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Metamath Proof Explorer

Theorem meetcom

Description: The meet of a poset is commutative. (The antecedent <. X , Y >. e. dom ./\ /\ <. Y , X >. e. dom ./\ i.e., "the meets exist" could be omitted as an artifact of our particular join definition, but other definitions may require it.) (Contributed by NM, 17-Sep-2011) (Revised by NM, 12-Sep-2018)

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

Proof

Step	Hyp	Ref	Expression
1		meetcom.b	\|- B = ( Base ` K )
2		meetcom.m	\|- ./\ = ( meet ` K )
3	1 2	meetcomALT	\|- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) = ( Y ./\ X ) )
4	3	adantr	\|- ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ ( <. X , Y >. e. dom ./\ /\ <. Y , X >. e. dom ./\ ) ) -> ( X ./\ Y ) = ( Y ./\ X ) )