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Theorem ople0 29682
Description: An element less than or equal to zero equals zero. (chle0 22906 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 29681 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )
54biantrud 494 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  ( X  .<_  .0.  /\  .0.  .<_  X ) ) )
6 opposet 29677 . . . 4  |-  ( K  e.  OP  ->  K  e.  Poset )
76adantr 452 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  K  e.  Poset )
8 simpr 448 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  e.  B )
91, 3op0cl 29679 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  B )
109adantr 452 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  e.  B )
111, 2posasymb 14372 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .0.  e.  B )  ->  (
( X  .<_  .0.  /\  .0.  .<_  X )  <->  X  =  .0.  ) )
127, 8, 10, 11syl3anc 1184 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( X  .<_  .0. 
/\  .0.  .<_  X )  <-> 
X  =  .0.  )
)
135, 12bitrd 245 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  X  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4180   ` cfv 5421   Basecbs 13432   lecple 13499   Posetcpo 14360   0.cp0 14429   OPcops 29667
This theorem is referenced by:  lub0N  29684  opoc1  29697  atlatmstc  29814  cvrat4  29937  lhpocnle  30510  cdleme22b  30835  tendoid  31267  tendoex  31469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-undef 6510  df-riota 6516  df-poset 14366  df-glb 14395  df-p0 14431  df-oposet 29671
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