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Theorem islindeps 32536
Description: The property of being a linearly dependent subset. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Hypotheses
Ref Expression
islindeps.b  |-  B  =  ( Base `  M
)
islindeps.z  |-  Z  =  ( 0g `  M
)
islindeps.r  |-  R  =  (Scalar `  M )
islindeps.e  |-  E  =  ( Base `  R
)
islindeps.0  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
islindeps  |-  ( ( M  e.  W  /\  S  e.  ~P B
)  ->  ( S linDepS  M  <->  E. f  e.  ( E  ^m  S ) ( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z  /\  E. 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)    W( x, f)    .0. ( x, f)    Z( x, f)

Proof of Theorem islindeps
StepHypRef Expression
1 lindepsnlininds 32535 . . 3  |-  ( ( S  e.  ~P B  /\  M  e.  W
)  ->  ( S linDepS  M  <->  -.  S linIndS  M ) )
21ancoms 453 . 2  |-  ( ( M  e.  W  /\  S  e.  ~P B
)  ->  ( S linDepS  M  <->  -.  S linIndS  M ) )
3 islindeps.b . . . . . 6  |-  B  =  ( Base `  M
)
4 islindeps.z . . . . . 6  |-  Z  =  ( 0g `  M
)
5 islindeps.r . . . . . 6  |-  R  =  (Scalar `  M )
6 islindeps.e . . . . . 6  |-  E  =  ( Base `  R
)
7 islindeps.0 . . . . . 6  |-  .0.  =  ( 0g `  R )
83, 4, 5, 6, 7islininds 32529 . . . . 5  |-  ( ( S  e.  ~P B  /\  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.  )
) ) )
98ancoms 453 . . . 4  |-  ( ( M  e.  W  /\  S  e.  ~P B
)  ->  ( 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.  )
) ) )
10 ibar 504 . . . . . 6  |-  ( 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 finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )
) ) )
1110bicomd 201 . . . . 5  |-  ( S  e.  ~P B  -> 
( ( 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.  ) )  <->  A. f  e.  ( E  ^m  S
) ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )
) )
1211adantl 466 . . . 4  |-  ( ( M  e.  W  /\  S  e.  ~P B
)  ->  ( ( 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.  )
)  <->  A. f  e.  ( E  ^m  S ) ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )
) )
139, 12bitrd 253 . . 3  |-  ( ( M  e.  W  /\  S  e.  ~P B
)  ->  ( S linIndS  M  <->  A. f  e.  ( E  ^m  S ) ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )
) )
1413notbid 294 . 2  |-  ( ( M  e.  W  /\  S  e.  ~P B
)  ->  ( -.  S linIndS  M  <->  -.  A. f  e.  ( E  ^m  S
) ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )
) )
15 rexnal 2915 . . . 4  |-  ( E. 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 finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )
)
16 df-ne 2664 . . . . . . . . 9  |-  ( ( f `  x )  =/=  .0.  <->  -.  (
f `  x )  =  .0.  )
1716rexbii 2969 . . . . . . . 8  |-  ( E. x  e.  S  ( f `  x )  =/=  .0.  <->  E. x  e.  S  -.  (
f `  x )  =  .0.  )
18 rexnal 2915 . . . . . . . 8  |-  ( E. x  e.  S  -.  ( f `  x
)  =  .0.  <->  -.  A. x  e.  S  ( f `  x )  =  .0.  )
1917, 18bitr2i 250 . . . . . . 7  |-  ( -. 
A. x  e.  S  ( f `  x
)  =  .0.  <->  E. x  e.  S  ( f `  x )  =/=  .0.  )
2019anbi2i 694 . . . . . 6  |-  ( ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  /\  -.  A. x  e.  S  ( f `  x
)  =  .0.  )  <->  ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  /\  E. x  e.  S  ( f `  x )  =/=  .0.  ) )
21 pm4.61 426 . . . . . 6  |-  ( -.  ( ( 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.  )
)
22 df-3an 975 . . . . . 6  |-  ( ( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z  /\  E. x  e.  S  ( f `  x )  =/=  .0.  ) 
<->  ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  /\  E. x  e.  S  ( f `  x )  =/=  .0.  ) )
2320, 21, 223bitr4i 277 . . . . 5  |-  ( -.  ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )  <->  ( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z  /\  E. x  e.  S  ( f `  x )  =/=  .0.  ) )
2423rexbii 2969 . . . 4  |-  ( E. f  e.  ( E  ^m  S )  -.  ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )  <->  E. f  e.  ( E  ^m  S ) ( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z  /\  E. x  e.  S  ( f `  x )  =/=  .0.  ) )
2515, 24bitr3i 251 . . 3  |-  ( -. 
A. f  e.  ( E  ^m  S ) ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x
)  =  .0.  )  <->  E. f  e.  ( E  ^m  S ) ( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z  /\  E. x  e.  S  ( f `  x )  =/=  .0.  ) )
2625a1i 11 . 2  |-  ( ( 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.  )  <->  E. f  e.  ( E  ^m  S ) ( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z  /\  E. x  e.  S  ( f `  x )  =/=  .0.  ) ) )
272, 14, 263bitrd 279 1  |-  ( ( M  e.  W  /\  S  e.  ~P B
)  ->  ( S linDepS  M  <->  E. f  e.  ( E  ^m  S ) ( f finSupp  .0.  /\  (
f ( linC  `  M
) S )  =  Z  /\  E. x  e.  S  ( f `  x )  =/=  .0.  ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   ~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   linDepS clindeps 32524
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-iota 5557  df-fv 5602  df-ov 6298  df-lininds 32525  df-lindeps 32527
This theorem is referenced by:  el0ldep  32549  ldepspr  32556  islindeps2  32566  isldepslvec2  32568  zlmodzxzldep  32587
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