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Theorem lcoel0 32128
Description: The zero vector is always a linear combination. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Assertion
Ref Expression
lcoel0  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  ( 0g `  M )  e.  ( M LinCo  V ) )

Proof of Theorem lcoel0
Dummy variables  s 
v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5876 . . . 4  |-  ( 0g
`  M )  e. 
_V
21snid 4055 . . 3  |-  ( 0g
`  M )  e. 
{ ( 0g `  M ) }
3 oveq2 6292 . . . 4  |-  ( V  =  (/)  ->  ( M LinCo 
V )  =  ( M LinCo  (/) ) )
4 lmodgrp 17319 . . . . . 6  |-  ( M  e.  LMod  ->  M  e. 
Grp )
5 grpmnd 15872 . . . . . 6  |-  ( M  e.  Grp  ->  M  e.  Mnd )
6 lco0 32127 . . . . . 6  |-  ( M  e.  Mnd  ->  ( M LinCo 
(/) )  =  {
( 0g `  M
) } )
74, 5, 63syl 20 . . . . 5  |-  ( M  e.  LMod  ->  ( M LinCo  (/) )  =  { ( 0g `  M ) } )
87adantr 465 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  ( M LinCo 
(/) )  =  {
( 0g `  M
) } )
93, 8sylan9eq 2528 . . 3  |-  ( ( V  =  (/)  /\  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) ) )  -> 
( M LinCo  V )  =  { ( 0g `  M ) } )
102, 9syl5eleqr 2562 . 2  |-  ( ( V  =  (/)  /\  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) ) )  -> 
( 0g `  M
)  e.  ( M LinCo 
V ) )
11 eqid 2467 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
12 eqid 2467 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  M
)
1311, 12lmod0vcl 17341 . . . . 5  |-  ( M  e.  LMod  ->  ( 0g
`  M )  e.  ( Base `  M
) )
1413adantr 465 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  ( 0g `  M )  e.  ( Base `  M
) )
1514adantl 466 . . 3  |-  ( ( -.  V  =  (/)  /\  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) ) )  ->  ( 0g `  M )  e.  (
Base `  M )
)
16 eqid 2467 . . . . . 6  |-  (Scalar `  M )  =  (Scalar `  M )
17 eqid 2467 . . . . . 6  |-  ( 0g
`  (Scalar `  M )
)  =  ( 0g
`  (Scalar `  M )
)
18 eqidd 2468 . . . . . . 7  |-  ( v  =  w  ->  ( 0g `  (Scalar `  M
) )  =  ( 0g `  (Scalar `  M ) ) )
1918cbvmptv 4538 . . . . . 6  |-  ( v  e.  V  |->  ( 0g
`  (Scalar `  M )
) )  =  ( w  e.  V  |->  ( 0g `  (Scalar `  M ) ) )
20 eqid 2467 . . . . . 6  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
2111, 16, 17, 12, 19, 20lcoc0 32122 . . . . 5  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  (
( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )  /\  ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) finSupp 
( 0g `  (Scalar `  M ) )  /\  ( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) ( linC  `  M
) V )  =  ( 0g `  M
) ) )
2221adantl 466 . . . 4  |-  ( ( -.  V  =  (/)  /\  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) ) )  ->  ( (
v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )  /\  ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) finSupp 
( 0g `  (Scalar `  M ) )  /\  ( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) ( linC  `  M
) V )  =  ( 0g `  M
) ) )
23 simpl 457 . . . . . . . 8  |-  ( ( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )  /\  ( -.  V  =  (/)  /\  ( M  e. 
LMod  /\  V  e.  ~P ( Base `  M )
) ) )  -> 
( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )
)
24 breq1 4450 . . . . . . . . . 10  |-  ( s  =  ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  ->  ( s finSupp  ( 0g `  (Scalar `  M ) )  <->  ( v  e.  V  |->  ( 0g
`  (Scalar `  M )
) ) finSupp  ( 0g `  (Scalar `  M )
) ) )
25 oveq1 6291 . . . . . . . . . . . 12  |-  ( s  =  ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  ->  ( s
( linC  `  M ) V )  =  ( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) ( linC  `  M ) V ) )
2625eqeq2d 2481 . . . . . . . . . . 11  |-  ( s  =  ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  ->  ( ( 0g `  M )  =  ( s ( linC  `  M ) V )  <-> 
( 0g `  M
)  =  ( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) ( linC  `  M ) V ) ) )
27 eqcom 2476 . . . . . . . . . . 11  |-  ( ( 0g `  M )  =  ( ( v  e.  V  |->  ( 0g
`  (Scalar `  M )
) ) ( linC  `  M ) V )  <-> 
( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) ( linC  `  M
) V )  =  ( 0g `  M
) )
2826, 27syl6bb 261 . . . . . . . . . 10  |-  ( s  =  ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  ->  ( ( 0g `  M )  =  ( s ( linC  `  M ) V )  <-> 
( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) ( linC  `  M
) V )  =  ( 0g `  M
) ) )
2924, 28anbi12d 710 . . . . . . . . 9  |-  ( s  =  ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  ->  ( (
s finSupp  ( 0g `  (Scalar `  M ) )  /\  ( 0g `  M )  =  ( s ( linC  `  M ) V ) )  <->  ( ( v  e.  V  |->  ( 0g
`  (Scalar `  M )
) ) finSupp  ( 0g `  (Scalar `  M )
)  /\  ( (
v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) ( linC  `  M ) V )  =  ( 0g `  M ) ) ) )
3029adantl 466 . . . . . . . 8  |-  ( ( ( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  e.  ( (
Base `  (Scalar `  M
) )  ^m  V
)  /\  ( -.  V  =  (/)  /\  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) ) ) )  /\  s  =  ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) )  ->  ( (
s finSupp  ( 0g `  (Scalar `  M ) )  /\  ( 0g `  M )  =  ( s ( linC  `  M ) V ) )  <->  ( ( v  e.  V  |->  ( 0g
`  (Scalar `  M )
) ) finSupp  ( 0g `  (Scalar `  M )
)  /\  ( (
v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) ( linC  `  M ) V )  =  ( 0g `  M ) ) ) )
3123, 30rspcedv 3218 . . . . . . 7  |-  ( ( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )  /\  ( -.  V  =  (/)  /\  ( M  e. 
