Theorem meetcomALT

Description: The meet of a poset is commutative. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 17-Sep-2011) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	meetcom.b	|- B = ( Base ` K )
	meetcom.m	|- ./\ = ( meet ` K )
Assertion	meetcomALT	|- ( ( K e. V /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) = ( Y ./\ X ) )

Proof

Step	Hyp	Ref	Expression
1		meetcom.b	|- B = ( Base ` K )
2		meetcom.m	|- ./\ = ( meet ` K )
3		prcom	|- { Y , X } = { X , Y }
4	3	fveq2i	|- ( ( glb ` K ) ` { Y , X } ) = ( ( glb ` K ) ` { X , Y } )
5	4	a1i	|- ( ( K e. V /\ X e. B /\ Y e. B ) -> ( ( glb ` K ) ` { Y , X } ) = ( ( glb ` K ) ` { X , Y } ) )
6		eqid	|- ( glb ` K ) = ( glb ` K )
7		simp1	|- ( ( K e. V /\ X e. B /\ Y e. B ) -> K e. V )
8		simp3	|- ( ( K e. V /\ X e. B /\ Y e. B ) -> Y e. B )
9		simp2	|- ( ( K e. V /\ X e. B /\ Y e. B ) -> X e. B )
10	6 2 7 8 9	meetval	|- ( ( K e. V /\ X e. B /\ Y e. B ) -> ( Y ./\ X ) = ( ( glb ` K ) ` { Y , X } ) )
11	6 2 7 9 8	meetval	|- ( ( K e. V /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) = ( ( glb ` K ) ` { X , Y } ) )
12	5 10 11	3eqtr4rd	|- ( ( K e. V /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) = ( Y ./\ X ) )