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Theorem mapdval2N 35287
Description: Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
mapdval2.h  |-  H  =  ( LHyp `  K
)
mapdval2.u  |-  U  =  ( ( DVecH `  K
) `  W )
mapdval2.s  |-  S  =  ( LSubSp `  U )
mapdval2.n  |-  N  =  ( LSpan `  U )
mapdval2.f  |-  F  =  (LFnl `  U )
mapdval2.l  |-  L  =  (LKer `  U )
mapdval2.o  |-  O  =  ( ( ocH `  K
) `  W )
mapdval2.m  |-  M  =  ( (mapd `  K
) `  W )
mapdval2.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
mapdval2.t  |-  ( ph  ->  T  e.  S )
mapdval2.c  |-  C  =  { g  e.  F  |  ( O `  ( O `  ( L `
 g ) ) )  =  ( L `
 g ) }
Assertion
Ref Expression
mapdval2N  |-  ( ph  ->  ( M `  T
)  =  { f  e.  C  |  E. v  e.  T  ( O `  ( L `  f ) )  =  ( N `  {
v } ) } )
Distinct variable groups:    v, C    f, g, F    f, K    v, g, L    v, N    g, O, v    v, f, T    v, U    f, W    ph, f, v
Allowed substitution hints:    ph( g)    C( f, g)    S( v, f, g)    T( g)    U( f, g)    F( v)    H( v, f, g)    K( v, g)    L( f)    M( v, f, g)    N( f, g)    O( f)    W( v, g)

Proof of Theorem mapdval2N
StepHypRef Expression
1 mapdval2.h . . 3  |-  H  =  ( LHyp `  K
)
2 mapdval2.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
3 mapdval2.s . . 3  |-  S  =  ( LSubSp `  U )
4 mapdval2.f . . 3  |-  F  =  (LFnl `  U )
5 mapdval2.l . . 3  |-  L  =  (LKer `  U )
6 mapdval2.o . . 3  |-  O  =  ( ( ocH `  K
) `  W )
7 mapdval2.m . . 3  |-  M  =  ( (mapd `  K
) `  W )
8 mapdval2.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
9 mapdval2.t . . 3  |-  ( ph  ->  T  e.  S )
10 mapdval2.c . . 3  |-  C  =  { g  e.  F  |  ( O `  ( O `  ( L `
 g ) ) )  =  ( L `
 g ) }
111, 2, 3, 4, 5, 6, 7, 8, 9, 10mapdvalc 35286 . 2  |-  ( ph  ->  ( M `  T
)  =  { f  e.  C  |  ( O `  ( L `
 f ) ) 
C_  T } )
121, 2, 8dvhlmod 34767 . . . . . . . . 9  |-  ( ph  ->  U  e.  LMod )
1312ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f )
)  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  ->  U  e.  LMod )
14 simplr 754 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f )
)  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  ->  ( O `  ( L `  f ) )  e.  (LSAtoms `  U )
)
15 eqid 2443 . . . . . . . . 9  |-  ( Base `  U )  =  (
Base `  U )
16 mapdval2.n . . . . . . . . 9  |-  N  =  ( LSpan `  U )
17 eqid 2443 . . . . . . . . 9  |-  (LSAtoms `  U
)  =  (LSAtoms `  U
)
1815, 16, 17islsati 32651 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  ( O `  ( L `  f ) )  e.  (LSAtoms `  U )
)  ->  E. v  e.  ( Base `  U
) ( O `  ( L `  f ) )  =  ( N `
 { v } ) )
1913, 14, 18syl2anc 661 . . . . . . 7  |-  ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f )
)  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  ->  E. v  e.  ( Base `  U
) ( O `  ( L `  f ) )  =  ( N `
 { v } ) )
20 simprr 756 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f ) )  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  /\  (
v  e.  ( Base `  U )  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) ) )  ->  ( O `  ( L `  f
) )  =  ( N `  { v } ) )
21 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f ) )  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  /\  (
v  e.  ( Base `  U )  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) ) )  ->  ( O `  ( L `  f
) )  C_  T
)
2220, 21eqsstr3d 3403 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f ) )  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  /\  (
v  e.  ( Base `  U )  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) ) )  ->  ( N `  { v } ) 
C_  T )
2312adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  f  e.  C )  ->  U  e.  LMod )
2423ad3antrrr 729 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f ) )  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  /\  (
v  e.  ( Base `  U )  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) ) )  ->  U  e.  LMod )
259adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  f  e.  C )  ->  T  e.  S )
2625ad3antrrr 729 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f ) )  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  /\  (
v  e.  ( Base `  U )  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) ) )  ->  T  e.  S )
