Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  el0ldep Structured version   Unicode version

Theorem el0ldep 31005
Description: A set containing the zero element of a module is always linearly dependent, if the underlying ring has at least two elements. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Assertion
Ref Expression
el0ldep  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  S linDepS  M )

Proof of Theorem el0ldep
Dummy variables  f 
s  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . . . 5  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2443 . . . . 5  |-  (Scalar `  M )  =  (Scalar `  M )
3 eqid 2443 . . . . 5  |-  ( 0g
`  (Scalar `  M )
)  =  ( 0g
`  (Scalar `  M )
)
4 eqid 2443 . . . . 5  |-  ( 1r
`  (Scalar `  M )
)  =  ( 1r
`  (Scalar `  M )
)
5 eqeq1 2449 . . . . . . 7  |-  ( s  =  y  ->  (
s  =  ( 0g
`  M )  <->  y  =  ( 0g `  M ) ) )
65ifbid 3816 . . . . . 6  |-  ( s  =  y  ->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) )  =  if ( y  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )
76cbvmptv 4388 . . . . 5  |-  ( s  e.  S  |->  if ( s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  =  ( y  e.  S  |->  if ( y  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )
81, 2, 3, 4, 7mptcfsupp 30799 . . . 4  |-  ( ( M  e.  LMod  /\  S  e.  ~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( s  e.  S  |->  if ( s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) finSupp  ( 0g `  (Scalar `  M ) ) )
983adant1r 1211 . . 3  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( s  e.  S  |->  if ( s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) finSupp  ( 0g `  (Scalar `  M ) ) )
10 simp1l 1012 . . . 4  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  M  e.  LMod )
11 simp2 989 . . . 4  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  S  e.  ~P ( Base `  M
) )
12 eqid 2443 . . . . 5  |-  ( 0g
`  M )  =  ( 0g `  M
)
13 eqid 2443 . . . . 5  |-  ( s  e.  S  |->  if ( s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  =  ( s  e.  S  |->  if ( s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )
141, 2, 3, 4, 12, 13linc0scn0 30962 . . . 4  |-  ( ( M  e.  LMod  /\  S  e.  ~P ( Base `  M
) )  ->  (
( s  e.  S  |->  if ( s  =  ( 0g `  M
) ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) ( linC  `  M ) S )  =  ( 0g `  M ) )
1510, 11, 14syl2anc 661 . . 3  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( (
s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) ( linC  `  M ) S )  =  ( 0g `  M ) )
16 simp3 990 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( 0g `  M )  e.  S
)
17 fvex 5706 . . . . . . 7  |-  ( 1r
`  (Scalar `  M )
)  e.  _V
1817a1i 11 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( 1r `  (Scalar `  M )
)  e.  _V )
19 iftrue 3802 . . . . . . 7  |-  ( s  =  ( 0g `  M )  ->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) )  =  ( 1r
`  (Scalar `  M )
) )
2019, 13fvmptg 5777 . . . . . 6  |-  ( ( ( 0g `  M
)  e.  S  /\  ( 1r `  (Scalar `  M ) )  e. 
_V )  ->  (
( s  e.  S  |->  if ( s  =  ( 0g `  M
) ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) `  ( 0g `  M ) )  =  ( 1r
`  (Scalar `  M )
) )
2116, 18, 20syl2anc 661 . . . . 5  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( (
s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) `  ( 0g `  M ) )  =  ( 1r `  (Scalar `  M ) ) )
222lmodrng 16961 . . . . . . . 8  |-  ( M  e.  LMod  ->  (Scalar `  M )  e.  Ring )
2322anim1i 568 . . . . . . 7  |-  ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M
) ) ) )  ->  ( (Scalar `  M )  e.  Ring  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) ) )
24233ad2ant1 1009 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( (Scalar `  M )  e.  Ring  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) ) )
25 eqid 2443 . . . . . . 7  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
2625, 4, 3rng1ne0 30778 . . . . . 6  |-  ( ( (Scalar `  M )  e.  Ring  /\  1  <  (
# `  ( Base `  (Scalar `  M )
) ) )  -> 
( 1r `  (Scalar `  M ) )  =/=  ( 0g `  (Scalar `  M ) ) )
2724, 26syl 16 . . . . 5  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( 1r `  (Scalar `  M )
)  =/=  ( 0g
`  (Scalar `  M )
) )
2821, 27eqnetrd 2631 . . . 4  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( (
s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) `  ( 0g `  M ) )  =/=  ( 0g `  (Scalar `  M ) ) )
29 fveq2 5696 . . . . . . 7  |-  ( x  =  ( 0g `  M )  ->  (
( s  e.  S  |->  if ( s  =  ( 0g `  M
) ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) `  x )  =  ( ( s  e.  S  |->  if ( s  =  ( 0g `  M
) ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) `  ( 0g `  M ) ) )
3029neeq1d 2626 . . . . . 6  |-  ( x  =  ( 0g `  M )  ->  (
( ( s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) `  x )  =/=  ( 0g `  (Scalar `  M ) )  <-> 
( ( s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) `  ( 0g
`  M ) )  =/=  ( 0g `  (Scalar `  M ) ) ) )
3130adantl 466 . . . . 5  |-  ( ( ( ( M  e. 
