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Theorem cdleme 30017
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
Dummy variable  z is distinct from all other variables.

Proof of Theorem cdleme
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 30016 . 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 2304 . . . . . 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 29664 . . . . 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 2636 . 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 5485 . . . 4  |-  ( f  =  z  ->  (
f `  P )  =  ( z `  P ) )
1716eqeq1d 2293 . . 3  |-  ( f  =  z  ->  (
( f `  P
)  =  Q  <->  ( z `  P )  =  Q ) )
1817reu4 2961 . 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 647 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 5    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   A.wral 2545   E.wrex 2546   E!wreu 2547   class class class wbr 4025   ` cfv 5222   lecple 13210   Atomscatm 28721   HLchlt 28808   LHypclh 29441   LTrncltrn 29558
This theorem is referenced by:  ltrniotaval  30038  cdlemeiota  30042  cdlemksv2  30304  cdlemkuv2  30324
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  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-rmo 2553  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 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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-map 6770  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-llines 28955  df-lplanes 28956  df-lvols 28957  df-lines 28958  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-lhyp 29445  df-laut 29446  df-ldil 29561  df-ltrn 29562  df-trl 29616
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