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Theorem cdlemk 31139
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 2380 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2380 . . 3  |-  ( join `  K )  =  (
join `  K )
3 eqid 2380 . . 3  |-  ( meet `  K )  =  (
meet `  K )
4 eqid 2380 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
5 eqid 2380 . . 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 2380 . . 3  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
10 eqid 2380 . . 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 2380 . . 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 2380 . . 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 2380 . . 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 31138 . 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 5660 . . . 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 2388 . . 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 2988 . 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 2543   A.wral 2642   E.wrex 2643   ifcif 3675    e. cmpt 4200    _I cid 4427   `'ccnv 4810    |` cres 4813    o. ccom 4815   ` cfv 5387  (class class class)co 6013   iota_crio 6471   Basecbs 13389   occoc 13457   joincjn 14321   meetcmee 14322   Atomscatm 29429   HLchlt 29516   LHypclh 30149   LTrncltrn 30266   trLctrl 30323   TEndoctendo 30917
This theorem is referenced by:  tendoex  31140  cdleml2N  31142
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-map 6949  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-clat 14457  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-llines 29663  df-lplanes 29664  df-lvols 29665  df-lines 29666  df-psubsp 29668  df-pmap 29669  df-padd 29961  df-lhyp 30153  df-laut 30154  df-ldil 30269  df-ltrn 30270  df-trl 30324  df-tendo 30920
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