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Theorem tendo1mulr 34723
Description: Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)
Hypotheses
Ref Expression
tendof.h  |-  H  =  ( LHyp `  K
)
tendof.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendof.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
tendo1mulr  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U  o.  (  _I  |`  T ) )  =  U )

Proof of Theorem tendo1mulr
StepHypRef Expression
1 tendof.h . . 3  |-  H  =  ( LHyp `  K
)
2 tendof.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
3 tendof.e . . 3  |-  E  =  ( ( TEndo `  K
) `  W )
41, 2, 3tendof 34715 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  U : T
--> T )
5 fcoi1 5685 . 2  |-  ( U : T --> T  -> 
( U  o.  (  _I  |`  T ) )  =  U )
64, 5syl 16 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U  o.  (  _I  |`  T ) )  =  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    _I cid 4731    |` cres 4942    o. ccom 4944   -->wf 5514   ` cfv 5518   HLchlt 33303   LHypclh 33936   LTrncltrn 34053   TEndoctendo 34704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-map 7318  df-tendo 34707
This theorem is referenced by:  erng1lem  34939  erngdvlem3  34942  erngdvlem3-rN  34950  erngdvlem4-rN  34951  dvhopN  35069  diclspsn  35147  dih1dimatlem0  35281
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