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Theorem lco0 32127
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 4616 . . 3  |-  (/)  e.  ~P ( Base `  M )
2 eqid 2467 . . . 4  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2467 . . . 4  |-  (Scalar `  M )  =  (Scalar `  M )
4 eqid 2467 . . . 4  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
52, 3, 4lcoop 32111 . . 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 5876 . . . . . . 7  |-  ( Base `  (Scalar `  M )
)  e.  _V
8 map0e 7456 . . . . . . 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 7142 . . . . . 6  |-  1o  =  { (/) }
119, 10syl6eq 2524 . . . . 5  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( ( Base `  (Scalar `  M ) )  ^m  (/) )  =  { (/) } )
1211rexeqdv 3065 . . . 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 32115 . . . . . . . 8  |-  ( M  e.  Mnd  ->  ( (/) ( linC  `  M ) (/) )  =  ( 0g
`  M ) )
1413adantr 465 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( (/) ( linC  `  M
) (/) )  =  ( 0g `  M ) )
1514eqeq2d 2481 . . . . . 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 4577 . . . . . 6  |-  (/)  e.  _V
18 breq1 4450 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w finSupp 
( 0g `  (Scalar `  M ) )  <->  (/) finSupp  ( 0g `  (Scalar `  M )
) ) )
19 fvex 5876 . . . . . . . . . . 11  |-  ( 0g
`  (Scalar `  M )
)  e.  _V
20 0fsupp 7851 . . . . . . . . . . 11  |-  ( ( 0g `  (Scalar `  M ) )  e. 
_V  ->  (/) finSupp  ( 0g `  (Scalar `  M ) ) )
2119, 20ax-mp 5 . . . . . . . . . 10  |-  (/) finSupp  ( 0g
`  (Scalar `  M )
)
22 0fin 7747 . . . . . . . . . 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 6291 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w ( linC  `  M ) (/) )  =  ( (/) ( linC  `  M ) (/) ) )
2625eqeq2d 2481 . . . . . . . 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 4063 . . . . . 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 3104 . 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 2467 . . . 4  |-  ( 0g
`  M )  =  ( 0g `  M
)
362, 35mndidcl 15756 . . 3  |-  ( M  e.  Mnd  ->  ( 0g `  M )  e.  ( Base `  M
) )
37 rabsn 4094 . . 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 2512 1  |-  ( M  e.  Mnd  ->  ( M LinCo 
(/) )  =  {
( 0g `  M
) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818   _Vcvv 3113   (/)c0 3785   ~Pcpw 4010   {csn 4027   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   1oc1o 7123    ^m cmap 7420   Fincfn 7516   finSupp cfsupp 7829   Basecbs 14490  Scalarcsca 14558   0gc0g 14695   Mndcmnd 15726   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-linc 32106  df-lco 32107
This theorem is referenced by:  lcoel0  32128
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