Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendo0cl Unicode version

Theorem tendo0cl 30904
Description: The additive identity is a trace-perserving endormorphism. (Contributed by NM, 12-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
tendo0cl  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
Distinct variable groups:    B, f    T, f
Allowed substitution hints:    E( f)    H( f)    K( f)    O( f)    W( f)

Proof of Theorem tendo0cl
Dummy variables  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2387 . 2  |-  ( le
`  K )  =  ( le `  K
)
2 tendo0.h . 2  |-  H  =  ( LHyp `  K
)
3 tendo0.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2387 . 2  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
5 tendo0.e . 2  |-  E  =  ( ( TEndo `  K
) `  W )
6 id 20 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 tendo0.b . . . . 5  |-  B  =  ( Base `  K
)
87, 2, 3idltrn 30264 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
98adantr 452 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T
)  ->  (  _I  |`  B )  e.  T
)
10 tendo0.o . . . 4  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
1110tendo0cbv 30900 . . 3  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
129, 11fmptd 5832 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O : T --> T )
137, 2, 3, 5, 10tendo0co2 30902 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  h  e.  T
)  ->  ( O `  ( g  o.  h
) )  =  ( ( O `  g
)  o.  ( O `
 h ) ) )
147, 2, 3, 5, 10, 1, 4tendo0tp 30903 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  ( O `  g
) ) ( le
`  K ) ( ( ( trL `  K
) `  W ) `  g ) )
151, 2, 3, 4, 5, 6, 12, 13, 14istendod 30876 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    e. cmpt 4207    _I cid 4434    |` cres 4820   ` cfv 5394   Basecbs 13396   lecple 13463   HLchlt 29465   LHypclh 30098   LTrncltrn 30215   trLctrl 30272   TEndoctendo 30866
This theorem is referenced by:  tendo0pl  30905  tendo0plr  30906  tendoipl  30911  tendoid0  30939  tendo0mul  30940  tendo0mulr  30941  tendoex  31089  cdleml5N  31094  erngdvlem1  31102  erngdvlem4  31105  erng0g  31108  erngdvlem1-rN  31110  erngdvlem4-rN  31113  dvh0g  31226  dvhopN  31231  dib1dim  31280  dib1dim2  31283  dibss  31284  diblss  31285  diblsmopel  31286  dicn0  31307  cdlemn4  31313  cdlemn4a  31314  cdlemn6  31317  dihopelvalcpre  31363  dihmeetlem4preN  31421  dihatlat  31449  dihatexv  31453
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-map 6956  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-lplanes 29613  df-lvols 29614  df-lines 29615  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-lhyp 30102  df-laut 30103  df-ldil 30218  df-ltrn 30219  df-trl 30273  df-tendo 30869
  Copyright terms: Public domain W3C validator