Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme35h Unicode version

Theorem cdleme35h 30718
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, 11-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
cdleme35h  |-  ( ( ( ( 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 )

Proof of Theorem cdleme35h
StepHypRef Expression
1 oveq1 5867 . . . . 5  |-  ( F  =  G  ->  ( F  .\/  U )  =  ( G  .\/  U
) )
2 oveq2 5868 . . . . . . 7  |-  ( F  =  G  ->  ( Q  .\/  F )  =  ( Q  .\/  G
) )
32oveq1d 5875 . . . . . 6  |-  ( F  =  G  ->  (
( Q  .\/  F
)  ./\  W )  =  ( ( Q 
.\/  G )  ./\  W ) )
43oveq2d 5876 . . . . 5  |-  ( F  =  G  ->  ( P  .\/  ( ( Q 
.\/  F )  ./\  W ) )  =  ( P  .\/  ( ( Q  .\/  G ) 
./\  W ) ) )
51, 4oveq12d 5878 . . . 4  |-  ( F  =  G  ->  (
( F  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  F )  ./\  W )
) )  =  ( ( G  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  G )  ./\  W )
) ) )
653ad2ant3 978 . . 3  |-  ( ( -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  ->  ( ( F  .\/  U )  ./\  ( P  .\/  ( ( Q  .\/  F )  ./\  W )
) )  =  ( ( G  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  G )  ./\  W )
) ) )
763ad2ant3 978 . 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 )  /\  F  =  G ) )  ->  (
( F  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  F )  ./\  W )
) )  =  ( ( G  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  G )  ./\  W )
) ) )
8 simp1 955 . . 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 )  /\  F  =  G ) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
9 simp21 988 . . 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 )  /\  F  =  G ) )  ->  P  =/=  Q )
10 simp22 989 . . 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 )  /\  F  =  G ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
11 simp31 991 . . 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 )  /\  F  =  G ) )  ->  -.  R  .<_  ( P  .\/  Q ) )
12 cdleme35.l . . . 4  |-  .<_  =  ( le `  K )
13 cdleme35.j . . . 4  |-  .\/  =  ( join `  K )
14 cdleme35.m . . . 4  |-  ./\  =  ( meet `  K )
15 cdleme35.a . . . 4  |-  A  =  ( Atoms `  K )
16 cdleme35.h . . . 4  |-  H  =  ( LHyp `  K
)
17 cdleme35.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
18 cdleme35.f . . . 4  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
1912, 13, 14, 15, 16, 17, 18cdleme35g 30717 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  (
( F  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  F )  ./\  W )
) )  =  R )
208, 9, 10, 11, 19syl121anc 1187 . 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 )  /\  F  =  G ) )  ->  (
( F  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  F )  ./\  W )
) )  =  R )
21 simp23 990 . . 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 )  /\  F  =  G ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
22 simp32 992 . . 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 )  /\  F  =  G ) )  ->  -.  S  .<_  ( P  .\/  Q ) )
23 cdleme35.g . . . 4  |-  G  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
2412, 13, 14, 15, 16, 17, 23cdleme35g 30717 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( G  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  G )  ./\  W )
) )  =  S )
258, 9, 21, 22, 24syl121anc 1187 . 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 )  /\  F  =  G ) )  ->  (
( G  .\/  U
)  ./\  ( P  .\/  ( ( Q  .\/  G )  ./\  W )
) )  =  S )
267, 20, 253eqtr3d 2325 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 )  /\  F  =  G ) )  ->  R  =  S )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   lecple 13217   joincjn 14080   meetcmee 14081   Atomscatm 29526   HLchlt 29613   LHypclh 30246
This theorem is referenced by:  cdleme35h2  30719  cdleme36m  30723
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-lines 29763  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250
  Copyright terms: Public domain W3C validator