Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvrat2 Structured version   Unicode version

Theorem cvrat2 34100
Description: A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 26968 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 2460 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 cvrat2.c . . . . . . . . 9  |-  C  =  (  <o  `  K )
5 cvrat2.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
61, 2, 3, 4, 5atcvrj0 34099 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  X C
( P  .\/  Q
) )  ->  ( X  =  ( 0. `  K )  <->  P  =  Q ) )
763expa 1191 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X C ( P  .\/  Q ) )  ->  ( X  =  ( 0. `  K )  <->  P  =  Q ) )
87necon3bid 2718 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X C ( P  .\/  Q ) )  ->  ( X  =/=  ( 0. `  K )  <->  P  =/=  Q ) )
9 simpl 457 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  HL )
10 simpr1 997 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  X  e.  B )
11 hllat 34035 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
1211adantr 465 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Lat )
13 simpr2 998 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  P  e.  A )
141, 5atbase 33961 . . . . . . . . . . 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 999 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  A )
171, 5atbase 33961 . . . . . . . . . . 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 15527 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
2012, 15, 18, 19syl3anc 1223 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  e.  B )
21 eqid 2460 . . . . . . . . . . 11  |-  ( lt
`  K )  =  ( lt `  K
)
221, 21, 4cvrlt 33942 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  /\  X C ( P  .\/  Q ) )  ->  X
( lt `  K
) ( P  .\/  Q ) )
2322ex 434 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  -> 
( X C ( P  .\/  Q )  ->  X ( lt
`  K ) ( P  .\/  Q ) ) )
249, 10, 20, 23syl3anc 1223 . . . . . . . 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 34093 . . . . . . . . 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 438 . . . . . . . 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 44 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X C ( P  .\/  Q )  ->  ( X  =/=  ( 0. `  K
)  ->  X  e.  A ) ) )
2827imp 429 . . . . . 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 434 . . . 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 431 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( P  =/=  Q  /\  X C ( P 
.\/  Q ) )  ->  X  e.  A
) )
33323impia 1188 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   ltcplt 15417   joincjn 15420   0.cp0 15513   Latclat 15521    <o ccvr 33934   Atomscatm 33935   HLchlt 34022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023
This theorem is referenced by:  cvrat3  34113  atcvrlln  34191  lncvrelatN  34452
  Copyright terms: Public domain W3C validator