Theorem lubsn

Description: The least upper bound of a singleton. ( chsupsn analog.) (Contributed by NM, 20-Oct-2011)

	Ref	Expression
Hypotheses	lubsn.b	|- B = ( Base ` K )
	lubsn.u	|- U = ( lub ` K )
Assertion	lubsn	|- ( ( K e. Lat /\ X e. B ) -> ( U ` { X } ) = X )

Proof

Step	Hyp	Ref	Expression
1		lubsn.b	|- B = ( Base ` K )
2		lubsn.u	|- U = ( lub ` K )
3		dfsn2	|- { X } = { X , X }
4	3	fveq2i	|- ( U ` { X } ) = ( U ` { X , X } )
5		eqid	|- ( join ` K ) = ( join ` K )
6		simpl	|- ( ( K e. Lat /\ X e. B ) -> K e. Lat )
7		simpr	|- ( ( K e. Lat /\ X e. B ) -> X e. B )
8	2 5 6 7 7	joinval	|- ( ( K e. Lat /\ X e. B ) -> ( X ( join ` K ) X ) = ( U ` { X , X } ) )
9	4 8	eqtr4id	|- ( ( K e. Lat /\ X e. B ) -> ( U ` { X } ) = ( X ( join ` K ) X ) )
10	1 5	latjidm	|- ( ( K e. Lat /\ X e. B ) -> ( X ( join ` K ) X ) = X )
11	9 10	eqtrd	|- ( ( K e. Lat /\ X e. B ) -> ( U ` { X } ) = X )