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Theorem linc0scn0 32322
Description: If a set contains the zero element of a module, there is a linear combination being 0 where not all scalars are 0. (Contributed by AV, 13-Apr-2019.)
Hypotheses
Ref Expression
linc0scn0.b  |-  B  =  ( Base `  M
)
linc0scn0.s  |-  S  =  (Scalar `  M )
linc0scn0.0  |-  .0.  =  ( 0g `  S )
linc0scn0.1  |-  .1.  =  ( 1r `  S )
linc0scn0.z  |-  Z  =  ( 0g `  M
)
linc0scn0.f  |-  F  =  ( x  e.  V  |->  if ( x  =  Z ,  .1.  ,  .0.  ) )
Assertion
Ref Expression
linc0scn0  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F ( linC  `  M ) V )  =  Z )
Distinct variable groups:    x, B    x, M    x, V    x, Z    x,  .0.    x,  .1.
Allowed substitution hints:    S( x)    F( x)

Proof of Theorem linc0scn0
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  M  e.  LMod )
2 linc0scn0.s . . . . . . . . 9  |-  S  =  (Scalar `  M )
32lmodrng 17332 . . . . . . . 8  |-  ( M  e.  LMod  ->  S  e. 
Ring )
42eqcomi 2480 . . . . . . . . . . 11  |-  (Scalar `  M )  =  S
54fveq2i 5869 . . . . . . . . . 10  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  S )
6 linc0scn0.1 . . . . . . . . . 10  |-  .1.  =  ( 1r `  S )
75, 6rngidcl 17032 . . . . . . . . 9  |-  ( S  e.  Ring  ->  .1.  e.  ( Base `  (Scalar `  M
) ) )
8 linc0scn0.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  S )
95, 8rng0cl 17033 . . . . . . . . 9  |-  ( S  e.  Ring  ->  .0.  e.  ( Base `  (Scalar `  M
) ) )
107, 9jca 532 . . . . . . . 8  |-  ( S  e.  Ring  ->  (  .1. 
e.  ( Base `  (Scalar `  M ) )  /\  .0.  e.  ( Base `  (Scalar `  M ) ) ) )
113, 10syl 16 . . . . . . 7  |-  ( M  e.  LMod  ->  (  .1. 
e.  ( Base `  (Scalar `  M ) )  /\  .0.  e.  ( Base `  (Scalar `  M ) ) ) )
1211ad2antrr 725 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  x  e.  V )  ->  (  .1.  e.  ( Base `  (Scalar `  M ) )  /\  .0.  e.  ( Base `  (Scalar `  M ) ) ) )
13 ifcl 3981 . . . . . 6  |-  ( (  .1.  e.  ( Base `  (Scalar `  M )
)  /\  .0.  e.  ( Base `  (Scalar `  M
) ) )  ->  if ( x  =  Z ,  .1.  ,  .0.  )  e.  ( Base `  (Scalar `  M )
) )
1412, 13syl 16 . . . . 5  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  x  e.  V )  ->  if ( x  =  Z ,  .1.  ,  .0.  )  e.  ( Base `  (Scalar `  M ) ) )
15 linc0scn0.f . . . . 5  |-  F  =  ( x  e.  V  |->  if ( x  =  Z ,  .1.  ,  .0.  ) )
1614, 15fmptd 6046 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  F : V --> ( Base `  (Scalar `  M )
) )
17 fvex 5876 . . . . . 6  |-  ( Base `  (Scalar `  M )
)  e.  _V
1817a1i 11 . . . . 5  |-  ( M  e.  LMod  ->  ( Base `  (Scalar `  M )
)  e.  _V )
19 elmapg 7434 . . . . 5  |-  ( ( ( Base `  (Scalar `  M ) )  e. 
