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Theorem tendo0pl 31662
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 445 . 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 31661 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
87adantr 453 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  O  e.  E )
9 simpr 449 . . 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 31652 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  O  e.  E  /\  S  e.  E
)  ->  ( O P S )  e.  E
)
121, 8, 9, 11syl3anc 1185 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( O P S )  e.  E
)
13 simpll 732 . . . . . 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 733 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  S  e.  E )
16 simpr 449 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  g  e.  T )
1710, 4tendopl2 31648 . . . . 5  |-  ( ( O  e.  E  /\  S  e.  E  /\  g  e.  T )  ->  ( ( O P S ) `  g
)  =  ( ( O `  g )  o.  ( S `  g ) ) )
1814, 15, 16, 17syl3anc 1185 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  (
( O P S ) `  g )  =  ( ( O `
 g )  o.  ( S `  g
) ) )
196, 2tendo02 31658 . . . . . 6  |-  ( g  e.  T  ->  ( O `  g )  =  (  _I  |`  B ) )
2019adantl 454 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  ( O `  g )  =  (  _I  |`  B ) )
2120coeq1d 5037 . . . 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 31638 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  g  e.  T
)  ->  ( S `  g )  e.  T
)
23223expa 1154 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  ( S `  g )  e.  T )
242, 3, 4ltrn1o 30995 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S `  g )  e.  T
)  ->  ( S `  g ) : B -1-1-onto-> B
)
2513, 23, 24syl2anc 644 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  ( S `  g ) : B -1-1-onto-> B )
26 f1of 5677 . . . . 5  |-  ( ( S `  g ) : B -1-1-onto-> B  ->  ( S `  g ) : B --> B )
27 fcoi2 5621 . . . . 5  |-  ( ( S `  g ) : B --> B  -> 
( (  _I  |`  B )  o.  ( S `  g ) )  =  ( S `  g
) )
2825, 26, 273syl 19 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  (
(  _I  |`  B )  o.  ( S `  g ) )  =  ( S `  g
) )
2918, 21, 283eqtrd 2474 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  (
( O P S ) `  g )  =  ( S `  g ) )
3029ralrimiva 2791 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  A. g  e.  T  ( ( O P S ) `  g )  =  ( S `  g ) )
313, 4, 5tendoeq1 31635 . 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 1190 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 360    = wceq 1653    e. wcel 1726   A.wral 2707    e. cmpt 4269    _I cid 4496    |` cres 4883    o. ccom 4885   -->wf 5453   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   Basecbs 13474   HLchlt 30222   LHypclh 30855   LTrncltrn 30972   TEndoctendo 31623
This theorem is referenced by:  tendo0plr  31663  erngdvlem1  31859  erngdvlem4  31862  erng0g  31865  erngdvlem1-rN  31867  erngdvlem4-rN  31870  dvh0g  31983  dvhopN  31988  diblss  32042  diblsmopel  32043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-map 7023  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-lvols 30371  df-lines 30372  df-psubsp 30374  df-pmap 30375  df-padd 30667  df-lhyp 30859  df-laut 30860  df-ldil 30975  df-ltrn 30976  df-trl 31030  df-tendo 31626
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