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Theorem linds0 32548
Description: The empty set is always a linearly independet subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Assertion
Ref Expression
linds0  |-  ( M  e.  V  ->  (/) linIndS  M )

Proof of Theorem linds0
Dummy variables  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 3938 . . . . . 6  |-  A. x  e.  (/)  ( (/) `  x
)  =  ( 0g
`  (Scalar `  M )
)
21a1ii 27 . . . . 5  |-  ( M  e.  V  ->  (
( (/) finSupp  ( 0g `  (Scalar `  M ) )  /\  ( (/) ( linC  `  M
) (/) )  =  ( 0g `  M ) )  ->  A. x  e.  (/)  ( (/) `  x
)  =  ( 0g
`  (Scalar `  M )
) ) )
3 0ex 4583 . . . . . 6  |-  (/)  e.  _V
4 breq1 4456 . . . . . . . . 9  |-  ( f  =  (/)  ->  ( f finSupp 
( 0g `  (Scalar `  M ) )  <->  (/) finSupp  ( 0g `  (Scalar `  M )
) ) )
5 oveq1 6302 . . . . . . . . . 10  |-  ( f  =  (/)  ->  ( f ( linC  `  M ) (/) )  =  ( (/) ( linC  `  M ) (/) ) )
65eqeq1d 2469 . . . . . . . . 9  |-  ( f  =  (/)  ->  ( ( f ( linC  `  M
) (/) )  =  ( 0g `  M )  <-> 
( (/) ( linC  `  M
) (/) )  =  ( 0g `  M ) ) )
74, 6anbi12d 710 . . . . . . . 8  |-  ( f  =  (/)  ->  ( ( f finSupp  ( 0g `  (Scalar `  M ) )  /\  ( f ( linC  `  M ) (/) )  =  ( 0g `  M
) )  <->  ( (/) finSupp  ( 0g
`  (Scalar `  M )
)  /\  ( (/) ( linC  `  M ) (/) )  =  ( 0g `  M
) ) ) )
8 fveq1 5871 . . . . . . . . . 10  |-  ( f  =  (/)  ->  ( f `
 x )  =  ( (/) `  x ) )
98eqeq1d 2469 . . . . . . . . 9  |-  ( f  =  (/)  ->  ( ( f `  x )  =  ( 0g `  (Scalar `  M ) )  <-> 
( (/) `  x )  =  ( 0g `  (Scalar `  M ) ) ) )
109ralbidv 2906 . . . . . . . 8  |-  ( f  =  (/)  ->  ( A. x  e.  (/)  ( f `
 x )  =  ( 0g `  (Scalar `  M ) )  <->  A. x  e.  (/)  ( (/) `  x
)  =  ( 0g
`  (Scalar `  M )
) ) )
117, 10imbi12d 320 . . . . . . 7  |-  ( f  =  (/)  ->  ( ( ( f finSupp  ( 0g
`  (Scalar `  M )
)  /\  ( f
( linC  `  M ) (/) )  =  ( 0g
`  M ) )  ->  A. x  e.  (/)  ( f `  x
)  =  ( 0g
`  (Scalar `  M )
) )  <->  ( ( (/) finSupp  ( 0g `  (Scalar `  M ) )  /\  ( (/) ( linC  `  M
) (/) )  =  ( 0g `  M ) )  ->  A. x  e.  (/)  ( (/) `  x
)  =  ( 0g
`  (Scalar `  M )
) ) ) )
1211ralsng 4068 . . . . . 6  |-  ( (/)  e.  _V  ->  ( A. f  e.  { (/) }  (
( f finSupp  ( 0g `  (Scalar `  M )
)  /\  ( f
( linC  `  M ) (/) )  =  ( 0g
`  M ) )  ->  A. x  e.  (/)  ( f `  x
)  =  ( 0g
`  (Scalar `  M )
) )  <->  ( ( (/) finSupp  ( 0g `  (Scalar `  M ) )  /\  ( (/) ( linC  `  M
) (/) )  =  ( 0g `  M ) )  ->  A. x  e.  (/)  ( (/) `  x
)  =  ( 0g
`  (Scalar `  M )
) ) ) )
133, 12mp1i 12 . . . . 5  |-  ( M  e.  V  ->  ( A. f  e.  { (/) }  ( ( f finSupp  ( 0g `  (Scalar `  M
) )  /\  (
f ( linC  `  M
) (/) )  =  ( 0g `  M ) )  ->  A. x  e.  (/)  ( f `  x )  =  ( 0g `  (Scalar `  M ) ) )  <-> 
( ( (/) finSupp  ( 0g
`  (Scalar `  M )
)  /\  ( (/) ( linC  `  M ) (/) )  =  ( 0g `  M
) )  ->  A. x  e.  (/)  ( (/) `  x
)  =  ( 0g
`  (Scalar `  M )
) ) ) )
142, 13mpbird 232 . . . 4  |-  ( M  e.  V  ->  A. f  e.  { (/) }  ( ( f finSupp  ( 0g `  (Scalar `  M ) )  /\  ( f ( linC  `  M ) (/) )  =  ( 0g `  M
) )  ->  A. x  e.  (/)  ( f `  x )  =  ( 0g `  (Scalar `  M ) ) ) )
15 fvex 5882 . . . . . . 7  |-  ( Base `  (Scalar `  M )
)  e.  _V
16 map0e 7468 . . . . . . 7  |-  ( (
Base `  (Scalar `  M
