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Theorem linc0scn0 31109
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 17082 . . . . . . . 8  |-  ( M  e.  LMod  ->  S  e. 
Ring )
42eqcomi 2467 . . . . . . . . . . 11  |-  (Scalar `  M )  =  S
54fveq2i 5805 . . . . . . . . . 10  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  S )
6 linc0scn0.1 . . . . . . . . . 10  |-  .1.  =  ( 1r `  S )
75, 6rngidcl 16791 . . . . . . . . 9  |-  ( S  e.  Ring  ->  .1.  e.  ( Base `  (Scalar `  M
) ) )
8 linc0scn0.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  S )
95, 8rng0cl 16792 . . . . . . . . 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 3942 . . . . . 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 5979 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  F : V --> ( Base `  (Scalar `  M )
) )
17 fvex 5812 . . . . . 6  |-  ( Base `  (Scalar `  M )
)  e.  _V
1817a1i 11 . . . . 5  |-  ( M  e.  LMod  ->  ( Base `  (Scalar `  M )
)  e.  _V )
19 elmapg 7340 . . . . 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 3975 . . . . . 6  |-  ~P B  =  ~P ( Base `  M
)
2423eleq2i 2532 . . . . 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 31095 . . 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 1219 . 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 5812 . . . . . . . . 9  |-  ( 1r
`  S )  e. 
_V
316, 30eqeltri 2538 . . . . . . . 8  |-  .1.  e.  _V
32 fvex 5812 . . . . . . . . 9  |-  ( 0g
`  S )  e. 
_V
338, 32eqeltri 2538 . . . . . . . 8  |-  .0.  e.  _V
3431, 33ifex 3969 . . . . . . 7  |-  if ( v  =  Z ,  .1.  ,  .0.  )  e. 
_V
35 eqeq1 2458 . . . . . . . . 9  |-  ( x  =  v  ->  (
x  =  Z  <->  v  =  Z ) )
3635ifbid 3922 . . . . . . . 8  |-  ( x  =  v  ->  if ( x  =  Z ,  .1.  ,  .0.  )  =  if ( v  =  Z ,  .1.  ,  .0.  ) )
3736, 15fvmptg 5884 . . . . . . 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 6218 . . . . 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 6280 . . . . . 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 6211 . . . . . . . 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 2454 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
452, 44, 6lmod1cl 17101 . . . . . . . . . . 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 2454 . . . . . . . . 9  |-  ( .s
`  M )  =  ( .s `  M
)
50 linc0scn0.z . . . . . . . . 9  |-  Z  =  ( 0g `  M
)
512, 49, 44, 50lmodvs0 17108 . . . . . . . 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 2495 . . . . . 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 3982 . . . . . . . . . . 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 17107 . . . . . . . 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 3933 . . . . 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 2499 . . . 4  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  (
( F `  v
) ( .s `  M ) v )  =  Z )
6463mpteq2dva 4489 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) )  =  ( v  e.  V  |->  Z ) )
6564oveq2d 6219 . 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 17081 . . . 4  |-  ( M  e.  LMod  ->  M  e. 
Grp )
67 grpmnd 15672 . . . 4  |-  ( M  e.  Grp  ->  M  e.  Mnd )
6866, 67syl 16 . . 3  |-  ( M  e.  LMod  ->  M  e. 
Mnd )
6950gsumz 15633 . . 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 2499 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 1370    e. wcel 1758   _Vcvv 3078   ifcif 3902   ~Pcpw 3971    |-> cmpt 4461   -->wf 5525   ` cfv 5529  (class class class)co 6203    ^m cmap 7327   Basecbs 14295  Scalarcsca 14363   .scvsca 14364   0gc0g 14500    gsumg cgsu 14501   Mndcmnd 15531   Grpcgrp 15532   1rcur 16728   Ringcrg 16771   LModclmod 17074   linC clinc 31090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-seq 11927  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-plusg 14373  df-0g 14502  df-gsum 14503  df-mnd 15537  df-grp 15667  df-mgp 16717  df-ur 16729  df-rng 16773  df-lmod 17076  df-linc 31092
This theorem is referenced by:  el0ldep  31152
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