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Theorem islininds 32120
Description: The property 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
islininds  |-  ( ( S  e.  V  /\  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.  ) ) ) )
Distinct variable groups:    f, E    f, M, x    S, f, x
Allowed substitution hints:    B( x, f)    R( x, f)    E( x)    V( x, f)    W( x, f)    .0. ( x, f)    Z( x, f)

Proof of Theorem islininds
Dummy variables  m  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . 4  |-  ( ( s  =  S  /\  m  =  M )  ->  s  =  S )
2 fveq2 5864 . . . . . . 7  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
3 islininds.b . . . . . . 7  |-  B  =  ( Base `  M
)
42, 3syl6eqr 2526 . . . . . 6  |-  ( m  =  M  ->  ( Base `  m )  =  B )
54adantl 466 . . . . 5  |-  ( ( s  =  S  /\  m  =  M )  ->  ( Base `  m
)  =  B )
65pweqd 4015 . . . 4  |-  ( ( s  =  S  /\  m  =  M )  ->  ~P ( Base `  m
)  =  ~P B
)
71, 6eleq12d 2549 . . 3  |-  ( ( s  =  S  /\  m  =  M )  ->  ( s  e.  ~P ( Base `  m )  <->  S  e.  ~P B ) )
8 fveq2 5864 . . . . . . . . 9  |-  ( m  =  M  ->  (Scalar `  m )  =  (Scalar `  M ) )
9 islininds.r . . . . . . . . 9  |-  R  =  (Scalar `  M )
108, 9syl6eqr 2526 . . . . . . . 8  |-  ( m  =  M  ->  (Scalar `  m )  =  R )
1110fveq2d 5868 . . . . . . 7  |-  ( m  =  M  ->  ( Base `  (Scalar `  m
) )  =  (
Base `  R )
)
12 islininds.e . . . . . . 7  |-  E  =  ( Base `  R
)
1311, 12syl6eqr 2526 . . . . . 6  |-  ( m  =  M  ->  ( Base `  (Scalar `  m
) )  =  E )
1413adantl 466 . . . . 5  |-  ( ( s  =  S  /\  m  =  M )  ->  ( Base `  (Scalar `  m ) )  =  E )
1514, 1oveq12d 6300 . . . 4  |-  ( ( s  =  S  /\  m  =  M )  ->  ( ( Base `  (Scalar `  m ) )  ^m  s )  =  ( E  ^m  S ) )
168adantl 466 . . . . . . . . . 10  |-  ( ( s  =  S  /\  m  =  M )  ->  (Scalar `  m )  =  (Scalar `  M )
)
1716, 9syl6eqr 2526 . . . . . . . . 9  |-  ( ( s  =  S  /\  m  =  M )  ->  (Scalar `  m )  =  R )
1817fveq2d 5868 . . . . . . . 8  |-  ( ( s  =  S  /\  m  =  M )  ->  ( 0g `  (Scalar `  m ) )  =  ( 0g `  R
) )
19 islininds.0 . . . . . . . 8  |-  .0.  =  ( 0g `  R )
2018, 19syl6eqr 2526 . . . . . . 7  |-  ( ( s  =  S  /\  m  =  M )  ->  ( 0g `  (Scalar `  m ) )  =  .0.  )
2120breq2d 4459 . . . . . 6  |-  ( ( s  =  S  /\  m  =  M )  ->  ( f finSupp  ( 0g
`  (Scalar `  m )
)  <->  f finSupp  .0.  ) )
22 fveq2 5864 . . . . . . . . 9  |-  ( m  =  M  ->  ( linC  `  m )  =  ( linC  `  M ) )
2322adantl 466 . . . . . . . 8  |-  ( ( s  =  S  /\  m  =  M )  ->  ( linC  `  m )  =  ( linC  `  M ) )
24 eqidd 2468 . . . . . . . 8  |-  ( ( s  =  S  /\  m  =  M )  ->  f  =  f )
2523, 24, 1oveq123d 6303 . . . . . . 7  |-  ( ( s  =  S  /\  m  =  M )  ->  ( f ( linC  `  m ) s )  =  ( f ( linC  `  M ) S ) )
26 fveq2 5864 . . . . . . . . 9  |-  ( m  =  M  ->  ( 0g `  m )  =  ( 0g `  M
) )
2726adantl 466 . . . . . . . 8  |-  ( ( s  =  S  /\  m  =  M )  ->  ( 0g `  m
)  =  ( 0g
`  M ) )
28 islininds.z . . . . . . . 8  |-  Z  =  ( 0g `  M
)
2927, 28syl6eqr 2526 . . . . . . 7  |-  ( ( s  =  S  /\  m  =  M )  ->  ( 0g `  m
)  =  Z )
3025, 29eqeq12d 2489 . . . . . 6  |-  ( ( s  =  S  /\  m  =  M )  ->  ( ( f ( linC  `  m ) s )  =  ( 0g `  m )  <->  ( f
( linC  `  M ) S )  =  Z ) )
3121, 30anbi12d 710 . . . . 5  |-  ( ( s  =  S  /\  m  =  M )  ->  ( ( f finSupp  ( 0g `  (Scalar `  m
) )  /\  (
f ( linC  `  m
) s )  =  ( 0g `  m
) )  <->  ( f finSupp  .0. 
