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Theorem cdleme 36429
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 36428 . 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 1026 . . . . 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 1022 . . . . 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 1023 . . . . 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 1027 . . . . 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 2485 . . . . . 6  |-  ( ( ( f `  P
)  =  Q  /\  ( z `  P
)  =  Q )  ->  ( f `  P )  =  ( z `  P ) )
11103ad2ant3 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 ) )  -> 
( f `  P
)  =  ( z `
 P ) )
121, 2, 3, 4cdlemd 36075 . . . . 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 1242 . . . 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 1195 . . 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 2877 . 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 5871 . . . 4  |-  ( f  =  z  ->  (
f `  P )  =  ( z `  P ) )
1716eqeq1d 2459 . . 3  |-  ( f  =  z  ->  (
( f `  P
)  =  Q  <->  ( z `  P )  =  Q ) )
1817reu4 3293 . 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   E!wreu 2809   class class class wbr 4456   ` cfv 5594   lecple 14719   Atomscatm 35131   HLchlt 35218   LHypclh 35851   LTrncltrn 35968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34827
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-map 7440  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-clat 15865  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-llines 35365  df-lplanes 35366  df-lvols 35367  df-lines 35368  df-psubsp 35370  df-pmap 35371  df-padd 35663  df-lhyp 35855  df-laut 35856  df-ldil 35971  df-ltrn 35972  df-trl 36027
This theorem is referenced by:  ltrniotaval  36450  cdlemeiota  36454  cdlemksv2  36716  cdlemkuv2  36736
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