LMod  /\  1  <  ( # `
 ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  /\  x  =  ( 0g `  M ) )  ->  ( (
( s  e.  S  |->  if ( s  =  ( 0g `  M
) ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) `  x )  =/=  ( 0g `  (Scalar `  M
) )  <->  ( (
s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) `  ( 0g `  M ) )  =/=  ( 0g `  (Scalar `  M ) ) ) )
3216, 31rspcedv 3082 . . . 4  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( (
( s  e.  S  |->  if ( s  =  ( 0g `  M
) ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) `  ( 0g `  M ) )  =/=  ( 0g
`  (Scalar `  M )
)  ->  E. x  e.  S  ( (
s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) `  x
)  =/=  ( 0g
`  (Scalar `  M )
) ) )
3328, 32mpd 15 . . 3  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  E. x  e.  S  ( (
s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) `  x
)  =/=  ( 0g
`  (Scalar `  M )
) )
342, 25, 4lmod1cl 16980 . . . . . . . . . 10  |-  ( M  e.  LMod  ->  ( 1r
`  (Scalar `  M )
)  e.  ( Base `  (Scalar `  M )
) )
352, 25, 3lmod0cl 16979 . . . . . . . . . 10  |-  ( M  e.  LMod  ->  ( 0g
`  (Scalar `  M )
)  e.  ( Base `  (Scalar `  M )
) )
3634, 35ifcld 3837 . . . . . . . . 9  |-  ( M  e.  LMod  ->  if ( s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) )  e.  ( Base `  (Scalar `  M ) ) )
3736adantr 465 . . . . . . . 8  |-  ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M
) ) ) )  ->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) )  e.  ( Base `  (Scalar `  M ) ) )
38373ad2ant1 1009 . . . . . . 7  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  if (
s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) )  e.  ( Base `  (Scalar `  M ) ) )
3938adantr 465 . . . . . 6  |-  ( ( ( ( M  e. 
LMod  /\  1  <  ( # `
 ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  /\  s  e.  S )  ->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) )  e.  ( Base `  (Scalar `  M )
) )
4039, 13fmptd 5872 . . . . 5  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( s  e.  S  |->  if ( s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) : S --> ( Base `  (Scalar `  M )
) )
41 fvex 5706 . . . . . . 7  |-  ( Base `  (Scalar `  M )
)  e.  _V
4241a1i 11 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( Base `  (Scalar `  M )
)  e.  _V )
43 elmapg 7232 . . . . . 6  |-  ( ( ( Base `  (Scalar `  M ) )  e. 
_V  /\  S  e.  ~P ( Base `  M
) )  ->  (
( s  e.  S  |->  if ( s  =  ( 0g `  M
) ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) )  e.  ( ( Base `  (Scalar `  M ) )  ^m  S )  <->  ( s  e.  S  |->  if ( s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) : S --> ( Base `  (Scalar `  M )
) ) )
4442, 11, 43syl2anc 661 . . . . 5  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( (
s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  e.  ( ( Base `  (Scalar `  M ) )  ^m  S )  <->  ( s  e.  S  |->  if ( s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) : S --> ( Base `  (Scalar `  M )
) ) )
4540, 44mpbird 232 . . . 4  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( s  e.  S  |->  if ( s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  e.  ( (
Base `  (Scalar `  M
) )  ^m  S
) )
46 breq1 4300 . . . . . 6  |-  ( f  =  ( s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  ->  ( f finSupp  ( 0g `  (Scalar `  M ) )  <->  ( s  e.  S  |->  if ( s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) finSupp  ( 0g `  (Scalar `  M ) ) ) )
47 oveq1 6103 . . . . . . 7  |-  ( f  =  ( s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  ->  ( f
( linC  `  M ) S )  =  ( ( s  e.  S  |->  if ( s  =  ( 0g `  M
) ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) ( linC  `  M ) S ) )
4847eqeq1d 2451 . . . . . 6  |-  ( f  =  ( s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  ->  ( (
f ( linC  `  M
) S )  =  ( 0g `  M
)  <->  ( ( s  e.  S  |->  if ( s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) ( linC  `  M
) S )  =  ( 0g `  M
) ) )
49 fveq1 5695 . . . . . . . 8  |-  ( f  =  ( s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  ->  ( f `  x )  =  ( ( s  e.  S  |->  if ( s  =  ( 0g `  M
) ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) `  x ) )
5049neeq1d 2626 . . . . . . 7  |-  ( f  =  ( s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  ->  ( (
f `  x )  =/=  ( 0g `  (Scalar `  M ) )  <->  ( (
s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) `  x
)  =/=  ( 0g
`  (Scalar `  M )
) ) )
5150rexbidv 2741 . . . . . 6  |-  ( f  =  ( s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  ->  ( E. x  e.  S  (
f `  x )  =/=  ( 0g `  (Scalar `  M ) )  <->  E. x  e.  S  ( (
s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) `  x
)  =/=  ( 0g
`  (Scalar `  M )
) ) )
5246, 48, 513anbi123d 1289 . . . . 5  |-  ( f  =  ( s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  ->  ( (
f finSupp  ( 0g `  (Scalar `  M ) )  /\  ( f ( linC  `  M ) S )  =  ( 0g `  M )  /\  E. x  e.  S  (
f `  x )  =/=  ( 0g `  (Scalar `  M ) ) )  <-> 
( ( s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) finSupp  ( 0g `  (Scalar `  M ) )  /\  ( ( s  e.  S  |->  if ( s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) ( linC  `  M
) S )  =  ( 0g `  M
)  /\  E. x  e.  S  ( (
s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) `  x
)  =/=  ( 0g
`  (Scalar `  M )
) ) ) )
5352adantl 466 . . . 4  |-  ( ( ( ( M  e. 
