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Metamath Proof Explorer

Theorem latabs2

Description: Lattice absorption law. From definition of lattice in Kalmbach p. 14. ( chabs2 analog.) (Contributed by NM, 8-Nov-2011)

Ref Expression
Hypotheses latabs1.b 
|- B = ( Base ` K )
latabs1.j 
|- .\/ = ( join ` K )
latabs1.m 
|- ./\ = ( meet ` K )
Assertion latabs2 
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ ( X .\/ Y ) ) = X )

Proof

Step Hyp Ref Expression
1 latabs1.b 
 |-  B = ( Base ` K )
2 latabs1.j 
 |-  .\/ = ( join ` K )
3 latabs1.m 
 |-  ./\ = ( meet ` K )
4 eqid 
 |-  ( le ` K ) = ( le ` K )
5 1 4 2 latlej1 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> X ( le ` K ) ( X .\/ Y ) )
6 1 2 latjcl 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) e. B )
7 1 4 3 latleeqm1 
 |-  ( ( K e. Lat /\ X e. B /\ ( X .\/ Y ) e. B ) -> ( X ( le ` K ) ( X .\/ Y ) <-> ( X ./\ ( X .\/ Y ) ) = X ) )
8 6 7 syld3an3 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ( le ` K ) ( X .\/ Y ) <-> ( X ./\ ( X .\/ Y ) ) = X ) )
9 5 8 mpbid 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ ( X .\/ Y ) ) = X )