Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mapdval4N Unicode version

Theorem mapdval4N 32115
Description: Value of projectivity from vector space H to dual space. TODO: 1. This is shorter than others - make it the official def? (but is not as obvious that it is  C_  C) 2. The unneeded direction of lcfl8a 31986 has awkward  E.- add another thm with only one direction of it? 3. Swap  O `  {
v } and  L `  f? (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
mapdval4.h  |-  H  =  ( LHyp `  K
)
mapdval4.u  |-  U  =  ( ( DVecH `  K
) `  W )
mapdval4.s  |-  S  =  ( LSubSp `  U )
mapdval4.f  |-  F  =  (LFnl `  U )
mapdval4.l  |-  L  =  (LKer `  U )
mapdval4.o  |-  O  =  ( ( ocH `  K
) `  W )
mapdval4.m  |-  M  =  ( (mapd `  K
) `  W )
mapdval4.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
mapdval4.t  |-  ( ph  ->  T  e.  S )
Assertion
Ref Expression
mapdval4N  |-  ( ph  ->  ( M `  T
)  =  { f  e.  F  |  E. v  e.  T  ( O `  { v } )  =  ( L `  f ) } )
Distinct variable groups:    v, f, F    f, K    v, L    v, O    T, f, v    v, U    f, W    ph, f, v
Allowed substitution hints:    S( v, f)    U( f)    H( v, f)    K( v)    L( f)    M( v, f)    O( f)    W( v)

Proof of Theorem mapdval4N
Dummy variables  g  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mapdval4.h . . 3  |-  H  =  ( LHyp `  K
)
2 mapdval4.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
3 mapdval4.s . . 3  |-  S  =  ( LSubSp `  U )
4 eqid 2404 . . 3  |-  ( LSpan `  U )  =  (
LSpan `  U )
5 mapdval4.f . . 3  |-  F  =  (LFnl `  U )
6 mapdval4.l . . 3  |-  L  =  (LKer `  U )
7 mapdval4.o . . 3  |-  O  =  ( ( ocH `  K
) `  W )
8 mapdval4.m . . 3  |-  M  =  ( (mapd `  K
) `  W )
9 mapdval4.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
10 mapdval4.t . . 3  |-  ( ph  ->  T  e.  S )
11 eqid 2404 . . 3  |-  { g  e.  F  |  ( O `  ( O `
 ( L `  g ) ) )  =  ( L `  g ) }  =  { g  e.  F  |  ( O `  ( O `  ( L `
 g ) ) )  =  ( L `
 g ) }
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11mapdval2N 32113 . 2  |-  ( ph  ->  ( M `  T
)  =  { f  e.  { g  e.  F  |  ( O `
 ( O `  ( L `  g ) ) )  =  ( L `  g ) }  |  E. v  e.  T  ( O `  ( L `  f
) )  =  ( ( LSpan `  U ) `  { v } ) } )
1311lcfl1lem 31974 . . . . . . 7  |-  ( f  e.  { g  e.  F  |  ( O `
 ( O `  ( L `  g ) ) )  =  ( L `  g ) }  <->  ( f  e.  F  /\  ( O `
 ( O `  ( L `  f ) ) )  =  ( L `  f ) ) )
1413anbi1i 677 . . . . . 6  |-  ( ( f  e.  { g  e.  F  |  ( O `  ( O `
 ( L `  g ) ) )  =  ( L `  g ) }  /\  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( ( LSpan `  U ) `  {
v } ) )  <-> 
( ( f  e.  F  /\  ( O `
 ( O `  ( L `  f ) ) )  =  ( L `  f ) )  /\  E. v  e.  T  ( O `  ( L `  f
) )  =  ( ( LSpan `  U ) `  { v } ) ) )
15 anass 631 . . . . . 6  |-  ( ( ( f  e.  F  /\  ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f ) )  /\  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( ( LSpan `  U ) `  {
v } ) )  <-> 
( f  e.  F  /\  ( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( ( LSpan `  U ) `  {
v } ) ) ) )
1614, 15bitri 241 . . . . 5  |-  ( ( f  e.  { g  e.  F  |  ( O `  ( O `
 ( L `  g ) ) )  =  ( L `  g ) }  /\  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( ( LSpan `  U ) `  {
v } ) )  <-> 
( f  e.  F  /\  ( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( ( LSpan `  U ) `  {
v } ) ) ) )
17 r19.42v 2822 . . . . . . 7  |-  ( E. v  e.  T  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  =  ( ( LSpan `  U
) `  { v } ) )  <->  ( ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
)  /\  E. v  e.  T  ( O `  ( L `  f
) )  =  ( ( LSpan `  U ) `  { v } ) ) )
