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Theorem lco0 30958
Description: The set of empty linear combinations over a monoid is the singleton with the identity element of the monoid. (Contributed by AV, 12-Apr-2019.)
Assertion
Ref Expression
lco0  |-  ( M  e.  Mnd  ->  ( M LinCo 
(/) )  =  {
( 0g `  M
) } )

Proof of Theorem lco0
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0elpw 4459 . . 3  |-  (/)  e.  ~P ( Base `  M )
2 eqid 2441 . . . 4  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2441 . . . 4  |-  (Scalar `  M )  =  (Scalar `  M )
4 eqid 2441 . . . 4  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
52, 3, 4lcoop 30942 . . 3  |-  ( ( M  e.  Mnd  /\  (/) 
e.  ~P ( Base `  M
) )  ->  ( M LinCo 
(/) )  =  {
v  e.  ( Base `  M )  |  E. w  e.  ( ( Base `  (Scalar `  M
) )  ^m  (/) ) ( w finSupp  ( 0g `  (Scalar `  M ) )  /\  v  =  ( w ( linC  `  M
) (/) ) ) } )
61, 5mpan2 671 . 2  |-  ( M  e.  Mnd  ->  ( M LinCo 
(/) )  =  {
v  e.  ( Base `  M )  |  E. w  e.  ( ( Base `  (Scalar `  M
) )  ^m  (/) ) ( w finSupp  ( 0g `  (Scalar `  M ) )  /\  v  =  ( w ( linC  `  M
) (/) ) ) } )
7 fvex 5699 . . . . . . 7  |-  ( Base `  (Scalar `  M )
)  e.  _V
8 map0e 7248 . . . . . . 7  |-  ( (
Base `  (Scalar `  M
) )  e.  _V  ->  ( ( Base `  (Scalar `  M ) )  ^m  (/) )  =  1o )
97, 8mp1i 12 . . . . . 6  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( ( Base `  (Scalar `  M ) )  ^m  (/) )  =  1o )
10 df1o2 6930 . . . . . 6  |-  1o  =  { (/) }
119, 10syl6eq 2489 . . . . 5  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( ( Base `  (Scalar `  M ) )  ^m  (/) )  =  { (/) } )
1211rexeqdv 2922 . . . 4  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( E. w  e.  ( ( Base `  (Scalar `  M ) )  ^m  (/) ) ( w finSupp  ( 0g `  (Scalar `  M
) )  /\  v  =  ( w ( linC  `  M ) (/) ) )  <->  E. w  e.  { (/) }  ( w finSupp  ( 0g
`  (Scalar `  M )
)  /\  v  =  ( w ( linC  `  M ) (/) ) ) ) )
13 lincval0 30946 . . . . . . . 8  |-  ( M  e.  Mnd  ->  ( (/) ( linC  `  M ) (/) )  =  ( 0g
`  M ) )
1413adantr 465 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( (/) ( linC  `  M
) (/) )  =  ( 0g `  M ) )
1514eqeq2d 2452 . . . . . 6  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( v  =  (
(/) ( linC  `  M )
(/) )  <->  v  =  ( 0g `  M ) ) )
1615anbi2d 703 . . . . 5  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( ( (/)  e.  Fin  /\  v  =  ( (/) ( linC  `  M ) (/) ) )  <->  ( (/)  e.  Fin  /\  v  =  ( 0g
`  M ) ) ) )
17 0ex 4420 . . . . . 6  |-  (/)  e.  _V
18 breq1 4293 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w finSupp 
( 0g `  (Scalar `  M ) )  <->  (/) finSupp  ( 0g `  (Scalar `  M )
) ) )
19 fvex 5699 . . . . . . . . . . 11  |-  ( 0g
`  (Scalar `  M )
)  e.  _V
20 0fsupp 7640 . . . . . . . . . . 11  |-  ( ( 0g `  (Scalar `  M ) )  e. 
