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Theorem ldual1dim 32816
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 2443 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
2 eqid 2443 . . . . . . . 8  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3 ldual1dim.d . . . . . . . 8  |-  D  =  (LDual `  W )
4 eqid 2443 . . . . . . . 8  |-  (Scalar `  D )  =  (Scalar `  D )
5 eqid 2443 . . . . . . . 8  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
6 ldual1dim.w . . . . . . . 8  |-  ( ph  ->  W  e.  LVec )
71, 2, 3, 4, 5, 6ldualsbase 32783 . . . . . . 7  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  (Scalar `  W ) ) )
87eleq2d 2510 . . . . . 6  |-  ( ph  ->  ( k  e.  (
Base `  (Scalar `  D
) )  <->  k  e.  ( Base `  (Scalar `  W
) ) ) )
98anbi1d 704 . . . . 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 2443 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
12 eqid 2443 . . . . . . . 8  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
13 eqid 2443 . . . . . . . 8  |-  ( .s
`  D )  =  ( .s `  D
)
146adantr 465 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  ->  W  e.  LVec )
15 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
k  e.  ( Base `  (Scalar `  W )
) )
16 ldual1dim.g . . . . . . . . 9  |-  ( ph  ->  G  e.  F )
1716adantr 465 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  ->  G  e.  F )
1810, 11, 1, 2, 12, 3, 13, 14, 15, 17ldualvs 32787 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
( k ( .s
`  D ) G )  =  ( G  oF ( .r
`  (Scalar `  W )
) ( ( Base `  W )  X.  {
k } ) ) )
1918eqeq2d 2454 . . . . . 6  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
( g  =  ( k ( .s `  D ) G )  <-> 
g  =  ( G  oF ( .r
`  (Scalar `  W )
) ( ( Base `  W )  X.  {
k } ) ) ) )
2019pm5.32da 641 . . . . 5  |-  ( ph  ->  ( ( k  e.  ( Base `  (Scalar `  W ) )  /\  g  =  ( k
( .s `  D
) G ) )  <-> 
( k  e.  (
Base `  (Scalar `  W
) )  /\  g  =  ( G  oF ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) ) ) )
219, 20bitrd 253 . . . 4  |-  ( ph  ->  ( ( k  e.  ( Base `  (Scalar `  D ) )  /\  g  =  ( k
( .s `  D
) G ) )  <-> 
( k  e.  (
Base `  (Scalar `  W
) )  /\  g  =  ( G  oF ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) ) ) )
2221rexbidv2 2743 . . 3  |-  ( ph  ->  ( E. k  e.  ( Base `  (Scalar `  D ) ) g  =  ( k ( .s `  D ) G )  <->  E. k  e.  ( Base `  (Scalar `  W ) ) g  =  ( G  oF ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) ) )
2322abbidv 2562 . 2  |-  ( ph  ->  { g  |  E. k  e.  ( Base `  (Scalar `  D )
) g  =  ( k ( .s `  D ) G ) }  =  { g  |  E. k  e.  ( Base `  (Scalar `  W ) ) g  =  ( G  oF ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) } )
24 lveclmod 17192 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
253, 24lduallmod 32803 . . . 4  |-  ( W  e.  LVec  ->  D  e. 
LMod )
266, 25syl 16 . . 3  |-  ( ph  ->  D  e.  LMod )
27 eqid 2443 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
2810, 3, 27, 6, 16ldualelvbase 32777 . . 3  |-  ( ph  ->  G  e.  ( Base `  D ) )
29 ldual1dim.n . . . 4  |-  N  =  ( LSpan `  D )
304, 5, 27, 13, 29lspsn 17088 . . 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 661 . 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 32771 . 2  |-  ( ph  ->  { g  e.  F  |  ( L `  G )  C_  ( L `  g ) }  =  { g  |  E. k  e.  (
Base `  (Scalar `  W
) ) g  =  ( G  oF ( .r `  (Scalar `  W ) ) ( ( Base `  W
)  X.  { k } ) ) } )
3423, 31, 333eqtr4d 2485 1  |-  ( ph  ->  ( N `  { G } )  =  {
g  e.  F  | 
( L `  G
)  C_  ( L `  g ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2721   {crab 2724    C_ wss 3333   {csn 3882    X. cxp 4843   ` cfv 5423  (class class class)co 6096    oFcof 6323   Basecbs 14179   .rcmulr 14244  Scalarcsca 14246   .scvsca 14247   LModclmod 16953   LSpanclspn 17057   LVecclvec 17188  LFnlclfn 32707  LKerclk 32735  LDualcld 32773
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-tpos 6750  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-0g 14385  df-mnd 15420  df-submnd 15470  df-grp 15550  df-minusg 15551  df-sbg 15552  df-subg 15683  df-cntz 15840  df-lsm 16140  df-cmn 16284  df-abl 16285  df-mgp 16597  df-ur 16609  df-rng 16652  df-oppr 16720  df-dvdsr 16738  df-unit 16739  df-invr 16769  df-drng 16839  df-lmod 16955  df-lss 17019  df-lsp 17058  df-lvec 17189  df-lshyp 32627  df-lfl 32708  df-lkr 32736  df-ldual 32774
This theorem is referenced by:  mapdsn3  35293
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