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Metamath Proof Explorer

Theorem latmidm

Description: Lattice meet is idempotent. Analogue of inidm . (Contributed by NM, 8-Nov-2011)

Ref Expression
Hypotheses latmidm.b 
|- B = ( Base ` K )
latmidm.m 
|- ./\ = ( meet ` K )
Assertion latmidm 
|- ( ( K e. Lat /\ X e. B ) -> ( X ./\ X ) = X )

Proof

Step Hyp Ref Expression
1 latmidm.b 
 |-  B = ( Base ` K )
2 latmidm.m 
 |-  ./\ = ( meet ` K )
3 eqid 
 |-  ( le ` K ) = ( le ` K )
4 simpl 
 |-  ( ( K e. Lat /\ X e. B ) -> K e. Lat )
5 1 2 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ X e. B ) -> ( X ./\ X ) e. B )
6 5 3anidm23 
 |-  ( ( K e. Lat /\ X e. B ) -> ( X ./\ X ) e. B )
7 simpr 
 |-  ( ( K e. Lat /\ X e. B ) -> X e. B )
8 1 3 2 latmle1 
 |-  ( ( K e. Lat /\ X e. B /\ X e. B ) -> ( X ./\ X ) ( le ` K ) X )
9 8 3anidm23 
 |-  ( ( K e. Lat /\ X e. B ) -> ( X ./\ X ) ( le ` K ) X )
10 1 3 latref 
 |-  ( ( K e. Lat /\ X e. B ) -> X ( le ` K ) X )
11 1 3 2 latlem12 
 |-  ( ( K e. Lat /\ ( X e. B /\ X e. B /\ X e. B ) ) -> ( ( X ( le ` K ) X /\ X ( le ` K ) X ) <-> X ( le ` K ) ( X ./\ X ) ) )
12 4 7 7 7 11 syl13anc 
 |-  ( ( K e. Lat /\ X e. B ) -> ( ( X ( le ` K ) X /\ X ( le ` K ) X ) <-> X ( le ` K ) ( X ./\ X ) ) )
13 10 10 12 mpbi2and 
 |-  ( ( K e. Lat /\ X e. B ) -> X ( le ` K ) ( X ./\ X ) )
14 1 3 4 6 7 9 13 latasymd 
 |-  ( ( K e. Lat /\ X e. B ) -> ( X ./\ X ) = X )