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Theorem ople0 30059
Description: An element less than or equal to zero equals zero. (chle0 22950 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 30058 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )
54biantrud 495 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  ( X  .<_  .0.  /\  .0.  .<_  X ) ) )
6 opposet 30054 . . . 4  |-  ( K  e.  OP  ->  K  e.  Poset )
76adantr 453 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  K  e.  Poset )
8 simpr 449 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  e.  B )
91, 3op0cl 30056 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
109adantr 453 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  e.  B )
111, 2posasymb 14414 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .0.  e.  B )  ->  (
( X  .<_  .0.  /\  .0.  .<_  X )  <->  X  =  .0.  ) )
127, 8, 10, 11syl3anc 1185 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( X  .<_  .0. 
/\  .0.  .<_  X )  <-> 
X  =  .0.  )
)
135, 12bitrd 246 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  X  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457   Basecbs 13474   lecple 13541   Posetcpo 14402   0.cp0 14471   OPcops 30044
This theorem is referenced by:  lub0N  30061  opoc1  30074  atlatmstc  30191  cvrat4  30314  lhpocnle  30887  cdleme22b  31212  tendoid  31644  tendoex  31846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-undef 6546  df-riota 6552  df-poset 14408  df-glb 14437  df-p0 14473  df-oposet 30048
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