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Theorem ldual1dim 29425
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 2358 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
2 eqid 2358 . . . . . . . 8  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3 ldual1dim.d . . . . . . . 8  |-  D  =  (LDual `  W )
4 eqid 2358 . . . . . . . 8  |-  (Scalar `  D )  =  (Scalar `  D )
5 eqid 2358 . . . . . . . 8  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
6 ldual1dim.w . . . . . . . 8  |-  ( ph  ->  W  e.  LVec )
71, 2, 3, 4, 5, 6ldualsbase 29392 . . . . . . 7  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  (Scalar `  W ) ) )
87eleq2d 2425 . . . . . 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 2358 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
12 eqid 2358 . . . . . . . 8  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
13 eqid 2358 . . . . . . . 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 29396 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
( k ( .s
`  D ) G )  =  ( G  o F ( .r
`  (Scalar `  W )
) ( ( Base `  W )  X.  {
k } ) ) )
1918eqeq2d 2369 . . . . . 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 2642 . . 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 2472 . 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 15958 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
253, 24lduallmod 29412 . . . 4  |-  ( W  e.  LVec  ->  D  e. 
LMod )
266, 25syl 15 . . 3  |-  ( ph  ->  D  e.  LMod )
27 eqid 2358 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
2810, 3, 27, 6, 16ldualelvbase 29386 . . 3  |-  ( ph  ->  G  e.  ( Base `  D ) )
29 ldual1dim.n . . . 4  |-  N  =  ( LSpan `  D )
304, 5, 27, 13, 29lspsn 15858 . . 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 29380 . 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 2400 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 1642    e. wcel 1710   {cab 2344   E.wrex 2620   {crab 2623    C_ wss 3228   {csn 3716    X. cxp 4769   ` cfv 5337  (class class class)co 5945    o Fcof 6163   Basecbs 13245   .rcmulr 13306  Scalarcsca 13308   .scvsca 13309   LModclmod 15726   LSpanclspn 15827   LVecclvec 15954  LFnlclfn 29316  LKerclk 29344  LDualcld 29382
This theorem is referenced by:  mapdsn3  31902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-tpos 6321  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-sca 13321  df-vsca 13322  df-0g 13503  df-mnd 14466  df-submnd 14515  df-grp 14588  df-minusg 14589  df-sbg 14590  df-subg 14717  df-cntz 14892  df-lsm 15046  df-cmn 15190  df-abl 15191  df-mgp 15425  df-rng 15439  df-ur 15441  df-oppr 15504  df-dvdsr 15522  df-unit 15523  df-invr 15553  df-drng 15613  df-lmod 15728  df-lss 15789  df-lsp 15828  df-lvec 15955  df-lshyp 29236  df-lfl 29317  df-lkr 29345  df-ldual 29383
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