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Theorem ldual1dim 30026
Description: Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
Hypotheses
Ref Expression
ldual1dim.f  |-  F  =  (LFnl `  W )
ldual1dim.l  |-  L  =  (LKer `  W )
ldual1dim.d  |-  D  =  (LDual `  W )
ldual1dim.n  |-  N  =  ( LSpan `  D )
ldual1dim.w  |-  ( ph  ->  W  e.  LVec )
ldual1dim.g  |-  ( ph  ->  G  e.  F )
Assertion
Ref Expression
ldual1dim  |-  ( ph  ->  ( N `  { G } )  =  {
g  e.  F  | 
( L `  G
)  C_  ( L `  g ) } )
Distinct variable groups:    D, g    g, G    g, N    ph, g
Allowed substitution hints:    F( g)    L( g)    W( g)

Proof of Theorem ldual1dim
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
2 eqid 2438 . . . . . . . 8  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3 ldual1dim.d . . . . . . . 8  |-  D  =  (LDual `  W )
4 eqid 2438 . . . . . . . 8  |-  (Scalar `  D )  =  (Scalar `  D )
5 eqid 2438 . . . . . . . 8  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
6 ldual1dim.w . . . . . . . 8  |-  ( ph  ->  W  e.  LVec )
71, 2, 3, 4, 5, 6ldualsbase 29993 . . . . . . 7  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  (Scalar `  W ) ) )
87eleq2d 2505 . . . . . 6  |-  ( ph  ->  ( k  e.  (
Base `  (Scalar `  D
) )  <->  k  e.  ( Base `  (Scalar `  W
) ) ) )
98anbi1d 687 . . . . 5  |-  ( ph  ->  ( ( k  e.  ( Base `  (Scalar `  D ) )  /\  g  =  ( k
( .s `  D
) G ) )  <-> 
( k  e.  (
Base `  (Scalar `  W
) )  /\  g  =  ( k ( .s `  D ) G ) ) ) )
10 ldual1dim.f . . . . . . . 8  |-  F  =  (LFnl `  W )
11 eqid 2438 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
12 eqid 2438 . . . . . . . 8  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
13 eqid 2438 . . . . . . . 8  |-  ( .s
`  D )  =  ( .s `  D
)
146adantr 453 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  ->  W  e.  LVec )
15 simpr 449 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
k  e.  ( Base `  (Scalar `  W )
) )
16 ldual1dim.g . . . . . . . . 9  |-  ( ph  ->  G  e.  F )
1716adantr 453 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  ->  G  e.  F )
1810, 11, 1, 2, 12, 3, 13, 14, 15, 17ldualvs 29997 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
( k ( .s
`  D ) G )  =  ( G  o F ( .r
`  (Scalar `  W )
) ( ( Base `  W )  X.  {
k } ) ) )
1918eqeq2d 2449 . . . . . 6  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
( g  =  ( k ( .s `  D ) G )  <-> 
g  =  ( G  o F ( .r
`  (Scalar `  W )
) ( ( Base `  W )  X.  {
k } ) ) ) )
2019pm5.32da 624 . . . . 5  |-  ( ph  ->  ( ( k  e.  ( Base `  (Scalar `  W ) )  /\  g  =  ( k
( .s `  D
) G ) )  <-> 
( k  e.  (
Base `  (Scalar `  W
) )  /\  g  =  ( G  o F ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) ) ) )
219, 20bitrd 246 . . . 4  |-  ( ph  ->  ( ( k  e.  ( Base `  (Scalar `  D ) )  /\  g  =  ( k
( .s `  D
) G ) )  <-> 
( k  e.  (
Base `  (Scalar `  W
) )  /\  g  =  ( G  o F ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) ) ) )
2221rexbidv2 2730 . . 3  |-  ( ph  ->  ( E. k  e.  ( Base `  (Scalar `  D ) ) g  =  ( k ( .s `  D ) G )  <->  E. k  e.  ( Base `  (Scalar `  W ) ) g  =  ( G  o F ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) ) )
2322abbidv 2552 . 2  |-  ( ph  ->  { g  |  E. k  e.  ( Base `  (Scalar `  D )
) g  =  ( k ( .s `  D ) G ) }  =  { g  |  E. k  e.  ( Base `  (Scalar `  W ) ) g  =  ( G  o F ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) } )
24 lveclmod 16180 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
253, 24lduallmod 30013 . . . 4  |-  ( W  e.  LVec  ->  D  e. 
LMod )
266, 25syl 16 . . 3  |-  ( ph  ->  D  e.  LMod )
27 eqid 2438 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
2810, 3, 27, 6, 16ldualelvbase 29987 . . 3  |-  ( ph  ->  G  e.  ( Base `  D ) )
29 ldual1dim.n . . . 4  |-  N  =  ( LSpan `  D )
304, 5, 27, 13, 29lspsn 16080 . . 3  |-  ( ( D  e.  LMod  /\  G  e.  ( Base `  D
) )  ->  ( N `  { G } )  =  {
g  |  E. k  e.  ( Base `  (Scalar `  D ) ) g  =  ( k ( .s `  D ) G ) } )
3126, 28, 30syl2anc 644 . 2  |-  ( ph  ->  ( N `  { G } )  =  {
g  |  E. k  e.  ( Base `  (Scalar `  D ) ) g  =  ( k ( .s `  D ) G ) } )
32 ldual1dim.l . . 3  |-  L  =  (LKer `  W )
3311, 1, 10, 32, 2, 12, 6, 16lfl1dim 29981 . 2  |-  ( ph  ->  { g  e.  F  |  ( L `  G )  C_  ( L `  g ) }  =  { g  |  E. k  e.  (
Base `  (Scalar `  W
) ) g  =  ( G  o F ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) } )
3423, 31, 333eqtr4d 2480 1  |-  ( ph  ->  ( N `  { G } )  =  {
g  e.  F  | 
( L `  G
)  C_  ( L `  g ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   E.wrex 2708   {crab 2711    C_ wss 3322   {csn 3816    X. cxp 4878   ` cfv 5456  (class class class)co 6083    o Fcof 6305   Basecbs 13471   .rcmulr 13532  Scalarcsca 13534   .scvsca 13535   LModclmod 15952   LSpanclspn 16049   LVecclvec 16176  LFnlclfn 29917  LKerclk 29945  LDualcld 29983
This theorem is referenced by:  mapdsn3  32503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-0g 13729  df-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-cntz 15118  df-lsm 15272  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-drng 15839  df-lmod 15954  df-lss 16011  df-lsp 16050  df-lvec 16177  df-lshyp 29837  df-lfl 29918  df-lkr 29946  df-ldual 29984
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