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Theorem ople0 32554
Description: An element less than or equal to zero equals zero. (chle0 24781 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 32553 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )
54biantrud 504 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  ( X  .<_  .0.  /\  .0.  .<_  X ) ) )
6 opposet 32548 . . . 4  |-  ( K  e.  OP  ->  K  e.  Poset )
76adantr 462 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  K  e.  Poset )
8 simpr 458 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  e.  B )
91, 3op0cl 32551 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
109adantr 462 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  e.  B )
111, 2posasymb 15118 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .0.  e.  B )  ->  (
( X  .<_  .0.  /\  .0.  .<_  X )  <->  X  =  .0.  ) )
127, 8, 10, 11syl3anc 1213 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( X  .<_  .0. 
/\  .0.  .<_  X )  <-> 
X  =  .0.  )
)
135, 12bitrd 253 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  X  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   class class class wbr 4289   ` cfv 5415   Basecbs 14170   lecple 14241   Posetcpo 15106   0.cp0 15203   OPcops 32539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-poset 15112  df-glb 15141  df-p0 15205  df-oposet 32543
This theorem is referenced by:  lub0N  32556  opoc1  32569  atlatmstc  32686  cvrat4  32809  lhpocnle  33382  cdleme22b  33707  tendoid  34139  tendoex  34341
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