Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atcmp Unicode version

Theorem atcmp 30123
Description: If two atoms are comparable, they are equal. (atsseq 22943 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 30113 . . 3  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
213ad2ant1 976 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  Poset )
3 eqid 2296 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
4 atcmp.a . . . 4  |-  A  =  ( Atoms `  K )
53, 4atbase 30101 . . 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 30101 . . 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 2296 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
103, 9atl0cl 30115 . . 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 2296 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
139, 12, 4atcvr0 30100 . . 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 30100 . . 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 30095 . 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Posetcpo 14090   0.cp0 14159    <o ccvr 30074   Atomscatm 30075   AtLatcal 30076
This theorem is referenced by:  atncmp  30124  atnlt  30125  atnle  30129  cvlsupr2  30155  cvratlem  30232  2atjm  30256  atbtwn  30257  2atm  30338  2llnmeqat  30382  dalem25  30509  dalem55  30538  dalem57  30540  snatpsubN  30561  pmapat  30574  2llnma1b  30597  cdlemblem  30604  lhp2at0nle  30846  lhpat3  30857  4atexlemcnd  30883  trlval3  30998  cdlemc5  31006  cdleme3  31048  cdleme7  31060  cdleme11k  31079  cdleme16b  31090  cdleme16e  31093  cdleme16f  31094  cdlemednpq  31110  cdleme20j  31129  cdleme22aa  31150  cdleme22cN  31153  cdleme22d  31154  cdlemf2  31373  cdlemb3  31417  cdlemg12e  31458  cdlemg17dALTN  31475  cdlemg19a  31494  cdlemg27b  31507  cdlemg31d  31511  trlcone  31539  cdlemi  31631  tendotr  31641  cdlemk17  31669  cdlemk52  31765  cdleml1N  31787  dia2dimlem1  31876  dia2dimlem2  31877  dia2dimlem3  31878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-poset 14096  df-plt 14108  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110
  Copyright terms: Public domain W3C validator