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Theorem tendo0mul 31308
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 31050 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. g  e.  T  g  =/=  (  _I  |`  B ) )
54adantr 452 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  E. g  e.  T  g  =/=  (  _I  |`  B ) )
6 simpll 731 . . 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 31272 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
109ad2antrr 707 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  O  e.  E )
11 simplr 732 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  U  e.  E )
122, 7tendococl 31254 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  O  e.  E  /\  U  e.  E
)  ->  ( O  o.  U )  e.  E
)
136, 10, 11, 12syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( O  o.  U )  e.  E )
14 simprl 733 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  g  e.  T )
152, 3, 7tendocl 31249 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  g  e.  T
)  ->  ( U `  g )  e.  T
)
166, 11, 14, 15syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( U `  g )  e.  T )
178, 1tendo02 31269 . . . . 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 31248 . . . . 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 1189 . . . 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 31269 . . . . 5  |-  ( g  e.  T  ->  ( O `  g )  =  (  _I  |`  B ) )
2221ad2antrl 709 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( O `  g )  =  (  _I  |`  B ) )
2318, 20, 223eqtr4d 2446 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
( O  o.  U
) `  g )  =  ( O `  g ) )
24 simpr 448 . . 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 31306 . . 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 1197 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( O  o.  U )  =  O )
275, 26rexlimddv 2794 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( O  o.  U )  =  O )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667    e. cmpt 4226    _I cid 4453    |` cres 4839    o. ccom 4841   ` cfv 5413   Basecbs 13424   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   TEndoctendo 31234
This theorem is referenced by:  cdleml5N  31462  cdleml9  31466
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tendo 31237
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