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 )