) -> <. C , D >. Cgr <. A , B >. )
12 11 ex 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. A , B >. Cgr <. C , D >. -> <. C , D >. Cgr <. A , B >. ) )