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Theorem lsslindf 27258
Description: Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
lsslindf.u  |-  U  =  ( LSubSp `  W )
lsslindf.x  |-  X  =  ( Ws  S )
Assertion
Ref Expression
lsslindf  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F LIndF  X  <->  F LIndF  W ) )

Proof of Theorem lsslindf
Dummy variables  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rellindf 27236 . . . 4  |-  Rel LIndF
21brrelexi 4910 . . 3  |-  ( F LIndF 
X  ->  F  e.  _V )
32a1i 11 . 2  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F LIndF  X  ->  F  e.  _V ) )
41brrelexi 4910 . . 3  |-  ( F LIndF 
W  ->  F  e.  _V )
54a1i 11 . 2  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F LIndF  W  ->  F  e.  _V ) )
6 simpr 448 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  X )
)  ->  F : dom  F --> ( Base `  X
) )
7 lsslindf.x . . . . . . . . 9  |-  X  =  ( Ws  S )
8 eqid 2435 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
97, 8ressbasss 13513 . . . . . . . 8  |-  ( Base `  X )  C_  ( Base `  W )
10 fss 5591 . . . . . . . 8  |-  ( ( F : dom  F --> ( Base `  X )  /\  ( Base `  X
)  C_  ( Base `  W ) )  ->  F : dom  F --> ( Base `  W ) )
116, 9, 10sylancl 644 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  X )
)  ->  F : dom  F --> ( Base `  W
) )
12 ffn 5583 . . . . . . . . 9  |-  ( F : dom  F --> ( Base `  W )  ->  F  Fn  dom  F )
1312adantl 453 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  W )
)  ->  F  Fn  dom  F )
14 simp3 959 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ran  F 
C_  S )
15 lsslindf.u . . . . . . . . . . . . 13  |-  U  =  ( LSubSp `  W )
168, 15lssss 16005 . . . . . . . . . . . 12  |-  ( S  e.  U  ->  S  C_  ( Base `  W
) )
17163ad2ant2 979 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  S  C_  ( Base `  W
) )
187, 8ressbas2 13512 . . . . . . . . . . 11  |-  ( S 
C_  ( Base `  W
)  ->  S  =  ( Base `  X )
)
1917, 18syl 16 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  S  =  ( Base `  X
) )
2014, 19sseqtrd 3376 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ran  F 
C_  ( Base `  X
) )
2120adantr 452 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  W )
)  ->  ran  F  C_  ( Base `  X )
)
22 df-f 5450 . . . . . . . 8  |-  ( F : dom  F --> ( Base `  X )  <->  ( F  Fn  dom  F  /\  ran  F 
C_  ( Base `  X
) ) )
2313, 21, 22sylanbrc 646 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  W )
)  ->  F : dom  F --> ( Base `  X
) )
2411, 23impbida 806 . . . . . 6  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F : dom  F --> ( Base `  X )  <->  F : dom  F --> ( Base `  W
) ) )
2524adantr 452 . . . . 5  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( F : dom  F --> ( Base `  X
)  <->  F : dom  F --> ( Base `  W )
) )
26 simpl2 961 . . . . . . . . . 10  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  S  e.  U
)
27 eqid 2435 . . . . . . . . . . . 12  |-  (Scalar `  W )  =  (Scalar `  W )
287, 27resssca 13596 . . . . . . . . . . 11  |-  ( S  e.  U  ->  (Scalar `  W )  =  (Scalar `  X ) )
2928eqcomd 2440 . . . . . . . . . 10  |-  ( S  e.  U  ->  (Scalar `  X )  =  (Scalar `  W ) )
3026, 29syl 16 . . . . . . . . 9  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  (Scalar `  X )  =  (Scalar `  W )
)
3130fveq2d 5724 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( Base `  (Scalar `  X ) )  =  ( Base `  (Scalar `  W ) ) )
3230fveq2d 5724 . . . . . . . . 9  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( 0g `  (Scalar `  X ) )  =  ( 0g `  (Scalar `  W ) ) )
3332sneqd 3819 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  { ( 0g
`  (Scalar `  X )
) }  =  {
( 0g `  (Scalar `  W ) ) } )
3431, 33difeq12d 3458 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( ( Base `  (Scalar `  X )
)  \  { ( 0g `  (Scalar `  X
