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Theorem ltrncoidN 33772
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 991 . . . . . . . 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 993 . . . . . . . 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 33768 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G : B
-1-1-onto-> B )
71, 2, 6syl2anc 661 . . . . . . 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 5669 . . . . . . 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 5002 . . . . 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 992 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  ->  F  e.  T )
123, 4, 5ltrn1o 33768 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
131, 11, 12syl2anc 661 . . . . . 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 5641 . . . . . 6  |-  ( F : B -1-1-onto-> B  ->  F : B
--> B )
15 fcoi1 5585 . . . . . 6  |-  ( F : B --> B  -> 
( F  o.  (  _I  |`  B ) )  =  F )
1613, 14, 153syl 20 . . . . 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 2476 . . . 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 5356 . . . 4  |-  ( ( F  o.  `' G
)  o.  G )  =  ( F  o.  ( `' G  o.  G
) )
1917, 18syl6eqr 2493 . . 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 461 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  -> 
( F  o.  `' G )  =  (  _I  |`  B )
)
2120coeq1d 5001 . . . 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 5641 . . . . 5  |-  ( G : B -1-1-onto-> B  ->  G : B
--> B )
23 fcoi2 5586 . . . . 5  |-  ( G : B --> B  -> 
( (  _I  |`  B )  o.  G )  =  G )
247, 22, 233syl 20 . . . 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 2475 . . 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 2475 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  ->  F  =  G )
27 simpr 461 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  F  =  G )
2827coeq1d 5001 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( F  o.  `' G
)  =  ( G  o.  `' G ) )
29 simpl1 991 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( K  e.  HL  /\  W  e.  H ) )
30 simpl3 993 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  G  e.  T )
3129, 30, 6syl2anc 661 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  G : B -1-1-onto-> B )
32 f1ococnv2 5667 . . . 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 2475 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( F  o.  `' G
)  =  (  _I  |`  B ) )
3526, 34impbida 828 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( ( F  o.  `' G
)  =  (  _I  |`  B )  <->  F  =  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    _I cid 4631   `'ccnv 4839    |` cres 4842    o. ccom 4844   -->wf 5414   -1-1-onto->wf1o 5417   ` cfv 5418   Basecbs 14174   HLchlt 32995   LHypclh 33628   LTrncltrn 33745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-map 7216  df-laut 33633  df-ldil 33748  df-ltrn 33749
This theorem is referenced by:  tendospcanN  34668
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