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Theorem tendo1mul 30935
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
tendo1mul  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( (  _I  |`  T )  o.  U )  =  U )

Proof of Theorem tendo1mul
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 30928 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  U : T
--> T )
5 fcoi2 5551 . 2  |-  ( U : T --> T  -> 
( (  _I  |`  T )  o.  U )  =  U )
64, 5syl 16 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( (  _I  |`  T )  o.  U )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    _I cid 4427    |` cres 4813    o. ccom 4815   -->wf 5383   ` cfv 5387   HLchlt 29516   LHypclh 30149   LTrncltrn 30266   TEndoctendo 30917
This theorem is referenced by:  erng1lem  31152  erngdvlem3  31155  erngdvlem3-rN  31163  erngdvlem4-rN  31164
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-map 6949  df-tendo 30920
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