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Metamath Proof Explorer

Theorem latabs1

Description: Lattice absorption law. From definition of lattice in Kalmbach p. 14. ( chabs1 analog.) (Contributed by NM, 8-Nov-2011)

Ref Expression
Hypotheses latabs1.b 
|- B = ( Base ` K )
latabs1.j 
|- .\/ = ( join ` K )
latabs1.m 
|- ./\ = ( meet ` K )
Assertion latabs1 
|- ( ( 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 3 latmle1 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) ( le ` K ) X )
6 1 3 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) e. B )
7 1 4 2 latleeqj2 
 |-  ( ( K e. Lat /\ ( X ./\ Y ) e. B /\ X e. B ) -> ( ( X ./\ Y ) ( le ` K ) X <-> ( X .\/ ( X ./\ Y ) ) = X ) )
8 7 3com23 
 |-  ( ( K e. Lat /\ X e. B /\ ( X ./\ Y ) e. B ) -> ( ( X ./\ Y ) ( le ` K ) X <-> ( X .\/ ( X ./\ Y ) ) = X ) )
9 6 8 syld3an3 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X ./\ Y ) ( le ` K ) X <-> ( X .\/ ( X ./\ Y ) ) = X ) )
10 5 9 mpbid 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ ( X ./\ Y ) ) = X )