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Theorem ltrn11at 31006
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 996 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  P  =/=  Q )
2 simp1 958 . . . 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 959 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  F  e.  T )
4 simp31 994 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  P  e.  A )
5 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
6 ltrneq2.a . . . . . 6  |-  A  =  ( Atoms `  K )
75, 6atbase 30149 . . . . 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 995 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  Q  e.  A )
105, 6atbase 30149 . . . . 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 30985 . . . 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 1189 . . 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 2638 . 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 225 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   ` cfv 5456   Basecbs 13471   Atomscatm 30123   HLchlt 30210   LHypclh 30843   LTrncltrn 30960
This theorem is referenced by:  cdlemg10a  31499  cdlemg12d  31505  cdlemg18a  31537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-ats 30127  df-laut 30848  df-ldil 30963  df-ltrn 30964
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