) )  e.  _V  ->  ( ( Base `  (Scalar `  M ) )  ^m  (/) )  =  1o )
1715, 16mp1i 12 . . . . . 6  |-  ( M  e.  V  ->  (
( Base `  (Scalar `  M
) )  ^m  (/) )  =  1o )
18 df1o2 7154 . . . . . 6  |-  1o  =  { (/) }
1917, 18syl6eq 2524 . . . . 5  |-  ( M  e.  V  ->  (
( Base `  (Scalar `  M
) )  ^m  (/) )  =  { (/) } )
2019raleqdv 3069 . . . 4  |-  ( M  e.  V  ->  ( A. f  e.  (
( Base `  (Scalar `  M
) )  ^m  (/) ) ( ( f finSupp  ( 0g
`  (Scalar `  M )
)  /\  ( f
( linC  `  M ) (/) )  =  ( 0g
`  M ) )  ->  A. x  e.  (/)  ( f `  x
)  =  ( 0g
`  (Scalar `  M )
) )  <->  A. f  e.  { (/) }  ( ( f finSupp  ( 0g `  (Scalar `  M ) )  /\  ( f ( linC  `  M ) (/) )  =  ( 0g `  M
) )  ->  A. x  e.  (/)  ( f `  x )  =  ( 0g `  (Scalar `  M ) ) ) ) )
2114, 20mpbird 232 . . 3  |-  ( M  e.  V  ->  A. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  (/) ) ( ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  ( f ( linC  `  M ) (/) )  =  ( 0g `  M
) )  ->  A. x  e.  (/)  ( f `  x )  =  ( 0g `  (Scalar `  M ) ) ) )
22 0elpw 4622 . . 3  |-  (/)  e.  ~P ( Base `  M )
2321, 22jctil 537 . 2  |-  ( M  e.  V  ->  ( (/) 
e.  ~P ( Base `  M
)  /\  A. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  (/) ) ( ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  ( f ( linC  `  M ) (/) )  =  ( 0g `  M
) )  ->  A. x  e.  (/)  ( f `  x )  =  ( 0g `  (Scalar `  M ) ) ) ) )
24 eqid 2467 . . . 4  |-  ( Base `  M )  =  (
Base `  M )
25 eqid 2467 . . . 4  |-  ( 0g
`  M )  =  ( 0g `  M
)
26 eqid 2467 . . . 4  |-  (Scalar `  M )  =  (Scalar `  M )
27 eqid 2467 . . . 4  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
28 eqid 2467 . . . 4  |-  ( 0g
`  (Scalar `  M )
)  =  ( 0g
`  (Scalar `  M )
)
2924, 25, 26, 27, 28islininds 32529 . . 3  |-  ( (
(/)  e.  _V  /\  M  e.  V )  ->  ( (/) linIndS  M 
<->  ( (/)  e.  ~P ( Base `  M )  /\  A. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  (/) ) ( ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  ( f ( linC  `  M ) (/) )  =  ( 0g `  M
) )  ->  A. x  e.  (/)  ( f `  x )  =  ( 0g `  (Scalar `  M ) ) ) ) ) )
303, 29mpan 670 . 2  |-  ( M  e.  V  ->  ( (/) linIndS  M 
<->  ( (/)  e.  ~P ( Base `  M )  /\  A. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  (/) ) ( ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  ( f ( linC  `  M ) (/) )  =  ( 0g `  M
) )  ->  A. x  e.  (/)  ( f `  x )  =  ( 0g `  (Scalar `  M ) ) ) ) ) )
3123, 30mpbird 232 1  |-  ( M  e.  V  ->  (/) linIndS  M )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   (/)c0 3790   ~Pcpw 4016   {csn 4033   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   1oc1o 7135    ^m cmap 7432   finSupp cfsupp 7841   Basecbs 14507  Scalarcsca 14575   0gc0g 14712   linC clinc 32487   linIndS clininds 32523
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-8 1769  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-pow 4631  ax-pr 4692  ax-un 6587
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-sbc 3337  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-id 4801  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1o 7142  df-map 7434  df-lininds 32525
This theorem is referenced by: (None)
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