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Theorem ellcoellss 31102
Description: Every linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Assertion
Ref Expression
ellcoellss  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  A. x  e.  ( M LinCo  V ) x  e.  S )
Distinct variable groups:    x, M    x, S    x, V

Proof of Theorem ellcoellss
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 simp1 988 . . . 4  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  M  e.  LMod )
2 eqid 2454 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2454 . . . . . . 7  |-  ( LSubSp `  M )  =  (
LSubSp `  M )
42, 3lssss 17142 . . . . . 6  |-  ( S  e.  ( LSubSp `  M
)  ->  S  C_  ( Base `  M ) )
543ad2ant2 1010 . . . . 5  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  S  C_  ( Base `  M ) )
6 sstr 3473 . . . . . . . 8  |-  ( ( V  C_  S  /\  S  C_  ( Base `  M
) )  ->  V  C_  ( Base `  M
) )
7 fvex 5810 . . . . . . . . . 10  |-  ( Base `  M )  e.  _V
87ssex 4545 . . . . . . . . 9  |-  ( V 
C_  ( Base `  M
)  ->  V  e.  _V )
9 elpwg 3977 . . . . . . . . . 10  |-  ( V  e.  _V  ->  ( V  e.  ~P ( Base `  M )  <->  V  C_  ( Base `  M ) ) )
109biimprd 223 . . . . . . . . 9  |-  ( V  e.  _V  ->  ( V  C_  ( Base `  M
)  ->  V  e.  ~P ( Base `  M
) ) )
118, 10mpcom 36 . . . . . . . 8  |-  ( V 
C_  ( Base `  M
)  ->  V  e.  ~P ( Base `  M
) )
126, 11syl 16 . . . . . . 7  |-  ( ( V  C_  S  /\  S  C_  ( Base `  M
) )  ->  V  e.  ~P ( Base `  M
) )
1312ex 434 . . . . . 6  |-  ( V 
C_  S  ->  ( S  C_  ( Base `  M
)  ->  V  e.  ~P ( Base `  M
) ) )
14133ad2ant3 1011 . . . . 5  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( S  C_  ( Base `  M )  ->  V  e.  ~P ( Base `  M ) ) )
155, 14mpd 15 . . . 4  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  V  e.  ~P ( Base `  M )
)
16 eqid 2454 . . . . 5  |-  (Scalar `  M )  =  (Scalar `  M )
17 eqid 2454 . . . . 5  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
182, 16, 17lcoval 31079 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  (
x  e.  ( M LinCo 
V )  <->  ( x  e.  ( Base `  M
)  /\  E. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  x  =  ( f
( linC  `  M ) V ) ) ) ) )
191, 15, 18syl2anc 661 . . 3  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( x  e.  ( M LinCo  V )  <-> 
( x  e.  (
Base `  M )  /\  E. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  x  =  ( f
( linC  `  M ) V ) ) ) ) )
20 lincellss 31093 . . . . . . . . . . . . 13  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( ( f  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )  /\  f finSupp  ( 0g `  (Scalar `  M ) ) )  ->  ( f
( linC  `  M ) V )  e.  S
) )
2120imp 429 . . . . . . . . . . . 12  |-  ( ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  (
f  e.  ( (
Base `  (Scalar `  M
) )  ^m  V
)  /\  f finSupp  ( 0g
`  (Scalar `  M )
) ) )  -> 
( f ( linC  `  M ) V )  e.  S )
22 eleq1 2526 . . . . . . . . . . . 12  |-  ( x  =  ( f ( linC  `  M ) V )  ->  ( x  e.  S  <->  ( f ( linC  `  M ) V )  e.  S ) )
2321, 22syl5ibr 221 . . . . . . . . . . 11  |-  ( x  =  ( f ( linC  `  M ) V )  ->  ( ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  ( f  e.  ( ( Base `  (Scalar `  M ) )  ^m  V )  /\  f finSupp  ( 0g `  (Scalar `  M ) ) ) )  ->  x  e.  S ) )
2423expd 436 . . . . . . . . . 10  |-  ( x  =  ( f ( linC  `  M ) V )  ->  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( ( f  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )  /\  f finSupp  ( 0g `  (Scalar `  M ) ) )  ->  x  e.  S ) ) )
2524com12 31 . . . . . . . . 9  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( x  =  ( f ( linC  `  M ) V )  ->  ( ( f  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )  /\  f finSupp  ( 0g `  (Scalar `  M ) ) )  ->  x  e.  S ) ) )
2625adantr 465 . . . . . . . 8  |-  ( ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  x  e.  ( Base `  M
) )  ->  (
x  =  ( f ( linC  `  M ) V )  ->  (
( f  e.  ( ( Base `  (Scalar `  M ) )  ^m  V )  /\  f finSupp  ( 0g `  (Scalar `  M ) ) )  ->  x  e.  S
) ) )
2726com13 80 . . . . . . 7  |-  ( ( f  e.  ( (
Base `  (Scalar `  M
) )  ^m  V
)  /\  f finSupp  ( 0g
`  (Scalar `  M )
) )  ->  (
x  =  ( f ( linC  `  M ) V )  ->  (
( ( M  e. 
LMod  /\  S  e.  (
LSubSp `  M )  /\  V  C_  S )  /\  x  e.  ( Base `  M ) )  ->  x  e.  S )
) )
2827impr 619 . . . . . 6  |-  ( ( f  e.  ( (
Base `  (Scalar `  M
) )  ^m  V
)  /\  ( f finSupp  ( 0g `  (Scalar `  M ) )  /\  x  =  ( f
( linC  `  M ) V ) ) )  ->  ( ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  x  e.  (
Base `  M )
)  ->  x  e.  S ) )
2928rexlimiva 2942 . . . . 5  |-  ( E. f  e.  ( (
Base `  (Scalar `  M
) )  ^m  V
) ( f finSupp  ( 0g `  (Scalar `  M
) )  /\  x  =  ( f ( linC  `  M ) V ) )  ->  ( (
( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  x  e.  ( Base `  M
) )  ->  x  e.  S ) )
3029com12 31 . . . 4  |-  ( ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  x  e.  ( Base `  M
) )  ->  ( E. f  e.  (
( Base `  (Scalar `  M
) )  ^m  V
) ( f finSupp  ( 0g `  (Scalar `  M
) )  /\  x  =  ( f ( linC  `  M ) V ) )  ->  x  e.  S ) )
3130expimpd 603 . . 3  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( ( x  e.  ( Base `  M
)  /\  E. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  x  =  ( f
( linC  `  M ) V ) ) )  ->  x  e.  S
) )
3219, 31sylbid 215 . 2  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( x  e.  ( M LinCo  V )  ->  x  e.  S
) )
3332ralrimiv 2828 1  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  A. x  e.  ( M LinCo  V ) x  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800   _Vcvv 3078    C_ wss 3437   ~Pcpw 3969   class class class wbr 4401   ` cfv 5527  (class class class)co 6201    ^m cmap 7325   finSupp cfsupp 7732   Basecbs 14293  Scalarcsca 14361   0gc0g 14498   LModclmod 17072   LSubSpclss 17137   linC clinc 31071   LinCo clinco 31072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-seq 11925  df-hash 12222  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-0g 14500  df-gsum 14501  df-mnd 15535  df-submnd 15585  df-grp 15665  df-minusg 15666  df-sbg 15667  df-subg 15798  df-cntz 15955  df-cmn 16401  df-abl 16402  df-mgp 16715  df-ur 16727  df-rng 16771  df-lmod 17074  df-lss 17138  df-linc 31073  df-lco 31074
This theorem is referenced by:  lcosslsp  31105
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