Theorem posjidm

Description: Poset join is idempotent. latjidm could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024)

	Ref	Expression
Hypotheses	posjidm.b	|- B = ( Base ` K )
	posjidm.j	|- .\/ = ( join ` K )
Assertion	posjidm	|- ( ( K e. Poset /\ X e. B ) -> ( X .\/ X ) = X )

Proof

Step	Hyp	Ref	Expression
1		posjidm.b	|- B = ( Base ` K )
2		posjidm.j	|- .\/ = ( join ` K )
3		eqid	|- ( lub ` K ) = ( lub ` K )
4		simpl	|- ( ( K e. Poset /\ X e. B ) -> K e. Poset )
5		simpr	|- ( ( K e. Poset /\ X e. B ) -> X e. B )
6	3 2 4 5 5	joinval	|- ( ( K e. Poset /\ X e. B ) -> ( X .\/ X ) = ( ( lub ` K ) ` { X , X } ) )
7		eqid	|- ( le ` K ) = ( le ` K )
8	1 7	posref	|- ( ( K e. Poset /\ X e. B ) -> X ( le ` K ) X )
9		eqidd	|- ( ( K e. Poset /\ X e. B ) -> { X , X } = { X , X } )
10	4 1 5 5 7 8 9 3	lubpr	|- ( ( K e. Poset /\ X e. B ) -> ( ( lub ` K ) ` { X , X } ) = X )
11	6 10	eqtrd	|- ( ( K e. Poset /\ X e. B ) -> ( X .\/ X ) = X )