) ) } )  =  ( ( Base `  (Scalar `  W )
)  \  { ( 0g `  (Scalar `  W
) ) } ) )
35 eqid 2435 . . . . . . . . . . . . 13  |-  ( .s
`  W )  =  ( .s `  W
)
367, 35ressvsca 13597 . . . . . . . . . . . 12  |-  ( S  e.  U  ->  ( .s `  W )  =  ( .s `  X
) )
3736eqcomd 2440 . . . . . . . . . . 11  |-  ( S  e.  U  ->  ( .s `  X )  =  ( .s `  W
) )
3826, 37syl 16 . . . . . . . . . 10  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( .s `  X )  =  ( .s `  W ) )
3938oveqd 6090 . . . . . . . . 9  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( k ( .s `  X ) ( F `  x
) )  =  ( k ( .s `  W ) ( F `
 x ) ) )
40 simpl1 960 . . . . . . . . . . 11  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  W  e.  LMod )
41 imassrn 5208 . . . . . . . . . . . 12  |-  ( F
" ( dom  F  \  { x } ) )  C_  ran  F
42 simpl3 962 . . . . . . . . . . . 12  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ran  F  C_  S
)
4341, 42syl5ss 3351 . . . . . . . . . . 11  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( F "
( dom  F  \  {
x } ) ) 
C_  S )
44 eqid 2435 . . . . . . . . . . . 12  |-  ( LSpan `  W )  =  (
LSpan `  W )
45 eqid 2435 . . . . . . . . . . . 12  |-  ( LSpan `  X )  =  (
LSpan `  X )
467, 44, 45, 15lsslsp 16083 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ( F " ( dom  F  \  { x } ) )  C_  S )  ->  ( ( LSpan `  W
) `  ( F " ( dom  F  \  { x } ) ) )  =  ( ( LSpan `  X ) `  ( F " ( dom  F  \  { x } ) ) ) )
4740, 26, 43, 46syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( ( LSpan `  W ) `  ( F " ( dom  F  \  { x } ) ) )  =  ( ( LSpan `  X ) `  ( F " ( dom  F  \  { x } ) ) ) )
4847eqcomd 2440 . . . . . . . . 9  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( ( LSpan `  X ) `  ( F " ( dom  F  \  { x } ) ) )  =  ( ( LSpan `  W ) `  ( F " ( dom  F  \  { x } ) ) ) )
4939, 48eleq12d 2503 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( ( k ( .s `  X
) ( F `  x ) )  e.  ( ( LSpan `  X
) `  ( F " ( dom  F  \  { x } ) ) )  <->  ( k
( .s `  W
) ( F `  x ) )  e.  ( ( LSpan `  W
) `  ( F " ( dom  F  \  { x } ) ) ) ) )
5049notbid 286 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( -.  (
k ( .s `  X ) ( F `
 x ) )  e.  ( ( LSpan `  X ) `  ( F " ( dom  F  \  { x } ) ) )  <->  -.  (
k ( .s `  W ) ( F `
 x ) )  e.  ( ( LSpan `  W ) `  ( F " ( dom  F  \  { x } ) ) ) ) )
5134, 50raleqbidv 2908 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( A. k  e.  ( ( Base `  (Scalar `  X ) )  \  { ( 0g `  (Scalar `  X ) ) } )  -.  (
k ( .s `  X ) ( F `
 x ) )  e.  ( ( LSpan `  X ) `  ( F " ( dom  F  \  { x } ) ) )  <->  A. k  e.  ( ( Base `  (Scalar `  W ) )  \  { ( 0g `  (Scalar `  W ) ) } )  -.  (
k ( .s `  W ) ( F `
 x ) )  e.  ( ( LSpan `  W ) `  ( F " ( dom  F  \  { x } ) ) ) ) )
5251ralbidv 2717 . . . . 5  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( A. x  e.  dom  F A. k  e.  ( ( Base `  (Scalar `  X ) )  \  { ( 0g `  (Scalar `  X ) ) } )  -.  (
k ( .s `  X ) ( F `
 x ) )  e.  ( ( LSpan `  X ) `  ( F " ( dom  F  \  { x } ) ) )  <->  A. x  e.  dom  F A. k  e.  ( ( Base `  (Scalar `  W ) )  \  { ( 0g `  (Scalar `  W ) ) } )  -.  (
k ( .s `  W ) ( F `
 x ) )  e.  ( ( LSpan `  W ) `  ( F " ( dom  F  \  { x } ) ) ) ) )
5325, 52anbi12d 692 . . . 4  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( ( F : dom  F --> ( Base `  X )  /\  A. x  e.  dom  F A. k  e.  ( ( Base `  (Scalar `  X
) )  \  {
( 0g `  (Scalar `  X ) ) } )  -.  ( k ( .s `  X
) ( F `  x ) )  e.  ( ( LSpan `  X
) `  ( F " ( dom  F  \  { x } ) ) ) )  <->  ( F : dom  F --> ( Base `  W )  /\  A. x  e.  dom  F A. k  e.  ( ( Base `  (Scalar `  W
) )  \  {
( 0g `  (Scalar `  W ) ) } )  -.  ( k ( .s `  W
) ( F `  x ) )  e.  ( ( LSpan `  W
) `  ( F " ( dom  F  \  { x } ) ) ) ) ) )
54 ovex 6098 . . . . . . 7  |-  ( Ws  S )  e.  _V
557, 54eqeltri 2505 . . . . . 6  |-  X  e. 
