Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  leatb Unicode version

Theorem leatb 29409
Description: A poset element less than or equal to an atom equals either zero or the atom. (atss 23699 analog.) (Contributed by NM, 17-Nov-2011.)
Hypotheses
Ref Expression
leatom.b  |-  B  =  ( Base `  K
)
leatom.l  |-  .<_  =  ( le `  K )
leatom.z  |-  .0.  =  ( 0. `  K )
leatom.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
leatb  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( X  .<_  P  <->  ( X  =  P  \/  X  =  .0.  ) ) )

Proof of Theorem leatb
StepHypRef Expression
1 leatom.b . . . . . 6  |-  B  =  ( Base `  K
)
2 leatom.l . . . . . 6  |-  .<_  =  ( le `  K )
3 leatom.z . . . . . 6  |-  .0.  =  ( 0. `  K )
41, 2, 3op0le 29303 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )
543adant3 977 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  .0.  .<_  X )
65biantrurd 495 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( X  .<_  P  <->  (  .0.  .<_  X  /\  X  .<_  P ) ) )
7 opposet 29299 . . . . . 6  |-  ( K  e.  OP  ->  K  e.  Poset )
873ad2ant1 978 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  Poset )
91, 3op0cl 29301 . . . . . . 7  |-  ( K  e.  OP  ->  .0.  e.  B )
10 leatom.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
111, 10atbase 29406 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
12 id 20 . . . . . . 7  |-  ( X  e.  B  ->  X  e.  B )
139, 11, 123anim123i 1139 . . . . . 6  |-  ( ( K  e.  OP  /\  P  e.  A  /\  X  e.  B )  ->  (  .0.  e.  B  /\  P  e.  B  /\  X  e.  B
) )
14133com23 1159 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  (  .0.  e.  B  /\  P  e.  B  /\  X  e.  B
) )
15 eqid 2389 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
163, 15, 10atcvr0 29405 . . . . . 6  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  (  <o  `  K
) P )
17163adant2 976 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  .0.  (  <o  `  K
) P )
181, 2, 15cvrnbtwn4 29396 . . . . 5  |-  ( ( K  e.  Poset  /\  (  .0.  e.  B  /\  P  e.  B  /\  X  e.  B )  /\  .0.  (  <o  `  K ) P )  ->  (
(  .0.  .<_  X  /\  X  .<_  P )  <->  (  .0.  =  X  \/  X  =  P ) ) )
198, 14, 17, 18syl3anc 1184 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( (  .0.  .<_  X  /\  X  .<_  P )  <-> 
(  .0.  =  X  \/  X  =  P ) ) )
20 eqcom 2391 . . . . 5  |-  (  .0.  =  X  <->  X  =  .0.  )
2120orbi1i 507 . . . 4  |-  ( (  .0.  =  X  \/  X  =  P )  <->  ( X  =  .0.  \/  X  =  P )
)
2219, 21syl6bb 253 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( (  .0.  .<_  X  /\  X  .<_  P )  <-> 
( X  =  .0. 
\/  X  =  P ) ) )
236, 22bitrd 245 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( X  .<_  P  <->  ( X  =  .0.  \/  X  =  P ) ) )
24 orcom 377 . 2  |-  ( ( X  =  .0.  \/  X  =  P )  <->  ( X  =  P  \/  X  =  .0.  )
)
2523, 24syl6bb 253 1  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( X  .<_  P  <->  ( X  =  P  \/  X  =  .0.  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4155   ` cfv 5396   Basecbs 13398   lecple 13465   Posetcpo 14326   0.cp0 14395   OPcops 29289    <o ccvr 29379   Atomscatm 29380
This theorem is referenced by:  leat  29410  leat2  29411  meetat  29413
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-glb 14361  df-p0 14397  df-oposet 29293  df-covers 29383  df-ats 29384
  Copyright terms: Public domain W3C validator