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Metamath Proof Explorer

Theorem latnlemlt

Description: Negation of "less than or equal to" expressed in terms of meet and less-than. ( nssinpss analog.) (Contributed by NM, 5-Feb-2012)

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

Proof

Step Hyp Ref Expression
1 latnlemlt.b 
 |-  B = ( Base ` K )
2 latnlemlt.l 
 |-  .<_ = ( le ` K )
3 latnlemlt.s 
 |-  .< = ( lt ` K )
4 latnlemlt.m 
 |-  ./\ = ( meet ` K )
5 1 2 4 latmle1 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) .<_ X )
6 5 biantrurd 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X ./\ Y ) =/= X <-> ( ( X ./\ Y ) .<_ X /\ ( X ./\ Y ) =/= X ) ) )
7 1 2 4 latleeqm1 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .<_ Y <-> ( X ./\ Y ) = X ) )
8 7 necon3bbid 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( -. X .<_ Y <-> ( X ./\ Y ) =/= X ) )
9 simp1 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> K e. Lat )
10 1 4 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) e. B )
11 simp2 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> X e. B )
12 2 3 pltval 
 |-  ( ( K e. Lat /\ ( X ./\ Y ) e. B /\ X e. B ) -> ( ( X ./\ Y ) .< X <-> ( ( X ./\ Y ) .<_ X /\ ( X ./\ Y ) =/= X ) ) )
13 9 10 11 12 syl3anc 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X ./\ Y ) .< X <-> ( ( X ./\ Y ) .<_ X /\ ( X ./\ Y ) =/= X ) ) )
14 6 8 13 3bitr4d 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( -. X .<_ Y <-> ( X ./\ Y ) .< X ) )