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Theorem op1le 32933
Description: If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 24868 analog.) (Contributed by NM, 5-Dec-2011.)
Hypotheses
Ref Expression
ople1.b  |-  B  =  ( Base `  K
)
ople1.l  |-  .<_  =  ( le `  K )
ople1.u  |-  .1.  =  ( 1. `  K )
Assertion
Ref Expression
op1le  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .1.  .<_  X  <->  X  =  .1.  ) )

Proof of Theorem op1le
StepHypRef Expression
1 ople1.b . . . 4  |-  B  =  ( Base `  K
)
2 ople1.l . . . 4  |-  .<_  =  ( le `  K )
3 ople1.u . . . 4  |-  .1.  =  ( 1. `  K )
41, 2, 3ople1 32932 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  .<_  .1.  )
54biantrurd 508 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .1.  .<_  X  <->  ( X  .<_  .1.  /\  .1.  .<_  X ) ) )
6 opposet 32922 . . . 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, 3op1cl 32926 . . . 4  |-  ( K  e.  OP  ->  .1.  e.  B )
109adantr 465 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .1.  e.  B )
111, 2posasymb 15143 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  (
( X  .<_  .1.  /\  .1.  .<_  X )  <->  X  =  .1.  ) )
127, 8, 10, 11syl3anc 1218 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( X  .<_  .1. 
/\  .1.  .<_  X )  <-> 
X  =  .1.  )
)
135, 12bitrd 253 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .1.  .<_  X  <->  X  =  .1.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4313   ` cfv 5439   Basecbs 14195   lecple 14266   Posetcpo 15131   1.cp1 15229   OPcops 32913
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552
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 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-poset 15137  df-lub 15165  df-p1 15231  df-oposet 32917
This theorem is referenced by:  glb0N  32934  lhpj1  33762
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