_V  /\  V  e.  ~P B )  ->  ( F  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
)  <->  F : V --> ( Base `  (Scalar `  M )
) ) )
2018, 19sylan 471 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F  e.  ( ( Base `  (Scalar `  M ) )  ^m  V )  <->  F : V
--> ( Base `  (Scalar `  M ) ) ) )
2116, 20mpbird 232 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  F  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
) )
22 linc0scn0.b . . . . . . 7  |-  B  =  ( Base `  M
)
2322pweqi 4014 . . . . . 6  |-  ~P B  =  ~P ( Base `  M
)
2423eleq2i 2545 . . . . 5  |-  ( V  e.  ~P B  <->  V  e.  ~P ( Base `  M
) )
2524biimpi 194 . . . 4  |-  ( V  e.  ~P B  ->  V  e.  ~P ( Base `  M ) )
2625adantl 466 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  V  e.  ~P ( Base `  M ) )
27 lincval 32308 . . 3  |-  ( ( M  e.  LMod  /\  F  e.  ( ( Base `  (Scalar `  M ) )  ^m  V )  /\  V  e.  ~P ( Base `  M
) )  ->  ( F ( linC  `  M ) V )  =  ( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) ) ) )
281, 21, 26, 27syl3anc 1228 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F ( linC  `  M ) V )  =  ( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s `  M
) v ) ) ) )
29 simpr 461 . . . . . . 7  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  v  e.  V )
30 fvex 5876 . . . . . . . . 9  |-  ( 1r
`  S )  e. 
_V
316, 30eqeltri 2551 . . . . . . . 8  |-  .1.  e.  _V
32 fvex 5876 . . . . . . . . 9  |-  ( 0g
`  S )  e. 
_V
338, 32eqeltri 2551 . . . . . . . 8  |-  .0.  e.  _V
3431, 33ifex 4008 . . . . . . 7  |-  if ( v  =  Z ,  .1.  ,  .0.  )  e. 
_V
35 eqeq1 2471 . . . . . . . . 9  |-  ( x  =  v  ->  (
x  =  Z  <->  v  =  Z ) )
3635ifbid 3961 . . . . . . . 8  |-  ( x  =  v  ->  if ( x  =  Z ,  .1.  ,  .0.  )  =  if ( v  =  Z ,  .1.  ,  .0.  ) )
3736, 15fvmptg 5949 . . . . . . 7  |-  ( ( v  e.  V  /\  if ( v  =  Z ,  .1.  ,  .0.  )  e.  _V )  ->  ( F `  v
)  =  if ( v  =  Z ,  .1.  ,  .0.  ) )
3829, 34, 37sylancl 662 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  ( F `  v )  =  if ( v  =  Z ,  .1.  ,  .0.  ) )
3938oveq1d 6300 . . . . 5  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  (
( F `  v
) ( .s `  M ) v )  =  ( if ( v  =  Z ,  .1.  ,  .0.  ) ( .s `  M ) v ) )
40 ovif 6364 . . . . . 6  |-  ( if ( v  =  Z ,  .1.  ,  .0.  ) ( .s `  M ) v )  =  if ( v  =  Z ,  (  .1.  ( .s `  M ) v ) ,  (  .0.  ( .s `  M ) v ) )
4140a1i 11 . . . . 5  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  ( if ( v  =  Z ,  .1.  ,  .0.  ) ( .s `  M ) v )  =  if ( v  =  Z ,  (  .1.  ( .s `  M ) v ) ,  (  .0.  ( .s `  M ) v ) ) )
42 oveq2 6293 . . . . . . . 8  |-  ( v  =  Z  ->  (  .1.  ( .s `  M
) v )  =  (  .1.  ( .s
`  M ) Z ) )
4342adantl 466 . . . . . . 7  |-  ( ( ( ( M  e. 
LMod  /\  V  e.  ~P B )  /\  v  e.  V )  /\  v  =  Z )  ->  (  .1.  ( .s `  M
) v )  =  (  .1.  ( .s
`  M ) Z ) )
44 eqid 2467 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
452, 44, 6lmod1cl 17351 . . . . . . . . . . 11  |-  ( M  e.  LMod  ->  .1.  e.  ( Base `  S )
)
4645ancli 551 . . . . . . . . . 10  |-  ( M  e.  LMod  ->  ( M  e.  LMod  /\  .1.  e.  ( Base `  S )
) )
4746adantr 465 . . . . . . . . 9  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( M  e.  LMod  /\  .1.  e.  ( Base `  S ) ) )
4847ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( M  e. 
