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Metamath Proof Explorer

Theorem atlltn0

Description: A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012)

Ref Expression
Hypotheses atlltne0.b 
|- B = ( Base ` K )
atlltne0.s 
|- .< = ( lt ` K )
atlltne0.z 
|- .0. = ( 0. ` K )
Assertion atlltn0 
|- ( ( K e. AtLat /\ X e. B ) -> ( .0. .< X <-> X =/= .0. ) )

Proof

Step Hyp Ref Expression
1 atlltne0.b 
 |-  B = ( Base ` K )
2 atlltne0.s 
 |-  .< = ( lt ` K )
3 atlltne0.z 
 |-  .0. = ( 0. ` K )
4 simpl 
 |-  ( ( K e. AtLat /\ X e. B ) -> K e. AtLat )
5 1 3 atl0cl 
 |-  ( K e. AtLat -> .0. e. B )
6 5 adantr 
 |-  ( ( K e. AtLat /\ X e. B ) -> .0. e. B )
7 simpr 
 |-  ( ( K e. AtLat /\ X e. B ) -> X e. B )
8 eqid 
 |-  ( le ` K ) = ( le ` K )
9 8 2 pltval 
 |-  ( ( K e. AtLat /\ .0. e. B /\ X e. B ) -> ( .0. .< X <-> ( .0. ( le ` K ) X /\ .0. =/= X ) ) )
10 4 6 7 9 syl3anc 
 |-  ( ( K e. AtLat /\ X e. B ) -> ( .0. .< X <-> ( .0. ( le ` K ) X /\ .0. =/= X ) ) )
11 necom 
 |-  ( X =/= .0. <-> .0. =/= X )
12 1 8 3 atl0le 
 |-  ( ( K e. AtLat /\ X e. B ) -> .0. ( le ` K ) X )
13 12 biantrurd 
 |-  ( ( K e. AtLat /\ X e. B ) -> ( .0. =/= X <-> ( .0. ( le ` K ) X /\ .0. =/= X ) ) )
14 11 13 bitr2id 
 |-  ( ( K e. AtLat /\ X e. B ) -> ( ( .0. ( le ` K ) X /\ .0. =/= X ) <-> X =/= .0. ) )
15 10 14 bitrd 
 |-  ( ( K e. AtLat /\ X e. B ) -> ( .0. .< X <-> X =/= .0. ) )