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Theorem tendo0co2 30954
Description: The additive identity trace-perserving endormorphism preserves composition of translations. TODO: why isn't this a special case of tendospdi1 31187? (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
tendo0.b  |-  B  =  ( Base `  K
)
tendo0.h  |-  H  =  ( LHyp `  K
)
tendo0.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendo0.e  |-  E  =  ( ( TEndo `  K
) `  W )
tendo0.o  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
Assertion
Ref Expression
tendo0co2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( O `  ( F  o.  G
) )  =  ( ( O `  F
)  o.  ( O `
 G ) ) )
Distinct variable groups:    B, f    T, f
Allowed substitution hints:    E( f)    F( f)    G( f)    H( f)    K( f)    O( f)    W( f)

Proof of Theorem tendo0co2
StepHypRef Expression
1 tendo0.h . . . 4  |-  H  =  ( LHyp `  K
)
2 tendo0.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
31, 2ltrnco 30885 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( F  o.  G )  e.  T
)
4 tendo0.o . . . 4  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
5 tendo0.b . . . 4  |-  B  =  ( Base `  K
)
64, 5tendo02 30953 . . 3  |-  ( ( F  o.  G )  e.  T  ->  ( O `  ( F  o.  G ) )  =  (  _I  |`  B ) )
73, 6syl 16 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( O `  ( F  o.  G
) )  =  (  _I  |`  B )
)
84, 5tendo02 30953 . . . . 5  |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
983ad2ant2 979 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( O `  F )  =  (  _I  |`  B )
)
104, 5tendo02 30953 . . . . 5  |-  ( G  e.  T  ->  ( O `  G )  =  (  _I  |`  B ) )
11103ad2ant3 980 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( O `  G )  =  (  _I  |`  B )
)
129, 11coeq12d 4971 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( ( O `  F )  o.  ( O `  G
) )  =  ( (  _I  |`  B )  o.  (  _I  |`  B ) ) )
13 f1oi 5647 . . . 4  |-  (  _I  |`  B ) : B -1-1-onto-> B
14 f1of 5608 . . . 4  |-  ( (  _I  |`  B ) : B -1-1-onto-> B  ->  (  _I  |`  B ) : B --> B )
15 fcoi1 5551 . . . 4  |-  ( (  _I  |`  B ) : B --> B  ->  (
(  _I  |`  B )  o.  (  _I  |`  B ) )  =  (  _I  |`  B ) )
1613, 14, 15mp2b 10 . . 3  |-  ( (  _I  |`  B )  o.  (  _I  |`  B ) )  =  (  _I  |`  B )
1712, 16syl6req 2430 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  (  _I  |`  B )  =  ( ( O `  F
)  o.  ( O `
 G ) ) )
187, 17eqtrd 2413 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( O `  ( F  o.  G
) )  =  ( ( O `  F
)  o.  ( O `
 G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    e. cmpt 4201    _I cid 4428    |` cres 4814    o. ccom 4816   -->wf 5384   -1-1-onto->wf1o 5387   ` cfv 5388   Basecbs 13390   HLchlt 29517   LHypclh 30150   LTrncltrn 30267   TEndoctendo 30918
This theorem is referenced by:  tendo0cl  30956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rmo 2651  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-iun 4031  df-iin 4032  df-br 4148  df-opab 4202  df-mpt 4203  df-id 4433  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-undef 6473  df-riota 6479  df-map 6950  df-poset 14324  df-plt 14336  df-lub 14352  df-glb 14353  df-join 14354  df-meet 14355  df-p0 14389  df-p1 14390  df-lat 14396  df-clat 14458  df-oposet 29343  df-ol 29345  df-oml 29346  df-covers 29433  df-ats 29434  df-atl 29465  df-cvlat 29489  df-hlat 29518  df-llines 29664  df-lplanes 29665  df-lvols 29666  df-lines 29667  df-psubsp 29669  df-pmap 29670  df-padd 29962  df-lhyp 30154  df-laut 30155  df-ldil 30270  df-ltrn 30271  df-trl 30325
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