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Theorem op1le 29200
Description: If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 22077 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 29199 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  .<_  .1.  )
54biantrurd 494 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .1.  .<_  X  <->  ( X  .<_  .1.  /\  .1.  .<_  X ) ) )
6 opposet 29190 . . . 4  |-  ( K  e.  OP  ->  K  e.  Poset )
76adantr 451 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  K  e.  Poset )
8 simpr 447 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  e.  B )
91, 3op1cl 29193 . . . 4  |-  ( K  e.  OP  ->  .1.  e.  B )
109adantr 451 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .1.  e.  B )
111, 2posasymb 14135 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  (
( X  .<_  .1.  /\  .1.  .<_  X )  <->  X  =  .1.  ) )
127, 8, 10, 11syl3anc 1182 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( X  .<_  .1. 
/\  .1.  .<_  X )  <-> 
X  =  .1.  )
)
135, 12bitrd 244 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .1.  .<_  X  <->  X  =  .1.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   class class class wbr 4060   ` cfv 5292   Basecbs 13195   lecple 13262   Posetcpo 14123   1.cp1 14193   OPcops 29180
This theorem is referenced by:  glb0N  29201  lhpj1  30029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-undef 6340  df-riota 6346  df-poset 14129  df-lub 14157  df-p1 14195  df-oposet 29184
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