Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcoel0 Structured version   Unicode version

Theorem lcoel0 30960
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 5700 . . . 4  |-  ( 0g
`  M )  e. 
_V
21snid 3904 . . 3  |-  ( 0g
`  M )  e. 
{ ( 0g `  M ) }
3 oveq2 6098 . . . 4  |-  ( V  =  (/)  ->  ( M LinCo 
V )  =  ( M LinCo  (/) ) )
4 lmodgrp 16954 . . . . . 6  |-  ( M  e.  LMod  ->  M  e. 
Grp )
5 grpmnd 15549 . . . . . 6  |-  ( M  e.  Grp  ->  M  e.  Mnd )
6 lco0 30959 . . . . . 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 2494 . . 3  |-  ( ( V  =  (/)  /\  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) ) )  -> 
( M LinCo  V )  =  { ( 0g `  M ) } )
102, 9syl5eleqr 2529 . 2  |-  ( ( V  =  (/)  /\  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) ) )  -> 
( 0g `  M
)  e.  ( M LinCo 
V ) )
11 eqid 2442 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
12 eqid 2442 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  M
)
1311, 12lmod0vcl 16976 . . . . 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 2442 . . . . . 6  |-  (Scalar `  M )  =  (Scalar `  M )
17 eqid 2442 . . . . . 6  |-  ( 0g
`  (Scalar `  M )
)  =  ( 0g
`  (Scalar `  M )
)
18 eqidd 2443 . . . . . . 7  |-  ( v  =  w  ->  ( 0g `  (Scalar `  M
) )  =  ( 0g `  (Scalar `  M ) ) )
1918cbvmptv 4382 . . . . . 6  |-  ( v  e.  V  |->  ( 0g
`  (Scalar `  M )
) )  =  ( w  e.  V  |->  ( 0g `  (Scalar `  M ) ) )
20 eqid 2442 . . . . . 6  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
2111, 16, 17, 12, 19, 20lcoc0 30954 . . . . 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 4294 . . . . . . . . . 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 6097 . . . . . . . . . . . 12  |-  ( s  =  ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  ->  ( s
( linC  `  M ) V )  =  ( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) ( linC  `  M ) V ) )
2625eqeq2d 2453 . . . . . . . . . . 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 2444 . . . . . . . . . . 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 3076 . . . . . . 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 1185 . . . 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 30944 . . . 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 913 . 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 965    = wceq 1369    e. wcel 1756   E.wrex 2715   (/)c0 3636   ~Pcpw 3859   {csn 3876   class class class wbr 4291    e. cmpt 4349   ` cfv 5417  (class class class)co 6090    ^m cmap 7213   finSupp cfsupp 7619   Basecbs 14173  Scalarcsca 14240   0gc0g 14377   Mndcmnd 15408   Grpcgrp 15409   LModclmod 16947   linC clinc 30936   LinCo clinco 30937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6831  df-rdg 6865  df-1o 6919  df-map 7215  df-en 7310  df-fin 7313  df-fsupp 7620  df-seq 11806  df-0g 14379  df-gsum 14380  df-mnd 15414  df-grp 15544  df-rng 16646  df-lmod 16949  df-linc 30938  df-lco 30939
This theorem is referenced by:  lincolss  30966
  Copyright terms: Public domain W3C validator