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Theorem tendo0cl 35987
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 2467 . 2  |-  ( le
`  K )  =  ( le `  K
)
2 tendo0.h . 2  |-  H  =  ( LHyp `  K
)
3 tendo0.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2467 . 2  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
5 tendo0.e . 2  |-  E  =  ( ( TEndo `  K
) `  W )
6 id 22 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 tendo0.b . . . . 5  |-  B  =  ( Base `  K
)
87, 2, 3idltrn 35347 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
98adantr 465 . . 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 35983 . . 3  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
129, 11fmptd 6056 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O : T --> T )
137, 2, 3, 5, 10tendo0co2 35985 . 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 35986 . 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 35959 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4511    _I cid 4796    |` cres 5007   ` cfv 5594   Basecbs 14507   lecple 14579   HLchlt 34548   LHypclh 35181   LTrncltrn 35298   trLctrl 35355   TEndoctendo 35949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-riotaBAD 34157
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-undef 7014  df-map 7434  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696  df-lvols 34697  df-lines 34698  df-psubsp 34700  df-pmap 34701  df-padd 34993  df-lhyp 35185  df-laut 35186  df-ldil 35301  df-ltrn 35302  df-trl 35356  df-tendo 35952
This theorem is referenced by:  tendo0pl  35988  tendo0plr  35989  tendoipl  35994  tendoid0  36022  tendo0mul  36023  tendo0mulr  36024  tendoex  36172  cdleml5N  36177  erngdvlem1  36185  erngdvlem4  36188  erng0g  36191  erngdvlem1-rN  36193  erngdvlem4-rN  36196  dvh0g  36309  dvhopN  36314  dib1dim  36363  dib1dim2  36366  dibss  36367  diblss  36368  diblsmopel  36369  dicn0  36390  cdlemn4  36396  cdlemn4a  36397  cdlemn6  36400  dihopelvalcpre  36446  dihmeetlem4preN  36504  dihatlat  36532  dihatexv  36536
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