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Theorem ltrncnvnid 30825
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 959 . 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 30822 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
653adant3 977 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  F : B -1-1-onto-> B
)
7 f1orel 5669 . . . . . . . 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 5313 . . . . . . 7  |-  ( Rel 
F  <->  `' `' F  =  F
)
108, 9sylib 189 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  `' `' F  =  F )
11 cnveq 5038 . . . . . 6  |-  ( `' F  =  (  _I  |`  B )  ->  `' `' F  =  `' (  _I  |`  B ) )
1210, 11sylan9req 2488 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  `' F  =  (  _I  |`  B ) )  ->  F  =  `' (  _I  |`  B ) )
13 cnvresid 5515 . . . . 5  |-  `' (  _I  |`  B )  =  (  _I  |`  B )
1412, 13syl6eq 2483 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  `' F  =  (  _I  |`  B ) )  ->  F  =  (  _I  |`  B ) )
1514ex 424 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( `' F  =  (  _I  |`  B )  ->  F  =  (  _I  |`  B )
) )
1615necon3d 2636 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    _I cid 4485   `'ccnv 4869    |` cres 4872   Rel wrel 4875   -1-1-onto->wf1o 5445   ` cfv 5446   Basecbs 13459   HLchlt 30049   LHypclh 30682   LTrncltrn 30799
This theorem is referenced by:  cdlemh2  31514  cdlemh  31515  cdlemkfid1N  31619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-laut 30687  df-ldil 30802  df-ltrn 30803
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