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Theorem ltrncnvnid 34110
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 990 . 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 34107 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
653adant3 1008 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  F : B -1-1-onto-> B
)
7 f1orel 5753 . . . . . . . 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 5397 . . . . . . 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 5122 . . . . . 6  |-  ( `' F  =  (  _I  |`  B )  ->  `' `' F  =  `' (  _I  |`  B ) )
1210, 11sylan9req 2516 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  `' F  =  (  _I  |`  B ) )  ->  F  =  `' (  _I  |`  B ) )
13 cnvresid 5597 . . . . 5  |-  `' (  _I  |`  B )  =  (  _I  |`  B )
1412, 13syl6eq 2511 . . . 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 2676 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648    _I cid 4740   `'ccnv 4948    |` cres 4951   Rel wrel 4954   -1-1-onto->wf1o 5526   ` cfv 5527   Basecbs 14293   HLchlt 33334   LHypclh 33967   LTrncltrn 34084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-map 7327  df-laut 33972  df-ldil 34087  df-ltrn 34088
This theorem is referenced by:  cdlemh2  34799  cdlemh  34800  cdlemkfid1N  34904
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