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Theorem atcvrj0 30299
Description: Two atoms covering the zero subspace are equal. (atcv1 23888 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 4218 . . . . . . . 8  |-  ( X  =  .0.  ->  ( X C ( P  .\/  Q )  <->  .0.  C ( P  .\/  Q ) ) )
21biimpd 200 . . . . . . 7  |-  ( X  =  .0.  ->  ( X C ( P  .\/  Q )  ->  .0.  C
( P  .\/  Q
) ) )
32adantl 454 . . . . . 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 30297 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  (  .0.  C ( P  .\/  Q )  <-> 
P  =  Q ) )
983adant3r1 1163 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (  .0.  C ( P  .\/  Q )  <->  P  =  Q
) )
109adantr 453 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X  =  .0.  )  ->  (  .0.  C ( P  .\/  Q )  <->  P  =  Q
) )
113, 10sylibd 207 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X  =  .0.  )  ->  ( X C ( P  .\/  Q )  ->  P  =  Q ) )
1211ex 425 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  =  .0.  ->  ( X C ( P 
.\/  Q )  ->  P  =  Q )
) )
1312com23 75 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( P  .\/  Q )  ->  ( X  =  .0.  ->  P  =  Q ) ) )
14133impia 1151 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  X C
( P  .\/  Q
) )  ->  ( X  =  .0.  ->  P  =  Q ) )
15 oveq1 6091 . . . . . . 7  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
1615breq2d 4227 . . . . . 6  |-  ( P  =  Q  ->  ( X C ( P  .\/  Q )  <->  X C ( Q 
.\/  Q ) ) )
1716biimpac 474 . . . . 5  |-  ( ( X C ( P 
.\/  Q )  /\  P  =  Q )  ->  X C ( Q 
.\/  Q ) )
18 simpr3 966 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  A )
194, 7hlatjidm 30240 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
2018, 19syldan 458 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( Q  .\/  Q )  =  Q )
2120breq2d 4227 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( Q  .\/  Q )  <->  X C Q ) )
22 hlatl 30232 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
2322adantr 453 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  AtLat )
24 simpr1 964 . . . . . . . 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 2438 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
2725, 26, 5, 6, 7atcvreq0 30186 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  Q  e.  A )  ->  ( X C Q  <->  X  =  .0.  ) )
2823, 24, 18, 27syl3anc 1185 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C Q  <->  X  =  .0.  ) )
2928biimpd 200 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C Q  ->  X  =  .0.  ) )
3021, 29sylbid 208 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( Q  .\/  Q )  ->  X  =  .0.  ) )
3117, 30syl5 31 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X C ( P  .\/  Q )  /\  P  =  Q )  ->  X  =  .0.  ) )
3231exp3a 427 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( P  .\/  Q )  ->  ( P  =  Q  ->  X  =  .0.  ) ) )
33323impia 1151 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  X C
( P  .\/  Q
) )  ->  ( P  =  Q  ->  X  =  .0.  ) )
3414, 33impbid 185 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   0.cp0 14471    <o ccvr 30134   Atomscatm 30135   AtLatcal 30136   HLchlt 30222
This theorem is referenced by:  cvrat2  30300  atcvrneN  30301  atcvrj2b  30303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223
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