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Theorem lcoss 30970
Description: A set of vectors of a module is a subset of the set of all linear combinations of the set. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Assertion
Ref Expression
lcoss  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  V  C_  ( M LinCo  V ) )

Proof of Theorem lcoss
Dummy variables  f 
v  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elelpwi 3871 . . . . . . 7  |-  ( ( v  e.  V  /\  V  e.  ~P ( Base `  M ) )  ->  v  e.  (
Base `  M )
)
21expcom 435 . . . . . 6  |-  ( V  e.  ~P ( Base `  M )  ->  (
v  e.  V  -> 
v  e.  ( Base `  M ) ) )
32adantl 466 . . . . 5  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  (
v  e.  V  -> 
v  e.  ( Base `  M ) ) )
43imp 429 . . . 4  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  v  e.  ( Base `  M )
)
5 eqid 2443 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
6 eqid 2443 . . . . . . 7  |-  (Scalar `  M )  =  (Scalar `  M )
7 eqid 2443 . . . . . . 7  |-  ( 0g
`  (Scalar `  M )
)  =  ( 0g
`  (Scalar `  M )
)
8 eqid 2443 . . . . . . 7  |-  ( 1r
`  (Scalar `  M )
)  =  ( 1r
`  (Scalar `  M )
)
9 equequ1 1736 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  =  v  <->  y  =  v ) )
109ifbid 3811 . . . . . . . 8  |-  ( x  =  y  ->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) )  =  if ( y  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )
1110cbvmptv 4383 . . . . . . 7  |-  ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  =  ( y  e.  V  |->  if ( y  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )
125, 6, 7, 8, 11mptcfsupp 30794 . . . . . 6  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
)  /\  v  e.  V )  ->  (
x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) finSupp  ( 0g
`  (Scalar `  M )
) )
13123expa 1187 . . . . 5  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) finSupp  ( 0g `  (Scalar `  M ) ) )
14 eqid 2443 . . . . . . . 8  |-  ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  =  ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )
155, 6, 7, 8, 14linc1 30959 . . . . . . 7  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
)  /\  v  e.  V )  ->  (
( x  e.  V  |->  if ( x  =  v ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) ( linC  `  M ) V )  =  v )
16153expa 1187 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  ( (
x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) ( linC  `  M ) V )  =  v )
1716eqcomd 2448 . . . . 5  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  v  =  ( ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) ( linC  `  M
) V ) )
18 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
196, 18, 8lmod1cl 16975 . . . . . . . . . 10  |-  ( M  e.  LMod  ->  ( 1r
`  (Scalar `  M )
)  e.  ( Base `  (Scalar `  M )
) )
206, 18, 7lmod0cl 16974 . . . . . . . . . 10  |-  ( M  e.  LMod  ->  ( 0g
`  (Scalar `  M )
)  e.  ( Base `  (Scalar `  M )
) )
2119, 20ifcld 3832 . . . . . . . . 9  |-  ( M  e.  LMod  ->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) )  e.  ( Base `  (Scalar `  M ) ) )
2221ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( M  e. 
LMod  /\  V  e.  ~P ( Base `  M )
)  /\  v  e.  V )  /\  x  e.  V )  ->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) )  e.  ( Base `  (Scalar `  M )
) )
2322, 14fmptd 5867 . . . . . . 7  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) : V --> ( Base `  (Scalar `  M )
) )
24 fvex 5701 . . . . . . . 8  |-  ( Base `  (Scalar `  M )
)  e.  _V
25 simplr 754 . . . . . . . 8  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  V  e.  ~P ( Base `  M
) )
26 elmapg 7227 . . . . . . . 8  |-  ( ( ( Base `  (Scalar `  M ) )  e. 
