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Theorem atl0le 30116
Description: Orthoposet zero is less than or equal to any element. (ch0le 22036 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 30115 . . 3  |-  ( K  e.  AtLat  ->  .0.  e.  B )
43adantr 451 . 2  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  .0.  e.  B )
5 atlpos 30113 . . . 4  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
6 atl0le.l . . . . 5  |-  .<_  =  ( le `  K )
71, 6, 2p0le 14165 . . . 4  |-  ( ( K  e.  Poset  /\  .0.  e.  B  /\  X  e.  B )  ->  .0.  .<_  X )
85, 7syl3an1 1215 . . 3  |-  ( ( K  e.  AtLat  /\  .0.  e.  B  /\  X  e.  B )  ->  .0.  .<_  X )
983com23 1157 . 2  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  .0.  e.  B )  ->  .0.  .<_  X )
104, 9mpd3an3 1278 1  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  .0.  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Posetcpo 14090   0.cp0 14159   AtLatcal 30076
This theorem is referenced by:  atlle0  30117  atlltn0  30118  atcvreq0  30126  trlval4  30999  dian0  31851  dia0  31864  dihmeetlem4preN  32118
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-undef 6314  df-riota 6320  df-glb 14125  df-p0 14161  df-lat 14168  df-atl 30110
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