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Theorem lincvalsc0 32892
Description: The linear combination where all scalars are 0. (Contributed by AV, 12-Apr-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.  )
Assertion
Ref Expression
lincvalsc0  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F ( linC  `  M ) V )  =  Z )
Distinct variable groups:    x, B    x, M    x, V    x,  .0.
Allowed substitution hints:    S( x)    F( x)    Z( x)

Proof of Theorem lincvalsc0
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 lincvalsc0.s . . . . . . . 8  |-  S  =  (Scalar `  M )
32eqcomi 2456 . . . . . . . . 9  |-  (Scalar `  M )  =  S
43fveq2i 5859 . . . . . . . 8  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  S )
5 lincvalsc0.0 . . . . . . . 8  |-  .0.  =  ( 0g `  S )
62, 4, 5lmod0cl 17517 . . . . . . 7  |-  ( M  e.  LMod  ->  .0.  e.  ( Base `  (Scalar `  M
) ) )
76adantr 465 . . . . . 6  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  .0.  e.  ( Base `  (Scalar `  M ) ) )
87adantr 465 . . . . 5  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  x  e.  V )  ->  .0.  e.  ( Base `  (Scalar `  M ) ) )
9 lincvalsc0.f . . . . 5  |-  F  =  ( x  e.  V  |->  .0.  )
108, 9fmptd 6040 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  F : V --> ( Base `  (Scalar `  M )
) )
11 fvex 5866 . . . . . 6  |-  ( Base `  (Scalar `  M )
)  e.  _V
1211a1i 11 . . . . 5  |-  ( M  e.  LMod  ->  ( Base `  (Scalar `  M )
)  e.  _V )
13 elmapg 7435 . . . . 5  |-  ( ( ( Base `  (Scalar `  M ) )  e. 
_V  /\  V  e.  ~P B )  ->  ( F  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
)  <->  F : V --> ( Base `  (Scalar `  M )
) ) )
1412, 13sylan 471 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F  e.  ( ( Base `  (Scalar `  M ) )  ^m  V )  <->  F : V
--> ( Base `  (Scalar `  M ) ) ) )
1510, 14mpbird 232 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  F  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
) )
16 lincvalsc0.b . . . . . . 7  |-  B  =  ( Base `  M
)
1716pweqi 4001 . . . . . 6  |-  ~P B  =  ~P ( Base `  M
)
1817eleq2i 2521 . . . . 5  |-  ( V  e.  ~P B  <->  V  e.  ~P ( Base `  M
) )
1918biimpi 194 . . . 4  |-  ( V  e.  ~P B  ->  V  e.  ~P ( Base `  M ) )
2019adantl 466 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  V  e.  ~P ( Base `  M ) )
21 lincval 32880 . . 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 ) ) ) )
221, 15, 20, 21syl3anc 1229 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F ( linC  `  M ) V )  =  ( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s `  M
) v ) ) ) )
23 simpr 461 . . . . . . 7  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  v  e.  V )
24 fvex 5866 . . . . . . . 8  |-  ( 0g
`  S )  e. 
_V
255, 24eqeltri 2527 . . . . . . 7  |-  .0.  e.  _V
26 eqidd 2444 . . . . . . . 8  |-  ( x  =  v  ->  .0.  =  .0.  )
2726, 9fvmptg 5939 . . . . . . 7  |-  ( ( v  e.  V  /\  .0.  e.  _V )  -> 
( F `  v
)  =  .0.  )
2823, 25, 27sylancl 662 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  ( F `  v )  =  .0.  )
2928oveq1d 6296 . . . . 5  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  (
( F `  v
) ( .s `  M ) v )  =  (  .0.  ( .s `  M ) v ) )
301adantr 465 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  M  e.  LMod )
31 elelpwi 4008 . . . . . . . . 9  |-  ( ( v  e.  V  /\  V  e.  ~P B
)  ->  v  e.  B )
3231expcom 435 . . . . . . . 8  |-  ( V  e.  ~P B  -> 
( v  e.  V  ->  v  e.  B ) )
3332adantl 466 . . . . . . 7  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( v  e.  V  ->  v  e.  B ) )
3433imp 429 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  v  e.  B )
35 eqid 2443 . . . . . . 7  |-  ( .s
`  M )  =  ( .s `  M
)
36 lincvalsc0.z . . . . . . 7  |-  Z  =  ( 0g `  M
)
3716, 2, 35, 5, 36lmod0vs 17524 . . . . . 6  |-  ( ( M  e.  LMod  /\  v  e.  B )  ->  (  .0.  ( .s `  M
) v )  =  Z )
3830, 34, 37syl2anc 661 . . . . 5  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  (  .0.  ( .s `  M
) v )  =  Z )
3929, 38eqtrd 2484 . . . 4  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  (
( F `  v
) ( .s `  M ) v )  =  Z )
4039mpteq2dva 4523 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) )  =  ( v  e.  V  |->  Z ) )
4140oveq2d 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 ) ) )
42 lmodgrp 17498 . . . 4  |-  ( M  e.  LMod  ->  M  e. 
Grp )
43 grpmnd 16041 . . . 4  |-  ( M  e.  Grp  ->  M  e.  Mnd )
4442, 43syl 16 . . 3  |-  ( M  e.  LMod  ->  M  e. 
Mnd )
4536gsumz 15984 . . 3  |-  ( ( M  e.  Mnd  /\  V  e.  ~P B
)  ->  ( M  gsumg  ( v  e.  V  |->  Z ) )  =  Z )
4644, 45sylan 471 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( M  gsumg  ( v  e.  V  |->  Z ) )  =  Z )
4722, 41, 463eqtrd 2488 1  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( F ( linC  `  M ) V )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095   ~Pcpw 3997    |-> cmpt 4495   -->wf 5574   ` cfv 5578  (class class class)co 6281    ^m cmap 7422   Basecbs 14614  Scalarcsca 14682   .scvsca 14683   0gc0g 14819    gsumg cgsu 14820   Mndcmnd 15898   Grpcgrp 16032   LModclmod 17491   linC clinc 32875
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-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-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  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-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-map 7424  df-seq 12090  df-0g 14821  df-gsum 14822  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-grp 16036  df-ring 17179  df-lmod 17493  df-linc 32877
This theorem is referenced by:  lcoc0  32893
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