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Theorem tendo0pl 30907
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 30906 . . . 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 30897 . . 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 30893 . . . . 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 30903 . . . . . 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 4976 . . . 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 30883 . . . . . . 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 30240 . . . . . 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 5616 . . . . 5  |-  ( ( S `  g ) : B -1-1-onto-> B  ->  ( S `  g ) : B --> B )
27 fcoi2 5560 . . . . 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 2425 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E )  /\  g  e.  T )  ->  (
( O P S ) `  g )  =  ( S `  g ) )
3029ralrimiva 2734 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  A. g  e.  T  ( ( O P S ) `  g )  =  ( S `  g ) )
313, 4, 5tendoeq1 30880 . 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 1649    e. wcel 1717   A.wral 2651    e. cmpt 4209    _I cid 4436    |` cres 4822    o. ccom 4824   -->wf 5392   -1-1-onto->wf1o 5395   ` cfv 5396  (class class class)co 6022    e. cmpt2 6024   Basecbs 13398   HLchlt 29467   LHypclh 30100   LTrncltrn 30217   TEndoctendo 30868
This theorem is referenced by:  tendo0plr  30908  erngdvlem1  31104  erngdvlem4  31107  erng0g  31110  erngdvlem1-rN  31112  erngdvlem4-rN  31115  dvh0g  31228  dvhopN  31233  diblss  31287  diblsmopel  31288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-map 6958  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-llines 29614  df-lplanes 29615  df-lvols 29616  df-lines 29617  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104  df-laut 30105  df-ldil 30220  df-ltrn 30221  df-trl 30275  df-tendo 30871
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