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Theorem tendocan 34111
Description: Cancellation law: if the values of two trace-preserving endormorphisms are equal, so are the endormorphisms. Lemma J of [Crawley] p. 118. (Contributed by NM, 21-Jun-2013.)
Hypotheses
Ref Expression
tendocan.b  |-  B  =  ( Base `  K
)
tendocan.h  |-  H  =  ( LHyp `  K
)
tendocan.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendocan.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
tendocan  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `
 F )  =  ( V `  F
) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  U  =  V )

Proof of Theorem tendocan
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 simp1l 1029 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `
 F )  =  ( V `  F
) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  K  e.  HL )
2 simp1r 1030 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `
 F )  =  ( V `  F
) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  W  e.  H )
3 simp21 1038 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `
 F )  =  ( V `  F
) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  U  e.  E )
4 simp22 1039 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `
 F )  =  ( V `  F
) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  V  e.  E )
5 simp11 1035 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  h  e.  T  /\  h  =/=  (  _I  |`  B ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
6 simp12 1036 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  h  e.  T  /\  h  =/=  (  _I  |`  B ) )  ->  ( U  e.  E  /\  V  e.  E  /\  ( U `
 F )  =  ( V `  F
) ) )
7 simp13l 1120 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  h  e.  T  /\  h  =/=  (  _I  |`  B ) )  ->  F  e.  T )
8 simp13r 1121 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  h  e.  T  /\  h  =/=  (  _I  |`  B ) )  ->  F  =/=  (  _I  |`  B ) )
9 simp2 1006 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  h  e.  T  /\  h  =/=  (  _I  |`  B ) )  ->  h  e.  T )
107, 8, 93jca 1185 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  h  e.  T  /\  h  =/=  (  _I  |`  B ) )  ->  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T
) )
11 simp3 1007 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  h  e.  T  /\  h  =/=  (  _I  |`  B ) )  ->  h  =/=  (  _I  |`  B ) )
12 tendocan.b . . . . . 6  |-  B  =  ( Base `  K
)
13 tendocan.h . . . . . 6  |-  H  =  ( LHyp `  K
)
14 tendocan.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
15 eqid 2429 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
16 tendocan.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
1712, 13, 14, 15, 16cdlemj3 34110 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  h  =/=  (  _I  |`  B ) )  ->  ( U `  h )  =  ( V `  h ) )
185, 6, 10, 11, 17syl31anc 1267 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  h  e.  T  /\  h  =/=  (  _I  |`  B ) )  ->  ( U `  h )  =  ( V `  h ) )
19183exp 1204 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `
 F )  =  ( V `  F
) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  (
h  e.  T  -> 
( h  =/=  (  _I  |`  B )  -> 
( U `  h
)  =  ( V `
 h ) ) ) )
2019ralrimiv 2844 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `
 F )  =  ( V `  F
) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  A. h  e.  T  ( h  =/=  (  _I  |`  B )  ->  ( U `  h )  =  ( V `  h ) ) )
2112, 13, 14, 16tendoeq2 34061 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. h  e.  T  (
h  =/=  (  _I  |`  B )  ->  ( U `  h )  =  ( V `  h ) ) )  ->  U  =  V )
221, 2, 3, 4, 20, 21syl221anc 1275 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `
 F )  =  ( V `  F
) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  U  =  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782    _I cid 4764    |` cres 4856   ` cfv 5601   Basecbs 15084   HLchlt 32636   LHypclh 33269   LTrncltrn 33386   trLctrl 33444   TEndoctendo 34039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-riotaBAD 32245
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-undef 7028  df-map 7482  df-preset 16128  df-poset 16146  df-plt 16159  df-lub 16175  df-glb 16176  df-join 16177  df-meet 16178  df-p0 16240  df-p1 16241  df-lat 16247  df-clat 16309  df-oposet 32462  df-ol 32464  df-oml 32465  df-covers 32552  df-ats 32553  df-atl 32584  df-cvlat 32608  df-hlat 32637  df-llines 32783  df-lplanes 32784  df-lvols 32785  df-lines 32786  df-psubsp 32788  df-pmap 32789  df-padd 33081  df-lhyp 33273  df-laut 33274  df-ldil 33389  df-ltrn 33390  df-trl 33445  df-tendo 34042
This theorem is referenced by:  tendoid0  34112  tendo0mul  34113  tendo0mulr  34114  cdleml3N  34265  cdleml8  34270
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