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Theorem atcvr0eq 29615
Description: The covers relation is not transitive. (atcv0eq 22959 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 29606 . . . . 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 29478 . . . . . . 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 29552 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
12113ad2ant1 976 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  OP )
13 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1413, 5op0cl 29374 . . . . . 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 29479 . . . . . 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 29556 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
1913, 2cvrntr 29614 . . . . 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 2507 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  (  .0.  C ( P  .\/  Q )  ->  P  =  Q ) )
231, 3hlatjidm 29558 . . . . 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 4049 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  .0.  C ( P 
.\/  P ) )
26 oveq2 5866 . . . 4  |-  ( P  =  Q  ->  ( P  .\/  P )  =  ( P  .\/  Q
) )
2726breq2d 4035 . . 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 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   joincjn 14078   0.cp0 14143   OPcops 29362    <o ccvr 29452   Atomscatm 29453   HLchlt 29540
This theorem is referenced by:  atcvrj0  29617
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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