18 simprr 734 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
)  /\  ( O `  ( L `  f
) )  =  ( ( LSpan `  U ) `  { v } ) ) )  ->  ( O `  ( L `  f ) )  =  ( ( LSpan `  U
) `  { v } ) )
1918fveq2d 5691 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
)  /\  ( O `  ( L `  f
) )  =  ( ( LSpan `  U ) `  { v } ) ) )  ->  ( O `  ( O `  ( L `  f
) ) )  =  ( O `  (
( LSpan `  U ) `  { v } ) ) )
20 simprl 733 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
)  /\  ( O `  ( L `  f
) )  =  ( ( LSpan `  U ) `  { v } ) ) )  ->  ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
) )
21 eqid 2404 . . . . . . . . . . 11  |-  ( Base `  U )  =  (
Base `  U )
229adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  f  e.  F )  ->  ( K  e.  HL  /\  W  e.  H ) )
2322adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  F )  /\  v  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
2423adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
)  /\  ( O `  ( L `  f
) )  =  ( ( LSpan `  U ) `  { v } ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2510adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  f  e.  F )  ->  T  e.  S )
2621, 3lssel 15969 . . . . . . . . . . . . . 14  |-  ( ( T  e.  S  /\  v  e.  T )  ->  v  e.  ( Base `  U ) )
2725, 26sylan 458 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f  e.  F )  /\  v  e.  T )  ->  v  e.  ( Base `  U
) )
2827snssd 3903 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  F )  /\  v  e.  T )  ->  { v }  C_  ( Base `  U ) )
2928adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
)  /\  ( O `  ( L `  f
) )  =  ( ( LSpan `  U ) `  { v } ) ) )  ->  { v }  C_  ( Base `  U ) )
301, 2, 7, 21, 4, 24, 29dochocsp 31862 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
)  /\  ( O `  ( L `  f
) )  =  ( ( LSpan `  U ) `  { v } ) ) )  ->  ( O `  ( ( LSpan `  U ) `  { v } ) )  =  ( O `
 { v } ) )
3119, 20, 303eqtr3rd 2445 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
)  /\  ( O `  ( L `  f
) )  =  ( ( LSpan `  U ) `  { v } ) ) )  ->  ( O `  { v } )  =  ( L `  f ) )
3227adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
v  e.  ( Base `  U ) )
33 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( O `  {
v } )  =  ( L `  f
) )
3433eqcomd 2409 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( L `  f
)  =  ( O `
 { v } ) )
35 sneq 3785 . . . . . . . . . . . . . . 15  |-  ( w  =  v  ->  { w }  =  { v } )
3635fveq2d 5691 . . . . . . . . . . . . . 14  |-  ( w  =  v  ->  ( O `  { w } )  =  ( O `  { v } ) )
3736eqeq2d 2415 . . . . . . . . . . . . 13  |-  ( w  =  v  ->  (
( L `  f
)  =  ( O `
 { w }
)  <->  ( L `  f )  =  ( O `  { v } ) ) )
3837rspcev 3012 . . . . . . . . . . . 12  |-  ( ( v  e.  ( Base `  U )  /\  ( L `  f )  =  ( O `  { v } ) )  ->  E. w  e.  ( Base `  U
) ( L `  f )  =  ( O `  { w } ) )
3932, 34, 38syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  ->  E. w  e.  ( Base `  U ) ( L `  f )  =  ( O `  { w } ) )
4023adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
41 simpllr 736 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
f  e.  F )
421, 7, 2, 21, 5, 6, 40, 41lcfl8a 31986 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  <->  E. w  e.  ( Base `  U
) ( L `  f )  =  ( O `  { w } ) ) )
4339, 42mpbird 224 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( O `  ( O `  ( L `  f ) ) )  =  ( L `  f ) )
441, 2, 7, 21, 4, 23, 27dochocsn 31864 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  F )  /\  v  e.  T )  ->  ( O `  ( O `  { v } ) )  =  ( (
LSpan `  U ) `  { v } ) )
45 fveq2 5687 . . . . . . . . . . . 12  |-  ( ( O `  { v } )  =  ( L `  f )  ->  ( O `  ( O `  { v } ) )  =  ( O `  ( L `  f )
