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Theorem lcoss 32127
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 4021 . . . . . . 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 2467 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
6 eqid 2467 . . . . . . 7  |-  (Scalar `  M )  =  (Scalar `  M )
7 eqid 2467 . . . . . . 7  |-  ( 0g
`  (Scalar `  M )
)  =  ( 0g
`  (Scalar `  M )
)
8 eqid 2467 . . . . . . 7  |-  ( 1r
`  (Scalar `  M )
)  =  ( 1r
`  (Scalar `  M )
)
9 equequ1 1747 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  =  v  <->  y  =  v ) )
109ifbid 3961 . . . . . . . 8  |-  ( x  =  y  ->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) )  =  if ( y  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )
1110cbvmptv 4538 . . . . . . 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 32063 . . . . . 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 1196 . . . . 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 2467 . . . . . . . 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 32116 . . . . . . 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 1196 . . . . . 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 2475 . . . . 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 2467 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
196, 18, 8lmod1cl 17334 . . . . . . . . . 10  |-  ( M  e.  LMod  ->  ( 1r
`  (Scalar `  M )
)  e.  ( Base `  (Scalar `  M )
) )
206, 18, 7lmod0cl 17333 . . . . . . . . . 10  |-  ( M  e.  LMod  ->  ( 0g
`  (Scalar `  M )
)  e.  ( Base `  (Scalar `  M )
) )
2119, 20ifcld 3982 . . . . . . . . 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 6044 . . . . . . 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 5875 . . . . . . . 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 7433 . . . . . . . 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 4450 . . . . . . . 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 6290 . . . . . . . . 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 2481 . . . . . . . 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 3218 . . . . 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 32103 . . . . 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 920 . . 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 3510 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 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113    C_ wss 3476   ifcif 3939   ~Pcpw 4010   class class class wbr 4447    |-> cmpt 4505   -->wf 5583   ` cfv 5587  (class class class)co 6283    ^m cmap 7420   finSupp cfsupp 7828   Basecbs 14489  Scalarcsca 14557   0gc0g 14694   1rcur 16952   LModclmod 17307   linC clinc 32095   LinCo clinco 32096
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 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
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-nel 2665  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-int 4283  df-iun 4327  df-iin 4328  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-se 4839  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 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-oi 7934  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-fzo 11792  df-seq 12075  df-hash 12373  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-0g 14696  df-gsum 14697  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-grp 15864  df-mulg 15867  df-cntz 16157  df-cmn 16603  df-mgp 16941  df-ur 16953  df-rng 16997  df-lmod 17309  df-linc 32097  df-lco 32098
This theorem is referenced by:  lspsslco  32128
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