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Theorem linindsi 33192
Description: The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Hypotheses
Ref Expression
islininds.b  |-  B  =  ( Base `  M
)
islininds.z  |-  Z  =  ( 0g `  M
)
islininds.r  |-  R  =  (Scalar `  M )
islininds.e  |-  E  =  ( Base `  R
)
islininds.0  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
linindsi  |-  ( S linIndS  M  ->  ( S  e. 
~P B  /\  A. f  e.  ( E  ^m  S ) ( ( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z )  ->  A. x  e.  S  ( f `  x )  =  .0.  ) ) )
Distinct variable groups:    f, E    f, M, x    S, f, x
Allowed substitution hints:    B( x, f)    R( x, f)    E( x)    .0. ( x, f)    Z( x, f)

Proof of Theorem linindsi
StepHypRef Expression
1 linindsv 33190 . . 3  |-  ( S linIndS  M  ->  ( S  e. 
_V  /\  M  e.  _V ) )
2 islininds.b . . . 4  |-  B  =  ( Base `  M
)
3 islininds.z . . . 4  |-  Z  =  ( 0g `  M
)
4 islininds.r . . . 4  |-  R  =  (Scalar `  M )
5 islininds.e . . . 4  |-  E  =  ( Base `  R
)
6 islininds.0 . . . 4  |-  .0.  =  ( 0g `  R )
72, 3, 4, 5, 6islininds 33191 . . 3  |-  ( ( S  e.  _V  /\  M  e.  _V )  ->  ( S linIndS  M  <->  ( S  e.  ~P B  /\  A. f  e.  ( E  ^m  S ) ( ( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z )  ->  A. x  e.  S  ( f `  x )  =  .0.  ) ) ) )
81, 7syl 16 . 2  |-  ( S linIndS  M  ->  ( S linIndS  M  <->  ( S  e.  ~P B  /\  A. f  e.  ( E  ^m  S ) ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )
) ) )
98ibi 241 1  |-  ( S linIndS  M  ->  ( S  e. 
~P B  /\  A. f  e.  ( E  ^m  S ) ( ( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z )  ->  A. x  e.  S  ( f `  x )  =  .0.  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109   ~Pcpw 4015   class class class wbr 4456   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   finSupp cfsupp 7847   Basecbs 14644  Scalarcsca 14715   0gc0g 14857   linC clinc 33149   linIndS clininds 33185
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-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-iota 5557  df-fv 5602  df-ov 6299  df-lininds 33187
This theorem is referenced by:  linindslinci  33193  linindscl  33196
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