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Theorem islinindfis 31097
Description: The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-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
islinindfis  |-  ( ( S  e.  Fin  /\  M  e.  W )  ->  ( S linIndS  M  <->  ( S  e.  ~P B  /\  A. f  e.  ( E  ^m  S ) ( ( f ( linC  `  M
) S )  =  Z  ->  A. x  e.  S  ( f `  x )  =  .0.  ) ) ) )
Distinct variable groups:    f, E    f, M, x    S, f, x    .0. , f    f, Z   
f, W
Allowed substitution hints:    B( x, f)    R( x, f)    E( x)    W( x)    .0. ( x)    Z( x)

Proof of Theorem islinindfis
StepHypRef Expression
1 islininds.b . . 3  |-  B  =  ( Base `  M
)
2 islininds.z . . 3  |-  Z  =  ( 0g `  M
)
3 islininds.r . . 3  |-  R  =  (Scalar `  M )
4 islininds.e . . 3  |-  E  =  ( Base `  R
)
5 islininds.0 . . 3  |-  .0.  =  ( 0g `  R )
61, 2, 3, 4, 5islininds 31094 . 2  |-  ( ( S  e.  Fin  /\  M  e.  W )  ->  ( 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 pm4.79 583 . . . . . . 7  |-  ( ( ( f finSupp  .0.  ->  A. x  e.  S  ( f `  x )  =  .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.  ) )
8 elmapi 7339 . . . . . . . . . . . . 13  |-  ( f  e.  ( E  ^m  S )  ->  f : S --> E )
98adantl 466 . . . . . . . . . . . 12  |-  ( ( ( S  e.  Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S ) )  ->  f : S
--> E )
10 simpll 753 . . . . . . . . . . . 12  |-  ( ( ( S  e.  Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S ) )  ->  S  e.  Fin )
11 fvex 5804 . . . . . . . . . . . . . 14  |-  ( 0g
`  R )  e. 
_V
125, 11eqeltri 2536 . . . . . . . . . . . . 13  |-  .0.  e.  _V
1312a1i 11 . . . . . . . . . . . 12  |-  ( ( ( S  e.  Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S ) )  ->  .0.  e.  _V )
149, 10, 13fdmfifsupp 7736 . . . . . . . . . . 11  |-  ( ( ( S  e.  Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S ) )  ->  f finSupp  .0.  )
1514adantr 465 . . . . . . . . . 10  |-  ( ( ( ( S  e. 
Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S
) )  /\  (
f ( linC  `  M
) S )  =  Z )  ->  f finSupp  .0.  )
1615imim1i 58 . . . . . . . . 9  |-  ( ( f finSupp  .0.  ->  A. x  e.  S  ( f `  x )  =  .0.  )  ->  ( (
( ( S  e. 
Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S
) )  /\  (
f ( linC  `  M
) S )  =  Z )  ->  A. x  e.  S  ( f `  x )  =  .0.  ) )
1716expd 436 . . . . . . . 8  |-  ( ( f finSupp  .0.  ->  A. x  e.  S  ( f `  x )  =  .0.  )  ->  ( (
( S  e.  Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S ) )  ->  ( (
f ( linC  `  M
) S )  =  Z  ->  A. x  e.  S  ( f `  x )  =  .0.  ) ) )
18 ax-1 6 . . . . . . . 8  |-  ( ( ( f ( linC  `  M ) S )  =  Z  ->  A. x  e.  S  ( f `  x )  =  .0.  )  ->  ( (
( S  e.  Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S ) )  ->  ( (
f ( linC  `  M
) S )  =  Z  ->  A. x  e.  S  ( f `  x )  =  .0.  ) ) )
1917, 18jaoi 379 . . . . . . 7  |-  ( ( ( f finSupp  .0.  ->  A. x  e.  S  ( f `  x )  =  .0.  )  \/  ( ( f ( linC  `  M ) S )  =  Z  ->  A. x  e.  S  ( f `  x )  =  .0.  ) )  ->  (
( ( S  e. 
Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S
) )  ->  (
( f ( linC  `  M ) S )  =  Z  ->  A. x  e.  S  ( f `  x )  =  .0.  ) ) )
207, 19sylbir 213 . . . . . 6  |-  ( ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )  ->  ( ( ( S  e.  Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S
) )  ->  (
( f ( linC  `  M ) S )  =  Z  ->  A. x  e.  S  ( f `  x )  =  .0.  ) ) )
2120com12 31 . . . . 5  |-  ( ( ( S  e.  Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S ) )  ->  ( (
( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z )  ->  A. x  e.  S  ( f `  x )  =  .0.  )  ->  ( (
f ( linC  `  M
) S )  =  Z  ->  A. x  e.  S  ( f `  x )  =  .0.  ) ) )
22 pm3.42 560 . . . . 5  |-  ( ( ( 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.  ) )
2321, 22impbid1 203 . . . 4  |-  ( ( ( S  e.  Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S ) )  ->  ( (
( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z )  ->  A. x  e.  S  ( f `  x )  =  .0.  )  <->  ( ( f ( linC  `  M ) S )  =  Z  ->  A. x  e.  S  ( f `  x
)  =  .0.  )
) )
2423ralbidva 2841 . . 3  |-  ( ( S  e.  Fin  /\  M  e.  W )  ->  ( A. f  e.  ( E  ^m  S
) ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )  <->  A. f  e.  ( E  ^m  S ) ( ( f ( linC  `  M ) S )  =  Z  ->  A. x  e.  S  ( f `  x )  =  .0.  ) ) )
2524anbi2d 703 . 2  |-  ( ( S  e.  Fin  /\  M  e.  W )  ->  ( ( 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.  ) )  <->  ( S  e.  ~P B  /\  A. f  e.  ( E  ^m  S ) ( ( f ( linC  `  M
) S )  =  Z  ->  A. x  e.  S  ( f `  x )  =  .0.  ) ) ) )
266, 25bitrd 253 1  |-  ( ( S  e.  Fin  /\  M  e.  W )  ->  ( S linIndS  M  <->  ( S  e.  ~P B  /\  A. f  e.  ( E  ^m  S ) ( ( f ( linC  `  M
) S )  =  Z  ->  A. x  e.  S  ( f `  x )  =  .0.  ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2796   _Vcvv 3072   ~Pcpw 3963   class class class wbr 4395   -->wf 5517   ` cfv 5521  (class class class)co 6195    ^m cmap 7319   Fincfn 7415   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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-supp 6796  df-er 7206  df-map 7321  df-en 7416  df-fin 7419  df-fsupp 7727  df-lininds 31090
This theorem is referenced by:  islinindfiss  31098
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