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Theorem ltrncnvnid 34798
Description: If a translation is different from the identity, so is its converse. (Contributed by NM, 17-Jun-2013.)
Hypotheses
Ref Expression
ltrn1o.b  |-  B  =  ( Base `  K
)
ltrn1o.h  |-  H  =  ( LHyp `  K
)
ltrn1o.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrncnvnid  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  `' F  =/=  (  _I  |`  B ) )

Proof of Theorem ltrncnvnid
StepHypRef Expression
1 simp3 993 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  F  =/=  (  _I  |`  B ) )
2 ltrn1o.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
3 ltrn1o.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
4 ltrn1o.t . . . . . . . . . 10  |-  T  =  ( ( LTrn `  K
) `  W )
52, 3, 4ltrn1o 34795 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
653adant3 1011 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  F : B -1-1-onto-> B
)
7 f1orel 5810 . . . . . . . 8  |-  ( F : B -1-1-onto-> B  ->  Rel  F )
86, 7syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  Rel  F )
9 dfrel2 5448 . . . . . . 7  |-  ( Rel 
F  <->  `' `' F  =  F
)
108, 9sylib 196 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  `' `' F  =  F )
11 cnveq 5167 . . . . . 6  |-  ( `' F  =  (  _I  |`  B )  ->  `' `' F  =  `' (  _I  |`  B ) )
1210, 11sylan9req 2522 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  `' F  =  (  _I  |`  B ) )  ->  F  =  `' (  _I  |`  B ) )
13 cnvresid 5649 . . . . 5  |-  `' (  _I  |`  B )  =  (  _I  |`  B )
1412, 13syl6eq 2517 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  `' F  =  (  _I  |`  B ) )  ->  F  =  (  _I  |`  B ) )
1514ex 434 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( `' F  =  (  _I  |`  B )  ->  F  =  (  _I  |`  B )
) )
1615necon3d 2684 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( F  =/=  (  _I  |`  B )  ->  `' F  =/=  (  _I  |`  B ) ) )
171, 16mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  `' F  =/=  (  _I  |`  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655    _I cid 4783   `'ccnv 4991    |` cres 4994   Rel wrel 4997   -1-1-onto->wf1o 5578   ` cfv 5579   Basecbs 14479   HLchlt 34022   LHypclh 34655   LTrncltrn 34772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-laut 34660  df-ldil 34775  df-ltrn 34776
This theorem is referenced by:  cdlemh2  35487  cdlemh  35488  cdlemkfid1N  35592
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