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Metamath Proof Explorer

Theorem latmle1

Description: A meet is less than or equal to its first argument. (Contributed by NM, 21-Oct-2011)

Ref Expression
Hypotheses latmle.b 
|- B = ( Base ` K )
latmle.l 
|- .<_ = ( le ` K )
latmle.m 
|- ./\ = ( meet ` K )
Assertion latmle1 
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) .<_ X )

Proof

Step Hyp Ref Expression
1 latmle.b 
 |-  B = ( Base ` K )
2 latmle.l 
 |-  .<_ = ( le ` K )
3 latmle.m 
 |-  ./\ = ( meet ` K )
4 simp1 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> K e. Lat )
5 simp2 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> X e. B )
6 simp3 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> Y e. B )
7 eqid 
 |-  ( join ` K ) = ( join ` K )
8 1 7 3 4 5 6 latcl2 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( <. X , Y >. e. dom ( join ` K ) /\ <. X , Y >. e. dom ./\ ) )
9 8 simprd 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> <. X , Y >. e. dom ./\ )
10 1 2 3 4 5 6 9 lemeet1 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) .<_ X )