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

Theorem cdleme 31196
Description: Lemma E in [Crawley] p. 113. If p,q are atoms not under hyperplane w, then there is a unique translation f such that f(p) = q. (Contributed by NM, 11-Apr-2013.)
Hypotheses
Ref Expression
cdleme.l  |-  .<_  =  ( le `  K )
cdleme.a  |-  A  =  ( Atoms `  K )
cdleme.h  |-  H  =  ( LHyp `  K
)
cdleme.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdleme  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  E! f  e.  T  ( f `  P )  =  Q )
Distinct variable groups:    A, f    f, K    .<_ , f    P, f    Q, f    T, f    f, W   
f, H

Proof of Theorem cdleme
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cdleme.l . . 3  |-  .<_  =  ( le `  K )
2 cdleme.a . . 3  |-  A  =  ( Atoms `  K )
3 cdleme.h . . 3  |-  H  =  ( LHyp `  K
)
4 cdleme.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
51, 2, 3, 4cdleme50ex 31195 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  E. f  e.  T  ( f `  P )  =  Q )
6 simp11 987 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  e.  T  /\  z  e.  T
)  /\  ( (
f `  P )  =  Q  /\  (
z `  P )  =  Q ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
7 simp2l 983 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  e.  T  /\  z  e.  T
)  /\  ( (
f `  P )  =  Q  /\  (
z `  P )  =  Q ) )  -> 
f  e.  T )
8 simp2r 984 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  e.  T  /\  z  e.  T
)  /\  ( (
f `  P )  =  Q  /\  (
z `  P )  =  Q ) )  -> 
z  e.  T )
9 simp12 988 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  e.  T  /\  z  e.  T
)  /\  ( (
f `  P )  =  Q  /\  (
z `  P )  =  Q ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
10 eqtr3 2454 . . . . . 6  |-  ( ( ( f `  P
)  =  Q  /\  ( z `  P
)  =  Q )  ->  ( f `  P )  =  ( z `  P ) )
11103ad2ant3 980 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  e.  T  /\  z  e.  T
)  /\  ( (
f `  P )  =  Q  /\  (
z `  P )  =  Q ) )  -> 
( f `  P
)  =  ( z `
 P ) )
121, 2, 3, 4cdlemd 30843 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  z  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( f `  P )  =  ( z `  P ) )  ->  f  =  z )
136, 7, 8, 9, 11, 12syl311anc 1198 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  e.  T  /\  z  e.  T
)  /\  ( (
f `  P )  =  Q  /\  (
z `  P )  =  Q ) )  -> 
f  =  z )
14133exp 1152 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( (
f  e.  T  /\  z  e.  T )  ->  ( ( ( f `
 P )  =  Q  /\  ( z `
 P )  =  Q )  ->  f  =  z ) ) )
1514ralrimivv 2789 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  A. f  e.  T  A. z  e.  T  ( (
( f `  P
)  =  Q  /\  ( z `  P
)  =  Q )  ->  f  =  z ) )
16 fveq1 5718 . . . 4  |-  ( f  =  z  ->  (
f `  P )  =  ( z `  P ) )
1716eqeq1d 2443 . . 3  |-  ( f  =  z  ->  (
( f `  P
)  =  Q  <->  ( z `  P )  =  Q ) )
1817reu4 3120 . 2  |-  ( E! f  e.  T  ( f `  P )  =  Q  <->  ( E. f  e.  T  (
f `  P )  =  Q  /\  A. f  e.  T  A. z  e.  T  ( (
( f `  P
)  =  Q  /\  ( z `  P
)  =  Q )  ->  f  =  z ) ) )
195, 15, 18sylanbrc 646 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  E! f  e.  T  ( f `  P )  =  Q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   E!wreu 2699   class class class wbr 4204   ` cfv 5445   lecple 13524   Atomscatm 29900   HLchlt 29987   LHypclh 30620   LTrncltrn 30737
This theorem is referenced by:  ltrniotaval  31217  cdlemeiota  31221  cdlemksv2  31483  cdlemkuv2  31503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-map 7011  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741  df-trl 30795
  Copyright terms: Public domain W3C validator