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Metamath Proof Explorer

Theorem lecm

Description: Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of Beran p. 40. (Contributed by NM, 13-Jun-2006) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 sseq1 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( A C_ B <-> if ( A e. CH , A , 0H ) C_ B ) )
2 breq1 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( A C_H B <-> if ( A e. CH , A , 0H ) C_H B ) )
3 1 2 imbi12d 
 |-  ( A = if ( A e. CH , A , 0H ) -> ( ( A C_ B -> A C_H B ) <-> ( if ( A e. CH , A , 0H ) C_ B -> if ( A e. CH , A , 0H ) C_H B ) ) )
4 sseq2 
 |-  ( B = if ( B e. CH , B , 0H ) -> ( if ( A e. CH , A , 0H ) C_ B <-> if ( A e. CH , A , 0H ) C_ if ( B e. CH , B , 0H ) ) )
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 4 5 imbi12d 
 |-  ( B = if ( B e. CH , B , 0H ) -> ( ( if ( A e. CH , A , 0H ) C_ B -> if ( A e. CH , A , 0H ) C_H B ) <-> ( if ( A e. CH , A , 0H ) C_ if ( B e. CH , B , 0H ) -> if ( A e. CH , A , 0H ) C_H if ( B e. CH , B , 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 lecmi 
 |-  ( if ( A e. CH , A , 0H ) C_ if ( B e. CH , B , 0H ) -> if ( A e. CH , A , 0H ) C_H if ( B e. CH , B , 0H ) )
11 3 6 10 dedth2h 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A C_ B -> A C_H B ) )
12 11 3impia 
 |-  ( ( A e. CH /\ B e. CH /\ A C_ B ) -> A C_H B )