_V  ->  (/) finSupp  ( 0g `  (Scalar `  M ) ) )
2119, 20ax-mp 5 . . . . . . . . . 10  |-  (/) finSupp  ( 0g
`  (Scalar `  M )
)
22 0fin 7538 . . . . . . . . . 10  |-  (/)  e.  Fin
2321, 222th 239 . . . . . . . . 9  |-  ( (/) finSupp  ( 0g `  (Scalar `  M ) )  <->  (/)  e.  Fin )
2418, 23syl6bb 261 . . . . . . . 8  |-  ( w  =  (/)  ->  ( w finSupp 
( 0g `  (Scalar `  M ) )  <->  (/)  e.  Fin ) )
25 oveq1 6096 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w ( linC  `  M ) (/) )  =  ( (/) ( linC  `  M ) (/) ) )
2625eqeq2d 2452 . . . . . . . 8  |-  ( w  =  (/)  ->  ( v  =  ( w ( linC  `  M ) (/) )  <->  v  =  ( (/) ( linC  `  M
) (/) ) ) )
2724, 26anbi12d 710 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( w finSupp  ( 0g `  (Scalar `  M ) )  /\  v  =  ( w ( linC  `  M
) (/) ) )  <->  ( (/)  e.  Fin  /\  v  =  ( (/) ( linC  `  M ) (/) ) ) ) )
2827rexsng 3911 . . . . . 6  |-  ( (/)  e.  _V  ->  ( E. w  e.  { (/) }  (
w finSupp  ( 0g `  (Scalar `  M ) )  /\  v  =  ( w
( linC  `  M ) (/) ) )  <->  ( (/)  e.  Fin  /\  v  =  ( (/) ( linC  `  M ) (/) ) ) ) )
2917, 28mp1i 12 . . . . 5  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( E. w  e. 
{ (/) }  ( w finSupp 
( 0g `  (Scalar `  M ) )  /\  v  =  ( w
( linC  `  M ) (/) ) )  <->  ( (/)  e.  Fin  /\  v  =  ( (/) ( linC  `  M ) (/) ) ) ) )
3022a1i 11 . . . . . 6  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  ->  (/) 
e.  Fin )
3130biantrurd 508 . . . . 5  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( v  =  ( 0g `  M )  <-> 
( (/)  e.  Fin  /\  v  =  ( 0g `  M ) ) ) )
3216, 29, 313bitr4d 285 . . . 4  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( E. w  e. 
{ (/) }  ( w finSupp 
( 0g `  (Scalar `  M ) )  /\  v  =  ( w
( linC  `  M ) (/) ) )  <->  v  =  ( 0g `  M ) ) )
3312, 32bitrd 253 . . 3  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( E. w  e.  ( ( Base `  (Scalar `  M ) )  ^m  (/) ) ( w finSupp  ( 0g `  (Scalar `  M
) )  /\  v  =  ( w ( linC  `  M ) (/) ) )  <-> 
v  =  ( 0g
`  M ) ) )
3433rabbidva 2961 . 2  |-  ( M  e.  Mnd  ->  { v  e.  ( Base `  M
)  |  E. w  e.  ( ( Base `  (Scalar `  M ) )  ^m  (/) ) ( w finSupp  ( 0g `  (Scalar `  M
) )  /\  v  =  ( w ( linC  `  M ) (/) ) ) }  =  { v  e.  ( Base `  M
)  |  v  =  ( 0g `  M
) } )
35 eqid 2441 . . . 4  |-  ( 0g
`  M )  =  ( 0g `  M
)
362, 35mndidcl 15437 . . 3  |-  ( M  e.  Mnd  ->  ( 0g `  M )  e.  ( Base `  M
) )
37 rabsn 3940 . . 3  |-  ( ( 0g `  M )  e.  ( Base `  M
)  ->  { v  e.  ( Base `  M
)  |  v  =  ( 0g `  M
) }  =  {
( 0g `  M
) } )
3836, 37syl 16 . 2  |-  ( M  e.  Mnd  ->  { v  e.  ( Base `  M
)  |  v  =  ( 0g `  M
) }  =  {
( 0g `  M
) } )
396, 34, 383eqtrd 2477 1  |-  ( M  e.  Mnd  ->  ( M LinCo 
(/) )  =  {
( 0g `  M
) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2714   {crab 2717   _Vcvv 2970   (/)c0 3635   ~Pcpw 3858   {csn 3875   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   1oc1o 6911    ^m cmap 7212   Fincfn 7308   finSupp cfsupp 7618   Basecbs 14172  Scalarcsca 14239   0gc0g 14376   Mndcmnd 15407   linC clinc 30935   LinCo clinco 30936
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-recs 6830  df-rdg 6864  df-1o 6918  df-map 7214  df-en 7309  df-fin 7312  df-fsupp 7619  df-seq 11805  df-0g 14378  df-gsum 14379  df-mnd 15413  df-linc 30937  df-lco 30938
This theorem is referenced by:  lcoel0  30959
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