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Theorem tendoid0 31622
Description: A trace-preserving endomorphism is the additive identity iff at least one of its values (at a non-identity translation) is the identity translation. (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
tendoid0  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  ( ( U `  F )  =  (  _I  |`  B )  <-> 
U  =  O ) )
Distinct variable groups:    B, f    T, f
Allowed substitution hints:    U( f)    E( f)    F( f)    H( f)    K( f)    O( f)    W( f)

Proof of Theorem tendoid0
StepHypRef Expression
1 simp3l 985 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  F  e.  T )
2 tendoid0.o . . . . . 6  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
3 tendoid0.b . . . . . 6  |-  B  =  ( Base `  K
)
42, 3tendo02 31584 . . . . 5  |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
51, 4syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  ( O `  F )  =  (  _I  |`  B )
)
65eqeq2d 2447 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  ( ( U `  F )  =  ( O `  F )  <->  ( U `  F )  =  (  _I  |`  B )
) )
7 simpl1 960 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  ( U `  F )  =  ( O `  F ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
8 simpl2 961 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  ( U `  F )  =  ( O `  F ) )  ->  U  e.  E )
9 tendoid0.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
10 tendoid0.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
11 tendoid0.e . . . . . . 7  |-  E  =  ( ( TEndo `  K
) `  W )
123, 9, 10, 11, 2tendo0cl 31587 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
137, 12syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  ( U `  F )  =  ( O `  F ) )  ->  O  e.  E )
14 simpr 448 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  ( U `  F )  =  ( O `  F ) )  -> 
( U `  F
)  =  ( O `
 F ) )
15 simpl3l 1012 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  ( U `  F )  =  ( O `  F ) )  ->  F  e.  T )
16 simpl3r 1013 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  ( U `  F )  =  ( O `  F ) )  ->  F  =/=  (  _I  |`  B ) )
173, 9, 10, 11tendocan 31621 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  O  e.  E  /\  ( U `
 F )  =  ( O `  F
) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  U  =  O )
187, 8, 13, 14, 15, 16, 17syl132anc 1202 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  ( U `  F )  =  ( O `  F ) )  ->  U  =  O )
1918ex 424 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  ( ( U `  F )  =  ( O `  F )  ->  U  =  O ) )
206, 19sylbird 227 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  ( ( U `  F )  =  (  _I  |`  B )  ->  U  =  O ) )
21 fveq1 5727 . . . 4  |-  ( U  =  O  ->  ( U `  F )  =  ( O `  F ) )
2221eqeq1d 2444 . . 3  |-  ( U  =  O  ->  (
( U `  F
)  =  (  _I  |`  B )  <->  ( O `  F )  =  (  _I  |`  B )
) )
235, 22syl5ibrcom 214 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  ( U  =  O  ->  ( U `
 F )  =  (  _I  |`  B ) ) )
2420, 23impbid 184 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  ( ( U `  F )  =  (  _I  |`  B )  <-> 
U  =  O ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599    e. cmpt 4266    _I cid 4493    |` cres 4880   ` cfv 5454   Basecbs 13469   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   TEndoctendo 31549
This theorem is referenced by:  tendoconid  31626  tendotr  31627  cdleml3N  31775  tendospcanN  31821
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956  df-tendo 31552
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