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Theorem tendo1ne0 36970
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 36710 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. g  e.  T  g  =/=  (  _I  |`  B ) )
5 simp3 996 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  ->  g  =/=  (  _I  |`  B ) )
6 fveq1 5847 . . . . . . . 8  |-  ( (  _I  |`  T )  =  O  ->  ( (  _I  |`  T ) `  g )  =  ( O `  g ) )
76adantl 464 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  (  _I  |`  T )  =  O )  ->  ( (  _I  |`  T ) `  g )  =  ( O `  g ) )
8 simpl2 998 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  (  _I  |`  T )  =  O )  ->  g  e.  T )
9 fvresi 6073 . . . . . . . 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 36929 . . . . . . . 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 432 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  ->  ( (  _I  |`  T )  =  O  ->  g  =  (  _I  |`  B )
) )
1615necon3d 2678 . . . 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 2948 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805    |-> cmpt 4497    _I cid 4779    |` cres 4990   ` cfv 5570   Basecbs 14719   HLchlt 35491   LHypclh 36124   LTrncltrn 36241   TEndoctendo 36894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-riotaBAD 35100
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-undef 6994  df-map 7414  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-p1 15872  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-llines 35638  df-lplanes 35639  df-lvols 35640  df-lines 35641  df-psubsp 35643  df-pmap 35644  df-padd 35936  df-lhyp 36128  df-laut 36129  df-ldil 36244  df-ltrn 36245  df-trl 36300
This theorem is referenced by:  cdleml9  37126  erngdvlem4  37133  erng1r  37137  erngdvlem4-rN  37141  dvalveclem  37168  dvheveccl  37255  dihord6apre  37399  dihatlat  37477
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