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Theorem gsumlsscl 30989
Description: Closure of a group sum in a linear subspace: A (finitely supported) sum of scalar multiplications of vectors of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Hypotheses
Ref Expression
gsumlsscl.s  |-  S  =  ( LSubSp `  M )
gsumlsscl.r  |-  R  =  (Scalar `  M )
gsumlsscl.b  |-  B  =  ( Base `  R
)
Assertion
Ref Expression
gsumlsscl  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  (
( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R ) )  -> 
( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) ) )  e.  Z ) )
Distinct variable groups:    v, B    v, F    v, M    v, R    v, S    v, V    v, Z

Proof of Theorem gsumlsscl
StepHypRef Expression
1 eqid 2454 . . 3  |-  ( 0g
`  M )  =  ( 0g `  M
)
2 lmodabl 17118 . . . . 5  |-  ( M  e.  LMod  ->  M  e. 
Abel )
323ad2ant1 1009 . . . 4  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  M  e.  Abel )
43adantr 465 . . 3  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  ->  M  e.  Abel )
5 ssexg 4549 . . . . . 6  |-  ( ( V  C_  Z  /\  Z  e.  S )  ->  V  e.  _V )
65ancoms 453 . . . . 5  |-  ( ( Z  e.  S  /\  V  C_  Z )  ->  V  e.  _V )
763adant1 1006 . . . 4  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  V  e.  _V )
87adantr 465 . . 3  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  ->  V  e.  _V )
9 3simpa 985 . . . . 5  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  ( M  e.  LMod  /\  Z  e.  S ) )
10 gsumlsscl.s . . . . . 6  |-  S  =  ( LSubSp `  M )
1110lsssubg 17164 . . . . 5  |-  ( ( M  e.  LMod  /\  Z  e.  S )  ->  Z  e.  (SubGrp `  M )
)
129, 11syl 16 . . . 4  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  Z  e.  (SubGrp `  M )
)
1312adantr 465 . . 3  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  ->  Z  e.  (SubGrp `  M
) )
149adantr 465 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  -> 
( M  e.  LMod  /\  Z  e.  S ) )
1514adantr 465 . . . . 5  |-  ( ( ( ( M  e. 
LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V
)  /\  F finSupp  ( 0g
`  R ) ) )  /\  v  e.  V )  ->  ( M  e.  LMod  /\  Z  e.  S ) )
16 elmapi 7347 . . . . . . . 8  |-  ( F  e.  ( B  ^m  V )  ->  F : V --> B )
17 ffvelrn 5953 . . . . . . . . 9  |-  ( ( F : V --> B  /\  v  e.  V )  ->  ( F `  v
)  e.  B )
1817ex 434 . . . . . . . 8  |-  ( F : V --> B  -> 
( v  e.  V  ->  ( F `  v
)  e.  B ) )
1916, 18syl 16 . . . . . . 7  |-  ( F  e.  ( B  ^m  V )  ->  (
v  e.  V  -> 
( F `  v
)  e.  B ) )
2019ad2antrl 727 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  -> 
( v  e.  V  ->  ( F `  v
)  e.  B ) )
2120imp 429 . . . . 5  |-  ( ( ( ( M  e. 
LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V
)  /\  F finSupp  ( 0g
`  R ) ) )  /\  v  e.  V )  ->  ( F `  v )  e.  B )
22 ssel 3461 . . . . . . . 8  |-  ( V 
C_  Z  ->  (
v  e.  V  -> 
v  e.  Z ) )
23223ad2ant3 1011 . . . . . . 7  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  (
v  e.  V  -> 
v  e.  Z ) )
2423adantr 465 . . . . . 6  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  -> 
( v  e.  V  ->  v  e.  Z ) )
2524imp 429 . . . . 5  |-  ( ( ( ( M  e. 
LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V
)  /\  F finSupp  ( 0g
`  R ) ) )  /\  v  e.  V )  ->  v  e.  Z )
26 gsumlsscl.r . . . . . 6  |-  R  =  (Scalar `  M )
27 eqid 2454 . . . . . 6  |-  ( .s
`  M )  =  ( .s `  M
)
28 gsumlsscl.b . . . . . 6  |-  B  =  ( Base `  R
)
2926, 27, 28, 10lssvscl 17162 . . . . 5  |-  ( ( ( M  e.  LMod  /\  Z  e.  S )  /\  ( ( F `
 v )  e.  B  /\  v  e.  Z ) )  -> 
( ( F `  v ) ( .s
`  M ) v )  e.  Z )
3015, 21, 25, 29syl12anc 1217 . . . 4  |-  ( ( ( ( M  e. 
LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V
)  /\  F finSupp  ( 0g
`  R ) ) )  /\  v  e.  V )  ->  (
( F `  v
) ( .s `  M ) v )  e.  Z )
31 eqid 2454 . . . 4  |-  ( v  e.  V  |->  ( ( F `  v ) ( .s `  M
) v ) )  =  ( v  e.  V  |->  ( ( F `
 v ) ( .s `  M ) v ) )
3230, 31fmptd 5979 . . 3  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  -> 
( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) ) : V --> Z )
33 simp1 988 . . . . . 6  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  M  e.  LMod )
34 eqid 2454 . . . . . . . . . . 11  |-  ( Base `  M )  =  (
Base `  M )
3534, 10lssss 17144 . . . . . . . . . 10  |-  ( Z  e.  S  ->  Z  C_  ( Base `  M
) )
36 sstr 3475 . . . . . . . . . . 11  |-  ( ( V  C_  Z  /\  Z  C_  ( Base `  M
) )  ->  V  C_  ( Base `  M
) )
3736expcom 435 . . . . . . . . . 10  |-  ( Z 
C_  ( Base `  M
)  ->  ( V  C_  Z  ->  V  C_  ( Base `  M ) ) )
3835, 37syl 16 . . . . . . . . 9  |-  ( Z  e.  S  ->  ( V  C_  Z  ->  V  C_  ( Base `  M
) ) )
3938a1i 11 . . . . . . . 8  |-  ( M  e.  LMod  ->  ( Z  e.  S  ->  ( V  C_  Z  ->  V  C_  ( Base `  M
) ) ) )
40393imp 1182 . . . . . . 7  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  V  C_  ( Base `  M
) )
41 elpwg 3979 . . . . . . . 8  |-  ( V  e.  _V  ->  ( V  e.  ~P ( Base `  M )  <->  V  C_  ( Base `  M ) ) )
427, 41syl 16 . . . . . . 7  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  ( V  e.  ~P ( Base `  M )  <->  V  C_  ( Base `  M ) ) )
4340, 42mpbird 232 . . . . . 6  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  V  e.  ~P ( Base `  M
) )
4433, 43jca 532 . . . . 5  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) ) )
4544adantr 465 . . . 4  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  -> 
( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) ) )
46 simprl 755 . . . 4  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  ->  F  e.  ( B  ^m  V ) )
47 simprr 756 . . . 4  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  ->  F finSupp  ( 0g `  R
) )
4826, 28scmfsupp 30960 . . . 4  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R ) )  -> 
( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) ) finSupp  ( 0g
`  M ) )
4945, 46, 47, 48syl3anc 1219 . . 3  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  -> 
( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) ) finSupp  ( 0g
`  M ) )
501, 4, 8, 13, 32, 49gsumsubgcl 16530 . 2  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  -> 
( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) ) )  e.  Z )
5150ex 434 1  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  (
( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R ) )  -> 
( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) ) )  e.  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3439   ~Pcpw 3971   class class class wbr 4403    |-> cmpt 4461   -->wf 5525   ` cfv 5529  (class class class)co 6203    ^m cmap 7327   finSupp cfsupp 7734   Basecbs 14295  Scalarcsca 14363   .scvsca 14364   0gc0g 14500    gsumg cgsu 14501  SubGrpcsubg 15797   Abelcabel 16402   LModclmod 17074   LSubSpclss 17139
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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
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 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-seq 11927  df-hash 12224  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-0g 14502  df-gsum 14503  df-mnd 15537  df-submnd 15587  df-grp 15667  df-minusg 15668  df-sbg 15669  df-subg 15800  df-cntz 15957  df-cmn 16403  df-abl 16404  df-mgp 16717  df-ur 16729  df-rng 16773  df-lmod 17076  df-lss 17140
This theorem is referenced by:  lincellss  31112
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