Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ltrncnvnid Structured version   Unicode version

Theorem ltrncnvnid 33125
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 999 . 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 33122 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
653adant3 1017 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  F : B -1-1-onto-> B
)
7 f1orel 5758 . . . . . . . 8  |-  ( F : B -1-1-onto-> B  ->  Rel  F )
86, 7syl 17 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  Rel  F )
9 dfrel2 5395 . . . . . . 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 5118 . . . . . 6  |-  ( `' F  =  (  _I  |`  B )  ->  `' `' F  =  `' (  _I  |`  B ) )
1210, 11sylan9req 2464 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  `' F  =  (  _I  |`  B ) )  ->  F  =  `' (  _I  |`  B ) )
13 cnvresid 5595 . . . . 5  |-  `' (  _I  |`  B )  =  (  _I  |`  B )
1412, 13syl6eq 2459 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  `' F  =  (  _I  |`  B ) )  ->  F  =  (  _I  |`  B ) )
1514ex 432 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( `' F  =  (  _I  |`  B )  ->  F  =  (  _I  |`  B )
) )
1615necon3d 2627 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598    _I cid 4732   `'ccnv 4941    |` cres 4944   Rel wrel 4947   -1-1-onto->wf1o 5524   ` cfv 5525   Basecbs 14733   HLchlt 32349   LHypclh 32982   LTrncltrn 33099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-map 7379  df-laut 32987  df-ldil 33102  df-ltrn 33103
This theorem is referenced by:  cdlemh2  33816  cdlemh  33817  cdlemkfid1N  33921
  Copyright terms: Public domain W3C validator