27 simprl 755 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f ) )  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  /\  (
v  e.  ( Base `  U )  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) ) )  ->  v  e.  ( Base `  U )
)
2815, 3, 16, 24, 26, 27lspsnel5 17088 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f ) )  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  /\  (
v  e.  ( Base `  U )  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) ) )  ->  ( v  e.  T  <->  ( N `  { v } ) 
C_  T ) )
2922, 28mpbird 232 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f ) )  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  /\  (
v  e.  ( Base `  U )  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) ) )  ->  v  e.  T )
3029, 20jca 532 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f ) )  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  /\  (
v  e.  ( Base `  U )  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) ) )  ->  ( v  e.  T  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) ) )
3130ex 434 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f )
)  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  ->  (
( v  e.  (
Base `  U )  /\  ( O `  ( L `  f )
)  =  ( N `
 { v } ) )  ->  (
v  e.  T  /\  ( O `  ( L `
 f ) )  =  ( N `  { v } ) ) ) )
3231reximdv2 2837 . . . . . . 7  |-  ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f )
)  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  ->  ( E. v  e.  ( Base `  U ) ( O `  ( L `
 f ) )  =  ( N `  { v } )  ->  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( N `  { v } ) ) )
3319, 32mpd 15 . . . . . 6  |-  ( ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f )
)  e.  (LSAtoms `  U
) )  /\  ( O `  ( L `  f ) )  C_  T )  ->  E. v  e.  T  ( O `  ( L `  f
) )  =  ( N `  { v } ) )
3433ex 434 . . . . 5  |-  ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f ) )  e.  (LSAtoms `  U )
)  ->  ( ( O `  ( L `  f ) )  C_  T  ->  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( N `  { v } ) ) )
35 eqid 2443 . . . . . . . . . . 11  |-  ( 0g
`  U )  =  ( 0g `  U
)
3635, 3lss0cl 17040 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  T  e.  S )  ->  ( 0g `  U )  e.  T )
3712, 9, 36syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  U
)  e.  T )
3837adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( O `  ( L `  f
) )  =  {
( 0g `  U
) } )  -> 
( 0g `  U
)  e.  T )
39 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  ( O `  ( L `  f
) )  =  {
( 0g `  U
) } )  -> 
( O `  ( L `  f )
)  =  { ( 0g `  U ) } )
4012adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( O `  ( L `  f
) )  =  {
( 0g `  U
) } )  ->  U  e.  LMod )
4135, 16lspsn0 17101 . . . . . . . . . 10  |-  ( U  e.  LMod  ->  ( N `
 { ( 0g
`  U ) } )  =  { ( 0g `  U ) } )
4240, 41syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( O `  ( L `  f
) )  =  {
( 0g `  U
) } )  -> 
( N `  {
( 0g `  U
) } )  =  { ( 0g `  U ) } )
4339, 42eqtr4d 2478 . . . . . . . 8  |-  ( (
ph  /\  ( O `  ( L `  f
) )  =  {
( 0g `  U
) } )  -> 
( O `  ( L `  f )
)  =  ( N `
 { ( 0g
`  U ) } ) )
44 sneq 3899 . . . . . . . . . . 11  |-  ( v  =  ( 0g `  U )  ->  { v }  =  { ( 0g `  U ) } )
4544fveq2d 5707 . . . . . . . . . 10  |-  ( v  =  ( 0g `  U )  ->  ( N `  { v } )  =  ( N `  { ( 0g `  U ) } ) )
4645eqeq2d 2454 . . . . . . . . 9  |-  ( v  =  ( 0g `  U )  ->  (
( O `  ( L `  f )
)  =  ( N `
 { v } )  <->  ( O `  ( L `  f ) )  =  ( N `
 { ( 0g
`  U ) } ) ) )
4746rspcev 3085 . . . . . . . 8  |-  ( ( ( 0g `  U
)  e.  T  /\  ( O `  ( L `
 f ) )  =  ( N `  { ( 0g `  U ) } ) )  ->  E. v  e.  T  ( O `  ( L `  f
) )  =  ( N `  { v } ) )
4838, 43, 47syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  ( O `  ( L `  f
) )  =  {
( 0g `  U
) } )  ->  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( N `  { v } ) )
4948adantlr 714 . . . . . 6  |-  ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f ) )  =  { ( 0g `  U ) } )  ->  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( N `  { v } ) )
5049a1d 25 . . . . 5  |-  ( ( ( ph  /\  f  e.  C )  /\  ( O `  ( L `  f ) )  =  { ( 0g `  U ) } )  ->  ( ( O `
