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Metamath Proof Explorer

Theorem latlej1

Description: A join's first argument is less than or equal to the join. ( chub1 analog.) (Contributed by NM, 17-Sep-2011)

Ref Expression
Hypotheses latlej.b 
|- B = ( Base ` K )
latlej.l 
|- .<_ = ( le ` K )
latlej.j 
|- .\/ = ( join ` K )
Assertion latlej1 
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> X .<_ ( X .\/ Y ) )

Proof

Step Hyp Ref Expression
1 latlej.b 
 |-  B = ( Base ` K )
2 latlej.l 
 |-  .<_ = ( le ` K )
3 latlej.j 
 |-  .\/ = ( join ` 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 
 |-  ( meet ` K ) = ( meet ` K )
8 1 3 7 4 5 6 latcl2 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( <. X , Y >. e. dom .\/ /\ <. X , Y >. e. dom ( meet ` K ) ) )
9 8 simpld 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> <. X , Y >. e. dom .\/ )
10 1 2 3 4 5 6 9 lejoin1 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> X .<_ ( X .\/ Y ) )