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Theorem lsslindf 27168
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 27146 . . . 4  |-  Rel LIndF
21brrelexi 4877 . . 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 4877 . . 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 2404 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
97, 8ressbasss 13476 . . . . . . . 8  |-  ( Base `  X )  C_  ( Base `  W )
10 fss 5558 . . . . . . . 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 5550 . . . . . . . . 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 15968 . . . . . . . . . . . 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 13475 . . . . . . . . . . 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 3344 . . . . . . . . 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 5417 . . . . . . . 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 2404 . . . . . . . . . . . 12  |-  (Scalar `  W )  =  (Scalar `  W )
287, 27resssca 13559 . . . . . . . . . . 11  |-  ( S  e.  U  ->  (Scalar `  W )  =  (Scalar `  X ) )
2928eqcomd 2409 . . . . . . . . . 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 5691 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( Base `  (Scalar `  X ) )  =  ( Base `  (Scalar `  W ) ) )
3230fveq2d 5691 . . . . . . . . 9  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( 0g `  (Scalar `  X ) )  =  ( 0g `  (Scalar `  W ) ) )
3332sneqd 3787 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  { ( 0g
`  (Scalar `  X )
) }  =  {
( 0g `  (Scalar `  W ) ) } )
3431, 33difeq12d 3426 . . . . . . 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 2404 . . . . . . . . . . . . 13  |-  ( .s
`  W )  =  ( .s `  W
)
367, 35ressvsca 13560 . . . . . . . . . . . 12  |-  ( S  e.  U  ->  ( .s `  W )  =  ( .s `  X
) )
3736eqcomd 2409 . . . . . . . . . . 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 6057 . . . . . . . . 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 5175 . . . . . . . . . . . 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 3319 . . . . . . . . . . 11  |-  ( ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  /\  F  e.  _V )  ->  ( F "
( dom  F  \  {
x } ) ) 
C_  S )
44 eqid 2404 . . . . . . . . . . . 12  |-  ( LSpan `  W )  =  (
LSpan `  W )
45 eqid 2404 . . . . . . . . . . . 12  |-  ( LSpan `  X )  =  (
LSpan `  X )
467, 44, 45, 15lsslsp 16046 . . . . . . . . . . 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 2409 . . . . . . . . 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 2472 . . . . . . . 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 2876 . . . . . 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 2686 . . . . 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 6065 . . . . . . 7  |-  ( Ws  S )  e.  _V
557, 54eqeltri 2474 . . . . . 6  |-  X  e. 
_V
5655a1i 11 . . . . 5  |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran  F  C_  S )  ->  X  e.  _V )
57 eqid 2404 . . . . . 6  |-  ( Base `  X )  =  (
Base `  X )
58 eqid 2404 . . . . . 6  |-  ( .s
`  X )  =  ( .s `  X
)
59 eqid 2404 . . . . . 6  |-  (Scalar `  X )  =  (Scalar `  X )
60 eqid 2404 . . . . . 6  |-  ( Base `  (Scalar `  X )
)  =  ( Base `  (Scalar `  X )
)
61 eqid 2404 . . . . . 6  |-  ( 0g
`  (Scalar `  X )
)  =  ( 0g
`  (Scalar `  X )
)
6257, 58, 45, 59, 60, 61islindf 27150 . . . . 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 2404 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
65 eqid 2404 . . . . . 6  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
668, 35, 44, 27, 64, 65islindf 27150 . . . . 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 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    \ cdif 3277    C_ wss 3280   {csn 3774   class class class wbr 4172   dom cdm 4837   ran crn 4838   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   Basecbs 13424   ↾s cress 13425  Scalarcsca 13487   .scvsca 13488   0gc0g 13678   LModclmod 15905   LSubSpclss 15963   LSpanclspn 16002   LIndF clindf 27142
This theorem is referenced by:  lsslinds  27169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-sca 13500  df-vsca 13501  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-mgp 15604  df-rng 15618  df-ur 15620  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lindf 27144
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