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 )