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Theorem op1le 35330
Description: If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 26478 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 35329 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  .<_  .1.  )
54biantrurd 506 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .1.  .<_  X  <->  ( X  .<_  .1.  /\  .1.  .<_  X ) ) )
6 opposet 35319 . . . 4  |-  ( K  e.  OP  ->  K  e.  Poset )
76adantr 463 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  K  e.  Poset )
8 simpr 459 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  e.  B )
91, 3op1cl 35323 . . . 4  |-  ( K  e.  OP  ->  .1.  e.  B )
109adantr 463 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .1.  e.  B )
111, 2posasymb 15699 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  (
( X  .<_  .1.  /\  .1.  .<_  X )  <->  X  =  .1.  ) )
127, 8, 10, 11syl3anc 1226 . 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 367    = wceq 1399    e. wcel 1826   class class class wbr 4367   ` cfv 5496   Basecbs 14634   lecple 14709   Posetcpo 15686   1.cp1 15785   OPcops 35310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-preset 15674  df-poset 15692  df-lub 15721  df-p1 15787  df-oposet 35314
This theorem is referenced by:  glb0N  35331  lhpj1  36159
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