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Theorem cdlemd 29547
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
StepHypRef Expression
1 simpl11 1035 . . . 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 1036 . . . . 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 1037 . . . . 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 520 . . . 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 449 . . . 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 964 . . . 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 965 . . . 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 2256 . . . . 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 29546 . . . 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 1201 . . 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 2599 . 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 29488 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( A. q  e.  A  ( F `  q )  =  ( G `  q )  <->  F  =  G ) )
17163ad2ant1 981 . 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 203 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 set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2516   class class class wbr 3983   ` cfv 4659   lecple 13163   joincjn 14026   Atomscatm 28604   HLchlt 28691   LHypclh 29324   LTrncltrn 29441
This theorem is referenced by:  ltrneq3  29548  cdleme  29900  cdlemg1a  29910  ltrniotavalbN  29924  cdlemg44  30073  cdlemk19  30209  cdlemk27-3  30247  cdlemk33N  30249  cdlemk34  30250  cdlemk53a  30295  cdlemk19u  30310  dia2dimlem4  30408  dih1dimatlem0  30669
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-map 6728  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-llines 28838  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328  df-laut 29329  df-ldil 29444  df-ltrn 29445  df-trl 29499
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