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Theorem cvrat2 32426
Description: A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 27705 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 2402 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 cvrat2.c . . . . . . . . 9  |-  C  =  (  <o  `  K )
5 cvrat2.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
61, 2, 3, 4, 5atcvrj0 32425 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  X C
( P  .\/  Q
) )  ->  ( X  =  ( 0. `  K )  <->  P  =  Q ) )
763expa 1197 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X C ( P  .\/  Q ) )  ->  ( X  =  ( 0. `  K )  <->  P  =  Q ) )
87necon3bid 2661 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X C ( P  .\/  Q ) )  ->  ( X  =/=  ( 0. `  K )  <->  P  =/=  Q ) )
9 simpl 455 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  HL )
10 simpr1 1003 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  X  e.  B )
11 hllat 32361 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
1211adantr 463 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Lat )
13 simpr2 1004 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  P  e.  A )
141, 5atbase 32287 . . . . . . . . . . 11  |-  ( P  e.  A  ->  P  e.  B )
1513, 14syl 17 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  P  e.  B )
16 simpr3 1005 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  A )
171, 5atbase 32287 . . . . . . . . . . 11  |-  ( Q  e.  A  ->  Q  e.  B )
1816, 17syl 17 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  B )
191, 2latjcl 16003 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
2012, 15, 18, 19syl3anc 1230 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  e.  B )
21 eqid 2402 . . . . . . . . . . 11  |-  ( lt
`  K )  =  ( lt `  K
)
221, 21, 4cvrlt 32268 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  /\  X C ( P  .\/  Q ) )  ->  X
( lt `  K
) ( P  .\/  Q ) )
2322ex 432 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  -> 
( X C ( P  .\/  Q )  ->  X ( lt
`  K ) ( P  .\/  Q ) ) )
249, 10, 20, 23syl3anc 1230 . . . . . . . 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 32419 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X  =/=  ( 0. `  K )  /\  X ( lt `  K ) ( P 
.\/  Q ) )  ->  X  e.  A
) )
2625expcomd 436 . . . . . . . 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 427 . . . . . 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 235 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X C ( P  .\/  Q ) )  ->  ( P  =/=  Q  ->  X  e.  A ) )
3029ex 432 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( P  .\/  Q )  ->  ( P  =/=  Q  ->  X  e.  A ) ) )
3130com23 78 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  =/=  Q  ->  ( X C ( P  .\/  Q )  ->  X  e.  A ) ) )
3231impd 429 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( P  =/=  Q  /\  X C ( P 
.\/  Q ) )  ->  X  e.  A
) )
33323impia 1194 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Basecbs 14839   ltcplt 15892   joincjn 15895   0.cp0 15989   Latclat 15997    <o ccvr 32260   Atomscatm 32261   HLchlt 32348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349
This theorem is referenced by:  cvrat3  32439  atcvrlln  32517  lncvrelatN  32778
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