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Theorem atlex 29431
Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 23711 analog.) (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
atlex.b  |-  B  =  ( Base `  K
)
atlex.l  |-  .<_  =  ( le `  K )
atlex.z  |-  .0.  =  ( 0. `  K )
atlex.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atlex  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  E. y  e.  A  y  .<_  X )
Distinct variable groups:    y, A    y, K    y, X
Allowed substitution hints:    B( y)    .<_ ( y)    .0. ( y)

Proof of Theorem atlex
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 atlex.b . . . . 5  |-  B  =  ( Base `  K
)
2 atlex.l . . . . 5  |-  .<_  =  ( le `  K )
3 atlex.z . . . . 5  |-  .0.  =  ( 0. `  K )
4 atlex.a . . . . 5  |-  A  =  ( Atoms `  K )
51, 2, 3, 4isatl 29414 . . . 4  |-  ( K  e.  AtLat 
<->  ( K  e.  Lat  /\  .0.  e.  B  /\  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y 
.<_  x ) ) )
65simp3bi 974 . . 3  |-  ( K  e.  AtLat  ->  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y  .<_  x ) )
7 neeq1 2558 . . . . 5  |-  ( x  =  X  ->  (
x  =/=  .0.  <->  X  =/=  .0.  ) )
8 breq2 4157 . . . . . 6  |-  ( x  =  X  ->  (
y  .<_  x  <->  y  .<_  X ) )
98rexbidv 2670 . . . . 5  |-  ( x  =  X  ->  ( E. y  e.  A  y  .<_  x  <->  E. y  e.  A  y  .<_  X ) )
107, 9imbi12d 312 . . . 4  |-  ( x  =  X  ->  (
( x  =/=  .0.  ->  E. y  e.  A  y  .<_  x )  <->  ( X  =/=  .0.  ->  E. y  e.  A  y  .<_  X ) ) )
1110rspccv 2992 . . 3  |-  ( A. x  e.  B  (
x  =/=  .0.  ->  E. y  e.  A  y 
.<_  x )  ->  ( X  e.  B  ->  ( X  =/=  .0.  ->  E. y  e.  A  y 
.<_  X ) ) )
126, 11syl 16 . 2  |-  ( K  e.  AtLat  ->  ( X  e.  B  ->  ( X  =/=  .0.  ->  E. y  e.  A  y  .<_  X ) ) )
13123imp 1147 1  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  E. y  e.  A  y  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650   class class class wbr 4153   ` cfv 5394   Basecbs 13396   lecple 13463   0.cp0 14393   Latclat 14401   Atomscatm 29378   AtLatcal 29379
This theorem is referenced by:  atnle  29432  atlatmstc  29434  cvratlem  29535  cvrat4  29557  2llnmat  29638  2lnat  29898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-atl 29413
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