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Theorem mapdval4N 35640
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 35511 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 2454 . . 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 2454 . . 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 35638 . 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 35499 . . . . . . 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 2981 . . . . . . 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 5806 . . . . . . . . . 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 2454 . . . . . . . . . . 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 17152 . . . . . . . . . . . . . 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 4129 . . . . . . . . . . . 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 35387 . . . . . . . . . 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 2504 . . . . . . . . 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 2462 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( L `  f
)  =  ( O `
 { v } ) )
35 sneq 3998 . . . . . . . . . . . . . . 15  |-  ( w  =  v  ->  { w }  =  { v } )
3635fveq2d 5806 . . . . . . . . . . . . . 14  |-  ( w  =  v  ->  ( O `  { w } )  =  ( O `  { v } ) )
3736eqeq2d 2468 . . . . . . . . . . . . 13  |-  ( w  =  v  ->  (
( L `  f
)  =  ( O `
 { w }
)  <->  ( L `  f )  =  ( O `  { v } ) ) )
3837rspcev 3179 . . . . . . . . . . . 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 35511 . . . . . . . . . . 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 35389 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  F )  /\  v  e.  T )  ->  ( O `  ( O `  { v } ) )  =  ( (
LSpan `  U ) `  { v } ) )
45 fveq2 5802 . . . . . . . . . . . 12  |-  ( ( O `  { v } )  =  ( L `  f )  ->  ( O `  ( O `  { v } ) )  =  ( O `  ( L `  f )
) )
4644, 45sylan9req 2516 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( ( LSpan `  U
) `  { v } )  =  ( O `  ( L `
 f ) ) )
4746eqcomd 2462 . . . . . . . . . 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 2865 . . . . . . 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 2590 . . 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 2808 . . 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 2808 . . 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 2520 . 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 2495 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 1370    e. wcel 1758   {cab 2439   E.wrex 2800   {crab 2803    C_ wss 3439   {csn 3988   ` cfv 5529   Basecbs 14296   LSubSpclss 17146   LSpanclspn 17185  LFnlclfn 33065  LKerclk 33093   HLchlt 33358   LHypclh 33991   DVecHcdvh 35086   ocHcoch 35355  mapdcmpd 35632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-riotaBAD 32967
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-tpos 6858  df-undef 6905  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-mulr 14375  df-sca 14377  df-vsca 14378  df-0g 14503  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-p1 15333  df-lat 15339  df-clat 15401  df-mnd 15538  df-submnd 15588  df-grp 15668  df-minusg 15669  df-sbg 15670  df-subg 15801  df-cntz 15958  df-lsm 16260  df-cmn 16404  df-abl 16405  df-mgp 16724  df-ur 16736  df-rng 16780  df-oppr 16848  df-dvdsr 16866  df-unit 16867  df-invr 16897  df-dvr 16908  df-drng 16967  df-lmod 17083  df-lss 17147  df-lsp 17186  df-lvec 17317  df-lsatoms 32984  df-lshyp 32985  df-lfl 33066  df-lkr 33094  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-llines 33505  df-lplanes 33506  df-lvols 33507  df-lines 33508  df-psubsp 33510  df-pmap 33511  df-padd 33803  df-lhyp 33995  df-laut 33996  df-ldil 34111  df-ltrn 34112  df-trl 34166  df-tgrp 34750  df-tendo 34762  df-edring 34764  df-dveca 35010  df-disoa 35037  df-dvech 35087  df-dib 35147  df-dic 35181  df-dih 35237  df-doch 35356  df-djh 35403  df-mapd 35633
This theorem is referenced by:  mapdval5N  35641  mapd1dim2lem1N  35652
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