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Theorem linindslinci 31001
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 31000 . . 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 4310 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f finSupp  .0.  <->  F finSupp  .0.  ) )
8 oveq1 6113 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f ( linC  `  M
) S )  =  ( F ( linC  `  M ) S ) )
98eqeq1d 2451 . . . . . . . . . 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 5705 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
1211eqeq1d 2451 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( f `  x
)  =  .0.  <->  ( F `  x )  =  .0.  ) )
1312ralbidv 2750 . . . . . . . . 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 3084 . . . . . . 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 1185 . . . . 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 965    = wceq 1369    e. wcel 1756   A.wral 2730   ~Pcpw 3875   class class class wbr 4307   ` cfv 5433  (class class class)co 6106    ^m cmap 7229   finSupp cfsupp 7635   Basecbs 14189  Scalarcsca 14256   0gc0g 14393   linC clinc 30957   linIndS clininds 30993
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pr 4546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-xp 4861  df-rel 4862  df-iota 5396  df-fv 5441  df-ov 6109  df-lininds 30995
This theorem is referenced by: (None)
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