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Theorem lincellss 39493
Description: A linear combination 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.)
Assertion
Ref Expression
lincellss  |-  ( ( 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
) )

Proof of Theorem lincellss
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 simpl1 1008 . . . 4  |-  ( ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  ( F  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
)  /\  F finSupp  ( 0g
`  (Scalar `  M )
) ) )  ->  M  e.  LMod )
2 simprl 762 . . . 4  |-  ( ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  ( F  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
)  /\  F finSupp  ( 0g
`  (Scalar `  M )
) ) )  ->  F  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
) )
3 ssexg 4567 . . . . . . . 8  |-  ( ( V  C_  S  /\  S  e.  ( LSubSp `  M ) )  ->  V  e.  _V )
43ancoms 454 . . . . . . 7  |-  ( ( S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  V  e.  _V )
5 eqid 2422 . . . . . . . . . 10  |-  ( Base `  M )  =  (
Base `  M )
6 eqid 2422 . . . . . . . . . 10  |-  ( LSubSp `  M )  =  (
LSubSp `  M )
75, 6lssss 18148 . . . . . . . . 9  |-  ( S  e.  ( LSubSp `  M
)  ->  S  C_  ( Base `  M ) )
8 sstr 3472 . . . . . . . . . . 11  |-  ( ( V  C_  S  /\  S  C_  ( Base `  M
) )  ->  V  C_  ( Base `  M
) )
9 elpwg 3987 . . . . . . . . . . 11  |-  ( V  e.  _V  ->  ( V  e.  ~P ( Base `  M )  <->  V  C_  ( Base `  M ) ) )
108, 9syl5ibrcom 225 . . . . . . . . . 10  |-  ( ( V  C_  S  /\  S  C_  ( Base `  M
) )  ->  ( V  e.  _V  ->  V  e.  ~P ( Base `  M ) ) )
1110expcom 436 . . . . . . . . 9  |-  ( S 
C_  ( Base `  M
)  ->  ( V  C_  S  ->  ( V  e.  _V  ->  V  e.  ~P ( Base `  M
) ) ) )
127, 11syl 17 . . . . . . . 8  |-  ( S  e.  ( LSubSp `  M
)  ->  ( V  C_  S  ->  ( V  e.  _V  ->  V  e.  ~P ( Base `  M
) ) ) )
1312imp 430 . . . . . . 7  |-  ( ( S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( V  e.  _V  ->  V  e.  ~P ( Base `  M ) ) )
144, 13mpd 15 . . . . . 6  |-  ( ( S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  V  e.  ~P ( Base `  M
) )
15143adant1 1023 . . . . 5  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  V  e.  ~P ( Base `  M )
)
1615adantr 466 . . . 4  |-  ( ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  ( F  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
)  /\  F finSupp  ( 0g
`  (Scalar `  M )
) ) )  ->  V  e.  ~P ( Base `  M ) )
17 lincval 39476 . . . 4  |-  ( ( M  e.  LMod  /\  F  e.  ( ( Base `  (Scalar `  M ) )  ^m  V )  /\  V  e.  ~P ( Base `  M
) )  ->  ( F ( linC  `  M ) V )  =  ( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) ) ) )
181, 2, 16, 17syl3anc 1264 . . 3  |-  ( ( ( 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 )  =  ( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s `  M
) v ) ) ) )
19 eqid 2422 . . . . 5  |-  (Scalar `  M )  =  (Scalar `  M )
20 eqid 2422 . . . . 5  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
216, 19, 20gsumlsscl 39442 . . . 4  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( ( F  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )  /\  F finSupp  ( 0g `  (Scalar `  M ) ) )  ->  ( M  gsumg  ( v  e.  V  |->  ( ( F `  v
) ( .s `  M ) v ) ) )  e.  S
) )
2221imp 430 . . 3  |-  ( ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  ( F  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
)  /\  F finSupp  ( 0g
`  (Scalar `  M )
) ) )  -> 
( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) ) )  e.  S )
2318, 22eqeltrd 2510 . 2  |-  ( ( ( 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 )
2423ex 435 1  |-  ( ( 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
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   _Vcvv 3081    C_ wss 3436   ~Pcpw 3979   class class class wbr 4420    |-> cmpt 4479   ` cfv 5598  (class class class)co 6302    ^m cmap 7477   finSupp cfsupp 7886   Basecbs 15109  Scalarcsca 15181   .scvsca 15182   0gc0g 15326    gsumg cgsu 15327   LModclmod 18079   LSubSpclss 18143   linC clinc 39471
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 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
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 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-2 10669  df-n0 10871  df-z 10939  df-uz 11161  df-fz 11786  df-fzo 11917  df-seq 12214  df-hash 12516  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-0g 15328  df-gsum 15329  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-grp 16661  df-minusg 16662  df-sbg 16663  df-subg 16802  df-cntz 16959  df-cmn 17420  df-abl 17421  df-mgp 17712  df-ur 17724  df-ring 17770  df-lmod 18081  df-lss 18144  df-linc 39473
This theorem is referenced by:  ellcoellss  39502
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