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Metamath Proof Explorer

Theorem latmlem2

Description: Add meet to both sides of a lattice ordering. ( sslin analog.) (Contributed by NM, 10-Nov-2011)

Ref Expression
Hypotheses latmle.b 
|- B = ( Base ` K )
latmle.l 
|- .<_ = ( le ` K )
latmle.m 
|- ./\ = ( meet ` K )
Assertion latmlem2 
|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .<_ Y -> ( Z ./\ X ) .<_ ( Z ./\ Y ) ) )

Proof

Step Hyp Ref Expression
1 latmle.b 
 |-  B = ( Base ` K )
2 latmle.l 
 |-  .<_ = ( le ` K )
3 latmle.m 
 |-  ./\ = ( meet ` K )
4 1 2 3 latmlem1 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .<_ Y -> ( X ./\ Z ) .<_ ( Y ./\ Z ) ) )
5 1 3 latmcom 
 |-  ( ( K e. Lat /\ X e. B /\ Z e. B ) -> ( X ./\ Z ) = ( Z ./\ X ) )
6 5 3adant3r2 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X ./\ Z ) = ( Z ./\ X ) )
7 1 3 latmcom 
 |-  ( ( K e. Lat /\ Y e. B /\ Z e. B ) -> ( Y ./\ Z ) = ( Z ./\ Y ) )
8 7 3adant3r1 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y ./\ Z ) = ( Z ./\ Y ) )
9 6 8 breq12d 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ./\ Z ) .<_ ( Y ./\ Z ) <-> ( Z ./\ X ) .<_ ( Z ./\ Y ) ) )
10 4 9 sylibd 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .<_ Y -> ( Z ./\ X ) .<_ ( Z ./\ Y ) ) )