_V
5655a1i 11 . . . . 5  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  X  e.  _V )
57 eqid 2435 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
58 eqid 2435 . . . . . 6  |-  ( .s
`  X )  =  ( .s `  X
)
59 eqid 2435 . . . . . 6  |-  (Scalar `  X )  =  (Scalar `  X )
60 eqid 2435 . . . . . 6  |-  ( Base `  (Scalar `  X )
)  =  ( Base `  (Scalar `  X )
)
61 eqid 2435 . . . . . 6  |-  ( 0g
`  (Scalar `  X )
)  =  ( 0g
`  (Scalar `  X )
)
6257, 58, 45, 59, 60, 61islindf 27240 . . . . 5  |-  ( ( X  e.  _V  /\  F  e.  _V )  ->  ( F LIndF  X  <->  ( F : dom  F --> ( Base `  X )  /\  A. x  e.  dom  F A. k  e.  ( ( Base `  (Scalar `  X
) )  \  {
( 0g `  (Scalar `  X ) ) } )  -.  ( k ( .s `  X
) ( F `  x ) )  e.  ( ( LSpan `  X
) `  ( F " ( dom  F  \  { x } ) ) ) ) ) )
6356, 62sylan 458 . . . 4  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( F LIndF  X  <->  ( F : dom  F --> ( Base `  X )  /\  A. x  e.  dom  F A. k  e.  ( ( Base `  (Scalar `  X ) )  \  { ( 0g `  (Scalar `  X ) ) } )  -.  (
k ( .s `  X ) ( F `
 x ) )  e.  ( ( LSpan `  X ) `  ( F " ( dom  F  \  { x } ) ) ) ) ) )
64 eqid 2435 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
65 eqid 2435 . . . . . 6  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
668, 35, 44, 27, 64, 65islindf 27240 . . . . 5  |-  ( ( W  e.  LMod  /\  F  e.  _V )  ->  ( F LIndF  W  <->  ( F : dom  F --> ( Base `  W
)  /\  A. x  e.  dom  F A. k  e.  ( ( Base `  (Scalar `  W ) )  \  { ( 0g `  (Scalar `  W ) ) } )  -.  (
k ( .s `  W ) ( F `
 x ) )  e.  ( ( LSpan `  W ) `  ( F " ( dom  F  \  { x } ) ) ) ) ) )
67663ad2antl1 1119 . . . 4  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( F LIndF  W  <->  ( F : dom  F --> ( Base `  W )  /\  A. x  e.  dom  F A. k  e.  ( ( Base `  (Scalar `  W ) )  \  { ( 0g `  (Scalar `  W ) ) } )  -.  (
k ( .s `  W ) ( F `
 x ) )  e.  ( ( LSpan `  W ) `  ( F " ( dom  F  \  { x } ) ) ) ) ) )
6853, 63, 673bitr4d 277 . . 3  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( F LIndF  X  <->  F LIndF 
W ) )
6968ex 424 . 2  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F  e.  _V  ->  ( F LIndF  X  <->  F LIndF  W ) ) )
703, 5, 69pm5.21ndd 344 1  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F LIndF  X  <->  F LIndF  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    \ cdif 3309    C_ wss 3312   {csn 3806   class class class wbr 4204   dom cdm 4870   ran crn 4871   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461   ↾s cress 13462  Scalarcsca 13524   .scvsca 13525   0gc0g 13715   LModclmod 15942   LSubSpclss 16000   LSpanclspn 16039   LIndF clindf 27232
This theorem is referenced by:  lsslinds  27259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-sca 13537  df-vsca 13538  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-mgp 15641  df-rng 15655  df-ur 15657  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lindf 27234
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