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Theorem cvrat2 30240
Description: A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 22983 analog.) (Contributed by NM, 30-Nov-2011.)
Hypotheses
Ref Expression
cvrat2.b  |-  B  =  ( Base `  K
)
cvrat2.j  |-  .\/  =  ( join `  K )
cvrat2.c  |-  C  =  (  <o  `  K )
cvrat2.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvrat2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  ( P  =/=  Q  /\  X C ( P  .\/  Q
) ) )  ->  X  e.  A )

Proof of Theorem cvrat2
StepHypRef Expression
1 cvrat2.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
2 cvrat2.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
3 eqid 2296 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 cvrat2.c . . . . . . . . 9  |-  C  =  (  <o  `  K )
5 cvrat2.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
61, 2, 3, 4, 5atcvrj0 30239 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  X C
( P  .\/  Q
) )  ->  ( X  =  ( 0. `  K )  <->  P  =  Q ) )
763expa 1151 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X C ( P  .\/  Q ) )  ->  ( X  =  ( 0. `  K )  <->  P  =  Q ) )
87necon3bid 2494 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X C ( P  .\/  Q ) )  ->  ( X  =/=  ( 0. `  K )  <->  P  =/=  Q ) )
9 simpl 443 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  HL )
10 simpr1 961 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  X  e.  B )
11 hllat 30175 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
1211adantr 451 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Lat )
13 simpr2 962 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  P  e.  A )
141, 5atbase 30101 . . . . . . . . . . 11  |-  ( P  e.  A  ->  P  e.  B )
1513, 14syl 15 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  P  e.  B )
16 simpr3 963 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  A )
171, 5atbase 30101 . . . . . . . . . . 11  |-  ( Q  e.  A  ->  Q  e.  B )
1816, 17syl 15 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  B )
191, 2latjcl 14172 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
2012, 15, 18, 19syl3anc 1182 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  e.  B )
21 eqid 2296 . . . . . . . . . . 11  |-  ( lt
`  K )  =  ( lt `  K
)
221, 21, 4cvrlt 30082 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  /\  X C ( P  .\/  Q ) )  ->  X
( lt `  K
) ( P  .\/  Q ) )
2322ex 423 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  -> 
( X C ( P  .\/  Q )  ->  X ( lt
`  K ) ( P  .\/  Q ) ) )
249, 10, 20, 23syl3anc 1182 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( P  .\/  Q )  ->  X ( lt `  K ) ( P  .\/  Q ) ) )
251, 21, 2, 3, 5cvrat 30233 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X  =/=  ( 0. `  K )  /\  X ( lt `  K ) ( P 
.\/  Q ) )  ->  X  e.  A
) )
2625exp3acom23 1362 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X ( lt `  K ) ( P 
.\/  Q )  -> 
( X  =/=  ( 0. `  K )  ->  X  e.  A )
) )
2724, 26syld 40 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( P  .\/  Q )  ->  ( X  =/=  ( 0. `  K
)  ->  X  e.  A ) ) )
2827imp 418 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X C ( P  .\/  Q ) )  ->  ( X  =/=  ( 0. `  K )  ->  X  e.  A ) )
298, 28sylbird 226 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X C ( P  .\/  Q ) )  ->  ( P  =/=  Q  ->  X  e.  A ) )
3029ex 423 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( P  .\/  Q )  ->  ( P  =/=  Q  ->  X  e.  A ) ) )
3130com23 72 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  =/=  Q  ->  ( X C ( P  .\/  Q )  ->  X  e.  A ) ) )
3231imp3a 420 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( P  =/=  Q  /\  X C ( P 
.\/  Q ) )  ->  X  e.  A
) )
33323impia 1148 1  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  ( P  =/=  Q  /\  X C ( P  .\/  Q
) ) )  ->  X  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ltcplt 14091   joincjn 14094   0.cp0 14159   Latclat 14167    <o ccvr 30074   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  cvrat3  30253  atcvrlln  30331  lncvrelatN  30592
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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