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

Theorem cvrat 32707
Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 27882 analog.) (Contributed by NM, 22-Nov-2011.)
Hypotheses
Ref Expression
cvrat.b  |-  B  =  ( Base `  K
)
cvrat.s  |-  .<  =  ( lt `  K )
cvrat.j  |-  .\/  =  ( join `  K )
cvrat.z  |-  .0.  =  ( 0. `  K )
cvrat.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvrat  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X  =/=  .0.  /\  X  .<  ( P  .\/  Q ) )  ->  X  e.  A )
)

Proof of Theorem cvrat
StepHypRef Expression
1 cvrat.b . . . 4  |-  B  =  ( Base `  K
)
2 cvrat.s . . . 4  |-  .<  =  ( lt `  K )
3 cvrat.j . . . 4  |-  .\/  =  ( join `  K )
4 cvrat.z . . . 4  |-  .0.  =  ( 0. `  K )
5 cvrat.a . . . 4  |-  A  =  ( Atoms `  K )
61, 2, 3, 4, 5cvratlem 32706 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  ( X  =/=  .0.  /\  X  .<  ( P  .\/  Q
) ) )  -> 
( -.  P ( le `  K ) X  ->  X  e.  A ) )
7 hllat 32649 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
87adantr 466 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Lat )
9 simpr2 1012 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  P  e.  A )
101, 5atbase 32575 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  B )
119, 10syl 17 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  P  e.  B )
12 simpr3 1013 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  A )
131, 5atbase 32575 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  B )
1412, 13syl 17 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  B )
151, 3latjcom 16260 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
168, 11, 14, 15syl3anc 1264 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
1716breq2d 4438 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  .<  ( P  .\/  Q )  <->  X  .<  ( Q 
.\/  P ) ) )
1817anbi2d 708 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X  =/=  .0.  /\  X  .<  ( P  .\/  Q ) )  <->  ( X  =/=  .0.  /\  X  .<  ( Q  .\/  P ) ) ) )
19 simpl 458 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  HL )
20 simpr1 1011 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  X  e.  B )
211, 2, 3, 4, 5cvratlem 32706 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  P  e.  A
) )  /\  ( X  =/=  .0.  /\  X  .<  ( Q  .\/  P
) ) )  -> 
( -.  Q ( le `  K ) X  ->  X  e.  A ) )
2221ex 435 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  P  e.  A
) )  ->  (
( X  =/=  .0.  /\  X  .<  ( Q  .\/  P ) )  -> 
( -.  Q ( le `  K ) X  ->  X  e.  A ) ) )
2319, 20, 12, 9, 22syl13anc 1266 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X  =/=  .0.  /\  X  .<  ( Q  .\/  P ) )  -> 
( -.  Q ( le `  K ) X  ->  X  e.  A ) ) )
2418, 23sylbid 218 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X  =/=  .0.  /\  X  .<  ( P  .\/  Q ) )  -> 
( -.  Q ( le `  K ) X  ->  X  e.  A ) ) )
2524imp 430 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  ( X  =/=  .0.  /\  X  .<  ( P  .\/  Q
) ) )  -> 
( -.  Q ( le `  K ) X  ->  X  e.  A ) )
26 hlpos 32651 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Poset )
2726adantr 466 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Poset )
281, 3latjcl 16252 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
298, 11, 14, 28syl3anc 1264 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  e.  B )
30 eqid 2429 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
311, 30, 2pltnle 16167 . . . . . . . . 9  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  /\  X  .<  ( P  .\/  Q ) )  ->  -.  ( P  .\/  Q ) ( le `  K
) X )
3231ex 435 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  ->  ( X  .<  ( P  .\/  Q )  ->  -.  ( P  .\/  Q ) ( le `  K ) X ) )
3327, 20, 29, 32syl3anc 1264 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  .<  ( P  .\/  Q )  ->  -.  ( P  .\/  Q ) ( le `  K ) X ) )
341, 30, 3latjle12 16263 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  B  /\  Q  e.  B  /\  X  e.  B
) )  ->  (
( P ( le
`  K ) X  /\  Q ( le
`  K ) X )  <->  ( P  .\/  Q ) ( le `  K ) X ) )
358, 11, 14, 20, 34syl13anc 1266 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( P ( le
`  K ) X  /\  Q ( le
`  K ) X )  <->  ( P  .\/  Q ) ( le `  K ) X ) )
3635biimpd 210 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( P ( le
`  K ) X  /\  Q ( le
`  K ) X )  ->  ( P  .\/  Q ) ( le
`  K ) X ) )
3733, 36nsyld 145 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  .<  ( P  .\/  Q )  ->  -.  ( P ( le `  K ) X  /\  Q ( le `  K ) X ) ) )
38 ianor 490 . . . . . 6  |-  ( -.  ( P ( le
`  K ) X  /\  Q ( le
`  K ) X )  <->  ( -.  P
( le `  K
) X  \/  -.  Q ( le `  K ) X ) )
3937, 38syl6ib 229 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  .<  ( P  .\/  Q )  ->  ( -.  P ( le `  K ) X  \/  -.  Q ( le `  K ) X ) ) )
4039imp 430 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X  .<  ( P  .\/  Q
) )  ->  ( -.  P ( le `  K ) X  \/  -.  Q ( le `  K ) X ) )
4140adantrl 720 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  ( X  =/=  .0.  /\  X  .<  ( P  .\/  Q
) ) )  -> 
( -.  P ( le `  K ) X  \/  -.  Q
( le `  K
) X ) )
426, 25, 41mpjaod 382 . 2  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  ( X  =/=  .0.  /\  X  .<  ( P  .\/  Q
) ) )  ->  X  e.  A )
4342ex 435 1  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X  =/=  .0.  /\  X  .<  ( P  .\/  Q ) )  ->  X  e.  A )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15160   Posetcpo 16140   ltcplt 16141   joincjn 16144   0.cp0 16238   Latclat 16246   Atomscatm 32549   HLchlt 32636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16128  df-poset 16146  df-plt 16159  df-lub 16175  df-glb 16176  df-join 16177  df-meet 16178  df-p0 16240  df-lat 16247  df-clat 16309  df-oposet 32462  df-ol 32464  df-oml 32465  df-covers 32552  df-ats 32553  df-atl 32584  df-cvlat 32608  df-hlat 32637
This theorem is referenced by:  cvrat2  32714
  Copyright terms: Public domain W3C validator