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Theorem ltrncoidN 34924
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 999 . . . . . . . 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 1001 . . . . . . . 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 34920 . . . . . . . 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 5842 . . . . . . 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 5163 . . . . 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 1000 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  ->  F  e.  T )
123, 4, 5ltrn1o 34920 . . . . . . 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 5814 . . . . . 6  |-  ( F : B -1-1-onto-> B  ->  F : B
--> B )
15 fcoi1 5757 . . . . . 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 2509 . . . 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 5524 . . . 4  |-  ( ( F  o.  `' G
)  o.  G )  =  ( F  o.  ( `' G  o.  G
) )
1917, 18syl6eqr 2526 . . 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 5162 . . . 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 5814 . . . . 5  |-  ( G : B -1-1-onto-> B  ->  G : B
--> B )
23 fcoi2 5758 . . . . 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 2508 . . 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 2508 . 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 5162 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( F  o.  `' G
)  =  ( G  o.  `' G ) )
29 simpl1 999 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( K  e.  HL  /\  W  e.  H ) )
30 simpl3 1001 . . . . 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 5840 . . . 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 2508 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( F  o.  `' G
)  =  (  _I  |`  B ) )
3526, 34impbida 830 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 973    = wceq 1379    e. wcel 1767    _I cid 4790   `'ccnv 4998    |` cres 5001    o. ccom 5003   -->wf 5582   -1-1-onto->wf1o 5585   ` cfv 5586   Basecbs 14483   HLchlt 34147   LHypclh 34780   LTrncltrn 34897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-laut 34785  df-ldil 34900  df-ltrn 34901
This theorem is referenced by:  tendospcanN  35820
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