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Theorem lincvalsc0 32121
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 2480 . . . . . . . . 9  |-  (Scalar `  M )  =  S
43fveq2i 5869 . . . . . . . 8  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  S )
5 lincvalsc0.0 . . . . . . . 8  |-  .0.  =  ( 0g `  S )
62, 4, 5lmod0cl 17338 . . . . . . 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 6045 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  ->  F : V --> ( Base `  (Scalar `  M )
) )
11 fvex 5876 . . . . . 6  |-  ( Base `  (Scalar `  M )
)  e.  _V
1211a1i 11 . . . . 5  |-  ( M  e.  LMod  ->  ( Base `  (Scalar `  M )
)  e.  _V )
13 elmapg 7433 . . . . 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 4014 . . . . . 6  |-  ~P B  =  ~P ( Base `  M
)
1817eleq2i 2545 . . . . 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 32109 . . 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 1228 . 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 5876 . . . . . . . 8  |-  ( 0g
`  S )  e. 
_V
255, 24eqeltri 2551 . . . . . . 7  |-  .0.  e.  _V
26 eqidd 2468 . . . . . . . 8  |-  ( x  =  v  ->  .0.  =  .0.  )
2726, 9fvmptg 5948 . . . . . . 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 6299 . . . . 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 4021 . . . . . . . . 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 2467 . . . . . . 7  |-  ( .s
`  M )  =  ( .s `  M
)
36 lincvalsc0.z . . . . . . 7  |-  Z  =  ( 0g `  M
)
3716, 2, 35, 5, 36lmod0vs 17345 . . . . . 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 2508 . . . 4  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P B
)  /\  v  e.  V )  ->  (
( F `  v
) ( .s `  M ) v )  =  Z )
4039mpteq2dva 4533 . . 3  |-  ( ( M  e.  LMod  /\  V  e.  ~P B )  -> 
( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) )  =  ( v  e.  V  |->  Z ) )
4140oveq2d 6300 . 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 17319 . . . 4  |-  ( M  e.  LMod  ->  M  e. 
Grp )
43 grpmnd 15872 . . . 4  |-  ( M  e.  Grp  ->  M  e.  Mnd )
4442, 43syl 16 . . 3  |-  ( M  e.  LMod  ->  M  e. 
Mnd )
4536gsumz 15833 . . 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 2512 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 1379    e. wcel 1767   _Vcvv 3113   ~Pcpw 4010    |-> cmpt 4505   -->wf 5584   ` cfv 5588  (class class class)co 6284    ^m cmap 7420   Basecbs 14490  Scalarcsca 14558   .scvsca 14559   0gc0g 14695    gsumg cgsu 14696   Mndcmnd 15726   Grpcgrp 15727   LModclmod 17312   linC clinc 32104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-map 7422  df-seq 12076  df-0g 14697  df-gsum 14698  df-mnd 15732  df-grp 15867  df-rng 17002  df-lmod 17314  df-linc 32106
This theorem is referenced by:  lcoc0  32122
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