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Theorem ple1 14478
Description: Any element is less than or equal to poset one (if defined). (Contributed by NM, 22-Oct-2011.)
Hypotheses
Ref Expression
ple1.b  |-  B  =  ( Base `  K
)
ple1.l  |-  .<_  =  ( le `  K )
ple1.u  |-  .1.  =  ( 1. `  K )
Assertion
Ref Expression
ple1  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  X  .<_  .1.  )

Proof of Theorem ple1
StepHypRef Expression
1 simp1 958 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  K  e.  Poset )
2 ssid 3369 . . . 4  |-  B  C_  B
32a1i 11 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  B  C_  B )
4 ple1.b . . . . . . 7  |-  B  =  ( Base `  K
)
5 eqid 2438 . . . . . . 7  |-  ( lub `  K )  =  ( lub `  K )
6 ple1.u . . . . . . 7  |-  .1.  =  ( 1. `  K )
74, 5, 6p1val 14476 . . . . . 6  |-  ( K  e.  Poset  ->  .1.  =  ( ( lub `  K
) `  B )
)
87adantr 453 . . . . 5  |-  ( ( K  e.  Poset  /\  .1.  e.  B )  ->  .1.  =  ( ( lub `  K ) `  B
) )
9 simpr 449 . . . . 5  |-  ( ( K  e.  Poset  /\  .1.  e.  B )  ->  .1.  e.  B )
108, 9eqeltrrd 2513 . . . 4  |-  ( ( K  e.  Poset  /\  .1.  e.  B )  ->  (
( lub `  K
) `  B )  e.  B )
11103adant2 977 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  (
( lub `  K
) `  B )  e.  B )
12 simp2 959 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  X  e.  B )
13 ple1.l . . . 4  |-  .<_  =  ( le `  K )
144, 13, 5luble 14443 . . 3  |-  ( ( ( K  e.  Poset  /\  B  C_  B )  /\  ( ( ( lub `  K ) `  B
)  e.  B  /\  X  e.  B )
)  ->  X  .<_  ( ( lub `  K
) `  B )
)
151, 3, 11, 12, 14syl22anc 1186 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  X  .<_  ( ( lub `  K
) `  B )
)
1673ad2ant1 979 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  .1.  =  ( ( lub `  K ) `  B
) )
1715, 16breqtrrd 4241 1  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  X  .<_  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322   class class class wbr 4215   ` cfv 5457   Basecbs 13474   lecple 13541   Posetcpo 14402   lubclub 14404   1.cp1 14472
This theorem is referenced by:  ople1  30063  lhp2lt  30872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-undef 6546  df-riota 6552  df-lub 14436  df-p1 14474
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