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Theorem atcvr0eq 29540
Description: The covers relation is not transitive. (atcv0eq 23730 analog.) (Contributed by NM, 29-Nov-2011.)
Hypotheses
Ref Expression
atcvr0eq.j  |-  .\/  =  ( join `  K )
atcvr0eq.z  |-  .0.  =  ( 0. `  K )
atcvr0eq.c  |-  C  =  (  <o  `  K )
atcvr0eq.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atcvr0eq  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  (  .0.  C ( P  .\/  Q )  <-> 
P  =  Q ) )

Proof of Theorem atcvr0eq
StepHypRef Expression
1 atcvr0eq.j . . . . . 6  |-  .\/  =  ( join `  K )
2 atcvr0eq.c . . . . . 6  |-  C  =  (  <o  `  K )
3 atcvr0eq.a . . . . . 6  |-  A  =  ( Atoms `  K )
41, 2, 3atcvr1 29531 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  P C ( P  .\/  Q ) ) )
5 atcvr0eq.z . . . . . . . 8  |-  .0.  =  ( 0. `  K )
65, 2, 3atcvr0 29403 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  .0.  C P )
763adant3 977 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  .0.  C P )
87biantrurd 495 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P C ( P  .\/  Q )  <-> 
(  .0.  C P  /\  P C ( P  .\/  Q ) ) ) )
94, 8bitrd 245 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  (  .0.  C P  /\  P C ( P  .\/  Q ) ) ) )
10 simp1 957 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  HL )
11 hlop 29477 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
12113ad2ant1 978 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  OP )
13 eqid 2387 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1413, 5op0cl 29299 . . . . . 6  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
1512, 14syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  .0.  e.  ( Base `  K ) )
1613, 3atbase 29404 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
17163ad2ant2 979 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  e.  ( Base `  K ) )
1813, 1, 3hlatjcl 29481 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
1913, 2cvrntr 29539 . . . . 5  |-  ( ( K  e.  HL  /\  (  .0.  e.  ( Base `  K )  /\  P  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
) )  ->  (
(  .0.  C P  /\  P C ( P  .\/  Q ) )  ->  -.  .0.  C ( P  .\/  Q ) ) )
2010, 15, 17, 18, 19syl13anc 1186 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( (  .0.  C P  /\  P C ( P  .\/  Q ) )  ->  -.  .0.  C ( P  .\/  Q ) ) )
219, 20sylbid 207 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  -.  .0.  C ( P  .\/  Q ) ) )
2221necon4ad 2611 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  (  .0.  C ( P  .\/  Q )  ->  P  =  Q ) )
231, 3hlatjidm 29483 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P  .\/  P
)  =  P )
24233adant3 977 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  P
)  =  P )
257, 24breqtrrd 4179 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  .0.  C ( P 
.\/  P ) )
26 oveq2 6028 . . . 4  |-  ( P  =  Q  ->  ( P  .\/  P )  =  ( P  .\/  Q
) )
2726breq2d 4165 . . 3  |-  ( P  =  Q  ->  (  .0.  C ( P  .\/  P )  <->  .0.  C ( P  .\/  Q ) ) )
2825, 27syl5ibcom 212 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =  Q  ->  .0.  C ( P  .\/  Q ) ) )
2922, 28impbid 184 1  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  (  .0.  C ( P  .\/  Q )  <-> 
P  =  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 2550   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   joincjn 14328   0.cp0 14393   OPcops 29287    <o ccvr 29377   Atomscatm 29378   HLchlt 29465
This theorem is referenced by:  atcvrj0  29542
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466
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