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

Theorem leatb 30104
Description: A poset element less than or equal to an atom equals either zero or the atom. (atss 22942 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 29998 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )
543adant3 975 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  .0.  .<_  X )
65biantrurd 494 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( X  .<_  P  <->  (  .0.  .<_  X  /\  X  .<_  P ) ) )
7 opposet 29994 . . . . . 6  |-  ( K  e.  OP  ->  K  e.  Poset )
873ad2ant1 976 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  Poset )
91, 3op0cl 29996 . . . . . . 7  |-  ( K  e.  OP  ->  .0.  e.  B )
10 leatom.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
111, 10atbase 30101 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
12 id 19 . . . . . . 7  |-  ( X  e.  B  ->  X  e.  B )
139, 11, 123anim123i 1137 . . . . . 6  |-  ( ( K  e.  OP  /\  P  e.  A  /\  X  e.  B )  ->  (  .0.  e.  B  /\  P  e.  B  /\  X  e.  B
) )
14133com23 1157 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  (  .0.  e.  B  /\  P  e.  B  /\  X  e.  B
) )
15 eqid 2296 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
163, 15, 10atcvr0 30100 . . . . . 6  |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  (  <o  `  K
) P )
17163adant2 974 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  .0.  (  <o  `  K
) P )
181, 2, 15cvrnbtwn4 30091 . . . . 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 1182 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( (  .0.  .<_  X  /\  X  .<_  P )  <-> 
(  .0.  =  X  \/  X  =  P ) ) )
20 eqcom 2298 . . . . 5  |-  (  .0.  =  X  <->  X  =  .0.  )
2120orbi1i 506 . . . 4  |-  ( (  .0.  =  X  \/  X  =  P )  <->  ( X  =  .0.  \/  X  =  P )
)
2219, 21syl6bb 252 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( (  .0.  .<_  X  /\  X  .<_  P )  <-> 
( X  =  .0. 
\/  X  =  P ) ) )
236, 22bitrd 244 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( X  .<_  P  <->  ( X  =  .0.  \/  X  =  P ) ) )
24 orcom 376 . 2  |-  ( ( X  =  .0.  \/  X  =  P )  <->  ( X  =  P  \/  X  =  .0.  )
)
2523, 24syl6bb 252 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Posetcpo 14090   0.cp0 14159   OPcops 29984    <o ccvr 30074   Atomscatm 30075
This theorem is referenced by:  leat  30105  leat2  30106  meetat  30108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-glb 14125  df-p0 14161  df-oposet 29988  df-covers 30078  df-ats 30079
  Copyright terms: Public domain W3C validator