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Theorem ple1 14150
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 955 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  K  e.  Poset )
2 ssid 3197 . . . 4  |-  B  C_  B
32a1i 10 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  B  C_  B )
4 ple1.b . . . . . . 7  |-  B  =  ( Base `  K
)
5 eqid 2283 . . . . . . 7  |-  ( lub `  K )  =  ( lub `  K )
6 ple1.u . . . . . . 7  |-  .1.  =  ( 1. `  K )
74, 5, 6p1val 14148 . . . . . 6  |-  ( K  e.  Poset  ->  .1.  =  ( ( lub `  K
) `  B )
)
87adantr 451 . . . . 5  |-  ( ( K  e.  Poset  /\  .1.  e.  B )  ->  .1.  =  ( ( lub `  K ) `  B
) )
9 simpr 447 . . . . 5  |-  ( ( K  e.  Poset  /\  .1.  e.  B )  ->  .1.  e.  B )
108, 9eqeltrrd 2358 . . . 4  |-  ( ( K  e.  Poset  /\  .1.  e.  B )  ->  (
( lub `  K
) `  B )  e.  B )
11103adant2 974 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  (
( lub `  K
) `  B )  e.  B )
12 simp2 956 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  X  e.  B )
13 ple1.l . . . 4  |-  .<_  =  ( le `  K )
144, 13, 5luble 14115 . . 3  |-  ( ( ( K  e.  Poset  /\  B  C_  B )  /\  ( ( ( lub `  K ) `  B
)  e.  B  /\  X  e.  B )
)  ->  X  .<_  ( ( lub `  K
) `  B )
)
151, 3, 11, 12, 14syl22anc 1183 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  X  .<_  ( ( lub `  K
) `  B )
)
1673ad2ant1 976 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  .1.  =  ( ( lub `  K ) `  B
) )
1715, 16breqtrrd 4049 1  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  X  .<_  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074   lubclub 14076   1.cp1 14144
This theorem is referenced by:  ople1  29381  lhp2lt  30190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-undef 6298  df-riota 6304  df-lub 14108  df-p1 14146
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