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Theorem ltrn11at 36287
Description: Frequently used one-to-one property of lattice translation atoms. (Contributed by NM, 5-May-2013.)
Hypotheses
Ref Expression
ltrneq2.a  |-  A  =  ( Atoms `  K )
ltrneq2.h  |-  H  =  ( LHyp `  K
)
ltrneq2.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrn11at  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  ( F `  P )  =/=  ( F `  Q
) )

Proof of Theorem ltrn11at
StepHypRef Expression
1 simp33 1032 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  P  =/=  Q )
2 simp1 994 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simp2 995 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  F  e.  T )
4 simp31 1030 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  P  e.  A )
5 eqid 2454 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
6 ltrneq2.a . . . . . 6  |-  A  =  ( Atoms `  K )
75, 6atbase 35430 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
84, 7syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  P  e.  ( Base `  K
) )
9 simp32 1031 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  Q  e.  A )
105, 6atbase 35430 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
119, 10syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  Q  e.  ( Base `  K
) )
12 ltrneq2.h . . . . 5  |-  H  =  ( LHyp `  K
)
13 ltrneq2.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
145, 12, 13ltrn11 36266 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  (
Base `  K )  /\  Q  e.  ( Base `  K ) ) )  ->  ( ( F `  P )  =  ( F `  Q )  <->  P  =  Q ) )
152, 3, 8, 11, 14syl112anc 1230 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  (
( F `  P
)  =  ( F `
 Q )  <->  P  =  Q ) )
1615necon3bid 2712 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  (
( F `  P
)  =/=  ( F `
 Q )  <->  P  =/=  Q ) )
171, 16mpbird 232 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  ( F `  P )  =/=  ( F `  Q
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   ` cfv 5570   Basecbs 14719   Atomscatm 35404   HLchlt 35491   LHypclh 36124   LTrncltrn 36241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-ats 35408  df-laut 36129  df-ldil 36244  df-ltrn 36245
This theorem is referenced by:  cdlemg10a  36782  cdlemg12d  36788  cdlemg18a  36820
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