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Theorem cdlemk 31785
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 2296 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2296 . . 3  |-  ( join `  K )  =  (
join `  K )
3 eqid 2296 . . 3  |-  ( meet `  K )  =  (
meet `  K )
4 eqid 2296 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
5 eqid 2296 . . 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 2296 . . 3  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
10 eqid 2296 . . 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 2296 . . 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 2296 . . 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 2296 . . 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 31784 . 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 5540 . . . 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 2304 . . 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 2897 . 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 15 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   ifcif 3578    e. cmpt 4093    _I cid 4320   `'ccnv 4704    |` cres 4707    o. ccom 4709   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   occoc 13232   joincjn 14094   meetcmee 14095   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   TEndoctendo 31563
This theorem is referenced by:  tendoex  31786  cdleml2N  31788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566
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