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Theorem linindsi 31095
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 31093 . . 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 31094 . . 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 1370    e. wcel 1758   A.wral 2796   _Vcvv 3072   ~Pcpw 3963   class class class wbr 4395   ` cfv 5521  (class class class)co 6195    ^m cmap 7319   finSupp cfsupp 7726   Basecbs 14287  Scalarcsca 14355   0gc0g 14492   linC clinc 31052   linIndS clininds 31088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-xp 4949  df-rel 4950  df-iota 5484  df-fv 5529  df-ov 6198  df-lininds 31090
This theorem is referenced by:  linindslinci  31096  linindscl  31099
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