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Theorem atcmp 30183
Description: If two atoms are comparable, they are equal. (atsseq 23855 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 30173 . . 3  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
213ad2ant1 979 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  Poset )
3 eqid 2438 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
4 atcmp.a . . . 4  |-  A  =  ( Atoms `  K )
53, 4atbase 30161 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
653ad2ant2 980 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  P  e.  ( Base `  K
) )
73, 4atbase 30161 . . 3  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
873ad2ant3 981 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  Q  e.  ( Base `  K
) )
9 eqid 2438 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
103, 9atl0cl 30175 . . 3  |-  ( K  e.  AtLat  ->  ( 0. `  K )  e.  (
Base `  K )
)
11103ad2ant1 979 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( 0. `  K )  e.  ( Base `  K
) )
12 eqid 2438 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
139, 12, 4atcvr0 30160 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) P )
14133adant3 978 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) P )
159, 12, 4atcvr0 30160 . . 3  |-  ( ( K  e.  AtLat  /\  Q  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) Q )
16153adant2 977 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) Q )
17 atcmp.l . . 3  |-  .<_  =  ( le `  K )
183, 17, 12cvrcmp 30155 . 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 1203 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 178    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457   Basecbs 13474   lecple 13541   Posetcpo 14402   0.cp0 14471    <o ccvr 30134   Atomscatm 30135   AtLatcal 30136
This theorem is referenced by:  atncmp  30184  atnlt  30185  atnle  30189  cvlsupr2  30215  cvratlem  30292  2atjm  30316  atbtwn  30317  2atm  30398  2llnmeqat  30442  dalem25  30569  dalem55  30598  dalem57  30600  snatpsubN  30621  pmapat  30634  2llnma1b  30657  cdlemblem  30664  lhp2at0nle  30906  lhpat3  30917  4atexlemcnd  30943  trlval3  31058  cdlemc5  31066  cdleme3  31108  cdleme7  31120  cdleme11k  31139  cdleme16b  31150  cdleme16e  31153  cdleme16f  31154  cdlemednpq  31170  cdleme20j  31189  cdleme22aa  31210  cdleme22cN  31213  cdleme22d  31214  cdlemf2  31433  cdlemb3  31477  cdlemg12e  31518  cdlemg17dALTN  31535  cdlemg19a  31554  cdlemg27b  31567  cdlemg31d  31571  trlcone  31599  cdlemi  31691  tendotr  31701  cdlemk17  31729  cdlemk52  31825  cdleml1N  31847  dia2dimlem1  31936  dia2dimlem2  31937  dia2dimlem3  31938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-poset 14408  df-plt 14420  df-lat 14480  df-covers 30138  df-ats 30139  df-atl 30170
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