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Theorem gsumlsscl 38420
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 2400 . . 3  |-  ( 0g
`  M )  =  ( 0g `  M
)
2 lmodabl 17767 . . . . 5  |-  ( M  e.  LMod  ->  M  e. 
Abel )
323ad2ant1 1016 . . . 4  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  M  e.  Abel )
43adantr 463 . . 3  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  ->  M  e.  Abel )
5 ssexg 4537 . . . . . 6  |-  ( ( V  C_  Z  /\  Z  e.  S )  ->  V  e.  _V )
65ancoms 451 . . . . 5  |-  ( ( Z  e.  S  /\  V  C_  Z )  ->  V  e.  _V )
763adant1 1013 . . . 4  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  V  e.  _V )
87adantr 463 . . 3  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  ->  V  e.  _V )
9 3simpa 992 . . . . 5  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  ( M  e.  LMod  /\  Z  e.  S ) )
10 gsumlsscl.s . . . . . 6  |-  S  =  ( LSubSp `  M )
1110lsssubg 17813 . . . . 5  |-  ( ( M  e.  LMod  /\  Z  e.  S )  ->  Z  e.  (SubGrp `  M )
)
129, 11syl 17 . . . 4  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  Z  e.  (SubGrp `  M )
)
1312adantr 463 . . 3  |-  ( ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  /\  ( F  e.  ( B  ^m  V )  /\  F finSupp  ( 0g `  R
) ) )  ->  Z  e.  (SubGrp `  M
) )
149adantr 463 . . . . . 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 463 . . . . 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 7396 . . . . . . . 8  |-  ( F  e.  ( B  ^m  V )  ->  F : V --> B )
17 ffvelrn 5961 . . . . . . . . 9  |-  ( ( F : V --> B  /\  v  e.  V )  ->  ( F `  v
)  e.  B )
1817ex 432 . . . . . . . 8  |-  ( F : V --> B  -> 
( v  e.  V  ->  ( F `  v
)  e.  B ) )
1916, 18syl 17 . . . . . . 7  |-  ( F  e.  ( B  ^m  V )  ->  (
v  e.  V  -> 
( F `  v
)  e.  B ) )
2019ad2antrl 726 . . . . . 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 427 . . . . 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 3433 . . . . . . . 8  |-  ( V 
C_  Z  ->  (
v  e.  V  -> 
v  e.  Z ) )
23223ad2ant3 1018 . . . . . . 7  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  (
v  e.  V  -> 
v  e.  Z ) )
2423adantr 463 . . . . . 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 427 . . . . 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 2400 . . . . . 6  |-  ( .s
`  M )  =  ( .s `  M
)
28 gsumlsscl.b . . . . . 6  |-  B  =  ( Base `  R
)
2926, 27, 28, 10lssvscl 17811 . . . . 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 1226 . . . 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 2400 . . . 4  |-  ( v  e.  V  |->  ( ( F `  v ) ( .s `  M
) v ) )  =  ( v  e.  V  |->  ( ( F `
 v ) ( .s `  M ) v ) )
3230, 31fmptd 5987 . . 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 995 . . . . . 6  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  M  e.  LMod )
34 eqid 2400 . . . . . . . . . . 11  |-  ( Base `  M )  =  (
Base `  M )
3534, 10lssss 17793 . . . . . . . . . 10  |-  ( Z  e.  S  ->  Z  C_  ( Base `  M
) )
36 sstr 3447 . . . . . . . . . . 11  |-  ( ( V  C_  Z  /\  Z  C_  ( Base `  M
) )  ->  V  C_  ( Base `  M
) )
3736expcom 433 . . . . . . . . . 10  |-  ( Z 
C_  ( Base `  M
)  ->  ( V  C_  Z  ->  V  C_  ( Base `  M ) ) )
3835, 37syl 17 . . . . . . . . 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 1189 . . . . . . 7  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  V  C_  ( Base `  M
) )
41 elpwg 3960 . . . . . . . 8  |-  ( V  e.  _V  ->  ( V  e.  ~P ( Base `  M )  <->  V  C_  ( Base `  M ) ) )
427, 41syl 17 . . . . . . 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 530 . . . . 5  |-  ( ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) ) )
4544adantr 463 . . . 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 38415 . . . 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 1228 . . 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 17146 . 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 432 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 367    /\ w3a 972    = wceq 1403    e. wcel 1840   _Vcvv 3056    C_ wss 3411   ~Pcpw 3952   class class class wbr 4392    |-> cmpt 4450   -->wf 5519   ` cfv 5523  (class class class)co 6232    ^m cmap 7375   finSupp cfsupp 7781   Basecbs 14731  Scalarcsca 14802   .scvsca 14803   0gc0g 14944    gsumg cgsu 14945  SubGrpcsubg 16409   Abelcabl 17013   LModclmod 17722   LSubSpclss 17788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-supp 6855  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-fsupp 7782  df-oi 7887  df-card 8270  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-n0 10755  df-z 10824  df-uz 11044  df-fz 11642  df-fzo 11766  df-seq 12060  df-hash 12358  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-0g 14946  df-gsum 14947  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-submnd 16181  df-grp 16271  df-minusg 16272  df-sbg 16273  df-subg 16412  df-cntz 16569  df-cmn 17014  df-abl 17015  df-mgp 17352  df-ur 17364  df-ring 17410  df-lmod 17724  df-lss 17789
This theorem is referenced by:  lincellss  38471
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