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Theorem atcvrj0 30239
Description: Two atoms covering the zero subspace are equal. (atcv1 22976 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 4042 . . . . . . . 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 30237 . . . . . . . 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 5881 . . . . . . 7  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
1615breq2d 4051 . . . . . 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 30180 . . . . . . . 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 4051 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( Q  .\/  Q )  <->  X C Q ) )
22 hlatl 30172 . . . . . . . . 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 2296 . . . . . . . . 9  |-  ( le
`  K )  =  ( le `  K
)
2725, 26, 5, 6, 7atcvreq0 30126 . . . . . . . 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   0.cp0 14159    <o ccvr 30074   Atomscatm 30075   AtLatcal 30076   HLchlt 30162
This theorem is referenced by:  cvrat2  30240  atcvrneN  30241  atcvrj2b  30243
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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