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Metamath Proof Explorer

Theorem posmidm

Description: Poset meet is idempotent. latmidm could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024)

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

Proof

Step Hyp Ref Expression
1 posjidm.b 
 |-  B = ( Base ` K )
2 posmidm.m 
 |-  ./\ = ( meet ` K )
3 eqid 
 |-  ( glb ` K ) = ( glb ` K )
4 simpl 
 |-  ( ( K e. Poset /\ X e. B ) -> K e. Poset )
5 simpr 
 |-  ( ( K e. Poset /\ X e. B ) -> X e. B )
6 3 2 4 5 5 meetval 
 |-  ( ( K e. Poset /\ X e. B ) -> ( X ./\ X ) = ( ( glb ` K ) ` { X , X } ) )
7 eqid 
 |-  ( le ` K ) = ( le ` K )
8 1 7 posref 
 |-  ( ( K e. Poset /\ X e. B ) -> X ( le ` K ) X )
9 eqidd 
 |-  ( ( K e. Poset /\ X e. B ) -> { X , X } = { X , X } )
10 4 1 5 5 7 8 9 3 glbpr 
 |-  ( ( K e. Poset /\ X e. B ) -> ( ( glb ` K ) ` { X , X } ) = X )
11 6 10 eqtrd 
 |-  ( ( K e. Poset /\ X e. B ) -> ( X ./\ X ) = X )