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Theorem lsslindf 18218
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 18196 . . . 4  |-  Rel LIndF
21brrelexi 4875 . . 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 4875 . . 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 458 . . . . . . . 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 2441 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
97, 8ressbasss 14226 . . . . . . . 8  |-  ( Base `  X )  C_  ( Base `  W )
10 fss 5564 . . . . . . . 8  |-  ( ( F : dom  F --> ( Base `  X )  /\  ( Base `  X
)  C_  ( Base `  W ) )  ->  F : dom  F --> ( Base `  W ) )
116, 9, 10sylancl 657 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  X )
)  ->  F : dom  F --> ( Base `  W
) )
12 ffn 5556 . . . . . . . . 9  |-  ( F : dom  F --> ( Base `  W )  ->  F  Fn  dom  F )
1312adantl 463 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  W )
)  ->  F  Fn  dom  F )
14 simp3 985 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ran  F 
C_  S )
15 lsslindf.u . . . . . . . . . . . . 13  |-  U  =  ( LSubSp `  W )
168, 15lssss 16996 . . . . . . . . . . . 12  |-  ( S  e.  U  ->  S  C_  ( Base `  W
) )
17163ad2ant2 1005 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  S  C_  ( Base `  W
) )
187, 8ressbas2 14225 . . . . . . . . . . 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 3389 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ran  F 
C_  ( Base `  X
) )
2120adantr 462 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  W )
)  ->  ran  F  C_  ( Base `  X )
)
22 df-f 5419 . . . . . . . 8  |-  ( F : dom  F --> ( Base `  X )  <->  ( F  Fn  dom  F  /\  ran  F 
C_  ( Base `  X
) ) )
2313, 21, 22sylanbrc 659 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  W )
)  ->  F : dom  F --> ( Base `  X
) )
2411, 23impbida 823 . . . . . 6  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F : dom  F --> ( Base `  X )  <->  F : dom  F --> ( Base `  W
) ) )
2524adantr 462 . . . . 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 987 . . . . . . . . . 10  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  S  e.  U
)
27 eqid 2441 . . . . . . . . . . . 12  |-  (Scalar `  W )  =  (Scalar `  W )
287, 27resssca 14312 . . . . . . . . . . 11  |-  ( S  e.  U  ->  (Scalar `  W )  =  (Scalar `  X ) )
2928eqcomd 2446 . . . . . . . . . 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 5692 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( Base `  (Scalar `  X ) )  =  ( Base `  (Scalar `  W ) ) )
3230fveq2d 5692 . . . . . . . . 9  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( 0g `  (Scalar `  X ) )  =  ( 0g `  (Scalar `  W ) ) )
3332sneqd 3886 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  { ( 0g
`  (Scalar `  X )
) }  =  {
( 0g `  (Scalar `  W ) ) } )
3431, 33difeq12d 3472 . . . . . . 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 2441 . . . . . . . . . . . . 13  |-  ( .s
`  W )  =  ( .s `  W
)
367, 35ressvsca 14313 . . . . . . . . . . . 12  |-  ( S  e.  U  ->  ( .s `  W )  =  ( .s `  X
) )
3736eqcomd 2446 . . . . . . . . . . 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 6107 . . . . . . . . 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 986 . . . . . . . . . . 11  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  W  e.  LMod )
41 imassrn 5177 . . . . . . . . . . . 12  |-  ( F
" ( dom  F  \  { x } ) )  C_  ran  F
42 simpl3 988 . . . . . . . . . . . 12  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ran  F  C_  S
)
4341, 42syl5ss 3364 . . . . . . . . . . 11  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( F "
( dom  F  \  {
x } ) ) 
C_  S )
44 eqid 2441 . . . . . . . . . . . 12  |-  ( LSpan `  W )  =  (
LSpan `  W )
45 eqid 2441 . . . . . . . . . . . 12  |-  ( LSpan `  X )  =  (
LSpan `  X )
467, 44, 45, 15lsslsp 17074 . . . . . . . . . . 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 1213 . . . . . . . . . 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 2446 . . . . . . . . 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 2509 . . . . . . . 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 294 . . . . . . 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 2929 . . . . . 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 2733 . . . . 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 705 . . . 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 6115 . . . . . . 7  |-  ( Ws  S )  e.  _V
557, 54eqeltri 2511 . . . . . 6  |-  X  e. 
_V
5655a1i 11 . . . . 5  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  X  e.  _V )
57 eqid 2441 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
58 eqid 2441 . . . . . 6  |-  ( .s
`  X )  =  ( .s `  X
)
59 eqid 2441 . . . . . 6  |-  (Scalar `  X )  =  (Scalar `  X )
60 eqid 2441 . . . . . 6  |-  ( Base `  (Scalar `  X )
)  =  ( Base `  (Scalar `  X )
)
61 eqid 2441 . . . . . 6  |-  ( 0g
`  (Scalar `  X )
)  =  ( 0g
`  (Scalar `  X )
)
6257, 58, 45, 59, 60, 61islindf 18200 . . . . 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 468 . . . 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 2441 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
65 eqid 2441 . . . . . 6  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
668, 35, 44, 27, 64, 65islindf 18200 . . . . 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 1145 . . . 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 285 . . 3  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( F LIndF  X  <->  F LIndF 
W ) )
6968ex 434 . 2  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F  e.  _V  ->  ( F LIndF  X  <->  F LIndF  W ) ) )
703, 5, 69pm5.21ndd 354 1  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F LIndF  X  <->  F LIndF  W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    \ cdif 3322    C_ wss 3325   {csn 3874   class class class wbr 4289   dom cdm 4836   ran crn 4837   "cima 4839    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   Basecbs 14170   ↾s cress 14171  Scalarcsca 14237   .scvsca 14238   0gc0g 14374   LModclmod 16928   LSubSpclss 16991   LSpanclspn 17030   LIndF clindf 18192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-sca 14250  df-vsca 14251  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-sbg 15540  df-subg 15671  df-mgp 16582  df-ur 16594  df-rng 16637  df-lmod 16930  df-lss 16992  df-lsp 17031  df-lindf 18194
This theorem is referenced by:  lsslinds  18219
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