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Theorem ople0 32832
Description: An element less than or equal to zero equals zero. (chle0 24846 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 32831 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )
54biantrud 507 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  ( X  .<_  .0.  /\  .0.  .<_  X ) ) )
6 opposet 32826 . . . 4  |-  ( K  e.  OP  ->  K  e.  Poset )
76adantr 465 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  K  e.  Poset )
8 simpr 461 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  e.  B )
91, 3op0cl 32829 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
109adantr 465 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  e.  B )
111, 2posasymb 15122 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .0.  e.  B )  ->  (
( X  .<_  .0.  /\  .0.  .<_  X )  <->  X  =  .0.  ) )
127, 8, 10, 11syl3anc 1218 . 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 1369    e. wcel 1756   class class class wbr 4292   ` cfv 5418   Basecbs 14174   lecple 14245   Posetcpo 15110   0.cp0 15207   OPcops 32817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-poset 15116  df-glb 15145  df-p0 15209  df-oposet 32821
This theorem is referenced by:  lub0N  32834  opoc1  32847  atlatmstc  32964  cvrat4  33087  lhpocnle  33660  cdleme22b  33985  tendoid  34417  tendoex  34619
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