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Theorem atcvrj0 30162
Description: Two atoms covering the zero subspace are equal. (atcv1 23875 analog.) (Contributed by NM, 29-Nov-2011.)
Hypotheses
Ref Expression
atcvrj0.b  |-  B  =  ( Base `  K
)
atcvrj0.j  |-  .\/  =  ( join `  K )
atcvrj0.z  |-  .0.  =  ( 0. `  K )
atcvrj0.c  |-  C  =  (  <o  `  K )
atcvrj0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atcvrj0  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  X C
( P  .\/  Q
) )  ->  ( X  =  .0.  <->  P  =  Q ) )

Proof of Theorem atcvrj0
StepHypRef Expression
1 breq1 4207 . . . . . . . 8  |-  ( X  =  .0.  ->  ( X C ( P  .\/  Q )  <->  .0.  C ( P  .\/  Q ) ) )
21biimpd 199 . . . . . . 7  |-  ( X  =  .0.  ->  ( X C ( P  .\/  Q )  ->  .0.  C
( P  .\/  Q
) ) )
32adantl 453 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X  =  .0.  )  ->  ( X C ( P  .\/  Q )  ->  .0.  C
( P  .\/  Q
) ) )
4 atcvrj0.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
5 atcvrj0.z . . . . . . . . 9  |-  .0.  =  ( 0. `  K )
6 atcvrj0.c . . . . . . . . 9  |-  C  =  (  <o  `  K )
7 atcvrj0.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
84, 5, 6, 7atcvr0eq 30160 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  (  .0.  C ( P  .\/  Q )  <-> 
P  =  Q ) )
983adant3r1 1162 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (  .0.  C ( P  .\/  Q )  <->  P  =  Q
) )
109adantr 452 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X  =  .0.  )  ->  (  .0.  C ( P  .\/  Q )  <->  P  =  Q
) )
113, 10sylibd 206 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X  =  .0.  )  ->  ( X C ( P  .\/  Q )  ->  P  =  Q ) )
1211ex 424 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  =  .0.  ->  ( X C ( P 
.\/  Q )  ->  P  =  Q )
) )
1312com23 74 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( P  .\/  Q )  ->  ( X  =  .0.  ->  P  =  Q ) ) )
14133impia 1150 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  X C
( P  .\/  Q
) )  ->  ( X  =  .0.  ->  P  =  Q ) )
15 oveq1 6080 . . . . . . 7  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
1615breq2d 4216 . . . . . 6  |-  ( P  =  Q  ->  ( X C ( P  .\/  Q )  <->  X C ( Q 
.\/  Q ) ) )
1716biimpac 473 . . . . 5  |-  ( ( X C ( P 
.\/  Q )  /\  P  =  Q )  ->  X C ( Q 
.\/  Q ) )
18 simpr3 965 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  A )
194, 7hlatjidm 30103 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
2018, 19syldan 457 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( Q  .\/  Q )  =  Q )
2120breq2d 4216 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( Q  .\/  Q )  <->  X C Q ) )
22 hlatl 30095 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
2322adantr 452 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  AtLat )
24 simpr1 963 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  X  e.  B )
25 atcvrj0.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
26 eqid 2435 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
2725, 26, 5, 6, 7atcvreq0 30049 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  Q  e.  A )  ->  ( X C Q  <->  X  =  .0.  ) )
2823, 24, 18, 27syl3anc 1184 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C Q  <->  X  =  .0.  ) )
2928biimpd 199 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C Q  ->  X  =  .0.  ) )
3021, 29sylbid 207 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( Q  .\/  Q )  ->  X  =  .0.  ) )
3117, 30syl5 30 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X C ( P  .\/  Q )  /\  P  =  Q )  ->  X  =  .0.  ) )
3231exp3a 426 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( P  .\/  Q )  ->  ( P  =  Q  ->  X  =  .0.  ) ) )
33323impia 1150 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  X C
( P  .\/  Q
) )  ->  ( P  =  Q  ->  X  =  .0.  ) )
3414, 33impbid 184 1  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  X C
( P  .\/  Q
) )  ->  ( X  =  .0.  <->  P  =  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   0.cp0 14458    <o ccvr 29997   Atomscatm 29998   AtLatcal 29999   HLchlt 30085
This theorem is referenced by:  cvrat2  30163  atcvrneN  30164  atcvrj2b  30166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086
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