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Theorem ltrncnvnid 30938
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 957 . 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 30935 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
653adant3 975 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  F : B -1-1-onto-> B
)
7 f1orel 5491 . . . . . . . 8  |-  ( F : B -1-1-onto-> B  ->  Rel  F )
86, 7syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  Rel  F )
9 dfrel2 5140 . . . . . . 7  |-  ( Rel 
F  <->  `' `' F  =  F
)
108, 9sylib 188 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  `' `' F  =  F )
11 cnveq 4871 . . . . . 6  |-  ( `' F  =  (  _I  |`  B )  ->  `' `' F  =  `' (  _I  |`  B ) )
1210, 11sylan9req 2349 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  `' F  =  (  _I  |`  B ) )  ->  F  =  `' (  _I  |`  B ) )
13 cnvresid 5338 . . . . 5  |-  `' (  _I  |`  B )  =  (  _I  |`  B )
1412, 13syl6eq 2344 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  `' F  =  (  _I  |`  B ) )  ->  F  =  (  _I  |`  B ) )
1514ex 423 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( `' F  =  (  _I  |`  B )  ->  F  =  (  _I  |`  B )
) )
1615necon3d 2497 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( F  =/=  (  _I  |`  B )  ->  `' F  =/=  (  _I  |`  B ) ) )
171, 16mpd 14 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  `' F  =/=  (  _I  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    _I cid 4320   `'ccnv 4704    |` cres 4707   Rel wrel 4710   -1-1-onto->wf1o 5270   ` cfv 5271   Basecbs 13164   HLchlt 30162   LHypclh 30795   LTrncltrn 30912
This theorem is referenced by:  cdlemh2  31627  cdlemh  31628  cdlemkfid1N  31732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-laut 30800  df-ldil 30915  df-ltrn 30916
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