Metamath Proof Explorer

Theorem cmcm3

Description: Commutation with orthocomplement. Remark in Kalmbach p. 23. (Contributed by NM, 13-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion cmcm3 
|- ( ( A e. CH /\ B e. CH ) -> ( A C_H B <-> ( _|_ ` A ) C_H B ) )

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 fveq2 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( _|_ ` A ) = ( _|_ ` if ( A e. CH , A , 0H ) ) )
3 2 breq1d 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( ( _|_ ` A ) C_H B <-> ( _|_ ` if ( A e. CH , A , 0H ) ) C_H B ) )
4 1 3 bibi12d 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( ( A C_H B <-> ( _|_ ` A ) C_H B ) <-> ( if ( A e. CH , A , 0H ) C_H B <-> ( _|_ ` if ( A e. CH , A , 0H ) ) C_H B ) ) )
5 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 ) ) )
6 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 ) ) )
7 5 6 bibi12d 
 |-  ( B = if ( B e. CH , B , 0H ) -> ( ( if ( A e. CH , A , 0H ) C_H B <-> ( _|_ ` if ( A e. CH , A , 0H ) ) C_H B ) <-> ( if ( A e. CH , A , 0H ) C_H if ( B e. CH , B , 0H ) <-> ( _|_ ` if ( A e. CH , A , 0H ) ) C_H if ( B e. CH , B , 0H ) ) ) )
8 h0elch 
 |-  0H e. CH
9 8 elimel 
 |-  if ( A e. CH , A , 0H ) e. CH
10 8 elimel 
 |-  if ( B e. CH , B , 0H ) e. CH
11 9 10 cmcm3i 
 |-  ( if ( A e. CH , A , 0H ) C_H if ( B e. CH , B , 0H ) <-> ( _|_ ` if ( A e. CH , A , 0H ) ) C_H if ( B e. CH , B , 0H ) )
12 4 7 11 dedth2h 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A C_H B <-> ( _|_ ` A ) C_H B ) )