) )
4644, 45sylan9req 2457 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( ( LSpan `  U
) `  { v } )  =  ( O `  ( L `
 f ) ) )
4746eqcomd 2409 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( O `  ( L `  f )
)  =  ( (
LSpan `  U ) `  { v } ) )
4843, 47jca 519 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) )  =  ( ( LSpan `  U ) `  {
v } ) ) )
4931, 48impbida 806 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  F )  /\  v  e.  T )  ->  (
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) )  =  ( ( LSpan `  U ) `  {
v } ) )  <-> 
( O `  {
v } )  =  ( L `  f
) ) )
5049rexbidva 2683 . . . . . . 7  |-  ( (
ph  /\  f  e.  F )  ->  ( E. v  e.  T  ( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) )  =  ( ( LSpan `  U ) `  {
v } ) )  <->  E. v  e.  T  ( O `  { v } )  =  ( L `  f ) ) )
5117, 50syl5bbr 251 . . . . . 6  |-  ( (
ph  /\  f  e.  F )  ->  (
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( ( LSpan `  U ) `  {
v } ) )  <->  E. v  e.  T  ( O `  { v } )  =  ( L `  f ) ) )
5251pm5.32da 623 . . . . 5  |-  ( ph  ->  ( ( f  e.  F  /\  ( ( O `  ( O `
 ( L `  f ) ) )  =  ( L `  f )  /\  E. v  e.  T  ( O `  ( L `  f ) )  =  ( ( LSpan `  U
) `  { v } ) ) )  <-> 
( f  e.  F  /\  E. v  e.  T  ( O `  { v } )  =  ( L `  f ) ) ) )
5316, 52syl5bb 249 . . . 4  |-  ( ph  ->  ( ( f  e. 
{ g  e.  F  |  ( O `  ( O `  ( L `
 g ) ) )  =  ( L `
 g ) }  /\  E. v  e.  T  ( O `  ( L `  f ) )  =  ( (
LSpan `  U ) `  { v } ) )  <->  ( f  e.  F  /\  E. v  e.  T  ( O `  { v } )  =  ( L `  f ) ) ) )
5453abbidv 2518 . . 3  |-  ( ph  ->  { f  |  ( f  e.  { g  e.  F  |  ( O `  ( O `
 ( L `  g ) ) )  =  ( L `  g ) }  /\  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( ( LSpan `  U ) `  {
v } ) ) }  =  { f  |  ( f  e.  F  /\  E. v  e.  T  ( O `  { v } )  =  ( L `  f ) ) } )
55 df-rab 2675 . . 3  |-  { f  e.  { g  e.  F  |  ( O `
 ( O `  ( L `  g ) ) )  =  ( L `  g ) }  |  E. v  e.  T  ( O `  ( L `  f
) )  =  ( ( LSpan `  U ) `  { v } ) }  =  { f  |  ( f  e. 
{ g  e.  F  |  ( O `  ( O `  ( L `
 g ) ) )  =  ( L `
 g ) }  /\  E. v  e.  T  ( O `  ( L `  f ) )  =  ( (
LSpan `  U ) `  { v } ) ) }
56 df-rab 2675 . . 3  |-  { f  e.  F  |  E. v  e.  T  ( O `  { v } )  =  ( L `  f ) }  =  { f  |  ( f  e.  F  /\  E. v  e.  T  ( O `  { v } )  =  ( L `  f ) ) }
5754, 55, 563eqtr4g 2461 . 2  |-  ( ph  ->  { f  e.  {
g  e.  F  | 
( O `  ( O `  ( L `  g ) ) )  =  ( L `  g ) }  |  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( ( LSpan `  U ) `  {
v } ) }  =  { f  e.  F  |  E. v  e.  T  ( O `  { v } )  =  ( L `  f ) } )
5812, 57eqtrd 2436 1  |-  ( ph  ->  ( M `  T
)  =  { f  e.  F  |  E. v  e.  T  ( O `  { v } )  =  ( L `  f ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   E.wrex 2667   {crab 2670    C_ wss 3280   {csn 3774   ` cfv 5413   Basecbs 13424   LSubSpclss 15963   LSpanclspn 16002  LFnlclfn 29540  LKerclk 29568   HLchlt 29833   LHypclh 30466   DVecHcdvh 31561   ocHcoch 31830  mapdcmpd 32107
This theorem is referenced by:  mapdval5N  32116  mapd1dim2lem1N  32127
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-undef 6502  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-0g 13682  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lvec 16130  df-lsatoms 29459  df-lshyp 29460  df-lfl 29541  df-lkr 29569  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tgrp 31225  df-tendo 31237  df-edring 31239  df-dveca 31485  df-disoa 31512  df-dvech 31562  df-dib 31622  df-dic 31656  df-dih 31712  df-doch 31831  df-djh 31878  df-mapd 32108
  Copyright terms: Public domain W3C validator