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Theorem tendo0cl 30979
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 2283 . 2  |-  ( le
`  K )  =  ( le `  K
)
2 tendo0.h . 2  |-  H  =  ( LHyp `  K
)
3 tendo0.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2283 . 2  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
5 tendo0.e . 2  |-  E  =  ( ( TEndo `  K
) `  W )
6 id 19 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 tendo0.b . . . . 5  |-  B  =  ( Base `  K
)
87, 2, 3idltrn 30339 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
98adantr 451 . . 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 30975 . . 3  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
129, 11fmptd 5684 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O : T --> T )
137, 2, 3, 5, 10tendo0co2 30977 . 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 30978 . 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 30951 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077    _I cid 4304    |` cres 4691   ` cfv 5255   Basecbs 13148   lecple 13215   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   TEndoctendo 30941
This theorem is referenced by:  tendo0pl  30980  tendo0plr  30981  tendoipl  30986  tendoid0  31014  tendo0mul  31015  tendo0mulr  31016  tendoex  31164  cdleml5N  31169  erngdvlem1  31177  erngdvlem4  31180  erng0g  31183  erngdvlem1-rN  31185  erngdvlem4-rN  31188  dvh0g  31301  dvhopN  31306  dib1dim  31355  dib1dim2  31358  dibss  31359  diblss  31360  diblsmopel  31361  dicn0  31382  cdlemn4  31388  cdlemn4a  31389  cdlemn6  31392  dihopelvalcpre  31438  dihmeetlem4preN  31496  dihatlat  31524  dihatexv  31528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944
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