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Theorem el0ldep 32441
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 2467 . . . . 5  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2467 . . . . 5  |-  (Scalar `  M )  =  (Scalar `  M )
3 eqid 2467 . . . . 5  |-  ( 0g
`  (Scalar `  M )
)  =  ( 0g
`  (Scalar `  M )
)
4 eqid 2467 . . . . 5  |-  ( 1r
`  (Scalar `  M )
)  =  ( 1r
`  (Scalar `  M )
)
5 eqeq1 2471 . . . . . . 7  |-  ( s  =  y  ->  (
s  =  ( 0g
`  M )  <->  y  =  ( 0g `  M ) ) )
65ifbid 3966 . . . . . 6  |-  ( s  =  y  ->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) )  =  if ( y  =  ( 0g
`  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )
76cbvmptv 4543 . . . . 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 32347 . . . 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 1221 . . 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 1020 . . . 4  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  M  e.  LMod )
11 simp2 997 . . . 4  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  S  e.  ~P ( Base `  M
) )
12 eqid 2467 . . . . 5  |-  ( 0g
`  M )  =  ( 0g `  M
)
13 eqid 2467 . . . . 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 32398 . . . 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 998 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) )  /\  S  e. 
~P ( Base `  M
)  /\  ( 0g `  M )  e.  S
)  ->  ( 0g `  M )  e.  S
)
17 fvex 5881 . . . . . . 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 3950 . . . . . . 7  |-  ( s  =  ( 0g `  M )  ->  if ( s  =  ( 0g `  M ) ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) )  =  ( 1r
`  (Scalar `  M )
) )
2019, 13fvmptg 5954 . . . . . 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 ) ) )
222lmodring 17368 . . . . . . . 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 1017 . . . . . 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 2467 . . . . . . 7  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
2625, 4, 3ring1ne0 17088 . . . . . 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 2760 . . . 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 5871 . . . . . . 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 2744 . . . . . 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 3223 . . . 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 17387 . . . . . . . . . 10  |-  ( M  e.  LMod  ->  ( 1r
`  (Scalar `  M )
)  e.  ( Base `  (Scalar `  M )
) )
352, 25, 3lmod0cl 17386 . . . . . . . . . 10  |-  ( M  e.  LMod  ->  ( 0g
`  (Scalar `  M )
)  e.  ( Base `  (Scalar `  M )
) )
3634, 35ifcld 3987 . . . . . . . . 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 1017 . . . . . . 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 6055 . . . . 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 5881 . . . . . . 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 7443 . . . . . 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 4455 . . . . . 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 6301 . . . . . . 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 2469 . . . . . 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 5870 . . . . . . . 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 2744 . . . . . . 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 2978 . . . . . 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 1299 . . . . 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 3223 . . 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 1327 . 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 32428 . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   _Vcvv 3118   ifcif 3944   ~Pcpw 4015   class class class wbr 4452    |-> cmpt 4510   -->wf 5589   ` cfv 5593  (class class class)co 6294    ^m cmap 7430   finSupp cfsupp 7839   1c1 9503    < clt 9638   #chash 12383   Basecbs 14502  Scalarcsca 14570   0gc0g 14707   1rcur 17002   Ringcrg 17047   LModclmod 17360   linC clinc 32379   linDepS clindeps 32416
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-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-supp 6912  df-recs 7052  df-rdg 7086  df-1o 7140  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fsupp 7840  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-seq 12086  df-hash 12384  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-plusg 14580  df-0g 14709  df-gsum 14710  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-grp 15906  df-minusg 15907  df-mgp 16991  df-ur 17003  df-ring 17049  df-lmod 17362  df-linc 32381  df-lininds 32417  df-lindeps 32419
This theorem is referenced by:  el0ldepsnzr  32442
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