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Theorem atcmp 29501
Description: If two atoms are comparable, they are equal. (atsseq 22927 analog.) (Contributed by NM, 13-Oct-2011.)
Hypotheses
Ref Expression
atcmp.l  |-  .<_  =  ( le `  K )
atcmp.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atcmp  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<_  Q  <->  P  =  Q ) )

Proof of Theorem atcmp
StepHypRef Expression
1 atlpos 29491 . . 3  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
213ad2ant1 976 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  Poset )
3 eqid 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
4 atcmp.a . . . 4  |-  A  =  ( Atoms `  K )
53, 4atbase 29479 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
653ad2ant2 977 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  P  e.  ( Base `  K
) )
73, 4atbase 29479 . . 3  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
873ad2ant3 978 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  Q  e.  ( Base `  K
) )
9 eqid 2283 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
103, 9atl0cl 29493 . . 3  |-  ( K  e.  AtLat  ->  ( 0. `  K )  e.  (
Base `  K )
)
11103ad2ant1 976 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( 0. `  K )  e.  ( Base `  K
) )
12 eqid 2283 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
139, 12, 4atcvr0 29478 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) P )
14133adant3 975 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) P )
159, 12, 4atcvr0 29478 . . 3  |-  ( ( K  e.  AtLat  /\  Q  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) Q )
16153adant2 974 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) Q )
17 atcmp.l . . 3  |-  .<_  =  ( le `  K )
183, 17, 12cvrcmp 29473 . 2  |-  ( ( K  e.  Poset  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
)  /\  ( 0. `  K )  e.  (
Base `  K )
)  /\  ( ( 0. `  K ) ( 
<o  `  K ) P  /\  ( 0. `  K ) (  <o  `  K ) Q ) )  ->  ( P  .<_  Q  <->  P  =  Q
) )
192, 6, 8, 11, 14, 16, 18syl132anc 1200 1  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<_  Q  <->  P  =  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074   0.cp0 14143    <o ccvr 29452   Atomscatm 29453   AtLatcal 29454
This theorem is referenced by:  atncmp  29502  atnlt  29503  atnle  29507  cvlsupr2  29533  cvratlem  29610  2atjm  29634  atbtwn  29635  2atm  29716  2llnmeqat  29760  dalem25  29887  dalem55  29916  dalem57  29918  snatpsubN  29939  pmapat  29952  2llnma1b  29975  cdlemblem  29982  lhp2at0nle  30224  lhpat3  30235  4atexlemcnd  30261  trlval3  30376  cdlemc5  30384  cdleme3  30426  cdleme7  30438  cdleme11k  30457  cdleme16b  30468  cdleme16e  30471  cdleme16f  30472  cdlemednpq  30488  cdleme20j  30507  cdleme22aa  30528  cdleme22cN  30531  cdleme22d  30532  cdlemf2  30751  cdlemb3  30795  cdlemg12e  30836  cdlemg17dALTN  30853  cdlemg19a  30872  cdlemg27b  30885  cdlemg31d  30889  trlcone  30917  cdlemi  31009  tendotr  31019  cdlemk17  31047  cdlemk52  31143  cdleml1N  31165  dia2dimlem1  31254  dia2dimlem2  31255  dia2dimlem3  31256
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-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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-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-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-poset 14080  df-plt 14092  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488
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