Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemk Unicode version

Theorem cdlemk 31090
Description: Lemma K of [Crawley] p. 118. Final result, lines 11 and 12 on p. 120: given two translations f and k with the same trace, there exists a trace-preserving endomorphism tau whose value at f is k. We use  F,  N, and  u to represent f, k, and tau. (Contributed by NM, 1-Aug-2013.)
Hypotheses
Ref Expression
cdlemk7.h  |-  H  =  ( LHyp `  K
)
cdlemk7.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk7.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk7.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
cdlemk  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  ->  E. u  e.  E  ( u `  F
)  =  N )
Distinct variable groups:    u, E    u, F    u, K    u, N    u, R    u, T    u, W
Allowed substitution hint:    H( u)

Proof of Theorem cdlemk
Dummy variables  f 
b  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2389 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2389 . . 3  |-  ( join `  K )  =  (
join `  K )
3 eqid 2389 . . 3  |-  ( meet `  K )  =  (
meet `  K )
4 eqid 2389 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
5 eqid 2389 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
6 cdlemk7.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdlemk7.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemk7.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
9 eqid 2389 . . 3  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
10 eqid 2389 . . 3  |-  ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  b ) ) (
meet `  K )
( ( N `  ( ( oc `  K ) `  W
) ) ( join `  K ) ( R `
 ( b  o.  `' F ) ) ) )  =  ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  b ) ) (
meet `  K )
( ( N `  ( ( oc `  K ) `  W
) ) ( join `  K ) ( R `
 ( b  o.  `' F ) ) ) )
11 eqid 2389 . . 3  |-  ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  f ) ) (
meet `  K )
( ( ( ( ( oc `  K
) `  W )
( join `  K )
( R `  b
) ) ( meet `  K ) ( ( N `  ( ( oc `  K ) `
 W ) ) ( join `  K
) ( R `  ( b  o.  `' F ) ) ) ) ( join `  K
) ( R `  ( f  o.  `' b ) ) ) )  =  ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  f ) ) (
meet `  K )
( ( ( ( ( oc `  K
) `  W )
( join `  K )
( R `  b
) ) ( meet `  K ) ( ( N `  ( ( oc `  K ) `
 W ) ) ( join `  K
) ( R `  ( b  o.  `' F ) ) ) ) ( join `  K
) ( R `  ( f  o.  `' b ) ) ) )
12 eqid 2389 . . 3  |-  ( iota_ z  e.  T A. b  e.  T  ( (
b  =/=  (  _I  |`  ( Base `  K
) )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  f )
)  ->  ( z `  ( ( oc `  K ) `  W
) )  =  ( ( ( ( oc
`  K ) `  W ) ( join `  K ) ( R `
 f ) ) ( meet `  K
) ( ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  b ) ) (
meet `  K )
( ( N `  ( ( oc `  K ) `  W
) ) ( join `  K ) ( R `
 ( b  o.  `' F ) ) ) ) ( join `  K
) ( R `  ( f  o.  `' b ) ) ) ) ) )  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  ( Base `  K ) )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 f ) )  ->  ( z `  ( ( oc `  K ) `  W
) )  =  ( ( ( ( oc
`  K ) `  W ) ( join `  K ) ( R `
 f ) ) ( meet `  K
) ( ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  b ) ) (
meet `  K )
( ( N `  ( ( oc `  K ) `  W
) ) ( join `  K ) ( R `
 ( b  o.  `' F ) ) ) ) ( join `  K
) ( R `  ( f  o.  `' b ) ) ) ) ) )
13 eqid 2389 . . 3  |-  ( f  e.  T  |->  if ( F  =  N , 
f ,  ( iota_ z  e.  T A. b  e.  T  ( (
b  =/=  (  _I  |`  ( Base `  K
) )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  f )
)  ->  ( z `  ( ( oc `  K ) `  W
) )  =  ( ( ( ( oc
`  K ) `  W ) ( join `  K ) ( R `
 f ) ) ( meet `  K
) ( ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  b ) ) (
meet `  K )
( ( N `  ( ( oc `  K ) `  W
) ) ( join `  K ) ( R `
 ( b  o.  `' F ) ) ) ) ( join `  K
) ( R `  ( f  o.  `' b ) ) ) ) ) ) ) )  =  ( f  e.  T  |->  if ( F  =  N , 
f ,  ( iota_ z  e.  T A. b  e.  T  ( (
b  =/=  (  _I  |`  ( Base `  K
) )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  f )
)  ->  ( z `  ( ( oc `  K ) `  W
) )  =  ( ( ( ( oc
`  K ) `  W ) ( join `  K ) ( R `
 f ) ) ( meet `  K
) ( ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  b ) ) (
meet `  K )
( ( N `  ( ( oc `  K ) `  W
) ) ( join `  K ) ( R `
 ( b  o.  `' F ) ) ) ) ( join `  K
) ( R `  ( f  o.  `' b ) ) ) ) ) ) ) )
14 cdlemk7.e . . 3  |-  E  =  ( ( TEndo `  K
) `  W )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cdlemk56w 31089 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  -> 
( ( f  e.  T  |->  if ( F  =  N ,  f ,  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  ( Base `  K ) )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 f ) )  ->  ( z `  ( ( oc `  K ) `  W
) )  =  ( ( ( ( oc
`  K ) `  W ) ( join `  K ) ( R `
 f ) ) ( meet `  K
) ( ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  b ) ) (
meet `  K )
( ( N `  ( ( oc `  K ) `  W
) ) ( join `  K ) ( R `
 ( b  o.  `' F ) ) ) ) ( join `  K
) ( R `  ( f  o.  `' b ) ) ) ) ) ) ) )  e.  E  /\  ( ( f  e.  T  |->  if ( F  =  N ,  f ,  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  ( Base `  K ) )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 f ) )  ->  ( z `  ( ( oc `  K ) `  W
) )  =  ( ( ( ( oc
`  K ) `  W ) ( join `  K ) ( R `
 f ) ) ( meet `  K
) ( ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  b ) ) (
meet `  K )
( ( N `  ( ( oc `  K ) `  W
) ) ( join `  K ) ( R `
 ( b  o.  `' F ) ) ) ) ( join `  K
) ( R `  ( f  o.  `' b ) ) ) ) ) ) ) ) `  F )  =  N ) )
16 fveq1 5669 . . . 4  |-  ( u  =  ( f  e.  T  |->  if ( F  =  N ,  f ,  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  ( Base `  K ) )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 f ) )  ->  ( z `  ( ( oc `  K ) `  W
) )  =  ( ( ( ( oc
`  K ) `  W ) ( join `  K ) ( R `
 f ) ) ( meet `  K
) ( ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  b ) ) (
meet `  K )
( ( N `  ( ( oc `  K ) `  W
) ) ( join `  K ) ( R `
 ( b  o.  `' F ) ) ) ) ( join `  K
) ( R `  ( f  o.  `' b ) ) ) ) ) ) ) )  ->  ( u `  F )  =  ( ( f  e.  T  |->  if ( F  =  N ,  f ,  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  ( Base `  K ) )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 f ) )  ->  ( z `  ( ( oc `  K ) `  W
) )  =  ( ( ( ( oc
`  K ) `  W ) ( join `  K ) ( R `
 f ) ) ( meet `  K
) ( ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  b ) ) (
meet `  K )
( ( N `  ( ( oc `  K ) `  W
) ) ( join `  K ) ( R `
 ( b  o.  `' F ) ) ) ) ( join `  K
) ( R `  ( f  o.  `' b ) ) ) ) ) ) ) ) `  F ) )
1716eqeq1d 2397 . . 3  |-  ( u  =  ( f  e.  T  |->  if ( F  =  N ,  f ,  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  ( Base `  K ) )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 f ) )  ->  ( z `  ( ( oc `  K ) `  W
) )  =  ( ( ( ( oc
`  K ) `  W ) ( join `  K ) ( R `
 f ) ) ( meet `  K
) ( ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  b ) ) (
meet `  K )
( ( N `  ( ( oc `  K ) `  W
) ) ( join `  K ) ( R `
 ( b  o.  `' F ) ) ) ) ( join `  K
) ( R `  ( f  o.  `' b ) ) ) ) ) ) ) )  ->  ( (
u `  F )  =  N  <->  ( ( f  e.  T  |->  if ( F  =  N , 
f ,  ( iota_ z  e.  T A. b  e.  T  ( (
b  =/=  (  _I  |`  ( Base `  K
) )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  f )
)  ->  ( z `  ( ( oc `  K ) `  W
) )  =  ( ( ( ( oc
`  K ) `  W ) ( join `  K ) ( R `
 f ) ) ( meet `  K
) ( ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  b ) ) (
meet `  K )
( ( N `  ( ( oc `  K ) `  W
) ) ( join `  K ) ( R `
 ( b  o.  `' F ) ) ) ) ( join `  K
) ( R `  ( f  o.  `' b ) ) ) ) ) ) ) ) `  F )  =  N ) )
1817rspcev 2997 . 2  |-  ( ( ( f  e.  T  |->  if ( F  =  N ,  f ,  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  ( Base `  K ) )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 f ) )  ->  ( z `  ( ( oc `  K ) `  W
) )  =  ( ( ( ( oc
`  K ) `  W ) ( join `  K ) ( R `
 f ) ) ( meet `  K
) ( ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  b ) ) (
meet `  K )
( ( N `  ( ( oc `  K ) `  W
) ) ( join `  K ) ( R `
 ( b  o.  `' F ) ) ) ) ( join `  K
) ( R `  ( f  o.  `' b ) ) ) ) ) ) ) )  e.  E  /\  ( ( f  e.  T  |->  if ( F  =  N ,  f ,  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  ( Base `  K ) )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 f ) )  ->  ( z `  ( ( oc `  K ) `  W
) )  =  ( ( ( ( oc
`  K ) `  W ) ( join `  K ) ( R `
 f ) ) ( meet `  K
) ( ( ( ( ( oc `  K ) `  W
) ( join `  K
) ( R `  b ) ) (
meet `  K )
( ( N `  ( ( oc `  K ) `  W
) ) ( join `  K ) ( R `
 ( b  o.  `' F ) ) ) ) ( join `  K
) ( R `  ( f  o.  `' b ) ) ) ) ) ) ) ) `  F )  =  N )  ->  E. u  e.  E  ( u `  F
)  =  N )
1915, 18syl 16 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N ) )  ->  E. u  e.  E  ( u `  F
)  =  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   E.wrex 2652   ifcif 3684    e. cmpt 4209    _I cid 4436   `'ccnv 4819    |` cres 4822    o. ccom 4824   ` cfv 5396  (class class class)co 6022   iota_crio 6480   Basecbs 13398   occoc 13466   joincjn 14330   meetcmee 14331   Atomscatm 29380   HLchlt 29467   LHypclh 30100   LTrncltrn 30217   trLctrl 30274   TEndoctendo 30868
This theorem is referenced by:  tendoex  31091  cdleml2N  31093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-map 6958  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-llines 29614  df-lplanes 29615  df-lvols 29616  df-lines 29617  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104  df-laut 30105  df-ldil 30220  df-ltrn 30221  df-trl 30275  df-tendo 30871
  Copyright terms: Public domain W3C validator