Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  linc0scn0 Structured version   Unicode version

Theorem linc0scn0 32759
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 )
32lmodring 17394 . . . . . . . 8  |-  ( M  e.  LMod  ->  S  e. 
Ring )
42eqcomi 2456 . . . . . . . . . . 11  |-  (Scalar `  M )  =  S
54fveq2i 5859 . . . . . . . . . 10  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  S )
6 linc0scn0.1 . . . . . . . . . 10  |-  .1.  =  ( 1r `  S )
75, 6ringidcl 17093 . . . . . . . . 9  |-  ( S  e.  Ring  ->  .1.  e.  ( Base `  (Scalar `  M
) ) )
8 linc0scn0.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  S )
95, 8ring0cl 17094 . . . . . . . . 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 3968 . . . . . 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 6040 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  F : V --> ( Base `  (Scalar `  M )
) )
17 fvex 5866 . . . . . 6  |-  ( Base `  (Scalar `  M )
)  e.  _V
1817a1i 11 . . . . 5  |-  ( M  e.  LMod  ->  ( Base `  (Scalar `  M )
)  e.  _V )
19 elmapg 7435 . . . . 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 4001 . . . . . 6  |-  ~P B  =  ~P ( Base `  M
)
2423eleq2i 2521 . . . . 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 32745 . . 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 1229 . 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 5866 . . . . . . . . 9  |-  ( 1r
`  S )  e. 
_V
316, 30eqeltri 2527 . . . . . . . 8  |-  .1.  e.  _V
32 fvex 5866 . . . . . . . . 9  |-  ( 0g
`  S )  e. 
_V
338, 32eqeltri 2527 . . . . . . . 8  |-  .0.  e.  _V
3431, 33ifex 3995 . . . . . . 7  |-  if ( v  =  Z ,  .1.  ,  .0.  )  e. 
_V
35 eqeq1 2447 . . . . . . . . 9  |-  ( x  =  v  ->  (
x  =  Z  <->  v  =  Z ) )
3635ifbid 3948 . . . . . . . 8  |-  ( x  =  v  ->  if ( x  =  Z ,  .1.  ,  .0.  )  =  if ( v  =  Z ,  .1.  ,  .0.  ) )
3736, 15fvmptg 5939 . . . . . . 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 6296 . . . . 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 6289 . . . . . . . 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 2443 . . . . . . . . . . . 12  |-  ( Base `  S )  =  (
Base `  S )
452, 44, 6lmod1cl 17413 . . . . . . . . . . 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 2443 . . . . . . . . 9  |-  ( .s
`  M )  =  ( .s `  M
)
50 linc0scn0.z . . . . . . . . 9  |-  Z  =  ( 0g `  M
)
512, 49, 44, 50lmodvs0 17420 . . . . . . . 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 2484 . . . . . 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 4008 . . . . . . . . . . 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 17419 . . . . . . . 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 3959 . . . . 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 2488 . . . 4  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  (
( F `  v
) ( .s `  M ) v )  =  Z )
6463mpteq2dva 4523 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) )  =  ( v  e.  V  |->  Z ) )
6564oveq2d 6297 . 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 17393 . . . 4  |-  ( M  e.  LMod  ->  M  e. 
Grp )
67 grpmnd 15936 . . . 4  |-  ( M  e.  Grp  ->  M  e.  Mnd )
6866, 67syl 16 . . 3  |-  ( M  e.  LMod  ->  M  e. 
Mnd )
6950gsumz 15879 . . 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 2488 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 1383    e. wcel 1804   _Vcvv 3095   ifcif 3926   ~Pcpw 3997    |-> cmpt 4495   -->wf 5574   ` cfv 5578  (class class class)co 6281    ^m cmap 7422   Basecbs 14509  Scalarcsca 14577   .scvsca 14578   0gc0g 14714    gsumg cgsu 14715   Mndcmnd 15793   Grpcgrp 15927   1rcur 17027   Ringcrg 17072   LModclmod 17386   linC clinc 32740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-seq 12087  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-plusg 14587  df-0g 14716  df-gsum 14717  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-mgp 17016  df-ur 17028  df-ring 17074  df-lmod 17388  df-linc 32742
This theorem is referenced by:  el0ldep  32802
  Copyright terms: Public domain W3C validator