LMod  /\  V  e.  ~P ( Base `  M )
) ) )  -> 
( ( ( v  e.  V  |->  ( 0g
`  (Scalar `  M )
) ) finSupp  ( 0g `  (Scalar `  M )
)  /\  ( (
v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) ( linC  `  M ) V )  =  ( 0g `  M ) )  ->  E. s  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( s finSupp 
( 0g `  (Scalar `  M ) )  /\  ( 0g `  M )  =  ( s ( linC  `  M ) V ) ) ) )
3231ex 434 . . . . . 6  |-  ( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )  ->  ( ( -.  V  =  (/)  /\  ( M  e.  LMod  /\  V  e. 
~P ( Base `  M
) ) )  -> 
( ( ( v  e.  V  |->  ( 0g
`  (Scalar `  M )
) ) finSupp  ( 0g `  (Scalar `  M )
)  /\  ( (
v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) ( linC  `  M ) V )  =  ( 0g `  M ) )  ->  E. s  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( s finSupp 
( 0g `  (Scalar `  M ) )  /\  ( 0g `  M )  =  ( s ( linC  `  M ) V ) ) ) ) )
3332com23 78 . . . . 5  |-  ( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )  ->  ( ( ( v  e.  V  |->  ( 0g
`  (Scalar `  M )
) ) finSupp  ( 0g `  (Scalar `  M )
)  /\  ( (
v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) ( linC  `  M ) V )  =  ( 0g `  M ) )  ->  ( ( -.  V  =  (/)  /\  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) ) )  ->  E. s  e.  (
( Base `  (Scalar `  M
) )  ^m  V
) ( s finSupp  ( 0g `  (Scalar `  M
) )  /\  ( 0g `  M )  =  ( s ( linC  `  M ) V ) ) ) ) )
34333impib 1194 . . . 4  |-  ( ( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )  /\  ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) finSupp 
( 0g `  (Scalar `  M ) )  /\  ( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) ( linC  `  M
) V )  =  ( 0g `  M
) )  ->  (
( -.  V  =  (/)  /\  ( M  e. 
LMod  /\  V  e.  ~P ( Base `  M )
) )  ->  E. s  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( s finSupp 
( 0g `  (Scalar `  M ) )  /\  ( 0g `  M )  =  ( s ( linC  `  M ) V ) ) ) )
3522, 34mpcom 36 . . 3  |-  ( ( -.  V  =  (/)  /\  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) ) )  ->  E. s  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( s finSupp 
( 0g `  (Scalar `  M ) )  /\  ( 0g `  M )  =  ( s ( linC  `  M ) V ) ) )
3611, 16, 20lcoval 32112 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  (
( 0g `  M
)  e.  ( M LinCo 
V )  <->  ( ( 0g `  M )  e.  ( Base `  M
)  /\  E. s  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( s finSupp 
( 0g `  (Scalar `  M ) )  /\  ( 0g `  M )  =  ( s ( linC  `  M ) V ) ) ) ) )
3736adantl 466 . . 3  |-  ( ( -.  V  =  (/)  /\  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) ) )  ->  ( ( 0g `  M )  e.  ( M LinCo  V )  <-> 
( ( 0g `  M )  e.  (
Base `  M )  /\  E. s  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( s finSupp 
( 0g `  (Scalar `  M ) )  /\  ( 0g `  M )  =  ( s ( linC  `  M ) V ) ) ) ) )
3815, 35, 37mpbir2and 920 . 2  |-  ( ( -.  V  =  (/)  /\  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) ) )  ->  ( 0g `  M )  e.  ( M LinCo  V ) )
3910, 38pm2.61ian 788 1  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  ( 0g `  M )  e.  ( M LinCo  V ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   (/)c0 3785   ~Pcpw 4010   {csn 4027   class class class wbr 4447    |-> cmpt 4505   ` cfv 5588  (class class class)co 6284    ^m cmap 7420   finSupp cfsupp 7829   Basecbs 14490  Scalarcsca 14558   0gc0g 14695   Mndcmnd 15726   Grpcgrp 15727   LModclmod 17312   linC clinc 32104   LinCo clinco 32105
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-3or 974  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-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  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-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-map 7422  df-en 7517  df-fin 7520  df-fsupp 7830  df-seq 12076  df-0g 14697  df-gsum 14698  df-mnd 15732  df-grp 15867  df-rng 17002  df-lmod 17314  df-linc 32106  df-lco 32107
This theorem is referenced by:  lincolss  32134
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