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Theorem ople0 32462
Description: An element less than or equal to zero equals zero. (chle0 26931 analog.) (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
op0le.b  |-  B  =  ( Base `  K
)
op0le.l  |-  .<_  =  ( le `  K )
op0le.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
ople0  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  X  =  .0.  ) )

Proof of Theorem ople0
StepHypRef Expression
1 op0le.b . . . 4  |-  B  =  ( Base `  K
)
2 op0le.l . . . 4  |-  .<_  =  ( le `  K )
3 op0le.z . . . 4  |-  .0.  =  ( 0. `  K )
41, 2, 3op0le 32461 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )
54biantrud 509 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  ( X  .<_  .0.  /\  .0.  .<_  X ) ) )
6 opposet 32456 . . . 4  |-  ( K  e.  OP  ->  K  e.  Poset )
76adantr 466 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  K  e.  Poset )
8 simpr 462 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  e.  B )
91, 3op0cl 32459 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
109adantr 466 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  e.  B )
111, 2posasymb 16149 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .0.  e.  B )  ->  (
( X  .<_  .0.  /\  .0.  .<_  X )  <->  X  =  .0.  ) )
127, 8, 10, 11syl3anc 1264 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( X  .<_  .0. 
/\  .0.  .<_  X )  <-> 
X  =  .0.  )
)
135, 12bitrd 256 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  X  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   class class class wbr 4426   ` cfv 5601   Basecbs 15084   lecple 15159   Posetcpo 16136   0.cp0 16234   OPcops 32447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-preset 16124  df-poset 16142  df-glb 16172  df-p0 16236  df-oposet 32451
This theorem is referenced by:  lub0N  32464  opoc1  32477  atlatmstc  32594  cvrat4  32717  lhpocnle  33290  cdleme22b  33617  tendoid  34049  tendoex  34251
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