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

Theorem mapdval4N 32119
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 31990 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 2408 . . 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 2408 . . 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 32117 . 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 31978 . . . . . . 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 2826 . . . . . . 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 5695 . . . . . . . . . 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 2408 . . . . . . . . . . 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 15973 . . . . . . . . . . . . . 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 3907 . . . . . . . . . . . 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 31866 . . . . . . . . . 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 2449 . . . . . . . . 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 2413 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( L `  f
)  =  ( O `
 { v } ) )
35 sneq 3789 . . . . . . . . . . . . . . 15  |-  ( w  =  v  ->  { w }  =  { v } )
3635fveq2d 5695 . . . . . . . . . . . . . 14  |-  ( w  =  v  ->  ( O `  { w } )  =  ( O `  { v } ) )
3736eqeq2d 2419 . . . . . . . . . . . . 13  |-  ( w  =  v  ->  (
( L `  f
)  =  ( O `
 { w }
)  <->  ( L `  f )  =  ( O `  { v } ) ) )
3837rspcev 3016 . . . . . . . . . . . 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 31990 . . . . . . . . . . 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 31868 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  F )  /\  v  e.  T )  ->  ( O `  ( O `  { v } ) )  =  ( (
LSpan `  U ) `  { v } ) )
45 fveq2 5691 . . . . . . . . . . . 12  |-  ( ( O `  { v } )  =  ( L `  f )  ->  ( O `  ( O `  { v } ) )  =  ( O `  ( L `  f )
) )
4644, 45sylan9req 2461 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  f  e.  F )  /\  v  e.  T
)  /\  ( O `  { v } )  =  ( L `  f ) )  -> 
( ( LSpan `  U
) `  { v } )  =  ( O `  ( L `
 f ) ) )
4746eqcomd 2413 . . . . . . . . . 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 2687 . . . . . . 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 2522 . . 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 2679 . . 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 2679 . . 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 2465 . 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 2440 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 2394   E.wrex 2671   {crab 2674    C_ wss 3284   {csn 3778   ` cfv 5417   Basecbs 13428   LSubSpclss 15967   LSpanclspn 16006  LFnlclfn 29544  LKerclk 29572   HLchlt 29837   LHypclh 30470   DVecHcdvh 31565   ocHcoch 31834  mapdcmpd 32111
This theorem is referenced by:  mapdval5N  32120  mapd1dim2lem1N  32131
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-tpos 6442  df-undef 6506  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-n0 10182  df-z 10243  df-uz 10449  df-fz 11004  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-sca 13504  df-vsca 13505  df-0g 13686  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-mnd 14649  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-subg 14900  df-cntz 15075  df-lsm 15229  df-cmn 15373  df-abl 15374  df-mgp 15608  df-rng 15622  df-ur 15624  df-oppr 15687  df-dvdsr 15705  df-unit 15706  df-invr 15736  df-dvr 15747  df-drng 15796  df-lmod 15911  df-lss 15968  df-lsp 16007  df-lvec 16134  df-lsatoms 29463  df-lshyp 29464  df-lfl 29545  df-lkr 29573  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986  df-lines 29987  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591  df-trl 30645  df-tgrp 31229  df-tendo 31241  df-edring 31243  df-dveca 31489  df-disoa 31516  df-dvech 31566  df-dib 31626  df-dic 31660  df-dih 31716  df-doch 31835  df-djh 31882  df-mapd 32112
  Copyright terms: Public domain W3C validator