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Theorem cvrat2 30226
Description: A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 23890 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 2436 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 cvrat2.c . . . . . . . . 9  |-  C  =  (  <o  `  K )
5 cvrat2.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
61, 2, 3, 4, 5atcvrj0 30225 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  X C
( P  .\/  Q
) )  ->  ( X  =  ( 0. `  K )  <->  P  =  Q ) )
763expa 1153 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X C ( P  .\/  Q ) )  ->  ( X  =  ( 0. `  K )  <->  P  =  Q ) )
87necon3bid 2636 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X C ( P  .\/  Q ) )  ->  ( X  =/=  ( 0. `  K )  <->  P  =/=  Q ) )
9 simpl 444 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  HL )
10 simpr1 963 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  X  e.  B )
11 hllat 30161 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
1211adantr 452 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Lat )
13 simpr2 964 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  P  e.  A )
141, 5atbase 30087 . . . . . . . . . . 11  |-  ( P  e.  A  ->  P  e.  B )
1513, 14syl 16 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  P  e.  B )
16 simpr3 965 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  A )
171, 5atbase 30087 . . . . . . . . . . 11  |-  ( Q  e.  A  ->  Q  e.  B )
1816, 17syl 16 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  B )
191, 2latjcl 14479 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
2012, 15, 18, 19syl3anc 1184 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  e.  B )
21 eqid 2436 . . . . . . . . . . 11  |-  ( lt
`  K )  =  ( lt `  K
)
221, 21, 4cvrlt 30068 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  /\  X C ( P  .\/  Q ) )  ->  X
( lt `  K
) ( P  .\/  Q ) )
2322ex 424 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  -> 
( X C ( P  .\/  Q )  ->  X ( lt
`  K ) ( P  .\/  Q ) ) )
249, 10, 20, 23syl3anc 1184 . . . . . . . 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 30219 . . . . . . . . 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 1381 . . . . . . . 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 42 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( P  .\/  Q )  ->  ( X  =/=  ( 0. `  K
)  ->  X  e.  A ) ) )
2827imp 419 . . . . . 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 227 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X C ( P  .\/  Q ) )  ->  ( P  =/=  Q  ->  X  e.  A ) )
3029ex 424 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( P  .\/  Q )  ->  ( P  =/=  Q  ->  X  e.  A ) ) )
3130com23 74 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  =/=  Q  ->  ( X C ( P  .\/  Q )  ->  X  e.  A ) ) )
3231imp3a 421 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( P  =/=  Q  /\  X C ( P 
.\/  Q ) )  ->  X  e.  A
) )
33323impia 1150 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ltcplt 14398   joincjn 14401   0.cp0 14466   Latclat 14474    <o ccvr 30060   Atomscatm 30061   HLchlt 30148
This theorem is referenced by:  cvrat3  30239  atcvrlln  30317  lncvrelatN  30578
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149
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