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Theorem tendo1ne0 34795
Description: The identity (unity) is not equal to the zero trace-preserving endomorphism. (Contributed by NM, 8-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
tendo1ne0  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  =/=  O )
Distinct variable groups:    B, f    T, f
Allowed substitution hints:    E( f)    H( f)    K( f)    O( f)    W( f)

Proof of Theorem tendo1ne0
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 tendoid0.b . . 3  |-  B  =  ( Base `  K
)
2 tendoid0.h . . 3  |-  H  =  ( LHyp `  K
)
3 tendoid0.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
41, 2, 3cdlemftr0 34535 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. g  e.  T  g  =/=  (  _I  |`  B ) )
5 simp3 990 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  ->  g  =/=  (  _I  |`  B ) )
6 fveq1 5797 . . . . . . . 8  |-  ( (  _I  |`  T )  =  O  ->  ( (  _I  |`  T ) `  g )  =  ( O `  g ) )
76adantl 466 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  (  _I  |`  T )  =  O )  ->  ( (  _I  |`  T ) `  g )  =  ( O `  g ) )
8 simpl2 992 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  (  _I  |`  T )  =  O )  ->  g  e.  T )
9 fvresi 6012 . . . . . . . 8  |-  ( g  e.  T  ->  (
(  _I  |`  T ) `
 g )  =  g )
108, 9syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  (  _I  |`  T )  =  O )  ->  ( (  _I  |`  T ) `  g )  =  g )
11 tendoid0.o . . . . . . . . 9  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
1211, 1tendo02 34754 . . . . . . . 8  |-  ( g  e.  T  ->  ( O `  g )  =  (  _I  |`  B ) )
138, 12syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  (  _I  |`  T )  =  O )  ->  ( O `  g )  =  (  _I  |`  B )
)
147, 10, 133eqtr3d 2503 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  (  _I  |`  T )  =  O )  ->  g  =  (  _I  |`  B ) )
1514ex 434 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  ->  ( (  _I  |`  T )  =  O  ->  g  =  (  _I  |`  B )
) )
1615necon3d 2675 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  ->  ( g  =/=  (  _I  |`  B )  ->  (  _I  |`  T )  =/=  O ) )
175, 16mpd 15 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  ->  (  _I  |`  T )  =/=  O )
1817rexlimdv3a 2947 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( E. g  e.  T  g  =/=  (  _I  |`  B )  -> 
(  _I  |`  T )  =/=  O ) )
194, 18mpd 15 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  =/=  O )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   E.wrex 2799    |-> cmpt 4457    _I cid 4738    |` cres 4949   ` cfv 5525   Basecbs 14291   HLchlt 33318   LHypclh 33951   LTrncltrn 34068   TEndoctendo 34719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-riotaBAD 32927
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-undef 6901  df-map 7325  df-poset 15234  df-plt 15246  df-lub 15262  df-glb 15263  df-join 15264  df-meet 15265  df-p0 15327  df-p1 15328  df-lat 15334  df-clat 15396  df-oposet 33144  df-ol 33146  df-oml 33147  df-covers 33234  df-ats 33235  df-atl 33266  df-cvlat 33290  df-hlat 33319  df-llines 33465  df-lplanes 33466  df-lvols 33467  df-lines 33468  df-psubsp 33470  df-pmap 33471  df-padd 33763  df-lhyp 33955  df-laut 33956  df-ldil 34071  df-ltrn 34072  df-trl 34126
This theorem is referenced by:  cdleml9  34951  erngdvlem4  34958  erng1r  34962  erngdvlem4-rN  34966  dvalveclem  34993  dvheveccl  35080  dihord6apre  35224  dihatlat  35302
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