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Theorem tendocan 30292
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 2284 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
16 tendocan.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
1712, 13, 14, 15, 16cdlemj3 30291 . . . . 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 2626 . 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 30242 . 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 1623    e. wcel 1685    =/= wne 2447   A.wral 2544    _I cid 4303    |` cres 4690   ` cfv 5221   Basecbs 13144   HLchlt 28819   LHypclh 29452   LTrncltrn 29569   trLctrl 29626   TEndoctendo 30220
This theorem is referenced by:  tendoid0  30293  tendo0mul  30294  tendo0mulr  30295  cdleml3N  30446  cdleml8  30451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-map 6770  df-poset 14076  df-plt 14088  df-lub 14104  df-glb 14105  df-join 14106  df-meet 14107  df-p0 14141  df-p1 14142  df-lat 14148  df-clat 14210  df-oposet 28645  df-ol 28647  df-oml 28648  df-covers 28735  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820  df-llines 28966  df-lplanes 28967  df-lvols 28968  df-lines 28969  df-psubsp 28971  df-pmap 28972  df-padd 29264  df-lhyp 29456  df-laut 29457  df-ldil 29572  df-ltrn 29573  df-trl 29627  df-tendo 30223
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