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Theorem atlltn0 29421
Description: A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012.)
Hypotheses
Ref Expression
atlltne0.b  |-  B  =  ( Base `  K
)
atlltne0.s  |-  .<  =  ( lt `  K )
atlltne0.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
atlltn0  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  (  .0.  .<  X  <->  X  =/=  .0.  ) )

Proof of Theorem atlltn0
StepHypRef Expression
1 simpl 444 . . 3  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  K  e.  AtLat )
2 atlltne0.b . . . . 5  |-  B  =  ( Base `  K
)
3 atlltne0.z . . . . 5  |-  .0.  =  ( 0. `  K )
42, 3atl0cl 29418 . . . 4  |-  ( K  e.  AtLat  ->  .0.  e.  B )
54adantr 452 . . 3  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  .0.  e.  B )
6 simpr 448 . . 3  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  X  e.  B )
7 eqid 2387 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
8 atlltne0.s . . . 4  |-  .<  =  ( lt `  K )
97, 8pltval 14344 . . 3  |-  ( ( K  e.  AtLat  /\  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  .<  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
101, 5, 6, 9syl3anc 1184 . 2  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  (  .0.  .<  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
11 necom 2631 . . 3  |-  ( X  =/=  .0.  <->  .0.  =/=  X )
122, 7, 3atl0le 29419 . . . 4  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  .0.  ( le `  K ) X )
1312biantrurd 495 . . 3  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  (  .0.  =/=  X  <->  (  .0.  ( le `  K ) X  /\  .0.  =/=  X ) ) )
1411, 13syl5rbb 250 . 2  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  (
(  .0.  ( le
`  K ) X  /\  .0.  =/=  X
)  <->  X  =/=  .0.  ) )
1510, 14bitrd 245 1  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  (  .0.  .<  X  <->  X  =/=  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394   Basecbs 13396   lecple 13463   ltcplt 14325   0.cp0 14393   AtLatcal 29379
This theorem is referenced by:  isat3  29422
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-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-undef 6479  df-riota 6485  df-plt 14342  df-glb 14359  df-p0 14395  df-lat 14402  df-atl 29413
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