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Theorem lcoc0 32758
Description: Properties of a linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Hypotheses
Ref Expression
lincvalsc0.b  |-  B  =  ( Base `  M
)
lincvalsc0.s  |-  S  =  (Scalar `  M )
lincvalsc0.0  |-  .0.  =  ( 0g `  S )
lincvalsc0.z  |-  Z  =  ( 0g `  M
)
lincvalsc0.f  |-  F  =  ( x  e.  V  |->  .0.  )
lcoc0.r  |-  R  =  ( Base `  S
)
Assertion
Ref Expression
lcoc0  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F  e.  ( R  ^m  V )  /\  F finSupp  .0.  /\  ( F ( linC  `  M ) V )  =  Z ) )
Distinct variable groups:    x, B    x, M    x, V    x,  .0.    x, F    x, R
Allowed substitution hints:    S( x)    Z( x)

Proof of Theorem lcoc0
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lincvalsc0.s . . . . . 6  |-  S  =  (Scalar `  M )
2 lcoc0.r . . . . . 6  |-  R  =  ( Base `  S
)
3 lincvalsc0.0 . . . . . 6  |-  .0.  =  ( 0g `  S )
41, 2, 3lmod0cl 17516 . . . . 5  |-  ( M  e.  LMod  ->  .0.  e.  R )
54ad2antrr 725 . . . 4  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  x  e.  V )  ->  .0.  e.  R )
6 lincvalsc0.f . . . 4  |-  F  =  ( x  e.  V  |->  .0.  )
75, 6fmptd 6040 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  F : V --> R )
8 fvex 5866 . . . . . 6  |-  ( Base `  S )  e.  _V
92, 8eqeltri 2527 . . . . 5  |-  R  e. 
_V
109a1i 11 . . . 4  |-  ( M  e.  LMod  ->  R  e. 
_V )
11 elmapg 7435 . . . 4  |-  ( ( R  e.  _V  /\  V  e.  ~P B
)  ->  ( F  e.  ( R  ^m  V
)  <->  F : V --> R ) )
1210, 11sylan 471 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F  e.  ( R  ^m  V )  <-> 
F : V --> R ) )
137, 12mpbird 232 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  F  e.  ( R  ^m  V ) )
14 eqidd 2444 . . . . . . 7  |-  ( x  =  v  ->  .0.  =  .0.  )
1514cbvmptv 4528 . . . . . 6  |-  ( x  e.  V  |->  .0.  )  =  ( v  e.  V  |->  .0.  )
166, 15eqtri 2472 . . . . 5  |-  F  =  ( v  e.  V  |->  .0.  )
17 simpr 461 . . . . 5  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  V  e.  ~P B
)
18 fvex 5866 . . . . . . 7  |-  ( 0g
`  S )  e. 
_V
193, 18eqeltri 2527 . . . . . 6  |-  .0.  e.  _V
2019a1i 11 . . . . 5  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  .0.  e.  _V )
2119a1i 11 . . . . 5  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  .0.  e.  _V )
2216, 17, 20, 21mptsuppd 6925 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F supp  .0.  )  =  { v  e.  V  |  .0.  =/=  .0.  }
)
23 neirr 2647 . . . . . . . 8  |-  -.  .0.  =/=  .0.
2423a1i 11 . . . . . . 7  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  -.  .0.  =/=  .0.  )
2524ralrimivw 2858 . . . . . 6  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  A. v  e.  V  -.  .0.  =/=  .0.  )
26 rabeq0 3793 . . . . . 6  |-  ( { v  e.  V  |  .0.  =/=  .0.  }  =  (/)  <->  A. v  e.  V  -.  .0.  =/=  .0.  )
2725, 26sylibr 212 . . . . 5  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  { v  e.  V  |  .0.  =/=  .0.  }  =  (/) )
28 0fin 7749 . . . . . 6  |-  (/)  e.  Fin
2928a1i 11 . . . . 5  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  (/) 
e.  Fin )
3027, 29eqeltrd 2531 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  { v  e.  V  |  .0.  =/=  .0.  }  e.  Fin )
3122, 30eqeltrd 2531 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F supp  .0.  )  e.  Fin )
326funmpt2 5615 . . . . 5  |-  Fun  F
3332a1i 11 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  Fun  F )
34 funisfsupp 7836 . . . 4  |-  ( ( Fun  F  /\  F  e.  ( R  ^m  V
)  /\  .0.  e.  _V )  ->  ( F finSupp  .0. 
<->  ( F supp  .0.  )  e.  Fin ) )
3533, 13, 20, 34syl3anc 1229 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F finSupp  .0.  <->  ( F supp  .0.  )  e.  Fin )
)
3631, 35mpbird 232 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  F finSupp  .0.  )
37 lincvalsc0.b . . 3  |-  B  =  ( Base `  M
)
38 lincvalsc0.z . . 3  |-  Z  =  ( 0g `  M
)
3937, 1, 3, 38, 6lincvalsc0 32757 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F ( linC  `  M ) V )  =  Z )
4013, 36, 393jca 1177 1  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F  e.  ( R  ^m  V )  /\  F finSupp  .0.  /\  ( F ( linC  `  M ) V )  =  Z ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   {crab 2797   _Vcvv 3095   (/)c0 3770   ~Pcpw 3997   class class class wbr 4437    |-> cmpt 4495   Fun wfun 5572   -->wf 5574   ` cfv 5578  (class class class)co 6281   supp csupp 6903    ^m cmap 7422   Fincfn 7518   finSupp cfsupp 7831   Basecbs 14613  Scalarcsca 14681   0gc0g 14818   LModclmod 17490   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
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-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-supp 6904  df-recs 7044  df-rdg 7078  df-map 7424  df-en 7519  df-fin 7522  df-fsupp 7832  df-seq 12089  df-0g 14820  df-gsum 14821  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-grp 16035  df-ring 17178  df-lmod 17492  df-linc 32742
This theorem is referenced by:  lcoel0  32764
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