Theorem glbpr

Description: The GLB of the set of two comparable elements in a poset is the less one of the two. (Contributed by Zhi Wang, 26-Sep-2024)

	Ref	Expression
Hypotheses	lubpr.k	|- ( ph -> K e. Poset )
	lubpr.b	|- B = ( Base ` K )
	lubpr.x	|- ( ph -> X e. B )
	lubpr.y	|- ( ph -> Y e. B )
	lubpr.l	|- .<_ = ( le ` K )
	lubpr.c	|- ( ph -> X .<_ Y )
	lubpr.s	|- ( ph -> S = { X , Y } )
	glbpr.g	|- G = ( glb ` K )
Assertion	glbpr	|- ( ph -> ( G ` S ) = X )

Proof

Step	Hyp	Ref	Expression
1		lubpr.k	|- ( ph -> K e. Poset )
2		lubpr.b	|- B = ( Base ` K )
3		lubpr.x	|- ( ph -> X e. B )
4		lubpr.y	|- ( ph -> Y e. B )
5		lubpr.l	|- .<_ = ( le ` K )
6		lubpr.c	|- ( ph -> X .<_ Y )
7		lubpr.s	|- ( ph -> S = { X , Y } )
8		glbpr.g	|- G = ( glb ` K )
9	1 2 3 4 5 6 7 8	glbprlem	|- ( ph -> ( S e. dom G /\ ( G ` S ) = X ) )
10	9	simprd	|- ( ph -> ( G ` S ) = X )