Theorem pmaplubN

Description: The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pmaplub.b	|- B = ( Base ` K )
	pmaplub.u	|- U = ( lub ` K )
	pmaplub.m	|- M = ( pmap ` K )
Assertion	pmaplubN	|- ( ( K e. HL /\ X e. B ) -> ( U ` ( M ` X ) ) = X )

Proof

Step	Hyp	Ref	Expression
1		pmaplub.b	|- B = ( Base ` K )
2		pmaplub.u	|- U = ( lub ` K )
3		pmaplub.m	|- M = ( pmap ` K )
4		eqid	|- ( le ` K ) = ( le ` K )
5		eqid	|- ( Atoms ` K ) = ( Atoms ` K )
6	1 4 5 3	pmapval	|- ( ( K e. HL /\ X e. B ) -> ( M ` X ) = { p e. ( Atoms ` K ) | p ( le ` K ) X } )
7	6	fveq2d	|- ( ( K e. HL /\ X e. B ) -> ( U ` ( M ` X ) ) = ( U ` { p e. ( Atoms ` K ) | p ( le ` K ) X } ) )
8		hlomcmat	|- ( K e. HL -> ( K e. OML /\ K e. CLat /\ K e. AtLat ) )
9	1 4 2 5	atlatmstc	|- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B ) -> ( U ` { p e. ( Atoms ` K ) | p ( le ` K ) X } ) = X )
10	8 9	sylan	|- ( ( K e. HL /\ X e. B ) -> ( U ` { p e. ( Atoms ` K ) | p ( le ` K ) X } ) = X )
11	7 10	eqtrd	|- ( ( K e. HL /\ X e. B ) -> ( U ` ( M ` X ) ) = X )