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Theorem cdlemd 33849
Description: If two translations agree at any atom not under the fiducial co-atom  W, then they are equal. Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)
Hypotheses
Ref Expression
cdlemd.l  |-  .<_  =  ( le `  K )
cdlemd.a  |-  A  =  ( Atoms `  K )
cdlemd.h  |-  H  =  ( LHyp `  K
)
cdlemd.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemd  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  ->  F  =  G )

Proof of Theorem cdlemd
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 simpl11 1063 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  q  e.  A )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simpl12 1064 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  q  e.  A )  ->  F  e.  T )
3 simpl13 1065 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  q  e.  A )  ->  G  e.  T )
42, 3jca 532 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  q  e.  A )  ->  ( F  e.  T  /\  G  e.  T )
)
5 simpr 461 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  q  e.  A )  ->  q  e.  A )
6 simpl2 992 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  q  e.  A )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
7 simpl3 993 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  q  e.  A )  ->  ( F `  P )  =  ( G `  P ) )
8 cdlemd.l . . . . 5  |-  .<_  =  ( le `  K )
9 eqid 2442 . . . . 5  |-  ( join `  K )  =  (
join `  K )
10 cdlemd.a . . . . 5  |-  A  =  ( Atoms `  K )
11 cdlemd.h . . . . 5  |-  H  =  ( LHyp `  K
)
12 cdlemd.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
138, 9, 10, 11, 12cdlemd9 33848 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  q  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( F `  q
)  =  ( G `
 q ) )
141, 4, 5, 6, 7, 13syl311anc 1232 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  q  e.  A )  ->  ( F `  q )  =  ( G `  q ) )
1514ralrimiva 2798 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  ->  A. q  e.  A  ( F `  q )  =  ( G `  q ) )
1610, 11, 12ltrneq2 33790 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( A. q  e.  A  ( F `  q )  =  ( G `  q )  <->  F  =  G ) )
17163ad2ant1 1009 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  ->  ( A. q  e.  A  ( F `  q )  =  ( G `  q )  <->  F  =  G ) )
1815, 17mpbid 210 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  ->  F  =  G )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2714   class class class wbr 4291   ` cfv 5417   lecple 14244   joincjn 15113   Atomscatm 32906   HLchlt 32993   LHypclh 33626   LTrncltrn 33743
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-map 7215  df-poset 15115  df-plt 15127  df-lub 15143  df-glb 15144  df-join 15145  df-meet 15146  df-p0 15208  df-p1 15209  df-lat 15215  df-clat 15277  df-oposet 32819  df-ol 32821  df-oml 32822  df-covers 32909  df-ats 32910  df-atl 32941  df-cvlat 32965  df-hlat 32994  df-llines 33140  df-psubsp 33145  df-pmap 33146  df-padd 33438  df-lhyp 33630  df-laut 33631  df-ldil 33746  df-ltrn 33747  df-trl 33801
This theorem is referenced by:  ltrneq3  33850  cdleme  34202  cdlemg1a  34212  ltrniotavalbN  34226  cdlemg44  34375  cdlemk19  34511  cdlemk27-3  34549  cdlemk33N  34551  cdlemk34  34552  cdlemk53a  34597  cdlemk19u  34612  dia2dimlem4  34710  dih1dimatlem0  34971
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