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Theorem cdleme 30749
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 30748 . 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 985 . . . . 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 981 . . . . 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 982 . . . . 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 986 . . . . 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 2302 . . . . . 6  |-  ( ( ( f `  P
)  =  Q  /\  ( z `  P
)  =  Q )  ->  ( f `  P )  =  ( z `  P ) )
11103ad2ant3 978 . . . . 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 30396 . . . . 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 1196 . . . 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 1150 . . 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 2634 . 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 5524 . . . 4  |-  ( f  =  z  ->  (
f `  P )  =  ( z `  P ) )
1716eqeq1d 2291 . . 3  |-  ( f  =  z  ->  (
( f `  P
)  =  Q  <->  ( z `  P )  =  Q ) )
1817reu4 2959 . 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 645 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   E!wreu 2545   class class class wbr 4023   ` cfv 5255   lecple 13215   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290
This theorem is referenced by:  ltrniotaval  30770  cdlemeiota  30774  cdlemksv2  31036  cdlemkuv2  31056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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