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Theorem tendo0pl 30980
Description: Property of the additive identity 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 ) )
tendo0pl.p  |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) )
Assertion
Ref Expression
tendo0pl  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( O P S )  =  S )
Distinct variable groups:    B, f    T, f    t, s, E    T, s, t, f    f, W, s, t
Allowed substitution hints:    B( t, s)    P( t, f, s)    S( t, f, s)    E( f)    H( t, f, s)    K( t, f, s)    O( t, f, s)

Proof of Theorem tendo0pl
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 simpl 443 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( K  e.  HL  /\  W  e.  H ) )
2 tendo0.b . . . . 5  |-  B  =  ( Base `  K
)
3 tendo0.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 tendo0.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
5 tendo0.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
6 tendo0.o . . . . 5  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
72, 3, 4, 5, 6tendo0cl 30979 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
87adantr 451 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  O  e.  E )
9 simpr 447 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  S  e.  E )
10 tendo0pl.p . . . 4  |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) )
113, 4, 5, 10tendoplcl 30970 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  O  e.  E  /\  S  e.  E
)  ->  ( O P S )  e.  E
)
121, 8, 9, 11syl3anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( O P S )  e.  E
)
13 simpll 730 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
1413, 7syl 15 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  O  e.  E )
15 simplr 731 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  S  e.  E )
16 simpr 447 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  g  e.  T )
1710, 4tendopl2 30966 . . . . 5  |-  ( ( O  e.  E  /\  S  e.  E  /\  g  e.  T )  ->  ( ( O P S ) `  g
)  =  ( ( O `  g )  o.  ( S `  g ) ) )
1814, 15, 16, 17syl3anc 1182 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  (
( O P S ) `  g )  =  ( ( O `
 g )  o.  ( S `  g
) ) )
196, 2tendo02 30976 . . . . . 6  |-  ( g  e.  T  ->  ( O `  g )  =  (  _I  |`  B ) )
2019adantl 452 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  ( O `  g )  =  (  _I  |`  B ) )
2120coeq1d 4845 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  (
( O `  g
)  o.  ( S `
 g ) )  =  ( (  _I  |`  B )  o.  ( S `  g )
) )
223, 4, 5tendocl 30956 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  g  e.  T
)  ->  ( S `  g )  e.  T
)
23223expa 1151 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  ( S `  g )  e.  T )
242, 3, 4ltrn1o 30313 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S `  g )  e.  T
)  ->  ( S `  g ) : B -1-1-onto-> B
)
2513, 23, 24syl2anc 642 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  ( S `  g ) : B -1-1-onto-> B )
26 f1of 5472 . . . . 5  |-  ( ( S `  g ) : B -1-1-onto-> B  ->  ( S `  g ) : B --> B )
27 fcoi2 5416 . . . . 5  |-  ( ( S `  g ) : B --> B  -> 
( (  _I  |`  B )  o.  ( S `  g ) )  =  ( S `  g
) )
2825, 26, 273syl 18 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  (
(  _I  |`  B )  o.  ( S `  g ) )  =  ( S `  g
) )
2918, 21, 283eqtrd 2319 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  (
( O P S ) `  g )  =  ( S `  g ) )
3029ralrimiva 2626 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  A. g  e.  T  ( ( O P S ) `  g )  =  ( S `  g ) )
313, 4, 5tendoeq1 30953 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( O P S )  e.  E  /\  S  e.  E )  /\  A. g  e.  T  (
( O P S ) `  g )  =  ( S `  g ) )  -> 
( O P S )  =  S )
321, 12, 9, 30, 31syl121anc 1187 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( O P S )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    e. cmpt 4077    _I cid 4304    |` cres 4691    o. ccom 4693   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941
This theorem is referenced by:  tendo0plr  30981  erngdvlem1  31177  erngdvlem4  31180  erng0g  31183  erngdvlem1-rN  31185  erngdvlem4-rN  31188  dvh0g  31301  dvhopN  31306  diblss  31360  diblsmopel  31361
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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