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Metamath Proof Explorer

Theorem latjidm

Description: Lattice join is idempotent. Analogue of unidm . (Contributed by NM, 8-Oct-2011)

Ref Expression
Hypotheses latjidm.b 
|- B = ( Base ` K )
latjidm.j 
|- .\/ = ( join ` K )
Assertion latjidm 
|- ( ( K e. Lat /\ X e. B ) -> ( X .\/ X ) = X )

Proof

Step Hyp Ref Expression
1 latjidm.b 
 |-  B = ( Base ` K )
2 latjidm.j 
 |-  .\/ = ( join ` K )
3 eqid 
 |-  ( le ` K ) = ( le ` K )
4 simpl 
 |-  ( ( K e. Lat /\ X e. B ) -> K e. Lat )
5 1 2 latjcl 
 |-  ( ( 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 latref 
 |-  ( ( K e. Lat /\ X e. B ) -> X ( le ` K ) X )
9 1 3 2 latjle12 
 |-  ( ( K e. Lat /\ ( X e. B /\ X e. B /\ X e. B ) ) -> ( ( X ( le ` K ) X /\ X ( le ` K ) X ) <-> ( X .\/ X ) ( le ` K ) X ) )
10 4 7 7 7 9 syl13anc 
 |-  ( ( K e. Lat /\ X e. B ) -> ( ( X ( le ` K ) X /\ X ( le ` K ) X ) <-> ( X .\/ X ) ( le ` K ) X ) )
11 8 8 10 mpbi2and 
 |-  ( ( K e. Lat /\ X e. B ) -> ( X .\/ X ) ( le ` K ) X )
12 1 3 2 latlej1 
 |-  ( ( K e. Lat /\ X e. B /\ X e. B ) -> X ( le ` K ) ( X .\/ X ) )
13 12 3anidm23 
 |-  ( ( K e. Lat /\ X e. B ) -> X ( le ` K ) ( X .\/ X ) )
14 1 3 4 6 7 11 13 latasymd 
 |-  ( ( K e. Lat /\ X e. B ) -> ( X .\/ X ) = X )