Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlema2N Unicode version

Theorem cdlema2N 30274
Description: A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlema2.b  |-  B  =  ( Base `  K
)
cdlema2.l  |-  .<_  =  ( le `  K )
cdlema2.j  |-  .\/  =  ( join `  K )
cdlema2.m  |-  ./\  =  ( meet `  K )
cdlema2.z  |-  .0.  =  ( 0. `  K )
cdlema2.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cdlema2N  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( R  ./\  X
)  =  .0.  )

Proof of Theorem cdlema2N
StepHypRef Expression
1 simp3ll 1028 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  R  =/=  P )
2 simp3rl 1030 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  P  .<_  X )
3 simp3rr 1031 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  -.  Q  .<_  X )
4 simp3lr 1029 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  R  .<_  ( P  .\/  Q ) )
52, 3, 43jca 1134 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )
6 cdlema2.b . . . . . 6  |-  B  =  ( Base `  K
)
7 cdlema2.l . . . . . 6  |-  .<_  =  ( le `  K )
8 cdlema2.j . . . . . 6  |-  .\/  =  ( join `  K )
9 cdlema2.a . . . . . 6  |-  A  =  ( Atoms `  K )
106, 7, 8, 9exatleN 29886 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q
) ) )  -> 
( R  .<_  X  <->  R  =  P ) )
115, 10syld3an3 1229 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( R  .<_  X  <->  R  =  P ) )
1211necon3bbid 2601 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( -.  R  .<_  X  <-> 
R  =/=  P ) )
131, 12mpbird 224 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  -.  R  .<_  X )
14 simp1l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  K  e.  HL )
15 hlatl 29843 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
1614, 15syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  K  e.  AtLat )
17 simp23 992 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  R  e.  A )
18 simp1r 982 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  X  e.  B )
19 cdlema2.m . . . 4  |-  ./\  =  ( meet `  K )
20 cdlema2.z . . . 4  |-  .0.  =  ( 0. `  K )
216, 7, 19, 20, 9atnle 29800 . . 3  |-  ( ( K  e.  AtLat  /\  R  e.  A  /\  X  e.  B )  ->  ( -.  R  .<_  X  <->  ( R  ./\ 
X )  =  .0.  ) )
2216, 17, 18, 21syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( -.  R  .<_  X  <-> 
( R  ./\  X
)  =  .0.  )
)
2313, 22mpbid 202 1  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( R  ./\  X
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   0.cp0 14421   Atomscatm 29746   AtLatcal 29747   HLchlt 29833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834
  Copyright terms: Public domain W3C validator