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Theorem lsslindf 18262
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 18240 . . . 4  |-  Rel LIndF
21brrelexi 4882 . . 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 4882 . . 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 461 . . . . . . . 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 2443 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
97, 8ressbasss 14233 . . . . . . . 8  |-  ( Base `  X )  C_  ( Base `  W )
10 fss 5570 . . . . . . . 8  |-  ( ( F : dom  F --> ( Base `  X )  /\  ( Base `  X
)  C_  ( Base `  W ) )  ->  F : dom  F --> ( Base `  W ) )
116, 9, 10sylancl 662 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  X )
)  ->  F : dom  F --> ( Base `  W
) )
12 ffn 5562 . . . . . . . . 9  |-  ( F : dom  F --> ( Base `  W )  ->  F  Fn  dom  F )
1312adantl 466 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  W )
)  ->  F  Fn  dom  F )
14 simp3 990 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ran  F 
C_  S )
15 lsslindf.u . . . . . . . . . . . . 13  |-  U  =  ( LSubSp `  W )
168, 15lssss 17021 . . . . . . . . . . . 12  |-  ( S  e.  U  ->  S  C_  ( Base `  W
) )
17163ad2ant2 1010 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  S  C_  ( Base `  W
) )
187, 8ressbas2 14232 . . . . . . . . . . 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 3395 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ran  F 
C_  ( Base `  X
) )
2120adantr 465 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  W )
)  ->  ran  F  C_  ( Base `  X )
)
22 df-f 5425 . . . . . . . 8  |-  ( F : dom  F --> ( Base `  X )  <->  ( F  Fn  dom  F  /\  ran  F 
C_  ( Base `  X
) ) )
2313, 21, 22sylanbrc 664 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F : dom  F --> ( Base `  W )
)  ->  F : dom  F --> ( Base `  X
) )
2411, 23impbida 828 . . . . . 6  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  ( F : dom  F --> ( Base `  X )  <->  F : dom  F --> ( Base `  W
) ) )
2524adantr 465 . . . . 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 992 . . . . . . . . . 10  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  S  e.  U
)
27 eqid 2443 . . . . . . . . . . . 12  |-  (Scalar `  W )  =  (Scalar `  W )
287, 27resssca 14319 . . . . . . . . . . 11  |-  ( S  e.  U  ->  (Scalar `  W )  =  (Scalar `  X ) )
2928eqcomd 2448 . . . . . . . . . 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 5698 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( Base `  (Scalar `  X ) )  =  ( Base `  (Scalar `  W ) ) )
3230fveq2d 5698 . . . . . . . . 9  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( 0g `  (Scalar `  X ) )  =  ( 0g `  (Scalar `  W ) ) )
3332sneqd 3892 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  { ( 0g
`  (Scalar `  X )
) }  =  {
( 0g `  (Scalar `  W ) ) } )
3431, 33difeq12d 3478 . . . . . . 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 2443 . . . . . . . . . . . . 13  |-  ( .s
`  W )  =  ( .s `  W
)
367, 35ressvsca 14320 . . . . . . . . . . . 12  |-  ( S  e.  U  ->  ( .s `  W )  =  ( .s `  X
) )
3736eqcomd 2448 . . . . . . . . . . 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 6111 . . . . . . . . 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 991 . . . . . . . . . . 11  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  W  e.  LMod )
41 imassrn 5183 . . . . . . . . . . . 12  |-  ( F
" ( dom  F  \  { x } ) )  C_  ran  F
42 simpl3 993 . . . . . . . . . . . 12  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ran  F  C_  S
)
4341, 42syl5ss 3370 . . . . . . . . . . 11  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( F "
( dom  F  \  {
x } ) ) 
C_  S )
44 eqid 2443 . . . . . . . . . . . 12  |-  ( LSpan `  W )  =  (
LSpan `  W )
45 eqid 2443 . . . . . . . . . . . 12  |-  ( LSpan `  X )  =  (
LSpan `  X )
467, 44, 45, 15lsslsp 17099 . . . . . . . . . . 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 1218 . . . . . . . . . 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 2448 . . . . . . . . 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 2511 . . . . . . . 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 2934 . . . . . 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 2738 . . . . 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 710 . . . 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 6119 . . . . . . 7  |-  ( Ws  S )  e.  _V
557, 54eqeltri 2513 . . . . . 6  |-  X  e. 
_V
5655a1i 11 . . . . 5  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  X  e.  _V )
57 eqid 2443 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
58 eqid 2443 . . . . . 6  |-  ( .s
`  X )  =  ( .s `  X
)
59 eqid 2443 . . . . . 6  |-  (Scalar `  X )  =  (Scalar `  X )
60 eqid 2443 . . . . . 6  |-  ( Base `  (Scalar `  X )
)  =  ( Base `  (Scalar `  X )
)
61 eqid 2443 . . . . . 6  |-  ( 0g
`  (Scalar `  X )
)  =  ( 0g
`  (Scalar `  X )
)
6257, 58, 45, 59, 60, 61islindf 18244 . . . . 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 471 . . . 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 2443 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
65 eqid 2443 . . . . . 6  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
668, 35, 44, 27, 64, 65islindf 18244 . . . . 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 1150 . . . 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 965    = wceq 1369    e. wcel 1756   A.wral 2718   _Vcvv 2975    \ cdif 3328    C_ wss 3331   {csn 3880   class class class wbr 4295   dom cdm 4843   ran crn 4844   "cima 4846    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6094   Basecbs 14177   ↾s cress 14178  Scalarcsca 14244   .scvsca 14245   0gc0g 14381   LModclmod 16951   LSubSpclss 17016   LSpanclspn 17055   LIndF clindf 18236
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 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
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 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-sca 14257  df-vsca 14258  df-0g 14383  df-mnd 15418  df-grp 15548  df-minusg 15549  df-sbg 15550  df-subg 15681  df-mgp 16595  df-ur 16607  df-rng 16650  df-lmod 16953  df-lss 17017  df-lsp 17056  df-lindf 18238
This theorem is referenced by:  lsslinds  18263
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