Metamath Proof Explorer

Theorem latnle

Description: Equivalent expressions for "not less than" in a lattice. ( chnle analog.) (Contributed by NM, 16-Nov-2011)

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

Proof

Step Hyp Ref Expression
1 latnle.b 
 |-  B = ( Base ` K )
2 latnle.l 
 |-  .<_ = ( le ` K )
3 latnle.s 
 |-  .< = ( lt ` K )
4 latnle.j 
 |-  .\/ = ( join ` K )
5 1 2 4 latlej1 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> X .<_ ( X .\/ Y ) )
6 5 biantrurd 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X =/= ( X .\/ Y ) <-> ( X .<_ ( X .\/ Y ) /\ X =/= ( X .\/ Y ) ) ) )
7 1 2 4 latleeqj1 
 |-  ( ( K e. Lat /\ Y e. B /\ X e. B ) -> ( Y .<_ X <-> ( Y .\/ X ) = X ) )
8 7 3com23 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( Y .<_ X <-> ( Y .\/ X ) = X ) )
9 eqcom 
 |-  ( ( Y .\/ X ) = X <-> X = ( Y .\/ X ) )
10 8 9 bitrdi 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( Y .<_ X <-> X = ( Y .\/ X ) ) )
11 1 4 latjcom 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) = ( Y .\/ X ) )
12 11 eqeq2d 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X = ( X .\/ Y ) <-> X = ( Y .\/ X ) ) )
13 10 12 bitr4d 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( Y .<_ X <-> X = ( X .\/ Y ) ) )
14 13 necon3bbid 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( -. Y .<_ X <-> X =/= ( X .\/ Y ) ) )
15 1 4 latjcl 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) e. B )
16 2 3 pltval 
 |-  ( ( K e. Lat /\ X e. B /\ ( X .\/ Y ) e. B ) -> ( X .< ( X .\/ Y ) <-> ( X .<_ ( X .\/ Y ) /\ X =/= ( X .\/ Y ) ) ) )
17 15 16 syld3an3 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .< ( X .\/ Y ) <-> ( X .<_ ( X .\/ Y ) /\ X =/= ( X .\/ Y ) ) ) )
18 6 14 17 3bitr4d 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( -. Y .<_ X <-> X .< ( X .\/ Y ) ) )