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Metamath Proof Explorer

Theorem atlex

Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. ( hatomic analog.) (Contributed by NM, 21-Oct-2011)

Ref Expression
Hypotheses atlex.b 
|- B = ( Base ` K )
atlex.l 
|- .<_ = ( le ` K )
atlex.z 
|- .0. = ( 0. ` K )
atlex.a 
|- A = ( Atoms ` K )
Assertion atlex 
|- ( ( K e. AtLat /\ X e. B /\ X =/= .0. ) -> E. y e. A y .<_ X )

Proof

Step Hyp Ref Expression
1 atlex.b 
 |-  B = ( Base ` K )
2 atlex.l 
 |-  .<_ = ( le ` K )
3 atlex.z 
 |-  .0. = ( 0. ` K )
4 atlex.a 
 |-  A = ( Atoms ` K )
5 eqid 
 |-  ( glb ` K ) = ( glb ` K )
6 1 5 2 3 4 isatl 
 |-  ( K e. AtLat <-> ( K e. Lat /\ B e. dom ( glb ` K ) /\ A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) ) )
7 6 simp3bi 
 |-  ( K e. AtLat -> A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) )
8 neeq1 
 |-  ( x = X -> ( x =/= .0. <-> X =/= .0. ) )
9 breq2 
 |-  ( x = X -> ( y .<_ x <-> y .<_ X ) )
10 9 rexbidv 
 |-  ( x = X -> ( E. y e. A y .<_ x <-> E. y e. A y .<_ X ) )
11 8 10 imbi12d 
 |-  ( x = X -> ( ( x =/= .0. -> E. y e. A y .<_ x ) <-> ( X =/= .0. -> E. y e. A y .<_ X ) ) )
12 11 rspccv 
 |-  ( A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) -> ( X e. B -> ( X =/= .0. -> E. y e. A y .<_ X ) ) )
13 7 12 syl 
 |-  ( K e. AtLat -> ( X e. B -> ( X =/= .0. -> E. y e. A y .<_ X ) ) )
14 13 3imp 
 |-  ( ( K e. AtLat /\ X e. B /\ X =/= .0. ) -> E. y e. A y .<_ X )