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

Theorem ltrncoidN 30999
Description: Two translations are equal if the composition of one with the converse of the other is the zero translation. This is an analog of vector subtraction. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
ltrn1o.b  |-  B  =  ( Base `  K
)
ltrn1o.h  |-  H  =  ( LHyp `  K
)
ltrn1o.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrncoidN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( ( F  o.  `' G
)  =  (  _I  |`  B )  <->  F  =  G ) )

Proof of Theorem ltrncoidN
StepHypRef Expression
1 simpl1 961 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simpl3 963 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  ->  G  e.  T )
3 ltrn1o.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
4 ltrn1o.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
5 ltrn1o.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
63, 4, 5ltrn1o 30995 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G : B
-1-1-onto-> B )
71, 2, 6syl2anc 644 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  ->  G : B -1-1-onto-> B )
8 f1ococnv1 5707 . . . . . . 7  |-  ( G : B -1-1-onto-> B  ->  ( `' G  o.  G )  =  (  _I  |`  B ) )
97, 8syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  -> 
( `' G  o.  G )  =  (  _I  |`  B )
)
109coeq2d 5038 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  -> 
( F  o.  ( `' G  o.  G
) )  =  ( F  o.  (  _I  |`  B ) ) )
11 simpl2 962 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  ->  F  e.  T )
123, 4, 5ltrn1o 30995 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
131, 11, 12syl2anc 644 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  ->  F : B -1-1-onto-> B )
14 f1of 5677 . . . . . 6  |-  ( F : B -1-1-onto-> B  ->  F : B
--> B )
15 fcoi1 5620 . . . . . 6  |-  ( F : B --> B  -> 
( F  o.  (  _I  |`  B ) )  =  F )
1613, 14, 153syl 19 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  -> 
( F  o.  (  _I  |`  B ) )  =  F )
1710, 16eqtr2d 2471 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  ->  F  =  ( F  o.  ( `' G  o.  G ) ) )
18 coass 5391 . . . 4  |-  ( ( F  o.  `' G
)  o.  G )  =  ( F  o.  ( `' G  o.  G
) )
1917, 18syl6eqr 2488 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  ->  F  =  ( ( F  o.  `' G
)  o.  G ) )
20 simpr 449 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  -> 
( F  o.  `' G )  =  (  _I  |`  B )
)
2120coeq1d 5037 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  -> 
( ( F  o.  `' G )  o.  G
)  =  ( (  _I  |`  B )  o.  G ) )
22 f1of 5677 . . . . 5  |-  ( G : B -1-1-onto-> B  ->  G : B
--> B )
23 fcoi2 5621 . . . . 5  |-  ( G : B --> B  -> 
( (  _I  |`  B )  o.  G )  =  G )
247, 22, 233syl 19 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  -> 
( (  _I  |`  B )  o.  G )  =  G )
2521, 24eqtrd 2470 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  -> 
( ( F  o.  `' G )  o.  G
)  =  G )
2619, 25eqtrd 2470 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  ->  F  =  G )
27 simpr 449 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  F  =  G )
2827coeq1d 5037 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( F  o.  `' G
)  =  ( G  o.  `' G ) )
29 simpl1 961 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( K  e.  HL  /\  W  e.  H ) )
30 simpl3 963 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  G  e.  T )
3129, 30, 6syl2anc 644 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  G : B -1-1-onto-> B )
32 f1ococnv2 5705 . . . 4  |-  ( G : B -1-1-onto-> B  ->  ( G  o.  `' G )  =  (  _I  |`  B )
)
3331, 32syl 16 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( G  o.  `' G
)  =  (  _I  |`  B ) )
3428, 33eqtrd 2470 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( F  o.  `' G
)  =  (  _I  |`  B ) )
3526, 34impbida 807 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( ( F  o.  `' G
)  =  (  _I  |`  B )  <->  F  =  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    _I cid 4496   `'ccnv 4880    |` cres 4883    o. ccom 4885   -->wf 5453   -1-1-onto->wf1o 5456   ` cfv 5457   Basecbs 13474   HLchlt 30222   LHypclh 30855   LTrncltrn 30972
This theorem is referenced by:  tendospcanN  31895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-laut 30860  df-ldil 30975  df-ltrn 30976
  Copyright terms: Public domain W3C validator