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Theorem tendocan 34826
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 1012 . 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 1013 . 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 1021 . 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 1022 . 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 1018 . . . . 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 1019 . . . . 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 1103 . . . . . 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 1104 . . . . . 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 989 . . . . . 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 1168 . . . . 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 990 . . . . 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 2454 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
16 tendocan.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
1712, 13, 14, 15, 16cdlemj3 34825 . . . . 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 1222 . . . 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 1187 . . 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 2828 . 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 34776 . 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 1230 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799    _I cid 4742    |` cres 4953   ` cfv 5529   Basecbs 14295   HLchlt 33353   LHypclh 33986   LTrncltrn 34103   trLctrl 34160   TEndoctendo 34754
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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-riotaBAD 32962
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  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 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-undef 6905  df-map 7329  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-p1 15332  df-lat 15338  df-clat 15400  df-oposet 33179  df-ol 33181  df-oml 33182  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354  df-llines 33500  df-lplanes 33501  df-lvols 33502  df-lines 33503  df-psubsp 33505  df-pmap 33506  df-padd 33798  df-lhyp 33990  df-laut 33991  df-ldil 34106  df-ltrn 34107  df-trl 34161  df-tendo 34757
This theorem is referenced by:  tendoid0  34827  tendo0mul  34828  tendo0mulr  34829  cdleml3N  34980  cdleml8  34985
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