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Theorem atcvr0eq 32910
Description: The covers relation is not transitive. (atcv0eq 25734 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 32901 . . . . 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 32773 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  .0.  C P )
763adant3 1008 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  .0.  C P )
87biantrurd 508 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P C ( P  .\/  Q )  <-> 
(  .0.  C P  /\  P C ( P  .\/  Q ) ) ) )
94, 8bitrd 253 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  (  .0.  C P  /\  P C ( P  .\/  Q ) ) ) )
10 simp1 988 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  HL )
11 hlop 32847 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
12113ad2ant1 1009 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  OP )
13 eqid 2438 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1413, 5op0cl 32669 . . . . . 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 32774 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
17163ad2ant2 1010 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  e.  ( Base `  K ) )
1813, 1, 3hlatjcl 32851 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
1913, 2cvrntr 32909 . . . . 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 1220 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( (  .0.  C P  /\  P C ( P  .\/  Q ) )  ->  -.  .0.  C ( P  .\/  Q ) ) )
219, 20sylbid 215 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  -.  .0.  C ( P  .\/  Q ) ) )
2221necon4ad 2667 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  (  .0.  C ( P  .\/  Q )  ->  P  =  Q ) )
231, 3hlatjidm 32853 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P  .\/  P
)  =  P )
24233adant3 1008 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  P
)  =  P )
257, 24breqtrrd 4313 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  .0.  C ( P 
.\/  P ) )
26 oveq2 6094 . . . 4  |-  ( P  =  Q  ->  ( P  .\/  P )  =  ( P  .\/  Q
) )
2726breq2d 4299 . . 3  |-  ( P  =  Q  ->  (  .0.  C ( P  .\/  P )  <->  .0.  C ( P  .\/  Q ) ) )
2825, 27syl5ibcom 220 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =  Q  ->  .0.  C ( P  .\/  Q ) ) )
2922, 28impbid 191 1  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  (  .0.  C ( P  .\/  Q )  <-> 
P  =  Q ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   joincjn 15106   0.cp0 15199   OPcops 32657    <o ccvr 32747   Atomscatm 32748   HLchlt 32835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836
This theorem is referenced by:  atcvrj0  32912
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