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Theorem tendocan 31086
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 979 . 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 980 . 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 988 . 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 989 . 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 985 . . . . 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 986 . . . . 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 1070 . . . . . 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 1071 . . . . . 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 956 . . . . . 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 1132 . . . . 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 957 . . . . 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 2285 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
16 tendocan.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
1712, 13, 14, 15, 16cdlemj3 31085 . . . . 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 1185 . . . 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 1150 . . 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 2627 . 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 31036 . 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 1193 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545    _I cid 4306    |` cres 4693   ` cfv 5257   Basecbs 13150   HLchlt 29613   LHypclh 30246   LTrncltrn 30363   trLctrl 30420   TEndoctendo 31014
This theorem is referenced by:  tendoid0  31087  tendo0mul  31088  tendo0mulr  31089  cdleml3N  31240  cdleml8  31245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-map 6776  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-llines 29760  df-lplanes 29761  df-lvols 29762  df-lines 29763  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250  df-laut 30251  df-ldil 30366  df-ltrn 30367  df-trl 30421  df-tendo 31017
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