Metamath Proof Explorer

Theorem cmmdi

Description: Commuting subspaces form a modular pair. (Contributed by NM, 16-Jan-2005) (New usage is discouraged.)

Ref Expression
Hypotheses sumdmdi.1 
|- A e. CH
sumdmdi.2 
|- B e. CH
Assertion cmmdi 
|- ( A C_H B -> A MH B )

Proof

Step Hyp Ref Expression
1 sumdmdi.1 
 |-  A e. CH
2 sumdmdi.2 
 |-  B e. CH
3 1 2 cmcm4i 
 |-  ( A C_H B <-> ( _|_ ` A ) C_H ( _|_ ` B ) )
4 1 choccli 
 |-  ( _|_ ` A ) e. CH
5 2 choccli 
 |-  ( _|_ ` B ) e. CH
6 4 5 osumcor2i 
 |-  ( ( _|_ ` A ) C_H ( _|_ ` B ) -> ( ( _|_ ` A ) +H ( _|_ ` B ) ) = ( ( _|_ ` A ) vH ( _|_ ` B ) ) )
7 3 6 sylbi 
 |-  ( A C_H B -> ( ( _|_ ` A ) +H ( _|_ ` B ) ) = ( ( _|_ ` A ) vH ( _|_ ` B ) ) )
8 4 5 sumdmdii 
 |-  ( ( ( _|_ ` A ) +H ( _|_ ` B ) ) = ( ( _|_ ` A ) vH ( _|_ ` B ) ) -> ( _|_ ` A ) MH* ( _|_ ` B ) )
9 7 8 syl 
 |-  ( A C_H B -> ( _|_ ` A ) MH* ( _|_ ` B ) )
10 mddmd 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> ( _|_ ` A ) MH* ( _|_ ` B ) ) )
11 1 2 10 mp2an 
 |-  ( A MH B <-> ( _|_ ` A ) MH* ( _|_ ` B ) )
12 9 11 sylibr 
 |-  ( A C_H B -> A MH B )