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Theorem cdleme 34204
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 34203 . 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 1018 . . . . 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 1014 . . . . 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 1015 . . . . 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 1019 . . . . 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 2462 . . . . . 6  |-  ( ( ( f `  P
)  =  Q  /\  ( z `  P
)  =  Q )  ->  ( f `  P )  =  ( z `  P ) )
11103ad2ant3 1011 . . . . 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 33851 . . . . 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 1232 . . . 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 1186 . . 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 2807 . 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 5690 . . . 4  |-  ( f  =  z  ->  (
f `  P )  =  ( z `  P ) )
1716eqeq1d 2451 . . 3  |-  ( f  =  z  ->  (
( f `  P
)  =  Q  <->  ( z `  P )  =  Q ) )
1817reu4 3153 . 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 664 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   E!wreu 2717   class class class wbr 4292   ` cfv 5418   lecple 14245   Atomscatm 32908   HLchlt 32995   LHypclh 33628   LTrncltrn 33745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-riotaBAD 32604
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-undef 6792  df-map 7216  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-p1 15210  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-llines 33142  df-lplanes 33143  df-lvols 33144  df-lines 33145  df-psubsp 33147  df-pmap 33148  df-padd 33440  df-lhyp 33632  df-laut 33633  df-ldil 33748  df-ltrn 33749  df-trl 33803
This theorem is referenced by:  ltrniotaval  34225  cdlemeiota  34229  cdlemksv2  34491  cdlemkuv2  34511
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