LMod  /\  1  <  ( # `
 ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  /\  f  =  ( s  e.  S  |->  if ( s  =  ( 0g `  M
) ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) )  ->  ( ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  ( f ( linC  `  M ) S )  =  ( 0g `  M )  /\  E. x  e.  S  (
f `  x )  =/=  ( 0g `  (Scalar `  M ) ) )  <-> 
( ( s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) finSupp  ( 0g `  (Scalar `  M ) )  /\  ( ( s  e.  S  |->  if ( s  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) ( linC  `  M
) S )  =  ( 0g `  M
)  /\  E. x  e.  S  ( (
s  e.  S  |->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) `  x
)  =/=  ( 0g
`  (Scalar `  M )
) ) ) )
5445, 53rspcedv 3082 . . 3  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( (
( s  e.  S  |->  if ( s  =  ( 0g `  M
) ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) finSupp  ( 0g `  (Scalar `  M
) )  /\  (
( s  e.  S  |->  if ( s  =  ( 0g `  M
) ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) ( linC  `  M ) S )  =  ( 0g `  M )  /\  E. x  e.  S  (
( s  e.  S  |->  if ( s  =  ( 0g `  M
) ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) `  x )  =/=  ( 0g `  (Scalar `  M
) ) )  ->  E. f  e.  (
( Base `  (Scalar `  M
) )  ^m  S
) ( f finSupp  ( 0g `  (Scalar `  M
) )  /\  (
f ( linC  `  M
) S )  =  ( 0g `  M
)  /\  E. x  e.  S  ( f `  x )  =/=  ( 0g `  (Scalar `  M
) ) ) ) )
559, 15, 33, 54mp3and 1317 . 2  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  E. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  S ) ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  ( f ( linC  `  M ) S )  =  ( 0g `  M )  /\  E. x  e.  S  (
f `  x )  =/=  ( 0g `  (Scalar `  M ) ) ) )
561, 12, 2, 25, 3islindeps 30992 . . 3  |-  ( ( M  e.  LMod  /\  S  e.  ~P ( Base `  M
) )  ->  ( S linDepS  M  <->  E. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  S ) ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  ( f ( linC  `  M ) S )  =  ( 0g `  M )  /\  E. x  e.  S  (
f `  x )  =/=  ( 0g `  (Scalar `  M ) ) ) ) )
5710, 11, 56syl2anc 661 . 2  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( S linDepS  M  <->  E. f  e.  (
( Base `  (Scalar `  M
) )  ^m  S
) ( f finSupp  ( 0g `  (Scalar `  M
) )  /\  (
f ( linC  `  M
) S )  =  ( 0g `  M
)  /\  E. x  e.  S  ( f `  x )  =/=  ( 0g `  (Scalar `  M
) ) ) ) )
5855, 57mpbird 232 1  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  S linDepS  M )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   E.wrex 2721   _Vcvv 2977   ifcif 3796   ~Pcpw 3865   class class class wbr 4297    e. cmpt 4355   -->wf 5419   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   finSupp cfsupp 7625   1c1 9288    < clt 9423   #chash 12108   Basecbs 14179  Scalarcsca 14246   0gc0g 14383   1rcur 16608   Ringcrg 16650   LModclmod 16953   linC clinc 30943   linDepS clindeps 30980
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-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-seq 11812  df-hash 12109  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-plusg 14256  df-0g 14385  df-gsum 14386  df-mnd 15420  df-grp 15550  df-minusg 15551  df-mgp 16597  df-ur 16609  df-rng 16652  df-lmod 16955  df-linc 30945  df-lininds 30981  df-lindeps 30983
This theorem is referenced by:  el0ldepsnzr  31006
  Copyright terms: Public domain W3C validator