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Theorem atcvr0eq 30237
Description: The covers relation is not transitive. (atcv0eq 22975 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 30228 . . . . 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 30100 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  .0.  C P )
763adant3 975 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  .0.  C P )
87biantrurd 494 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P C ( P  .\/  Q )  <-> 
(  .0.  C P  /\  P C ( P  .\/  Q ) ) ) )
94, 8bitrd 244 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  (  .0.  C P  /\  P C ( P  .\/  Q ) ) ) )
10 simp1 955 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  HL )
11 hlop 30174 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
12113ad2ant1 976 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  OP )
13 eqid 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1413, 5op0cl 29996 . . . . . 6  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
1512, 14syl 15 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  .0.  e.  ( Base `  K ) )
1613, 3atbase 30101 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
17163ad2ant2 977 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  e.  ( Base `  K ) )
1813, 1, 3hlatjcl 30178 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
1913, 2cvrntr 30236 . . . . 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 1184 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( (  .0.  C P  /\  P C ( P  .\/  Q ) )  ->  -.  .0.  C ( P  .\/  Q ) ) )
219, 20sylbid 206 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  -.  .0.  C ( P  .\/  Q ) ) )
2221necon4ad 2520 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  (  .0.  C ( P  .\/  Q )  ->  P  =  Q ) )
231, 3hlatjidm 30180 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P  .\/  P
)  =  P )
24233adant3 975 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  P
)  =  P )
257, 24breqtrrd 4065 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  .0.  C ( P 
.\/  P ) )
26 oveq2 5882 . . . 4  |-  ( P  =  Q  ->  ( P  .\/  P )  =  ( P  .\/  Q
) )
2726breq2d 4051 . . 3  |-  ( P  =  Q  ->  (  .0.  C ( P  .\/  P )  <->  .0.  C ( P  .\/  Q ) ) )
2825, 27syl5ibcom 211 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =  Q  ->  .0.  C ( P  .\/  Q ) ) )
2922, 28impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   joincjn 14094   0.cp0 14159   OPcops 29984    <o ccvr 30074   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  atcvrj0  30239
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-rep 4147  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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  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-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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