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Theorem ltrncoidN 30764
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 960 . . . . . . . 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 962 . . . . . . . 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 30760 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G : B
-1-1-onto-> B )
71, 2, 6syl2anc 643 . . . . . . 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 5695 . . . . . . 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 5026 . . . . 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 961 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  ->  F  e.  T )
123, 4, 5ltrn1o 30760 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
131, 11, 12syl2anc 643 . . . . . 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 5665 . . . . . 6  |-  ( F : B -1-1-onto-> B  ->  F : B
--> B )
15 fcoi1 5608 . . . . . 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 2468 . . . 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 5379 . . . 4  |-  ( ( F  o.  `' G
)  o.  G )  =  ( F  o.  ( `' G  o.  G
) )
1917, 18syl6eqr 2485 . . 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 448 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  -> 
( F  o.  `' G )  =  (  _I  |`  B )
)
2120coeq1d 5025 . . . 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 5665 . . . . 5  |-  ( G : B -1-1-onto-> B  ->  G : B
--> B )
23 fcoi2 5609 . . . . 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 2467 . . 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 2467 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  ->  F  =  G )
27 simpr 448 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  F  =  G )
2827coeq1d 5025 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( F  o.  `' G
)  =  ( G  o.  `' G ) )
29 simpl1 960 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( K  e.  HL  /\  W  e.  H ) )
30 simpl3 962 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  G  e.  T )
3129, 30, 6syl2anc 643 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  G : B -1-1-onto-> B )
32 f1ococnv2 5693 . . . 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 2467 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( F  o.  `' G
)  =  (  _I  |`  B ) )
3526, 34impbida 806 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    _I cid 4485   `'ccnv 4868    |` cres 4871    o. ccom 4873   -->wf 5441   -1-1-onto->wf1o 5444   ` cfv 5445   Basecbs 13457   HLchlt 29987   LHypclh 30620   LTrncltrn 30737
This theorem is referenced by:  tendospcanN  31660
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-map 7011  df-laut 30625  df-ldil 30740  df-ltrn 30741
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