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Theorem leatb 30027
Description: A poset element less than or equal to an atom equals either zero or the atom. (atss 23841 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 29921 . . . . 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 29917 . . . . . 6  |-  ( K  e.  OP  ->  K  e.  Poset )
873ad2ant1 978 . . . . 5  |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  Poset )
91, 3op0cl 29919 . . . . . . 7  |-  ( K  e.  OP  ->  .0.  e.  B )
10 leatom.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
111, 10atbase 30024 . . . . . . 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 2435 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
163, 15, 10atcvr0 30023 . . . . . 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 30014 . . . . 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 2437 . . . . 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 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   Posetcpo 14389   0.cp0 14458   OPcops 29907    <o ccvr 29997   Atomscatm 29998
This theorem is referenced by:  leat  30028  leat2  30029  meetat  30031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-glb 14424  df-p0 14460  df-oposet 29911  df-covers 30001  df-ats 30002
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