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Theorem ltrn11at 34094
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 1026 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  P  =/=  Q )
2 simp1 988 . . . 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 989 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  F  e.  T )
4 simp31 1024 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  P  e.  A )
5 eqid 2451 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
6 ltrneq2.a . . . . . 6  |-  A  =  ( Atoms `  K )
75, 6atbase 33237 . . . . 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 1025 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  Q  e.  A )
105, 6atbase 33237 . . . . 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 34073 . . . 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 1223 . . 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 2704 . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2642   ` cfv 5513   Basecbs 14273   Atomscatm 33211   HLchlt 33298   LHypclh 33931   LTrncltrn 34048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-map 7313  df-ats 33215  df-laut 33936  df-ldil 34051  df-ltrn 34052
This theorem is referenced by:  cdlemg10a  34587  cdlemg12d  34593  cdlemg18a  34625
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