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Theorem linindslinci 32531
Description: The implications of being a linearly independent subset and a linear combination of this subset being 0. (Contributed by AV, 24-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
linindslinci  |-  ( ( S linIndS  M  /\  ( F  e.  ( E  ^m  S )  /\  F finSupp  .0. 
/\  ( F ( linC  `  M ) S )  =  Z ) )  ->  A. x  e.  S  ( F `  x )  =  .0.  )
Distinct variable groups:    x, M    x, S    x, F
Allowed substitution hints:    B( x)    R( x)    E( x)    .0. ( x)    Z( x)

Proof of Theorem linindslinci
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 islininds.b . . . 4  |-  B  =  ( Base `  M
)
2 islininds.z . . . 4  |-  Z  =  ( 0g `  M
)
3 islininds.r . . . 4  |-  R  =  (Scalar `  M )
4 islininds.e . . . 4  |-  E  =  ( Base `  R
)
5 islininds.0 . . . 4  |-  .0.  =  ( 0g `  R )
61, 2, 3, 4, 5linindsi 32530 . . 3  |-  ( 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.  ) ) )
7 breq1 4456 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f finSupp  .0.  <->  F finSupp  .0.  ) )
8 oveq1 6302 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f ( linC  `  M
) S )  =  ( F ( linC  `  M ) S ) )
98eqeq1d 2469 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( f ( linC  `  M ) S )  =  Z  <->  ( F
( linC  `  M ) S )  =  Z ) )
107, 9anbi12d 710 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z )  <->  ( F finSupp  .0. 
/\  ( F ( linC  `  M ) S )  =  Z ) ) )
11 fveq1 5871 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
1211eqeq1d 2469 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( f `  x
)  =  .0.  <->  ( F `  x )  =  .0.  ) )
1312ralbidv 2906 . . . . . . . . 9  |-  ( f  =  F  ->  ( A. x  e.  S  ( f `  x
)  =  .0.  <->  A. x  e.  S  ( F `  x )  =  .0.  ) )
1410, 13imbi12d 320 . . . . . . . 8  |-  ( f  =  F  ->  (
( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )  <->  ( ( F finSupp  .0.  /\  ( F ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( F `  x )  =  .0.  ) ) )
1514rspcv 3215 . . . . . . 7  |-  ( F  e.  ( E  ^m  S )  ->  ( A. f  e.  ( E  ^m  S ) ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )  ->  ( ( F finSupp  .0.  /\  ( F ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( F `  x )  =  .0.  ) ) )
1615com23 78 . . . . . 6  |-  ( F  e.  ( E  ^m  S )  ->  (
( F finSupp  .0.  /\  ( F ( linC  `  M ) S )  =  Z )  ->  ( A. f  e.  ( E  ^m  S ) ( ( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z )  ->  A. x  e.  S  ( f `  x )  =  .0.  )  ->  A. x  e.  S  ( F `  x )  =  .0.  ) ) )
17163impib 1194 . . . . 5  |-  ( ( F  e.  ( E  ^m  S )  /\  F finSupp  .0.  /\  ( F ( linC  `  M ) S )  =  Z )  ->  ( A. f  e.  ( E  ^m  S ) ( ( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z )  ->  A. x  e.  S  ( f `  x )  =  .0.  )  ->  A. x  e.  S  ( F `  x )  =  .0.  ) )
1817com12 31 . . . 4  |-  ( A. f  e.  ( E  ^m  S ) ( ( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z )  ->  A. x  e.  S  ( f `  x )  =  .0.  )  ->  ( ( F  e.  ( E  ^m  S )  /\  F finSupp  .0. 
/\  ( F ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( F `  x )  =  .0.  ) )
1918adantl 466 . . 3  |-  ( ( 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.  )
)  ->  ( ( F  e.  ( E  ^m  S )  /\  F finSupp  .0. 
/\  ( F ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( F `  x )  =  .0.  ) )
206, 19syl 16 . 2  |-  ( S linIndS  M  ->  ( ( F  e.  ( E  ^m  S )  /\  F finSupp  .0. 
/\  ( F ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( F `  x )  =  .0.  ) )
2120imp 429 1  |-  ( ( S linIndS  M  /\  ( 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    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   ~Pcpw 4016   class class class wbr 4453   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   finSupp cfsupp 7841   Basecbs 14507  Scalarcsca 14575   0gc0g 14712   linC clinc 32487   linIndS clininds 32523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-iota 5557  df-fv 5602  df-ov 6298  df-lininds 32525
This theorem is referenced by: (None)
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