/\  ( f ( linC  `  M ) S )  =  Z ) ) )
3210fveq2d 5868 . . . . . . . . 9  |-  ( m  =  M  ->  ( 0g `  (Scalar `  m
) )  =  ( 0g `  R ) )
3332, 19syl6eqr 2526 . . . . . . . 8  |-  ( m  =  M  ->  ( 0g `  (Scalar `  m
) )  =  .0.  )
3433adantl 466 . . . . . . 7  |-  ( ( s  =  S  /\  m  =  M )  ->  ( 0g `  (Scalar `  m ) )  =  .0.  )
3534eqeq2d 2481 . . . . . 6  |-  ( ( s  =  S  /\  m  =  M )  ->  ( ( f `  x )  =  ( 0g `  (Scalar `  m ) )  <->  ( f `  x )  =  .0.  ) )
361, 35raleqbidv 3072 . . . . 5  |-  ( ( s  =  S  /\  m  =  M )  ->  ( A. x  e.  s  ( f `  x )  =  ( 0g `  (Scalar `  m ) )  <->  A. x  e.  S  ( f `  x )  =  .0.  ) )
3731, 36imbi12d 320 . . . 4  |-  ( ( s  =  S  /\  m  =  M )  ->  ( ( ( f finSupp 
( 0g `  (Scalar `  m ) )  /\  ( f ( linC  `  m ) s )  =  ( 0g `  m ) )  ->  A. x  e.  s 
( f `  x
)  =  ( 0g
`  (Scalar `  m )
) )  <->  ( (
f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x )  =  .0.  ) ) )
3815, 37raleqbidv 3072 . . 3  |-  ( ( s  =  S  /\  m  =  M )  ->  ( A. f  e.  ( ( Base `  (Scalar `  m ) )  ^m  s ) ( ( f finSupp  ( 0g `  (Scalar `  m ) )  /\  ( f ( linC  `  m ) s )  =  ( 0g `  m ) )  ->  A. x  e.  s 
( f `  x
)  =  ( 0g
`  (Scalar `  m )
) )  <->  A. f  e.  ( E  ^m  S
) ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )
) )
397, 38anbi12d 710 . 2  |-  ( ( s  =  S  /\  m  =  M )  ->  ( ( s  e. 
~P ( Base `  m
)  /\  A. f  e.  ( ( Base `  (Scalar `  m ) )  ^m  s ) ( ( f finSupp  ( 0g `  (Scalar `  m ) )  /\  ( f ( linC  `  m ) s )  =  ( 0g `  m ) )  ->  A. x  e.  s 
( f `  x
)  =  ( 0g
`  (Scalar `  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.  ) ) ) )
40 df-lininds 32116 . 2  |- linIndS  =  { <. s ,  m >.  |  ( s  e.  ~P ( Base `  m )  /\  A. f  e.  ( ( Base `  (Scalar `  m ) )  ^m  s ) ( ( f finSupp  ( 0g `  (Scalar `  m ) )  /\  ( f ( linC  `  m ) s )  =  ( 0g `  m ) )  ->  A. x  e.  s 
( f `  x
)  =  ( 0g
`  (Scalar `  m )
) ) ) }
4139, 40brabga 4761 1  |-  ( ( S  e.  V  /\  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.  ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   ~Pcpw 4010   class class class wbr 4447   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   finSupp cfsupp 7825   Basecbs 14483  Scalarcsca 14551   0gc0g 14688   linC clinc 32078   linIndS clininds 32114
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 4568  ax-nul 4576  ax-pr 4686
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-iota 5549  df-fv 5594  df-ov 6285  df-lininds 32116
This theorem is referenced by:  linindsi  32121  islinindfis  32123  islindeps  32127  lindslininds  32138  linds0  32139  lindsrng01  32142  snlindsntor  32145
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