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Theorem lco0 38989
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 4594 . . 3  |-  (/)  e.  ~P ( Base `  M )
2 eqid 2429 . . . 4  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2429 . . . 4  |-  (Scalar `  M )  =  (Scalar `  M )
4 eqid 2429 . . . 4  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
52, 3, 4lcoop 38973 . . 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 675 . 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 5891 . . . . . . 7  |-  ( Base `  (Scalar `  M )
)  e.  _V
8 map0e 7517 . . . . . . 7  |-  ( (
Base `  (Scalar `  M
) )  e.  _V  ->  ( ( Base `  (Scalar `  M ) )  ^m  (/) )  =  1o )
97, 8mp1i 13 . . . . . 6  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( ( Base `  (Scalar `  M ) )  ^m  (/) )  =  1o )
10 df1o2 7202 . . . . . 6  |-  1o  =  { (/) }
119, 10syl6eq 2486 . . . . 5  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( ( Base `  (Scalar `  M ) )  ^m  (/) )  =  { (/) } )
1211rexeqdv 3039 . . . 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 38977 . . . . . . . 8  |-  ( M  e.  Mnd  ->  ( (/) ( linC  `  M ) (/) )  =  ( 0g
`  M ) )
1413adantr 466 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( (/) ( linC  `  M
) (/) )  =  ( 0g `  M ) )
1514eqeq2d 2443 . . . . . 6  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( v  =  (
(/) ( linC  `  M )
(/) )  <->  v  =  ( 0g `  M ) ) )
1615anbi2d 708 . . . . 5  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( ( (/)  e.  Fin  /\  v  =  ( (/) ( linC  `  M ) (/) ) )  <->  ( (/)  e.  Fin  /\  v  =  ( 0g
`  M ) ) ) )
17 0ex 4557 . . . . . 6  |-  (/)  e.  _V
18 breq1 4429 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w finSupp 
( 0g `  (Scalar `  M ) )  <->  (/) finSupp  ( 0g `  (Scalar `  M )
) ) )
19 fvex 5891 . . . . . . . . . . 11  |-  ( 0g
`  (Scalar `  M )
)  e.  _V
20 0fsupp 7911 . . . . . . . . . . 11  |-  ( ( 0g `  (Scalar `  M ) )  e. 
_V  ->  (/) finSupp  ( 0g `  (Scalar `  M ) ) )
2119, 20ax-mp 5 . . . . . . . . . 10  |-  (/) finSupp  ( 0g
`  (Scalar `  M )
)
22 0fin 7805 . . . . . . . . . 10  |-  (/)  e.  Fin
2321, 222th 242 . . . . . . . . 9  |-  ( (/) finSupp  ( 0g `  (Scalar `  M ) )  <->  (/)  e.  Fin )
2418, 23syl6bb 264 . . . . . . . 8  |-  ( w  =  (/)  ->  ( w finSupp 
( 0g `  (Scalar `  M ) )  <->  (/)  e.  Fin ) )
25 oveq1 6312 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w ( linC  `  M ) (/) )  =  ( (/) ( linC  `  M ) (/) ) )
2625eqeq2d 2443 . . . . . . . 8  |-  ( w  =  (/)  ->  ( v  =  ( w ( linC  `  M ) (/) )  <->  v  =  ( (/) ( linC  `  M
) (/) ) ) )
2724, 26anbi12d 715 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( w finSupp  ( 0g `  (Scalar `  M ) )  /\  v  =  ( w ( linC  `  M
) (/) ) )  <->  ( (/)  e.  Fin  /\  v  =  ( (/) ( linC  `  M ) (/) ) ) ) )
2827rexsng 4038 . . . . . 6  |-  ( (/)  e.  _V  ->  ( E. w  e.  { (/) }  (
w finSupp  ( 0g `  (Scalar `  M ) )  /\  v  =  ( w
( linC  `  M ) (/) ) )  <->  ( (/)  e.  Fin  /\  v  =  ( (/) ( linC  `  M ) (/) ) ) ) )
2917, 28mp1i 13 . . . . 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 510 . . . . 5  |-  ( ( M  e.  Mnd  /\  v  e.  ( Base `  M ) )  -> 
( v  =  ( 0g `  M )  <-> 
( (/)  e.  Fin  /\  v  =  ( 0g `  M ) ) ) )
3216, 29, 313bitr4d 288 . . . 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 256 . . 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 3078 . 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 2429 . . . 4  |-  ( 0g
`  M )  =  ( 0g `  M
)
362, 35mndidcl 16505 . . 3  |-  ( M  e.  Mnd  ->  ( 0g `  M )  e.  ( Base `  M
) )
37 rabsn 4070 . . 3  |-  ( ( 0g `  M )  e.  ( Base `  M
)  ->  { v  e.  ( Base `  M
)  |  v  =  ( 0g `  M
) }  =  {
( 0g `  M
) } )
3836, 37syl 17 . 2  |-  ( M  e.  Mnd  ->  { v  e.  ( Base `  M
)  |  v  =  ( 0g `  M
) }  =  {
( 0g `  M
) } )
396, 34, 383eqtrd 2474 1  |-  ( M  e.  Mnd  ->  ( M LinCo 
(/) )  =  {
( 0g `  M
) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783   {crab 2786   _Vcvv 3087   (/)c0 3767   ~Pcpw 3985   {csn 4002   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   1oc1o 7183    ^m cmap 7480   Fincfn 7577   finSupp cfsupp 7889   Basecbs 15084  Scalarcsca 15155   0gc0g 15297   Mndcmnd 16486   linC clinc 38966   LinCo clinco 38967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-map 7482  df-en 7578  df-fin 7581  df-fsupp 7890  df-seq 12211  df-0g 15299  df-gsum 15300  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-linc 38968  df-lco 38969
This theorem is referenced by:  lcoel0  38990
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