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Theorem atcmp 29948
Description: If two atoms are comparable, they are equal. (atsseq 23838 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 29938 . . 3  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
213ad2ant1 978 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  Poset )
3 eqid 2435 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
4 atcmp.a . . . 4  |-  A  =  ( Atoms `  K )
53, 4atbase 29926 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
653ad2ant2 979 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  P  e.  ( Base `  K
) )
73, 4atbase 29926 . . 3  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
873ad2ant3 980 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  Q  e.  ( Base `  K
) )
9 eqid 2435 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
103, 9atl0cl 29940 . . 3  |-  ( K  e.  AtLat  ->  ( 0. `  K )  e.  (
Base `  K )
)
11103ad2ant1 978 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( 0. `  K )  e.  ( Base `  K
) )
12 eqid 2435 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
139, 12, 4atcvr0 29925 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) P )
14133adant3 977 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) P )
159, 12, 4atcvr0 29925 . . 3  |-  ( ( K  e.  AtLat  /\  Q  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) Q )
16153adant2 976 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) Q )
17 atcmp.l . . 3  |-  .<_  =  ( le `  K )
183, 17, 12cvrcmp 29920 . 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 1202 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 177    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5445   Basecbs 13457   lecple 13524   Posetcpo 14385   0.cp0 14454    <o ccvr 29899   Atomscatm 29900   AtLatcal 29901
This theorem is referenced by:  atncmp  29949  atnlt  29950  atnle  29954  cvlsupr2  29980  cvratlem  30057  2atjm  30081  atbtwn  30082  2atm  30163  2llnmeqat  30207  dalem25  30334  dalem55  30363  dalem57  30365  snatpsubN  30386  pmapat  30399  2llnma1b  30422  cdlemblem  30429  lhp2at0nle  30671  lhpat3  30682  4atexlemcnd  30708  trlval3  30823  cdlemc5  30831  cdleme3  30873  cdleme7  30885  cdleme11k  30904  cdleme16b  30915  cdleme16e  30918  cdleme16f  30919  cdlemednpq  30935  cdleme20j  30954  cdleme22aa  30975  cdleme22cN  30978  cdleme22d  30979  cdlemf2  31198  cdlemb3  31242  cdlemg12e  31283  cdlemg17dALTN  31300  cdlemg19a  31319  cdlemg27b  31332  cdlemg31d  31336  trlcone  31364  cdlemi  31456  tendotr  31466  cdlemk17  31494  cdlemk52  31590  cdleml1N  31612  dia2dimlem1  31701  dia2dimlem2  31702  dia2dimlem3  31703
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-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
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-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  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-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-iota 5409  df-fun 5447  df-fv 5453  df-ov 6075  df-poset 14391  df-plt 14403  df-lat 14463  df-covers 29903  df-ats 29904  df-atl 29935
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