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Theorem leatb 29482
Description: A poset element less than or equal to an atom equals either zero or the atom. (atss 22926 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 29376 . . . . 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 29372 . . . . . 6  |-  ( K  e.  OP  ->  K  e.  Poset )
873ad2ant1 976 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  Poset )
91, 3op0cl 29374 . . . . . . 7  |-  ( K  e.  OP  ->  .0.  e.  B )
10 leatom.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
111, 10atbase 29479 . . . . . . 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 2283 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
163, 15, 10atcvr0 29478 . . . . . 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 29469 . . . . 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 2285 . . . . 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 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074   0.cp0 14143   OPcops 29362    <o ccvr 29452   Atomscatm 29453
This theorem is referenced by:  leat  29483  leat2  29484  meetat  29486
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-ov 5861  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-glb 14109  df-p0 14145  df-oposet 29366  df-covers 29456  df-ats 29457
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