Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ldual1dim Unicode version

Theorem ldual1dim 29429
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 2285 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
2 eqid 2285 . . . . . . . 8  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3 ldual1dim.d . . . . . . . 8  |-  D  =  (LDual `  W )
4 eqid 2285 . . . . . . . 8  |-  (Scalar `  D )  =  (Scalar `  D )
5 eqid 2285 . . . . . . . 8  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
6 ldual1dim.w . . . . . . . 8  |-  ( ph  ->  W  e.  LVec )
71, 2, 3, 4, 5, 6ldualsbase 29396 . . . . . . 7  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  (Scalar `  W ) ) )
87eleq2d 2352 . . . . . 6  |-  ( ph  ->  ( k  e.  (
Base `  (Scalar `  D
) )  <->  k  e.  ( Base `  (Scalar `  W
) ) ) )
98anbi1d 685 . . . . 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 2285 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
12 eqid 2285 . . . . . . . 8  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
13 eqid 2285 . . . . . . . 8  |-  ( .s
`  D )  =  ( .s `  D
)
146adantr 451 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  ->  W  e.  LVec )
15 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
k  e.  ( Base `  (Scalar `  W )
) )
16 ldual1dim.g . . . . . . . . 9  |-  ( ph  ->  G  e.  F )
1716adantr 451 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  ->  G  e.  F )
1810, 11, 1, 2, 12, 3, 13, 14, 15, 17ldualvs 29400 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
( k ( .s
`  D ) G )  =  ( G  o F ( .r
`  (Scalar `  W )
) ( ( Base `  W )  X.  {
k } ) ) )
1918eqeq2d 2296 . . . . . 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 622 . . . . 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 244 . . . 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 2568 . . 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 2399 . 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 15861 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
253, 24lduallmod 29416 . . . 4  |-  ( W  e.  LVec  ->  D  e. 
LMod )
266, 25syl 15 . . 3  |-  ( ph  ->  D  e.  LMod )
27 eqid 2285 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
2810, 3, 27, 6, 16ldualelvbase 29390 . . 3  |-  ( ph  ->  G  e.  ( Base `  D ) )
29 ldual1dim.n . . . 4  |-  N  =  ( LSpan `  D )
304, 5, 27, 13, 29lspsn 15761 . . 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 642 . 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 29384 . 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 2327 1  |-  ( ph  ->  ( N `  { G } )  =  {
g  e.  F  | 
( L `  G
)  C_  ( L `  g ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686   {cab 2271   E.wrex 2546   {crab 2549    C_ wss 3154   {csn 3642    X. cxp 4689   ` cfv 5257  (class class class)co 5860    o Fcof 6078   Basecbs 13150   .rcmulr 13211  Scalarcsca 13213   .scvsca 13214   LModclmod 15629   LSpanclspn 15730   LVecclvec 15857  LFnlclfn 29320  LKerclk 29348  LDualcld 29386
This theorem is referenced by:  mapdsn3  31906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-tpos 6236  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-sca 13226  df-vsca 13227  df-0g 13406  df-mnd 14369  df-submnd 14418  df-grp 14491  df-minusg 14492  df-sbg 14493  df-subg 14620  df-cntz 14795  df-lsm 14949  df-cmn 15093  df-abl 15094  df-mgp 15328  df-rng 15342  df-ur 15344  df-oppr 15407  df-dvdsr 15425  df-unit 15426  df-invr 15456  df-drng 15516  df-lmod 15631  df-lss 15692  df-lsp 15731  df-lvec 15858  df-lshyp 29240  df-lfl 29321  df-lkr 29349  df-ldual 29387
  Copyright terms: Public domain W3C validator