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Metamath Proof Explorer

Theorem latlem12

Description: An element is less than or equal to a meet iff the element is less than or equal to each argument of the meet. (Contributed by NM, 21-Oct-2011)

Ref Expression
Hypotheses latmle.b 
|- B = ( Base ` K )
latmle.l 
|- .<_ = ( le ` K )
latmle.m 
|- ./\ = ( meet ` K )
Assertion latlem12 
|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .<_ Y /\ X .<_ Z ) <-> X .<_ ( Y ./\ Z ) ) )

Proof

Step Hyp Ref Expression
1 latmle.b 
 |-  B = ( Base ` K )
2 latmle.l 
 |-  .<_ = ( le ` K )
3 latmle.m 
 |-  ./\ = ( meet ` K )
4 latpos 
 |-  ( K e. Lat -> K e. Poset )
5 4 adantr 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. Poset )
6 simpr2 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
7 simpr3 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
8 simpr1 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )
9 eqid 
 |-  ( join ` K ) = ( join ` K )
10 simpl 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. Lat )
11 1 9 3 10 6 7 latcl2 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( <. Y , Z >. e. dom ( join ` K ) /\ <. Y , Z >. e. dom ./\ ) )
12 11 simprd 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> <. Y , Z >. e. dom ./\ )
13 1 2 3 5 6 7 8 12 meetle 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .<_ Y /\ X .<_ Z ) <-> X .<_ ( Y ./\ Z ) ) )