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Theorem atl0le 30029
Description: Orthoposet zero is less than or equal to any element. (ch0le 22935 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 30028 . . 3  |-  ( K  e.  AtLat  ->  .0.  e.  B )
43adantr 452 . 2  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  .0.  e.  B )
5 atlpos 30026 . . . 4  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
6 atl0le.l . . . . 5  |-  .<_  =  ( le `  K )
71, 6, 2p0le 14464 . . . 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 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   Posetcpo 14389   0.cp0 14458   AtLatcal 29989
This theorem is referenced by:  atlle0  30030  atlltn0  30031  atcvreq0  30039  trlval4  30912  dian0  31764  dia0  31777  dihmeetlem4preN  32031
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-undef 6535  df-riota 6541  df-glb 14424  df-p0 14460  df-lat 14467  df-atl 30023
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