 ( L `  f ) )  C_  T  ->  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( N `  { v } ) ) )
518adantr 465 . . . . . 6  |-  ( (
ph  /\  f  e.  C )  ->  ( K  e.  HL  /\  W  e.  H ) )
5210lcfl1lem 35148 . . . . . . . 8  |-  ( f  e.  C  <->  ( f  e.  F  /\  ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
) ) )
5352simplbi 460 . . . . . . 7  |-  ( f  e.  C  ->  f  e.  F )
5453adantl 466 . . . . . 6  |-  ( (
ph  /\  f  e.  C )  ->  f  e.  F )
551, 6, 2, 35, 17, 4, 5, 51, 54dochsat0 35114 . . . . 5  |-  ( (
ph  /\  f  e.  C )  ->  (
( O `  ( L `  f )
)  e.  (LSAtoms `  U
)  \/  ( O `
 ( L `  f ) )  =  { ( 0g `  U ) } ) )
5634, 50, 55mpjaodan 784 . . . 4  |-  ( (
ph  /\  f  e.  C )  ->  (
( O `  ( L `  f )
)  C_  T  ->  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( N `  { v } ) ) )
57 simp3 990 . . . . . 6  |-  ( ( ( ph  /\  f  e.  C )  /\  v  e.  T  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) )  ->  ( O `  ( L `  f ) )  =  ( N `
 { v } ) )
58233ad2ant1 1009 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  C )  /\  v  e.  T  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) )  ->  U  e.  LMod )
59253ad2ant1 1009 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  C )  /\  v  e.  T  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) )  ->  T  e.  S
)
60 simp2 989 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  C )  /\  v  e.  T  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) )  ->  v  e.  T
)
613, 16, 58, 59, 60lspsnel5a 17089 . . . . . 6  |-  ( ( ( ph  /\  f  e.  C )  /\  v  e.  T  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) )  ->  ( N `  { v } ) 
C_  T )
6257, 61eqsstrd 3402 . . . . 5  |-  ( ( ( ph  /\  f  e.  C )  /\  v  e.  T  /\  ( O `  ( L `  f ) )  =  ( N `  {
v } ) )  ->  ( O `  ( L `  f ) )  C_  T )
6362rexlimdv3a 2855 . . . 4  |-  ( (
ph  /\  f  e.  C )  ->  ( E. v  e.  T  ( O `  ( L `
 f ) )  =  ( N `  { v } )  ->  ( O `  ( L `  f ) )  C_  T )
)
6456, 63impbid 191 . . 3  |-  ( (
ph  /\  f  e.  C )  ->  (
( O `  ( L `  f )
)  C_  T  <->  E. v  e.  T  ( O `  ( L `  f
) )  =  ( N `  { v } ) ) )
6564rabbidva 2975 . 2  |-  ( ph  ->  { f  e.  C  |  ( O `  ( L `  f ) )  C_  T }  =  { f  e.  C  |  E. v  e.  T  ( O `  ( L `
 f ) )  =  ( N `  { v } ) } )
6611, 65eqtrd 2475 1  |-  ( ph  ->  ( M `  T
)  =  { f  e.  C  |  E. v  e.  T  ( O `  ( L `  f ) )  =  ( N `  {
v } ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2728   {crab 2731    C_ wss 3340   {csn 3889   ` cfv 5430   Basecbs 14186   0gc0g 14390   LModclmod 16960   LSubSpclss 17025   LSpanclspn 17064  LSAtomsclsa 32631  LFnlclfn 32714  LKerclk 32742   HLchlt 33007   LHypclh 33640   DVecHcdvh 34735   ocHcoch 35004  mapdcmpd 35281
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 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-riotaBAD 32616
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-tpos 6757  df-undef 6804  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-sca 14266  df-vsca 14267  df-0g 14392  df-poset 15128  df-plt 15140  df-lub 15156  df-glb 15157  df-join 15158  df-meet 15159  df-p0 15221  df-p1 15222  df-lat 15228  df-clat 15290  df-mnd 15427  df-submnd 15477  df-grp 15557  df-minusg 15558  df-sbg 15559  df-subg 15690  df-cntz 15847  df-lsm 16147  df-cmn 16291  df-abl 16292  df-mgp 16604  df-ur 16616  df-rng 16659  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-dvr 16787  df-drng 16846  df-lmod 16962  df-lss 17026  df-lsp 17065  df-lvec 17196  df-lsatoms 32633  df-lshyp 32634  df-lfl 32715  df-lkr 32743  df-oposet 32833  df-ol 32835  df-oml 32836  df-covers 32923  df-ats 32924  df-atl 32955  df-cvlat 32979  df-hlat 33008  df-llines 33154  df-lplanes 33155  df-lvols 33156  df-lines 33157  df-psubsp 33159  df-pmap 33160  df-padd 33452  df-lhyp 33644  df-laut 33645  df-ldil 33760  df-ltrn 33761  df-trl 33815  df-tgrp 34399  df-tendo 34411  df-edring 34413  df-dveca 34659  df-disoa 34686  df-dvech 34736  df-dib 34796  df-dic 34830  df-dih 34886  df-doch 35005  df-djh 35052  df-mapd 35282
This theorem is referenced by:  mapdval3N  35288  mapdval4N  35289
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