Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atlex Unicode version

Theorem atlex 30128
Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 22956 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 30111 . . . 4  |-  ( K  e.  AtLat 
<->  ( K  e.  Lat  /\  .0.  e.  B  /\  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y 
.<_  x ) ) )
65simp3bi 972 . . 3  |-  ( K  e.  AtLat  ->  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y  .<_  x ) )
7 neeq1 2467 . . . . 5  |-  ( x  =  X  ->  (
x  =/=  .0.  <->  X  =/=  .0.  ) )
8 breq2 4043 . . . . . 6  |-  ( x  =  X  ->  (
y  .<_  x  <->  y  .<_  X ) )
98rexbidv 2577 . . . . 5  |-  ( x  =  X  ->  ( E. y  e.  A  y  .<_  x  <->  E. y  e.  A  y  .<_  X ) )
107, 9imbi12d 311 . . . 4  |-  ( x  =  X  ->  (
( x  =/=  .0.  ->  E. y  e.  A  y  .<_  x )  <->  ( X  =/=  .0.  ->  E. y  e.  A  y  .<_  X ) ) )
1110rspccv 2894 . . 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 15 . 2  |-  ( K  e.  AtLat  ->  ( X  e.  B  ->  ( X  =/=  .0.  ->  E. y  e.  A  y  .<_  X ) ) )
13123imp 1145 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 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   0.cp0 14159   Latclat 14167   Atomscatm 30075   AtLatcal 30076
This theorem is referenced by:  atnle  30129  atlatmstc  30131  cvratlem  30232  cvrat4  30254  2llnmat  30335  2lnat  30595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-atl 30110
  Copyright terms: Public domain W3C validator