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Theorem tendo0pl 34440
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 457 . 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 34439 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
87adantr 465 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  O  e.  E )
9 simpr 461 . . 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 34430 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  O  e.  E  /\  S  e.  E
)  ->  ( O P S )  e.  E
)
121, 8, 9, 11syl3anc 1218 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( O P S )  e.  E
)
13 simpll 753 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
1413, 7syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  O  e.  E )
15 simplr 754 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  S  e.  E )
16 simpr 461 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  g  e.  T )
1710, 4tendopl2 34426 . . . . 5  |-  ( ( O  e.  E  /\  S  e.  E  /\  g  e.  T )  ->  ( ( O P S ) `  g
)  =  ( ( O `  g )  o.  ( S `  g ) ) )
1814, 15, 16, 17syl3anc 1218 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  (
( O P S ) `  g )  =  ( ( O `
 g )  o.  ( S `  g
) ) )
196, 2tendo02 34436 . . . . . 6  |-  ( g  e.  T  ->  ( O `  g )  =  (  _I  |`  B ) )
2019adantl 466 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  ( O `  g )  =  (  _I  |`  B ) )
2120coeq1d 5006 . . . 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 34416 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  g  e.  T
)  ->  ( S `  g )  e.  T
)
23223expa 1187 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  ( S `  g )  e.  T )
242, 3, 4ltrn1o 33773 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S `  g )  e.  T
)  ->  ( S `  g ) : B -1-1-onto-> B
)
2513, 23, 24syl2anc 661 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  ( S `  g ) : B -1-1-onto-> B )
26 f1of 5646 . . . . 5  |-  ( ( S `  g ) : B -1-1-onto-> B  ->  ( S `  g ) : B --> B )
27 fcoi2 5591 . . . . 5  |-  ( ( S `  g ) : B --> B  -> 
( (  _I  |`  B )  o.  ( S `  g ) )  =  ( S `  g
) )
2825, 26, 273syl 20 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  (
(  _I  |`  B )  o.  ( S `  g ) )  =  ( S `  g
) )
2918, 21, 283eqtrd 2479 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  (
( O P S ) `  g )  =  ( S `  g ) )
3029ralrimiva 2804 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  A. g  e.  T  ( ( O P S ) `  g )  =  ( S `  g ) )
313, 4, 5tendoeq1 34413 . 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 1223 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( O P S )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720    e. cmpt 4355    _I cid 4636    |` cres 4847    o. ccom 4849   -->wf 5419   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   Basecbs 14179   HLchlt 33000   LHypclh 33633   LTrncltrn 33750   TEndoctendo 34401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-riotaBAD 32609
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-undef 6797  df-map 7221  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-p1 15215  df-lat 15221  df-clat 15283  df-oposet 32826  df-ol 32828  df-oml 32829  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001  df-llines 33147  df-lplanes 33148  df-lvols 33149  df-lines 33150  df-psubsp 33152  df-pmap 33153  df-padd 33445  df-lhyp 33637  df-laut 33638  df-ldil 33753  df-ltrn 33754  df-trl 33808  df-tendo 34404
This theorem is referenced by:  tendo0plr  34441  erngdvlem1  34637  erngdvlem4  34640  erng0g  34643  erngdvlem1-rN  34645  erngdvlem4-rN  34648  dvh0g  34761  dvhopN  34766  diblss  34820  diblsmopel  34821
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