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Theorem tendoid0 33825
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 1025 . . . . 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 33787 . . . . 5  |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
51, 4syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  ( O `  F )  =  (  _I  |`  B )
)
65eqeq2d 2416 . . 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 1000 . . . . 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 1001 . . . . 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 33790 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
137, 12syl 17 . . . . 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 459 . . . . 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 1052 . . . . 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 1053 . . . . 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 33824 . . . . 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 1248 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  ( U `  F )  =  ( O `  F ) )  ->  U  =  O )
1918ex 432 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  ( ( U `  F )  =  ( O `  F )  ->  U  =  O ) )
206, 19sylbird 235 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  ( ( U `  F )  =  (  _I  |`  B )  ->  U  =  O ) )
21 fveq1 5804 . . . 4  |-  ( U  =  O  ->  ( U `  F )  =  ( O `  F ) )
2221eqeq1d 2404 . . 3  |-  ( U  =  O  ->  (
( U `  F
)  =  (  _I  |`  B )  <->  ( O `  F )  =  (  _I  |`  B )
) )
235, 22syl5ibrcom 222 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  ( U  =  O  ->  ( U `
 F )  =  (  _I  |`  B ) ) )
2420, 23impbid 191 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598    |-> cmpt 4452    _I cid 4732    |` cres 4944   ` cfv 5525   Basecbs 14733   HLchlt 32349   LHypclh 32982   LTrncltrn 33099   TEndoctendo 33752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-riotaBAD 31958
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-undef 6959  df-map 7379  df-preset 15773  df-poset 15791  df-plt 15804  df-lub 15820  df-glb 15821  df-join 15822  df-meet 15823  df-p0 15885  df-p1 15886  df-lat 15892  df-clat 15954  df-oposet 32175  df-ol 32177  df-oml 32178  df-covers 32265  df-ats 32266  df-atl 32297  df-cvlat 32321  df-hlat 32350  df-llines 32496  df-lplanes 32497  df-lvols 32498  df-lines 32499  df-psubsp 32501  df-pmap 32502  df-padd 32794  df-lhyp 32986  df-laut 32987  df-ldil 33102  df-ltrn 33103  df-trl 33158  df-tendo 33755
This theorem is referenced by:  tendoconid  33829  tendotr  33830  cdleml3N  33978  tendospcanN  34024
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