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Theorem linc0scn0 33278
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 455 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  M  e.  LMod )
2 linc0scn0.s . . . . . . . . 9  |-  S  =  (Scalar `  M )
32lmodring 17715 . . . . . . . 8  |-  ( M  e.  LMod  ->  S  e. 
Ring )
42eqcomi 2467 . . . . . . . . . . 11  |-  (Scalar `  M )  =  S
54fveq2i 5851 . . . . . . . . . 10  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  S )
6 linc0scn0.1 . . . . . . . . . 10  |-  .1.  =  ( 1r `  S )
75, 6ringidcl 17414 . . . . . . . . 9  |-  ( S  e.  Ring  ->  .1.  e.  ( Base `  (Scalar `  M
) ) )
8 linc0scn0.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  S )
95, 8ring0cl 17415 . . . . . . . . 9  |-  ( S  e.  Ring  ->  .0.  e.  ( Base `  (Scalar `  M
) ) )
107, 9jca 530 . . . . . . . 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 723 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  x  e.  V )  ->  (  .1.  e.  ( Base `  (Scalar `  M ) )  /\  .0.  e.  ( Base `  (Scalar `  M ) ) ) )
13 ifcl 3971 . . . . . 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 6031 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  F : V --> ( Base `  (Scalar `  M )
) )
17 fvex 5858 . . . . . 6  |-  ( Base `  (Scalar `  M )
)  e.  _V
1817a1i 11 . . . . 5  |-  ( M  e.  LMod  ->  ( Base `  (Scalar `  M )
)  e.  _V )
19 elmapg 7425 . . . . 5  |-  ( ( ( Base `  (Scalar `  M ) )  e. 
_V  /\  V  e.  ~P B )  ->  ( F  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
)  <->  F : V --> ( Base `  (Scalar `  M )
) ) )
2018, 19sylan 469 . . . 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 4003 . . . . . 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 464 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  V  e.  ~P ( Base `  M ) )
27 lincval 33264 . . 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 1226 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F ( linC  `  M ) V )  =  ( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s `  M
) v ) ) ) )
29 simpr 459 . . . . . . 7  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  v  e.  V )
30 fvex 5858 . . . . . . . . 9  |-  ( 1r
`  S )  e. 
_V
316, 30eqeltri 2538 . . . . . . . 8  |-  .1.  e.  _V
32 fvex 5858 . . . . . . . . 9  |-  ( 0g
`  S )  e. 
_V
338, 32eqeltri 2538 . . . . . . . 8  |-  .0.  e.  _V
3431, 33ifex 3997 . . . . . . 7  |-  if ( v  =  Z ,  .1.  ,  .0.  )  e. 
_V
35 eqeq1 2458 . . . . . . . . 9  |-  ( x  =  v  ->  (
x  =  Z  <->  v  =  Z ) )
3635ifbid 3951 . . . . . . . 8  |-  ( x  =  v  ->  if ( x  =  Z ,  .1.  ,  .0.  )  =  if ( v  =  Z ,  .1.  ,  .0.  ) )
3736, 15fvmptg 5929 . . . . . . 7  |-  ( ( v  e.  V  /\  if ( v  =  Z ,  .1.  ,  .0.  )  e.  _V )  ->  ( F `  v
)  =  if ( v  =  Z ,  .1.  ,  .0.  ) )
3829, 34, 37sylancl 660 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  ( F `  v )  =  if ( v  =  Z ,  .1.  ,  .0.  ) )
3938oveq1d 6285 . . . . 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 6352 . . . . . 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 6278 . . . . . . . 8  |-  ( v  =  Z  ->  (  .1.  ( .s `  M
) v )  =  (  .1.  ( .s
`  M ) Z ) )
4342adantl 464 . . . . . . 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 17734 . . . . . . . . . . 11  |-  ( M  e.  LMod  ->  .1.  e.  ( Base `  S )
)
4645ancli 549 . . . . . . . . . 10  |-  ( M  e.  LMod  ->  ( M  e.  LMod  /\  .1.  e.  ( Base `  S )
) )
4746adantr 463 . . . . . . . . 9  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( M  e.  LMod  /\  .1.  e.  ( Base `  S ) ) )
4847ad2antrr 723 . . . . . . . 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 17741 . . . . . . . 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 463 . . . . . . . 8  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  M  e.  LMod )
55 elelpwi 4010 . . . . . . . . . . 11  |-  ( ( v  e.  V  /\  V  e.  ~P B
)  ->  v  e.  B )
5655expcom 433 . . . . . . . . . 10  |-  ( V  e.  ~P B  -> 
( v  e.  V  ->  v  e.  B ) )
5756adantl 464 . . . . . . . . 9  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( v  e.  V  ->  v  e.  B ) )
5857imp 427 . . . . . . . 8  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  v  e.  B )
5922, 2, 49, 8, 50lmod0vs 17740 . . . . . . . 8  |-  ( ( M  e.  LMod  /\  v  e.  B )  ->  (  .0.  ( .s `  M
) v )  =  Z )
6054, 58, 59syl2anc 659 . . . . . . 7  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  (  .0.  ( .s `  M
) v )  =  Z )
6160adantr 463 . . . . . 6  |-  ( ( ( ( M  e. 
LMod  /\  V  e.  ~P B )  /\  v  e.  V )  /\  -.  v  =  Z )  ->  (  .0.  ( .s
`  M ) v )  =  Z )
6253, 61ifeqda 3962 . . . . 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 4525 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) )  =  ( v  e.  V  |->  Z ) )
6564oveq2d 6286 . 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 17714 . . . 4  |-  ( M  e.  LMod  ->  M  e. 
Grp )
67 grpmnd 16261 . . . 4  |-  ( M  e.  Grp  ->  M  e.  Mnd )
6866, 67syl 16 . . 3  |-  ( M  e.  LMod  ->  M  e. 
Mnd )
6950gsumz 16204 . . 3  |-  ( ( M  e.  Mnd  /\  V  e.  ~P B
)  ->  ( M  gsumg  ( v  e.  V  |->  Z ) )  =  Z )
7068, 69sylan 469 . 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 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   ifcif 3929   ~Pcpw 3999    |-> cmpt 4497   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   Basecbs 14716  Scalarcsca 14787   .scvsca 14788   0gc0g 14929    gsumg cgsu 14930   Mndcmnd 16118   Grpcgrp 16252   1rcur 17348   Ringcrg 17393   LModclmod 17707   linC clinc 33259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-seq 12090  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-plusg 14797  df-0g 14931  df-gsum 14932  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-mgp 17337  df-ur 17349  df-ring 17395  df-lmod 17709  df-linc 33261
This theorem is referenced by:  el0ldep  33321
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