LMod  /\  V  e.  ~P B )  /\  v  e.  V )  /\  v  =  Z )  ->  ( M  e.  LMod  /\  .1.  e.  ( Base `  S
) ) )
49 eqid 2467 . . . . . . . . 9  |-  ( .s
`  M )  =  ( .s `  M
)
50 linc0scn0.z . . . . . . . . 9  |-  Z  =  ( 0g `  M
)
512, 49, 44, 50lmodvs0 17358 . . . . . . . 8  |-  ( ( M  e.  LMod  /\  .1.  e.  ( Base `  S
) )  ->  (  .1.  ( .s `  M
) Z )  =  Z )
5248, 51syl 16 . . . . . . 7  |-  ( ( ( ( M  e. 
LMod  /\  V  e.  ~P B )  /\  v  e.  V )  /\  v  =  Z )  ->  (  .1.  ( .s `  M
) Z )  =  Z )
5343, 52eqtrd 2508 . . . . . 6  |-  ( ( ( ( M  e. 
LMod  /\  V  e.  ~P B )  /\  v  e.  V )  /\  v  =  Z )  ->  (  .1.  ( .s `  M
) v )  =  Z )
541adantr 465 . . . . . . . 8  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  M  e.  LMod )
55 elelpwi 4021 . . . . . . . . . . 11  |-  ( ( v  e.  V  /\  V  e.  ~P B
)  ->  v  e.  B )
5655expcom 435 . . . . . . . . . 10  |-  ( V  e.  ~P B  -> 
( v  e.  V  ->  v  e.  B ) )
5756adantl 466 . . . . . . . . 9  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( v  e.  V  ->  v  e.  B ) )
5857imp 429 . . . . . . . 8  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  v  e.  B )
5922, 2, 49, 8, 50lmod0vs 17357 . . . . . . . 8  |-  ( ( M  e.  LMod  /\  v  e.  B )  ->  (  .0.  ( .s `  M
) v )  =  Z )
6054, 58, 59syl2anc 661 . . . . . . 7  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  (  .0.  ( .s `  M
) v )  =  Z )
6160adantr 465 . . . . . 6  |-  ( ( ( ( M  e. 
LMod  /\  V  e.  ~P B )  /\  v  e.  V )  /\  -.  v  =  Z )  ->  (  .0.  ( .s
`  M ) v )  =  Z )
6253, 61ifeqda 3972 . . . . 5  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  if ( v  =  Z ,  (  .1.  ( .s `  M ) v ) ,  (  .0.  ( .s `  M
) v ) )  =  Z )
6339, 41, 623eqtrd 2512 . . . 4  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  (
( F `  v
) ( .s `  M ) v )  =  Z )
6463mpteq2dva 4533 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) )  =  ( v  e.  V  |->  Z ) )
6564oveq2d 6301 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) ) )  =  ( M  gsumg  ( v  e.  V  |->  Z ) ) )
66 lmodgrp 17331 . . . 4  |-  ( M  e.  LMod  ->  M  e. 
Grp )
67 grpmnd 15876 . . . 4  |-  ( M  e.  Grp  ->  M  e.  Mnd )
6866, 67syl 16 . . 3  |-  ( M  e.  LMod  ->  M  e. 
Mnd )
6950gsumz 15836 . . 3  |-  ( ( M  e.  Mnd  /\  V  e.  ~P B
)  ->  ( M  gsumg  ( v  e.  V  |->  Z ) )  =  Z )
7068, 69sylan 471 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( M  gsumg  ( v  e.  V  |->  Z ) )  =  Z )
7128, 65, 703eqtrd 2512 1  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F ( linC  `  M ) V )  =  Z )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   ifcif 3939   ~Pcpw 4010    |-> cmpt 4505   -->wf 5584   ` cfv 5588  (class class class)co 6285    ^m cmap 7421   Basecbs 14493  Scalarcsca 14561   .scvsca 14562   0gc0g 14698    gsumg cgsu 14699   Mndcmnd 15729   Grpcgrp 15730   1rcur 16967   Ringcrg 17012   LModclmod 17324   linC clinc 32303
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-seq 12077  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-plusg 14571  df-0g 14700  df-gsum 14701  df-mnd 15735  df-grp 15871  df-mgp 16956  df-ur 16968  df-rng 17014  df-lmod 17326  df-linc 32305
This theorem is referenced by:  el0ldep  32365
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