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Theorem cdlemk 31708
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 2435 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2435 . . 3  |-  ( join `  K )  =  (
join `  K )
3 eqid 2435 . . 3  |-  ( meet `  K )  =  (
meet `  K )
4 eqid 2435 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
5 eqid 2435 . . 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 2435 . . 3  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
10 eqid 2435 . . 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 2435 . . 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 2435 . . 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 2435 . . 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 31707 . 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 5719 . . . 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 2443 . . 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 3044 . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   ifcif 3731    e. cmpt 4258    _I cid 4485   `'ccnv 4869    |` cres 4872    o. ccom 4874   ` cfv 5446  (class class class)co 6073   iota_crio 6534   Basecbs 13461   occoc 13529   joincjn 14393   meetcmee 14394   Atomscatm 29998   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   trLctrl 30892   TEndoctendo 31486
This theorem is referenced by:  tendoex  31709  cdleml2N  31711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-tendo 31489
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