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Metamath Proof Explorer

Theorem atlle0

Description: An element less than or equal to zero equals zero. ( chle0 analog.) (Contributed by NM, 21-Oct-2011)

Ref Expression
Hypotheses atl0le.b 
|- B = ( Base ` K )
atl0le.l 
|- .<_ = ( le ` K )
atl0le.z 
|- .0. = ( 0. ` K )
Assertion atlle0 
|- ( ( K e. AtLat /\ X e. B ) -> ( X .<_ .0. <-> X = .0. ) )

Proof

Step Hyp Ref Expression
1 atl0le.b 
 |-  B = ( Base ` K )
2 atl0le.l 
 |-  .<_ = ( le ` K )
3 atl0le.z 
 |-  .0. = ( 0. ` K )
4 1 2 3 atl0le 
 |-  ( ( K e. AtLat /\ X e. B ) -> .0. .<_ X )
5 4 biantrud 
 |-  ( ( K e. AtLat /\ X e. B ) -> ( X .<_ .0. <-> ( X .<_ .0. /\ .0. .<_ X ) ) )
6 atlpos 
 |-  ( K e. AtLat -> K e. Poset )
7 6 adantr 
 |-  ( ( K e. AtLat /\ X e. B ) -> K e. Poset )
8 simpr 
 |-  ( ( K e. AtLat /\ X e. B ) -> X e. B )
9 1 3 atl0cl 
 |-  ( K e. AtLat -> .0. e. B )
10 9 adantr 
 |-  ( ( K e. AtLat /\ X e. B ) -> .0. e. B )
11 1 2 posasymb 
 |-  ( ( K e. Poset /\ X e. B /\ .0. e. B ) -> ( ( X .<_ .0. /\ .0. .<_ X ) <-> X = .0. ) )
12 7 8 10 11 syl3anc 
 |-  ( ( K e. AtLat /\ X e. B ) -> ( ( X .<_ .0. /\ .0. .<_ X ) <-> X = .0. ) )
13 5 12 bitrd 
 |-  ( ( K e. AtLat /\ X e. B ) -> ( X .<_ .0. <-> X = .0. ) )