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Theorem cvlatexchb1 29450
Description: A version of cvlexchb1 29446 for atoms. (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
cvlatexch.l  |-  .<_  =  ( le `  K )
cvlatexch.j  |-  .\/  =  ( join `  K )
cvlatexch.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvlatexchb1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( R  .\/  Q
)  <->  ( R  .\/  P )  =  ( R 
.\/  Q ) ) )

Proof of Theorem cvlatexchb1
StepHypRef Expression
1 cvlatl 29441 . . . . 5  |-  ( K  e.  CvLat  ->  K  e.  AtLat
)
21adantr 452 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
)  ->  K  e.  AtLat
)
3 simpr1 963 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
)  ->  P  e.  A )
4 simpr3 965 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
)  ->  R  e.  A )
5 cvlatexch.l . . . . 5  |-  .<_  =  ( le `  K )
6 cvlatexch.a . . . . 5  |-  A  =  ( Atoms `  K )
75, 6atncmp 29428 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  R  e.  A )  ->  ( -.  P  .<_  R  <->  P  =/=  R ) )
82, 3, 4, 7syl3anc 1184 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
)  ->  ( -.  P  .<_  R  <->  P  =/=  R ) )
9 eqid 2388 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
109, 6atbase 29405 . . . 4  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
11 cvlatexch.j . . . . . 6  |-  .\/  =  ( join `  K )
129, 5, 11, 6cvlexchb1 29446 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  ( Base `  K ) )  /\  -.  P  .<_  R )  ->  ( P  .<_  ( R  .\/  Q )  <-> 
( R  .\/  P
)  =  ( R 
.\/  Q ) ) )
13123expia 1155 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  ( Base `  K ) ) )  ->  ( -.  P  .<_  R  ->  ( P  .<_  ( R  .\/  Q
)  <->  ( R  .\/  P )  =  ( R 
.\/  Q ) ) ) )
1410, 13syl3anr3 1238 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
)  ->  ( -.  P  .<_  R  ->  ( P  .<_  ( R  .\/  Q )  <->  ( R  .\/  P )  =  ( R 
.\/  Q ) ) ) )
158, 14sylbird 227 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
)  ->  ( P  =/=  R  ->  ( P  .<_  ( R  .\/  Q
)  <->  ( R  .\/  P )  =  ( R 
.\/  Q ) ) ) )
16153impia 1150 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R
)  ->  ( P  .<_  ( R  .\/  Q
)  <->  ( R  .\/  P )  =  ( R 
.\/  Q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   Basecbs 13397   lecple 13464   joincjn 14329   Atomscatm 29379   AtLatcal 29380   CvLatclc 29381
This theorem is referenced by:  cvlatexchb2  29451  cvlatexch1  29452  cvlatexch3  29454  hlatexchb1  29508  llnexchb2lem  29983  4atexlemunv  30181  cdleme19d  30421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-poset 14331  df-plt 14343  df-lub 14359  df-join 14361  df-lat 14403  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438
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