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

Theorem atlex 30041
Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 23855 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 30024 . . . 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 2606 . . . . 5  |-  ( x  =  X  ->  (
x  =/=  .0.  <->  X  =/=  .0.  ) )
8 breq2 4208 . . . . . 6  |-  ( x  =  X  ->  (
y  .<_  x  <->  y  .<_  X ) )
98rexbidv 2718 . . . . 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 3041 . . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   0.cp0 14458   Latclat 14466   Atomscatm 29988   AtLatcal 29989
This theorem is referenced by:  atnle  30042  atlatmstc  30044  cvratlem  30145  cvrat4  30167  2llnmat  30248  2lnat  30508
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-atl 30023
  Copyright terms: Public domain W3C validator