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Theorem cdlemk 29964
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
StepHypRef Expression
1 eqid 2253 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2253 . . 3  |-  ( join `  K )  =  (
join `  K )
3 eqid 2253 . . 3  |-  ( meet `  K )  =  (
meet `  K )
4 eqid 2253 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
5 eqid 2253 . . 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 2253 . . 3  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
10 eqid 2253 . . 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 2253 . . 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 2253 . . 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 2253 . . 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 29963 . 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 5376 . . . 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 2261 . . 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 ) )
1817rcla4ev 2821 . 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 17 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   A.wral 2509   E.wrex 2510   ifcif 3470    e. cmpt 3974    _I cid 4197   `'ccnv 4579    |` cres 4582    o. ccom 4584   ` cfv 4592  (class class class)co 5710   iota_crio 6181   Basecbs 13022   occoc 13090   joincjn 13922   meetcmee 13923   Atomscatm 28254   HLchlt 28341   LHypclh 28974   LTrncltrn 29091   trLctrl 29148   TEndoctendo 29742
This theorem is referenced by:  tendoex  29965  cdleml2N  29967
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28167  df-ol 28169  df-oml 28170  df-covers 28257  df-ats 28258  df-atl 28289  df-cvlat 28313  df-hlat 28342  df-llines 28488  df-lplanes 28489  df-lvols 28490  df-lines 28491  df-psubsp 28493  df-pmap 28494  df-padd 28786  df-lhyp 28978  df-laut 28979  df-ldil 29094  df-ltrn 29095  df-trl 29149  df-tendo 29745
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