Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  linds0 Structured version   Unicode version

Theorem linds0 30999
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 3784 . . . . . 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 4422 . . . . . 6  |-  (/)  e.  _V
4 breq1 4295 . . . . . . . . 9  |-  ( f  =  (/)  ->  ( f finSupp 
( 0g `  (Scalar `  M ) )  <->  (/) finSupp  ( 0g `  (Scalar `  M )
) ) )
5 oveq1 6098 . . . . . . . . . 10  |-  ( f  =  (/)  ->  ( f ( linC  `  M ) (/) )  =  ( (/) ( linC  `  M ) (/) ) )
65eqeq1d 2451 . . . . . . . . 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 5690 . . . . . . . . . 10  |-  ( f  =  (/)  ->  ( f `
 x )  =  ( (/) `  x ) )
98eqeq1d 2451 . . . . . . . . 9  |-  ( f  =  (/)  ->  ( ( f `  x )  =  ( 0g `  (Scalar `  M ) )  <-> 
( (/) `  x )  =  ( 0g `  (Scalar `  M ) ) ) )
109ralbidv 2735 . . . . . . . 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 3912 . . . . . 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 5701 . . . . . . 7  |-  ( Base `  (Scalar `  M )
)  e.  _V
16 map0e 7250 . . . . . . 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 6932 . . . . . 6  |-  1o  =  { (/) }
1917, 18syl6eq 2491 . . . . 5  |-  ( M  e.  V  ->  (
( Base `  (Scalar `  M
) )  ^m  (/) )  =  { (/) } )
2019raleqdv 2923 . . . 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 4461 . . 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 2443 . . . 4  |-  ( Base `  M )  =  (
Base `  M )
25 eqid 2443 . . . 4  |-  ( 0g
`  M )  =  ( 0g `  M
)
26 eqid 2443 . . . 4  |-  (Scalar `  M )  =  (Scalar `  M )
27 eqid 2443 . . . 4  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
28 eqid 2443 . . . 4  |-  ( 0g
`  (Scalar `  M )
)  =  ( 0g
`  (Scalar `  M )
)
2924, 25, 26, 27, 28islininds 30980 . . 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 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972   (/)c0 3637   ~Pcpw 3860   {csn 3877   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   1oc1o 6913    ^m cmap 7214   finSupp cfsupp 7620   Basecbs 14174  Scalarcsca 14241   0gc0g 14378   linC clinc 30938   linIndS clininds 30974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1o 6920  df-map 7216  df-lininds 30976
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator