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Theorem cdlemd 33225
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 1072 . . . 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 1073 . . . . 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 1074 . . . . 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 530 . . . 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 459 . . . 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 1001 . . . 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 1002 . . . 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 2402 . . . . 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 33224 . . . 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 1244 . . 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 2818 . 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 33165 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( A. q  e.  A  ( F `  q )  =  ( G `  q )  <->  F  =  G ) )
17163ad2ant1 1018 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   class class class wbr 4395   ` cfv 5569   lecple 14916   joincjn 15897   Atomscatm 32281   HLchlt 32368   LHypclh 33001   LTrncltrn 33118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-map 7459  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515  df-psubsp 32520  df-pmap 32521  df-padd 32813  df-lhyp 33005  df-laut 33006  df-ldil 33121  df-ltrn 33122  df-trl 33177
This theorem is referenced by:  ltrneq3  33226  cdleme  33579  cdlemg1a  33589  ltrniotavalbN  33603  cdlemg44  33752  cdlemk19  33888  cdlemk27-3  33926  cdlemk33N  33928  cdlemk34  33929  cdlemk53a  33974  cdlemk19u  33989  dia2dimlem4  34087  dih1dimatlem0  34348
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