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Theorem tendo0mul 36023
Description: Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 1-Aug-2013.)
Hypotheses
Ref Expression
tendoid0.b  |-  B  =  ( Base `  K
)
tendoid0.h  |-  H  =  ( LHyp `  K
)
tendoid0.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendoid0.e  |-  E  =  ( ( TEndo `  K
) `  W )
tendoid0.o  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
Assertion
Ref Expression
tendo0mul  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( O  o.  U )  =  O )
Distinct variable groups:    B, f    T, f
Allowed substitution hints:    U( f)    E( f)    H( f)    K( f)    O( f)    W( f)

Proof of Theorem tendo0mul
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 tendoid0.b . . . 4  |-  B  =  ( Base `  K
)
2 tendoid0.h . . . 4  |-  H  =  ( LHyp `  K
)
3 tendoid0.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
41, 2, 3cdlemftr0 35765 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. g  e.  T  g  =/=  (  _I  |`  B ) )
54adantr 465 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  E. g  e.  T  g  =/=  (  _I  |`  B ) )
6 simpll 753 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 tendoid0.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
8 tendoid0.o . . . . . 6  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
91, 2, 3, 7, 8tendo0cl 35987 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
109ad2antrr 725 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  O  e.  E )
11 simplr 754 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  U  e.  E )
122, 7tendococl 35969 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  O  e.  E  /\  U  e.  E
)  ->  ( O  o.  U )  e.  E
)
136, 10, 11, 12syl3anc 1228 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( O  o.  U )  e.  E )
14 simprl 755 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  g  e.  T )
152, 3, 7tendocl 35964 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  g  e.  T
)  ->  ( U `  g )  e.  T
)
166, 11, 14, 15syl3anc 1228 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( U `  g )  e.  T )
178, 1tendo02 35984 . . . . 5  |-  ( ( U `  g )  e.  T  ->  ( O `  ( U `  g ) )  =  (  _I  |`  B ) )
1816, 17syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( O `  ( U `  g ) )  =  (  _I  |`  B ) )
192, 3, 7tendocoval 35963 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( O  e.  E  /\  U  e.  E )  /\  g  e.  T )  ->  (
( O  o.  U
) `  g )  =  ( O `  ( U `  g ) ) )
206, 10, 11, 14, 19syl121anc 1233 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
( O  o.  U
) `  g )  =  ( O `  ( U `  g ) ) )
218, 1tendo02 35984 . . . . 5  |-  ( g  e.  T  ->  ( O `  g )  =  (  _I  |`  B ) )
2221ad2antrl 727 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( O `  g )  =  (  _I  |`  B ) )
2318, 20, 223eqtr4d 2518 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
( O  o.  U
) `  g )  =  ( O `  g ) )
24 simpr 461 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )
251, 2, 3, 7tendocan 36021 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( O  o.  U )  e.  E  /\  O  e.  E  /\  ( ( O  o.  U ) `
 g )  =  ( O `  g
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( O  o.  U )  =  O )
266, 13, 10, 23, 24, 25syl131anc 1241 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( O  o.  U )  =  O )
275, 26rexlimddv 2963 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( O  o.  U )  =  O )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818    |-> cmpt 4511    _I cid 4796    |` cres 5007    o. ccom 5009   ` cfv 5594   Basecbs 14507   HLchlt 34548   LHypclh 35181   LTrncltrn 35298   TEndoctendo 35949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-riotaBAD 34157
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-undef 7014  df-map 7434  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696  df-lvols 34697  df-lines 34698  df-psubsp 34700  df-pmap 34701  df-padd 34993  df-lhyp 35185  df-laut 35186  df-ldil 35301  df-ltrn 35302  df-trl 35356  df-tendo 35952
This theorem is referenced by:  cdleml5N  36177  cdleml9  36181
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