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Theorem mapdval4N 35117
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 34988 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 2438 . . 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 2438 . . 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 35115 . 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 34976 . . . . . . 7  |-  ( f  e.  { g  e.  F  |  ( O `
 ( O `  ( L `  g ) ) )  =  ( L `  g ) }  <->  ( f  e.  F  /\  ( O `
 ( O `  ( L `  f ) ) )  =  ( L `  f ) ) )
1413anbi1i 695 . . . . . 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 649 . . . . . 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 249 . . . . 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 2870 . . . . . . 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 756 . . . . . . . . . . 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 5690 . . . . . . . . . 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 755 . . . . . . . . . 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 2438 . . . . . . . . . . 11  |-  ( Base `  U )  =  (
Base `  U )
229adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  f  e.  F )  ->  ( K  e.  HL  /\  W  e.  H ) )
2322adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  F )  /\  v  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
2423adantr 465 . . . . . . . . . . 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 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  f  e.  F )  ->  T  e.  S )
2621, 3lssel 16996 . . . . . . . . . . . . . 14  |-  ( ( T  e.  S  /\  v  e.  T )  ->  v  e.  ( Base `  U ) )
2725, 26sylan 471 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f  e.  F )  /\  v  e.  T )  ->  v  e.  ( Base `  U
) )
2827snssd 4013 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  F )  /\  v  e.  T )  ->  { v }  C_  ( Base `  U ) )
2928adantr 465 . . . . . . . . . . 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 34864 . . . . . . . . . 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 2479 . . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
v  e.  ( Base `  U ) )
33 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( O `  {
v } )  =  ( L `  f
) )
3433eqcomd 2443 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( L `  f
)  =  ( O `
 { v } ) )
35 sneq 3882 . . . . . . . . . . . . . . 15  |-  ( w  =  v  ->  { w }  =  { v } )
3635fveq2d 5690 . . . . . . . . . . . . . 14  |-  ( w  =  v  ->  ( O `  { w } )  =  ( O `  { v } ) )
3736eqeq2d 2449 . . . . . . . . . . . . 13  |-  ( w  =  v  ->  (
( L `  f
)  =  ( O `
 { w }
)  <->  ( L `  f )  =  ( O `  { v } ) ) )
3837rspcev 3068 . . . . . . . . . . . 12  |-  ( ( v  e.  ( Base `  U )  /\  ( L `  f )  =  ( O `  { v } ) )  ->  E. w  e.  ( Base `  U
) ( L `  f )  =  ( O `  { w } ) )
3932, 34, 38syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  ->  E. w  e.  ( Base `  U ) ( L `  f )  =  ( O `  { w } ) )
4023adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
41 simpllr 758 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
f  e.  F )
421, 7, 2, 21, 5, 6, 40, 41lcfl8a 34988 . . . . . . . . . . 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 232 . . . . . . . . . 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 34866 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  F )  /\  v  e.  T )  ->  ( O `  ( O `  { v } ) )  =  ( (
LSpan `  U ) `  { v } ) )
45 fveq2 5686 . . . . . . . . . . . 12  |-  ( ( O `  { v } )  =  ( L `  f )  ->  ( O `  ( O `  { v } ) )  =  ( O `  ( L `  f )
) )
4644, 45sylan9req 2491 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( ( LSpan `  U
) `  { v } )  =  ( O `  ( L `
 f ) ) )
4746eqcomd 2443 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( O `  ( L `  f )
)  =  ( (
LSpan `  U ) `  { v } ) )
4843, 47jca 532 . . . . . . . . 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 828 . . . . . . . 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 2727 . . . . . . 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 259 . . . . . 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 641 . . . . 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 257 . . . 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 2552 . . 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 2719 . . 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 2719 . . 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 2495 . 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 2470 1  |-  ( ph  ->  ( M `  T
)  =  { f  e.  F  |  E. v  e.  T  ( O `  { v } )  =  ( L `  f ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2424   E.wrex 2711   {crab 2714    C_ wss 3323   {csn 3872   ` cfv 5413   Basecbs 14166   LSubSpclss 16990   LSpanclspn 17029  LFnlclfn 32542  LKerclk 32570   HLchlt 32835   LHypclh 33468   DVecHcdvh 34563   ocHcoch 34832  mapdcmpd 35109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-riotaBAD 32444
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-undef 6784  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-sca 14246  df-vsca 14247  df-0g 14372  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-mnd 15407  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-subg 15669  df-cntz 15826  df-lsm 16126  df-cmn 16270  df-abl 16271  df-mgp 16580  df-ur 16592  df-rng 16635  df-oppr 16703  df-dvdsr 16721  df-unit 16722  df-invr 16752  df-dvr 16763  df-drng 16812  df-lmod 16928  df-lss 16991  df-lsp 17030  df-lvec 17161  df-lsatoms 32461  df-lshyp 32462  df-lfl 32543  df-lkr 32571  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-llines 32982  df-lplanes 32983  df-lvols 32984  df-lines 32985  df-psubsp 32987  df-pmap 32988  df-padd 33280  df-lhyp 33472  df-laut 33473  df-ldil 33588  df-ltrn 33589  df-trl 33643  df-tgrp 34227  df-tendo 34239  df-edring 34241  df-dveca 34487  df-disoa 34514  df-dvech 34564  df-dib 34624  df-dic 34658  df-dih 34714  df-doch 34833  df-djh 34880  df-mapd 35110
This theorem is referenced by:  mapdval5N  35118  mapd1dim2lem1N  35129
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