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Theorem atcvrj0 29617
Description: Two atoms covering the zero subspace are equal. (atcv1 22960 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 4026 . . . . . . . 8  |-  ( X  =  .0.  ->  ( X C ( P  .\/  Q )  <->  .0.  C ( P  .\/  Q ) ) )
21biimpd 198 . . . . . . 7  |-  ( X  =  .0.  ->  ( X C ( P  .\/  Q )  ->  .0.  C
( P  .\/  Q
) ) )
32adantl 452 . . . . . 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 29615 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  (  .0.  C ( P  .\/  Q )  <-> 
P  =  Q ) )
983adant3r1 1160 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (  .0.  C ( P  .\/  Q )  <->  P  =  Q
) )
109adantr 451 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X  =  .0.  )  ->  (  .0.  C ( P  .\/  Q )  <->  P  =  Q
) )
113, 10sylibd 205 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X  =  .0.  )  ->  ( X C ( P  .\/  Q )  ->  P  =  Q ) )
1211ex 423 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  =  .0.  ->  ( X C ( P 
.\/  Q )  ->  P  =  Q )
) )
1312com23 72 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( P  .\/  Q )  ->  ( X  =  .0.  ->  P  =  Q ) ) )
14133impia 1148 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  X C
( P  .\/  Q
) )  ->  ( X  =  .0.  ->  P  =  Q ) )
15 oveq1 5865 . . . . . . 7  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
1615breq2d 4035 . . . . . 6  |-  ( P  =  Q  ->  ( X C ( P  .\/  Q )  <->  X C ( Q 
.\/  Q ) ) )
1716biimpac 472 . . . . 5  |-  ( ( X C ( P 
.\/  Q )  /\  P  =  Q )  ->  X C ( Q 
.\/  Q ) )
18 simpr3 963 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  A )
194, 7hlatjidm 29558 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
2018, 19syldan 456 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( Q  .\/  Q )  =  Q )
2120breq2d 4035 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( Q  .\/  Q )  <->  X C Q ) )
22 hlatl 29550 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
2322adantr 451 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  AtLat )
24 simpr1 961 . . . . . . . 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 2283 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
2725, 26, 5, 6, 7atcvreq0 29504 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  Q  e.  A )  ->  ( X C Q  <->  X  =  .0.  ) )
2823, 24, 18, 27syl3anc 1182 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C Q  <->  X  =  .0.  ) )
2928biimpd 198 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C Q  ->  X  =  .0.  ) )
3021, 29sylbid 206 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( Q  .\/  Q )  ->  X  =  .0.  ) )
3117, 30syl5 28 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X C ( P  .\/  Q )  /\  P  =  Q )  ->  X  =  .0.  ) )
3231exp3a 425 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( P  .\/  Q )  ->  ( P  =  Q  ->  X  =  .0.  ) ) )
33323impia 1148 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  X C
( P  .\/  Q
) )  ->  ( P  =  Q  ->  X  =  .0.  ) )
3414, 33impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   0.cp0 14143    <o ccvr 29452   Atomscatm 29453   AtLatcal 29454   HLchlt 29540
This theorem is referenced by:  cvrat2  29618  atcvrneN  29619  atcvrj2b  29621
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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