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Theorem tendo0mulr 31698
Description: Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 13-Feb-2014.)
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
tendo0mulr  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U  o.  O )  =  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 tendo0mulr
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 31439 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. g  e.  T  g  =/=  (  _I  |`  B ) )
54adantr 453 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  E. g  e.  T  g  =/=  (  _I  |`  B ) )
6 simpll 732 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 simplr 733 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  U  e.  E )
8 tendoid0.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
9 tendoid0.o . . . . . 6  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
101, 2, 3, 8, 9tendo0cl 31661 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
1110ad2antrr 708 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  O  e.  E )
122, 8tendococl 31643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  O  e.  E
)  ->  ( U  o.  O )  e.  E
)
136, 7, 11, 12syl3anc 1185 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( U  o.  O )  e.  E )
149, 1tendo02 31658 . . . . . . 7  |-  ( g  e.  T  ->  ( O `  g )  =  (  _I  |`  B ) )
1514ad2antrl 710 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( O `  g )  =  (  _I  |`  B ) )
1615fveq2d 5735 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( U `  ( O `  g ) )  =  ( U `  (  _I  |`  B ) ) )
171, 2, 8tendoid 31644 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
1817adantr 453 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
1916, 18eqtrd 2470 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( U `  ( O `  g ) )  =  (  _I  |`  B ) )
20 simprl 734 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  g  e.  T )
212, 3, 8tendocoval 31637 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  O  e.  E )  /\  g  e.  T )  ->  (
( U  o.  O
) `  g )  =  ( U `  ( O `  g ) ) )
226, 7, 11, 20, 21syl121anc 1190 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
( U  o.  O
) `  g )  =  ( U `  ( O `  g ) ) )
2319, 22, 153eqtr4d 2480 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  (
( U  o.  O
) `  g )  =  ( O `  g ) )
24 simpr 449 . . 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, 8tendocan 31695 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( U  o.  O )  e.  E  /\  O  e.  E  /\  ( ( U  o.  O ) `
 g )  =  ( O `  g
) )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( U  o.  O )  =  O )
266, 13, 11, 23, 24, 25syl131anc 1198 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E )  /\  (
g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( U  o.  O )  =  O )
275, 26rexlimddv 2836 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U  o.  O )  =  O )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    e. cmpt 4269    _I cid 4496    |` cres 4883    o. ccom 4885   ` cfv 5457   Basecbs 13474   HLchlt 30222   LHypclh 30855   LTrncltrn 30972   TEndoctendo 31623
This theorem is referenced by:  dib1dim2  32040  diblss  32042
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-fal 1330  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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