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Theorem tendo0cl 31524
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 2435 . 2  |-  ( le
`  K )  =  ( le `  K
)
2 tendo0.h . 2  |-  H  =  ( LHyp `  K
)
3 tendo0.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2435 . 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 30884 . . . 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 31520 . . 3  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
129, 11fmptd 5885 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O : T --> T )
137, 2, 3, 5, 10tendo0co2 31522 . 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 31523 . 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 31496 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    e. cmpt 4258    _I cid 4485    |` cres 4872   ` cfv 5446   Basecbs 13461   lecple 13528   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   trLctrl 30892   TEndoctendo 31486
This theorem is referenced by:  tendo0pl  31525  tendo0plr  31526  tendoipl  31531  tendoid0  31559  tendo0mul  31560  tendo0mulr  31561  tendoex  31709  cdleml5N  31714  erngdvlem1  31722  erngdvlem4  31725  erng0g  31728  erngdvlem1-rN  31730  erngdvlem4-rN  31733  dvh0g  31846  dvhopN  31851  dib1dim  31900  dib1dim2  31903  dibss  31904  diblss  31905  diblsmopel  31906  dicn0  31927  cdlemn4  31933  cdlemn4a  31934  cdlemn6  31937  dihopelvalcpre  31983  dihmeetlem4preN  32041  dihatlat  32069  dihatexv  32073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-tendo 31489
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