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Metamath Proof Explorer

Theorem cmcm2

Description: Commutation with orthocomplement. Theorem 2.3(i) of Beran p. 39. (Contributed by NM, 14-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion cmcm2 
|- ( ( A e. CH /\ B e. CH ) -> ( A C_H B <-> A C_H ( _|_ ` B ) ) )

Proof

Step Hyp Ref Expression
1 cmcm3 
 |-  ( ( B e. CH /\ A e. CH ) -> ( B C_H A <-> ( _|_ ` B ) C_H A ) )
2 1 ancoms 
 |-  ( ( A e. CH /\ B e. CH ) -> ( B C_H A <-> ( _|_ ` B ) C_H A ) )
3 cmcm 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A C_H B <-> B C_H A ) )
4 choccl 
 |-  ( B e. CH -> ( _|_ ` B ) e. CH )
5 cmcm 
 |-  ( ( A e. CH /\ ( _|_ ` B ) e. CH ) -> ( A C_H ( _|_ ` B ) <-> ( _|_ ` B ) C_H A ) )
6 4 5 sylan2 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A C_H ( _|_ ` B ) <-> ( _|_ ` B ) C_H A ) )
7 2 3 6 3bitr4d 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A C_H B <-> A C_H ( _|_ ` B ) ) )