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Theorem tendo0cl 31601
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 2296 . 2  |-  ( le
`  K )  =  ( le `  K
)
2 tendo0.h . 2  |-  H  =  ( LHyp `  K
)
3 tendo0.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2296 . 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 30961 . . . 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 31597 . . 3  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
129, 11fmptd 5700 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O : T --> T )
137, 2, 3, 5, 10tendo0co2 31599 . 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 31600 . 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 31573 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093    _I cid 4320    |` cres 4707   ` cfv 5271   Basecbs 13164   lecple 13231   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   TEndoctendo 31563
This theorem is referenced by:  tendo0pl  31602  tendo0plr  31603  tendoipl  31608  tendoid0  31636  tendo0mul  31637  tendo0mulr  31638  tendoex  31786  cdleml5N  31791  erngdvlem1  31799  erngdvlem4  31802  erng0g  31805  erngdvlem1-rN  31807  erngdvlem4-rN  31810  dvh0g  31923  dvhopN  31928  dib1dim  31977  dib1dim2  31980  dibss  31981  diblss  31982  diblsmopel  31983  dicn0  32004  cdlemn4  32010  cdlemn4a  32011  cdlemn6  32014  dihopelvalcpre  32060  dihmeetlem4preN  32118  dihatlat  32146  dihatexv  32150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566
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