Metamath Proof Explorer

Theorem cmcm

Description: Commutation is symmetric. Theorem 2(v) of Kalmbach p. 22. (Contributed by NM, 13-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion cmcm 
|- ( ( A e. CH /\ B e. CH ) -> ( A C_H B <-> B C_H A ) )

Proof

Step Hyp Ref Expression
1 breq1 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( A C_H B <-> if ( A e. CH , A , 0H ) C_H B ) )
2 breq2 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( B C_H A <-> B C_H if ( A e. CH , A , 0H ) ) )
3 1 2 bibi12d 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( ( A C_H B <-> B C_H A ) <-> ( if ( A e. CH , A , 0H ) C_H B <-> B C_H if ( A e. CH , A , 0H ) ) ) )
4 breq2 
 |-  ( B = if ( B e. CH , B , 0H ) -> ( if ( A e. CH , A , 0H ) C_H B <-> if ( A e. CH , A , 0H ) C_H if ( B e. CH , B , 0H ) ) )
5 breq1 
 |-  ( B = if ( B e. CH , B , 0H ) -> ( B C_H if ( A e. CH , A , 0H ) <-> if ( B e. CH , B , 0H ) C_H if ( A e. CH , A , 0H ) ) )
6 4 5 bibi12d 
 |-  ( B = if ( B e. CH , B , 0H ) -> ( ( if ( A e. CH , A , 0H ) C_H B <-> B C_H if ( A e. CH , A , 0H ) ) <-> ( if ( A e. CH , A , 0H ) C_H if ( B e. CH , B , 0H ) <-> if ( B e. CH , B , 0H ) C_H if ( A e. CH , A , 0H ) ) ) )
7 h0elch 
 |-  0H e. CH
8 7 elimel 
 |-  if ( A e. CH , A , 0H ) e. CH
9 7 elimel 
 |-  if ( B e. CH , B , 0H ) e. CH
10 8 9 cmcmi 
 |-  ( if ( A e. CH , A , 0H ) C_H if ( B e. CH , B , 0H ) <-> if ( B e. CH , B , 0H ) C_H if ( A e. CH , A , 0H ) )
11 3 6 10 dedth2h 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A C_H B <-> B C_H A ) )