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Theorem ltrncoidN 33606
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 1008 . . . . . . . 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 1010 . . . . . . . 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 33602 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G : B
-1-1-onto-> B )
71, 2, 6syl2anc 665 . . . . . . 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 5851 . . . . . . 7  |-  ( G : B -1-1-onto-> B  ->  ( `' G  o.  G )  =  (  _I  |`  B ) )
97, 8syl 17 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  -> 
( `' G  o.  G )  =  (  _I  |`  B )
)
109coeq2d 5009 . . . . 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 1009 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  ->  F  e.  T )
123, 4, 5ltrn1o 33602 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
131, 11, 12syl2anc 665 . . . . . 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 5823 . . . . . 6  |-  ( F : B -1-1-onto-> B  ->  F : B
--> B )
15 fcoi1 5766 . . . . . 6  |-  ( F : B --> B  -> 
( F  o.  (  _I  |`  B ) )  =  F )
1613, 14, 153syl 18 . . . . 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 2462 . . . 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 5366 . . . 4  |-  ( ( F  o.  `' G
)  o.  G )  =  ( F  o.  ( `' G  o.  G
) )
1917, 18syl6eqr 2479 . . 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 462 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  -> 
( F  o.  `' G )  =  (  _I  |`  B )
)
2120coeq1d 5008 . . . 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 5823 . . . . 5  |-  ( G : B -1-1-onto-> B  ->  G : B
--> B )
23 fcoi2 5767 . . . . 5  |-  ( G : B --> B  -> 
( (  _I  |`  B )  o.  G )  =  G )
247, 22, 233syl 18 . . . 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 2461 . . 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 2461 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( F  o.  `' G
)  =  (  _I  |`  B ) )  ->  F  =  G )
27 simpr 462 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  F  =  G )
2827coeq1d 5008 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( F  o.  `' G
)  =  ( G  o.  `' G ) )
29 simpl1 1008 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( K  e.  HL  /\  W  e.  H ) )
30 simpl3 1010 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  G  e.  T )
3129, 30, 6syl2anc 665 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  G : B -1-1-onto-> B )
32 f1ococnv2 5849 . . . 4  |-  ( G : B -1-1-onto-> B  ->  ( G  o.  `' G )  =  (  _I  |`  B )
)
3331, 32syl 17 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( G  o.  `' G
)  =  (  _I  |`  B ) )
3428, 33eqtrd 2461 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  G )  ->  ( F  o.  `' G
)  =  (  _I  |`  B ) )
3526, 34impbida 840 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    _I cid 4756   `'ccnv 4845    |` cres 4848    o. ccom 4850   -->wf 5589   -1-1-onto->wf1o 5592   ` cfv 5593   Basecbs 15099   HLchlt 32829   LHypclh 33462   LTrncltrn 33579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6300  df-oprab 6301  df-mpt2 6302  df-map 7474  df-laut 33467  df-ldil 33582  df-ltrn 33583
This theorem is referenced by:  tendospcanN  34504
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