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Theorem lcoss 38988
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 3996 . . . . . . 7  |-  ( ( v  e.  V  /\  V  e.  ~P ( Base `  M ) )  ->  v  e.  (
Base `  M )
)
21expcom 436 . . . . . 6  |-  ( V  e.  ~P ( Base `  M )  ->  (
v  e.  V  -> 
v  e.  ( Base `  M ) ) )
32adantl 467 . . . . 5  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  (
v  e.  V  -> 
v  e.  ( Base `  M ) ) )
43imp 430 . . . 4  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  v  e.  ( Base `  M )
)
5 eqid 2429 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
6 eqid 2429 . . . . . . 7  |-  (Scalar `  M )  =  (Scalar `  M )
7 eqid 2429 . . . . . . 7  |-  ( 0g
`  (Scalar `  M )
)  =  ( 0g
`  (Scalar `  M )
)
8 eqid 2429 . . . . . . 7  |-  ( 1r
`  (Scalar `  M )
)  =  ( 1r
`  (Scalar `  M )
)
9 equequ1 1850 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  =  v  <->  y  =  v ) )
109ifbid 3937 . . . . . . . 8  |-  ( x  =  y  ->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) )  =  if ( y  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) ) )
1110cbvmptv 4518 . . . . . . 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 38924 . . . . . 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 1205 . . . . 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 2429 . . . . . . . 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 38977 . . . . . . 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 1205 . . . . . 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 2437 . . . . 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 2429 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
196, 18, 8lmod1cl 18053 . . . . . . . . . 10  |-  ( M  e.  LMod  ->  ( 1r
`  (Scalar `  M )
)  e.  ( Base `  (Scalar `  M )
) )
206, 18, 7lmod0cl 18052 . . . . . . . . . 10  |-  ( M  e.  LMod  ->  ( 0g
`  (Scalar `  M )
)  e.  ( Base `  (Scalar `  M )
) )
2119, 20ifcld 3958 . . . . . . . . 9  |-  ( M  e.  LMod  ->  if ( x  =  v ,  ( 1r `  (Scalar `  M ) ) ,  ( 0g `  (Scalar `  M ) ) )  e.  ( Base `  (Scalar `  M ) ) )
2221ad3antrrr 734 . . . . . . . 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 6061 . . . . . . 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 5891 . . . . . . . 8  |-  ( Base `  (Scalar `  M )
)  e.  _V
25 simplr 760 . . . . . . . 8  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  V  e.  ~P ( Base `  M
) )
26 elmapg 7493 . . . . . . . 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 667 . . . . . . 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 235 . . . . . 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 4429 . . . . . . . 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 6312 . . . . . . . . 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 2443 . . . . . . . 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 715 . . . . . . 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 467 . . . . . 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 3192 . . . . 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 683 . . . 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 38964 . . . . 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 466 . . . 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 930 . . 3  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  v  e.  V
)  ->  v  e.  ( M LinCo  V ) )
3938ex 435 . 2  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  (
v  e.  V  -> 
v  e.  ( M LinCo 
V ) ) )
4039ssrdv 3476 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 187    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783   _Vcvv 3087    C_ wss 3442   ifcif 3915   ~Pcpw 3985   class class class wbr 4426    |-> cmpt 4484   -->wf 5597   ` cfv 5601  (class class class)co 6305    ^m cmap 7480   finSupp cfsupp 7889   Basecbs 15084  Scalarcsca 15155   0gc0g 15297   1rcur 17670   LModclmod 18026   linC clinc 38956   LinCo clinco 38957
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  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
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-nel 2628  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-int 4259  df-iun 4304  df-iin 4305  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-se 4814  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-isom 5610  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-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-0g 15299  df-gsum 15300  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-mgp 17659  df-ur 17671  df-ring 17717  df-lmod 18028  df-linc 38958  df-lco 38959
This theorem is referenced by:  lspsslco  38989
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