_V  /\  V  e.  ~P ( Base `  M
) )  ->  (
( x  e.  V  |->  if ( x  =  v ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) )  e.  ( ( Base `  (Scalar `  M ) )  ^m  V )  <->  ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) : V --> ( Base `  (Scalar `  M )
) ) )
2724, 25, 26sylancr 663 . . . . . . 7  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  ( (
x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  e.  ( ( Base `  (Scalar `  M ) )  ^m  V )  <->  ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) : V --> ( Base `  (Scalar `  M )
) ) )
2823, 27mpbird 232 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  e.  ( (
Base `  (Scalar `  M
) )  ^m  V
) )
29 breq1 4295 . . . . . . . 8  |-  ( f  =  ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  ->  ( f finSupp  ( 0g `  (Scalar `  M ) )  <->  ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) finSupp  ( 0g `  (Scalar `  M ) ) ) )
30 oveq1 6098 . . . . . . . . 9  |-  ( f  =  ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  ->  ( f
( linC  `  M ) V )  =  ( ( x  e.  V  |->  if ( x  =  v ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) ( linC  `  M ) V ) )
3130eqeq2d 2454 . . . . . . . 8  |-  ( f  =  ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  ->  ( v  =  ( f ( linC  `  M ) V )  <-> 
v  =  ( ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) ( linC  `  M ) V ) ) )
3229, 31anbi12d 710 . . . . . . 7  |-  ( f  =  ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )  ->  ( (
f finSupp  ( 0g `  (Scalar `  M ) )  /\  v  =  ( f
( linC  `  M ) V ) )  <->  ( (
x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) finSupp  ( 0g
`  (Scalar `  M )
)  /\  v  =  ( ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) ( linC  `  M
) V ) ) ) )
3332adantl 466 . . . . . 6  |-  ( ( ( ( M  e. 
LMod  /\  V  e.  ~P ( Base `  M )
)  /\  v  e.  V )  /\  f  =  ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) )  ->  (
( f finSupp  ( 0g `  (Scalar `  M )
)  /\  v  =  ( f ( linC  `  M ) V ) )  <->  ( ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) finSupp  ( 0g `  (Scalar `  M ) )  /\  v  =  ( ( x  e.  V  |->  if ( x  =  v ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) ( linC  `  M ) V ) ) ) )
3428, 33rspcedv 3077 . . . . 5  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  ( (
( x  e.  V  |->  if ( x  =  v ,  ( 1r
`  (Scalar `  M )
) ,  ( 0g
`  (Scalar `  M )
) ) ) finSupp  ( 0g `  (Scalar `  M
) )  /\  v  =  ( ( x  e.  V  |->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) ) ( linC  `  M
) V ) )  ->  E. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  v  =  ( f
( linC  `  M ) V ) ) ) )
3513, 17, 34mp2and 679 . . . 4  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  E. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  v  =  ( f
( linC  `  M ) V ) ) )
365, 6, 18lcoval 30946 . . . . 5  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  (
v  e.  ( M LinCo 
V )  <->  ( v  e.  ( Base `  M
)  /\  E. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  v  =  ( f
( linC  `  M ) V ) ) ) ) )
3736adantr 465 . . . 4  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  ( v  e.  ( M LinCo  V )  <-> 
( v  e.  (
Base `  M )  /\  E. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  v  =  ( f
( linC  `  M ) V ) ) ) ) )
384, 35, 37mpbir2and 913 . . 3  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  v  e.  ( M LinCo  V ) )
3938ex 434 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  (
v  e.  V  -> 
v  e.  ( M LinCo 
V ) ) )
4039ssrdv 3362 1  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  V  C_  ( M LinCo  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716   _Vcvv 2972    C_ wss 3328   ifcif 3791   ~Pcpw 3860   class class class wbr 4292    e. cmpt 4350   -->wf 5414   ` cfv 5418  (class class class)co 6091    ^m cmap 7214   finSupp cfsupp 7620   Basecbs 14174  Scalarcsca 14241   0gc0g 14378   1rcur 16603   LModclmod 16948   linC clinc 30938   LinCo clinco 30939
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-0g 14380  df-gsum 14381  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-grp 15545  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-mgp 16592  df-ur 16604  df-rng 16647  df-lmod 16950  df-linc 30940  df-lco 30941
This theorem is referenced by:  lspsslco  30971
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