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Theorem atl0le 29419
Description: Orthoposet zero is less than or equal to any element. (ch0le 22791 analog.) (Contributed by NM, 12-Oct-2011.)
Hypotheses
Ref Expression
atl0le.b  |-  B  =  ( Base `  K
)
atl0le.l  |-  .<_  =  ( le `  K )
atl0le.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
atl0le  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  .0.  .<_  X )

Proof of Theorem atl0le
StepHypRef Expression
1 atl0le.b . . . 4  |-  B  =  ( Base `  K
)
2 atl0le.z . . . 4  |-  .0.  =  ( 0. `  K )
31, 2atl0cl 29418 . . 3  |-  ( K  e.  AtLat  ->  .0.  e.  B )
43adantr 452 . 2  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  .0.  e.  B )
5 atlpos 29416 . . . 4  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
6 atl0le.l . . . . 5  |-  .<_  =  ( le `  K )
71, 6, 2p0le 14399 . . . 4  |-  ( ( K  e.  Poset  /\  .0.  e.  B  /\  X  e.  B )  ->  .0.  .<_  X )
85, 7syl3an1 1217 . . 3  |-  ( ( K  e.  AtLat  /\  .0.  e.  B  /\  X  e.  B )  ->  .0.  .<_  X )
983com23 1159 . 2  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  .0.  e.  B )  ->  .0.  .<_  X )
104, 9mpd3an3 1280 1  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  .0.  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4153   ` cfv 5394   Basecbs 13396   lecple 13463   Posetcpo 14324   0.cp0 14393   AtLatcal 29379
This theorem is referenced by:  atlle0  29420  atlltn0  29421  atcvreq0  29429  trlval4  30302  dian0  31154  dia0  31167  dihmeetlem4preN  31421
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-glb 14359  df-p0 14395  df-lat 14402  df-atl 29413
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