Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme35h2 Unicode version

Theorem cdleme35h2 30573
Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one outside of  P  .\/  Q line. TODO: FIX COMMENT (Contributed by NM, 18-Mar-2013.)
Hypotheses
Ref Expression
cdleme35.l  |-  .<_  =  ( le `  K )
cdleme35.j  |-  .\/  =  ( join `  K )
cdleme35.m  |-  ./\  =  ( meet `  K )
cdleme35.a  |-  A  =  ( Atoms `  K )
cdleme35.h  |-  H  =  ( LHyp `  K
)
cdleme35.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme35.f  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
cdleme35.g  |-  G  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
Assertion
Ref Expression
cdleme35h2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  R  =/=  S
) )  ->  F  =/=  G )

Proof of Theorem cdleme35h2
StepHypRef Expression
1 simp33 995 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  R  =/=  S
) )  ->  R  =/=  S )
2 simpl1 960 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  R  =/=  S
) )  /\  F  =  G )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
3 simpl2 961 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  R  =/=  S
) )  /\  F  =  G )  ->  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )
4 simpl31 1038 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  R  =/=  S
) )  /\  F  =  G )  ->  -.  R  .<_  ( P  .\/  Q ) )
5 simpl32 1039 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  R  =/=  S
) )  /\  F  =  G )  ->  -.  S  .<_  ( P  .\/  Q ) )
6 simpr 448 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  R  =/=  S
) )  /\  F  =  G )  ->  F  =  G )
7 cdleme35.l . . . . . 6  |-  .<_  =  ( le `  K )
8 cdleme35.j . . . . . 6  |-  .\/  =  ( join `  K )
9 cdleme35.m . . . . . 6  |-  ./\  =  ( meet `  K )
10 cdleme35.a . . . . . 6  |-  A  =  ( Atoms `  K )
11 cdleme35.h . . . . . 6  |-  H  =  ( LHyp `  K
)
12 cdleme35.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
13 cdleme35.f . . . . . 6  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
14 cdleme35.g . . . . . 6  |-  G  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
157, 8, 9, 10, 11, 12, 13, 14cdleme35h 30572 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  F  =  G ) )  ->  R  =  S )
162, 3, 4, 5, 6, 15syl113anc 1196 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  R  =/=  S
) )  /\  F  =  G )  ->  R  =  S )
1716ex 424 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  R  =/=  S
) )  ->  ( F  =  G  ->  R  =  S ) )
1817necon3d 2590 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  R  =/=  S
) )  ->  ( R  =/=  S  ->  F  =/=  G ) )
191, 18mpd 15 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  R  =/=  S
) )  ->  F  =/=  G )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   lecple 13465   joincjn 14330   meetcmee 14331   Atomscatm 29380   HLchlt 29467   LHypclh 30100
This theorem is referenced by:  cdleme35sn2aw  30574
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-lines 29617  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104
  Copyright terms: Public domain W3C validator