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Theorem ltrnatneq 33673
Description: If any atom (under  W) is not equal to its translation, so is any other atom. TODO:  -.  P  .<_  W isn't needed to prove this. Will removing it shorten (and not lengthen) proofs using it? (Contributed by NM, 6-May-2013.)
Hypotheses
Ref Expression
ltrn2eq.l  |-  .<_  =  ( le `  K )
ltrn2eq.a  |-  A  =  ( Atoms `  K )
ltrn2eq.h  |-  H  =  ( LHyp `  K
)
ltrn2eq.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrnatneq  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( F `  Q )  =/=  Q )

Proof of Theorem ltrnatneq
StepHypRef Expression
1 ltrn2eq.l . . . 4  |-  .<_  =  ( le `  K )
2 ltrn2eq.a . . . 4  |-  A  =  ( Atoms `  K )
3 ltrn2eq.h . . . 4  |-  H  =  ( LHyp `  K
)
4 ltrn2eq.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
51, 2, 3, 4ltrn2ateq 33671 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( F `  P
)  =  P  <->  ( F `  Q )  =  Q ) )
65necon3bid 2683 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( F `  P
)  =/=  P  <->  ( F `  Q )  =/=  Q
) )
76biimp3a 1365 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( F `  Q )  =/=  Q )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   class class class wbr 4421   ` cfv 5599   lecple 15190   Atomscatm 32754   HLchlt 32841   LHypclh 33474   LTrncltrn 33591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-map 7480  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-p1 16279  df-lat 16285  df-clat 16347  df-oposet 32667  df-ol 32669  df-oml 32670  df-covers 32757  df-ats 32758  df-atl 32789  df-cvlat 32813  df-hlat 32842  df-lhyp 33478  df-laut 33479  df-ldil 33594  df-ltrn 33595  df-trl 33650
This theorem is referenced by:  ltrnatlw  33674  cdlemg13  34144  cdlemg17i  34161  cdlemg17pq  34164  cdlemg19  34176  cdlemg21  34178
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