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Theorem ple1 14428
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 957 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  K  e.  Poset )
2 ssid 3327 . . . 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 2404 . . . . . . 7  |-  ( lub `  K )  =  ( lub `  K )
6 ple1.u . . . . . . 7  |-  .1.  =  ( 1. `  K )
74, 5, 6p1val 14426 . . . . . 6  |-  ( K  e.  Poset  ->  .1.  =  ( ( lub `  K
) `  B )
)
87adantr 452 . . . . 5  |-  ( ( K  e.  Poset  /\  .1.  e.  B )  ->  .1.  =  ( ( lub `  K ) `  B
) )
9 simpr 448 . . . . 5  |-  ( ( K  e.  Poset  /\  .1.  e.  B )  ->  .1.  e.  B )
108, 9eqeltrrd 2479 . . . 4  |-  ( ( K  e.  Poset  /\  .1.  e.  B )  ->  (
( lub `  K
) `  B )  e.  B )
11103adant2 976 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  (
( lub `  K
) `  B )  e.  B )
12 simp2 958 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  X  e.  B )
13 ple1.l . . . 4  |-  .<_  =  ( le `  K )
144, 13, 5luble 14393 . . 3  |-  ( ( ( K  e.  Poset  /\  B  C_  B )  /\  ( ( ( lub `  K ) `  B
)  e.  B  /\  X  e.  B )
)  ->  X  .<_  ( ( lub `  K
) `  B )
)
151, 3, 11, 12, 14syl22anc 1185 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  X  .<_  ( ( lub `  K
) `  B )
)
1673ad2ant1 978 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  .1.  =  ( ( lub `  K ) `  B
) )
1715, 16breqtrrd 4198 1  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  X  .<_  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3280   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   Posetcpo 14352   lubclub 14354   1.cp1 14422
This theorem is referenced by:  ople1  29674  lhp2lt  30483
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-undef 6502  df-riota 6508  df-lub 14386  df-p1 14424
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