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Theorem mapdval4N 36830
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 36701 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 2467 . . 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 2467 . . 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 36828 . 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 36689 . . . . . . 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 3021 . . . . . . 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 5876 . . . . . . . . . 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 2467 . . . . . . . . . . 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 17455 . . . . . . . . . . . . . 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 4178 . . . . . . . . . . . 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 36577 . . . . . . . . . 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 2517 . . . . . . . . 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 2475 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( L `  f
)  =  ( O `
 { v } ) )
35 sneq 4043 . . . . . . . . . . . . . . 15  |-  ( w  =  v  ->  { w }  =  { v } )
3635fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( w  =  v  ->  ( O `  { w } )  =  ( O `  { v } ) )
3736eqeq2d 2481 . . . . . . . . . . . . 13  |-  ( w  =  v  ->  (
( L `  f
)  =  ( O `
 { w }
)  <->  ( L `  f )  =  ( O `  { v } ) ) )
3837rspcev 3219 . . . . . . . . . . . 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 36701 . . . . . . . . . . 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 36579 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  F )  /\  v  e.  T )  ->  ( O `  ( O `  { v } ) )  =  ( (
LSpan `  U ) `  { v } ) )
45 fveq2 5872 . . . . . . . . . . . 12  |-  ( ( O `  { v } )  =  ( L `  f )  ->  ( O `  ( O `  { v } ) )  =  ( O `  ( L `  f )
) )
4644, 45sylan9req 2529 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( ( LSpan `  U
) `  { v } )  =  ( O `  ( L `
 f ) ) )
4746eqcomd 2475 . . . . . . . . . 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 830 . . . . . . . 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 2975 . . . . . . 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 2603 . . 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 2826 . . 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 2826 . . 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 2533 . 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 2508 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 1379    e. wcel 1767   {cab 2452   E.wrex 2818   {crab 2821    C_ wss 3481   {csn 4033   ` cfv 5594   Basecbs 14507   LSubSpclss 17449   LSpanclspn 17488  LFnlclfn 34255  LKerclk 34283   HLchlt 34548   LHypclh 35181   DVecHcdvh 36276   ocHcoch 36545  mapdcmpd 36822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-riotaBAD 34157
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-undef 7014  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-0g 14714  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227  df-lsm 16529  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-lsatoms 34174  df-lshyp 34175  df-lfl 34256  df-lkr 34284  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696  df-lvols 34697  df-lines 34698  df-psubsp 34700  df-pmap 34701  df-padd 34993  df-lhyp 35185  df-laut 35186  df-ldil 35301  df-ltrn 35302  df-trl 35356  df-tgrp 35940  df-tendo 35952  df-edring 35954  df-dveca 36200  df-disoa 36227  df-dvech 36277  df-dib 36337  df-dic 36371  df-dih 36427  df-doch 36546  df-djh 36593  df-mapd 36823
This theorem is referenced by:  mapdval5N  36831  mapd1dim2lem1N  36842
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