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 33131
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 5882 . . . 4  |-  ( 0g
`  M )  e. 
_V
21snid 4060 . . 3  |-  ( 0g
`  M )  e. 
{ ( 0g `  M ) }
3 oveq2 6304 . . . 4  |-  ( V  =  (/)  ->  ( M LinCo 
V )  =  ( M LinCo  (/) ) )
4 lmodgrp 17645 . . . . . 6  |-  ( M  e.  LMod  ->  M  e. 
Grp )
5 grpmnd 16188 . . . . . 6  |-  ( M  e.  Grp  ->  M  e.  Mnd )
6 lco0 33130 . . . . . 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 2518 . . 3  |-  ( ( V  =  (/)  /\  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) ) )  -> 
( M LinCo  V )  =  { ( 0g `  M ) } )
102, 9syl5eleqr 2552 . 2  |-  ( ( V  =  (/)  /\  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) ) )  -> 
( 0g `  M
)  e.  ( M LinCo 
V ) )
11 eqid 2457 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
12 eqid 2457 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  M
)
1311, 12lmod0vcl 17667 . . . . 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 2457 . . . . . 6  |-  (Scalar `  M )  =  (Scalar `  M )
17 eqid 2457 . . . . . 6  |-  ( 0g
`  (Scalar `  M )
)  =  ( 0g
`  (Scalar `  M )
)
18 eqidd 2458 . . . . . . 7  |-  ( v  =  w  ->  ( 0g `  (Scalar `  M
) )  =  ( 0g `  (Scalar `  M ) ) )
1918cbvmptv 4548 . . . . . 6  |-  ( v  e.  V  |->  ( 0g
`  (Scalar `  M )
) )  =  ( w  e.  V  |->  ( 0g `  (Scalar `  M ) ) )
20 eqid 2457 . . . . . 6  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
2111, 16, 17, 12, 19, 20lcoc0 33125 . . . . 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 4459 . . . . . . . . . 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 6303 . . . . . . . . . . . 12  |-  ( s  =  ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) )  ->  ( s
( linC  `  M ) V )  =  ( ( v  e.  V  |->  ( 0g `  (Scalar `  M ) ) ) ( linC  `  M ) V ) )
2625eqeq2d 2471 . . . . . . . . . . 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 2466 . . . . . . . . . . 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 3214 . . . . . . 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 33115 . . . 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 922 . 2  |-  ( ( -.  V  =  (/)  /\  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) ) )  ->  ( 0g `  M )  e.  ( M LinCo  V ) )
3910, 38pm2.61ian 790 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 1395    e. wcel 1819   E.wrex 2808   (/)c0 3793   ~Pcpw 4015   {csn 4032   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   finSupp cfsupp 7847   Basecbs 14643  Scalarcsca 14714   0gc0g 14856   Mndcmnd 16045   Grpcgrp 16179   LModclmod 17638   linC clinc 33107   LinCo clinco 33108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-map 7440  df-en 7536  df-fin 7539  df-fsupp 7848  df-seq 12110  df-0g 14858  df-gsum 14859  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-grp 16183  df-ring 17326  df-lmod 17640  df-linc 33109  df-lco 33110
This theorem is referenced by:  lincolss  33137
  Copyright terms: Public domain W3C validator