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Theorem islinindfis 33304
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 33301 . 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 pm4.79 581 . . . . . . 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 7433 . . . . . . . . . . . . 13  |-  ( f  e.  ( E  ^m  S )  ->  f : S --> E )
98adantl 464 . . . . . . . . . . . 12  |-  ( ( ( S  e.  Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S ) )  ->  f : S
--> E )
10 simpll 751 . . . . . . . . . . . 12  |-  ( ( ( S  e.  Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S ) )  ->  S  e.  Fin )
11 fvex 5858 . . . . . . . . . . . . . 14  |-  ( 0g
`  R )  e. 
_V
125, 11eqeltri 2538 . . . . . . . . . . . . 13  |-  .0.  e.  _V
1312a1i 11 . . . . . . . . . . . 12  |-  ( ( ( S  e.  Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S ) )  ->  .0.  e.  _V )
149, 10, 13fdmfifsupp 7831 . . . . . . . . . . 11  |-  ( ( ( S  e.  Fin  /\  M  e.  W )  /\  f  e.  ( E  ^m  S ) )  ->  f finSupp  .0.  )
1514adantr 463 . . . . . . . . . 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 434 . . . . . . . 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 377 . . . . . . 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 pm3.42 558 . . . . 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 2890 . . 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 701 . 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 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106   ~Pcpw 3999   class class class wbr 4439   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   Fincfn 7509   finSupp cfsupp 7821   Basecbs 14716  Scalarcsca 14787   0gc0g 14929   linC clinc 33259   linIndS clininds 33295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-er 7303  df-map 7414  df-en 7510  df-fin 7513  df-fsupp 7822  df-lininds 33297
This theorem is referenced by:  islinindfiss  33305
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