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Theorem ldual1dim 34992
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 2457 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
2 eqid 2457 . . . . . . . 8  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3 ldual1dim.d . . . . . . . 8  |-  D  =  (LDual `  W )
4 eqid 2457 . . . . . . . 8  |-  (Scalar `  D )  =  (Scalar `  D )
5 eqid 2457 . . . . . . . 8  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
6 ldual1dim.w . . . . . . . 8  |-  ( ph  ->  W  e.  LVec )
71, 2, 3, 4, 5, 6ldualsbase 34959 . . . . . . 7  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  (Scalar `  W ) ) )
87eleq2d 2527 . . . . . 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 2457 . . . . . . . 8  |-  ( Base `  W )  =  (
Base `  W )
12 eqid 2457 . . . . . . . 8  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
13 eqid 2457 . . . . . . . 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 34963 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
( k ( .s
`  D ) G )  =  ( G  oF ( .r
`  (Scalar `  W )
) ( ( Base `  W )  X.  {
k } ) ) )
1918eqeq2d 2471 . . . . . 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 2964 . . 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 2593 . 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 17878 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
253, 24lduallmod 34979 . . . 4  |-  ( W  e.  LVec  ->  D  e. 
LMod )
266, 25syl 16 . . 3  |-  ( ph  ->  D  e.  LMod )
27 eqid 2457 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
2810, 3, 27, 6, 16ldualelvbase 34953 . . 3  |-  ( ph  ->  G  e.  ( Base `  D ) )
29 ldual1dim.n . . . 4  |-  N  =  ( LSpan `  D )
304, 5, 27, 13, 29lspsn 17774 . . 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 34947 . 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 2508 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 1395    e. wcel 1819   {cab 2442   E.wrex 2808   {crab 2811    C_ wss 3471   {csn 4032    X. cxp 5006   ` cfv 5594  (class class class)co 6296    oFcof 6537   Basecbs 14643   .rcmulr 14712  Scalarcsca 14714   .scvsca 14715   LModclmod 17638   LSpanclspn 17743   LVecclvec 17874  LFnlclfn 34883  LKerclk 34911  LDualcld 34949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-sca 14727  df-vsca 14728  df-0g 14858  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-subg 16324  df-cntz 16481  df-lsm 16782  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-oppr 17398  df-dvdsr 17416  df-unit 17417  df-invr 17447  df-drng 17524  df-lmod 17640  df-lss 17705  df-lsp 17744  df-lvec 17875  df-lshyp 34803  df-lfl 34884  df-lkr 34912  df-ldual 34950
This theorem is referenced by:  mapdsn3  37471
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