MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ple1 Unicode version

Theorem ple1 14249
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 3273 . . . 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 2358 . . . . . . 7  |-  ( lub `  K )  =  ( lub `  K )
6 ple1.u . . . . . . 7  |-  .1.  =  ( 1. `  K )
74, 5, 6p1val 14247 . . . . . 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 2433 . . . 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 14214 . . 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 4130 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 1642    e. wcel 1710    C_ wss 3228   class class class wbr 4104   ` cfv 5337   Basecbs 13245   lecple 13312   Posetcpo 14173   lubclub 14175   1.cp1 14243
This theorem is referenced by:  ople1  29450  lhp2lt  30259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-undef 6385  df-riota 6391  df-lub 14207  df-p1 14245
  Copyright terms: Public domain W3C validator