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Theorem tendo0pl 31427
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 444 . 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 31426 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
87adantr 452 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  O  e.  E )
9 simpr 448 . . 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 31417 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  O  e.  E  /\  S  e.  E
)  ->  ( O P S )  e.  E
)
121, 8, 9, 11syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( O P S )  e.  E
)
13 simpll 731 . . . . . 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 732 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  S  e.  E )
16 simpr 448 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  g  e.  T )
1710, 4tendopl2 31413 . . . . 5  |-  ( ( O  e.  E  /\  S  e.  E  /\  g  e.  T )  ->  ( ( O P S ) `  g
)  =  ( ( O `  g )  o.  ( S `  g ) ) )
1814, 15, 16, 17syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  (
( O P S ) `  g )  =  ( ( O `
 g )  o.  ( S `  g
) ) )
196, 2tendo02 31423 . . . . . 6  |-  ( g  e.  T  ->  ( O `  g )  =  (  _I  |`  B ) )
2019adantl 453 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  ( O `  g )  =  (  _I  |`  B ) )
2120coeq1d 5025 . . . 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 31403 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  g  e.  T
)  ->  ( S `  g )  e.  T
)
23223expa 1153 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  ( S `  g )  e.  T )
242, 3, 4ltrn1o 30760 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S `  g )  e.  T
)  ->  ( S `  g ) : B -1-1-onto-> B
)
2513, 23, 24syl2anc 643 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  ( S `  g ) : B -1-1-onto-> B )
26 f1of 5665 . . . . 5  |-  ( ( S `  g ) : B -1-1-onto-> B  ->  ( S `  g ) : B --> B )
27 fcoi2 5609 . . . . 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 2471 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  (
( O P S ) `  g )  =  ( S `  g ) )
3029ralrimiva 2781 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  A. g  e.  T  ( ( O P S ) `  g )  =  ( S `  g ) )
313, 4, 5tendoeq1 31400 . 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 1189 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 359    = wceq 1652    e. wcel 1725   A.wral 2697    e. cmpt 4258    _I cid 4485    |` cres 4871    o. ccom 4873   -->wf 5441   -1-1-onto->wf1o 5444   ` cfv 5445  (class class class)co 6072    e. cmpt2 6074   Basecbs 13457   HLchlt 29987   LHypclh 30620   LTrncltrn 30737   TEndoctendo 31388
This theorem is referenced by:  tendo0plr  31428  erngdvlem1  31624  erngdvlem4  31627  erng0g  31630  erngdvlem1-rN  31632  erngdvlem4-rN  31635  dvh0g  31748  dvhopN  31753  diblss  31807  diblsmopel  31808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-map 7011  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741  df-trl 30795  df-tendo 31391
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