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Theorem lincfsuppcl 30788
Description: A linear combination of vectors (with finite support) is a vector. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Hypotheses
Ref Expression
lincfsuppcl.b  |-  B  =  ( Base `  M
)
lincfsuppcl.r  |-  R  =  (Scalar `  M )
lincfsuppcl.s  |-  S  =  ( Base `  R
)
lincfsuppcl.0  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
lincfsuppcl  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  -> 
( F ( linC  `  M ) V )  e.  B )

Proof of Theorem lincfsuppcl
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 simp1 983 . . 3  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  ->  M  e.  LMod )
2 lincfsuppcl.s . . . . . . . . 9  |-  S  =  ( Base `  R
)
3 lincfsuppcl.r . . . . . . . . . 10  |-  R  =  (Scalar `  M )
43fveq2i 5691 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  (Scalar `  M
) )
52, 4eqtri 2461 . . . . . . . 8  |-  S  =  ( Base `  (Scalar `  M ) )
65oveq1i 6100 . . . . . . 7  |-  ( S  ^m  V )  =  ( ( Base `  (Scalar `  M ) )  ^m  V )
76eleq2i 2505 . . . . . 6  |-  ( F  e.  ( S  ^m  V )  <->  F  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) )
87biimpi 194 . . . . 5  |-  ( F  e.  ( S  ^m  V )  ->  F  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) )
98adantr 462 . . . 4  |-  ( ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  )  ->  F  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) )
1093ad2ant3 1006 . . 3  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  ->  F  e.  ( ( Base `  (Scalar `  M
) )  ^m  V
) )
11 elpwg 3865 . . . . . 6  |-  ( V  e.  W  ->  ( V  e.  ~P ( Base `  M )  <->  V  C_  ( Base `  M ) ) )
12 lincfsuppcl.b . . . . . . . . 9  |-  B  =  ( Base `  M
)
1312a1i 11 . . . . . . . 8  |-  ( V  e.  W  ->  B  =  ( Base `  M
) )
1413eqcomd 2446 . . . . . . 7  |-  ( V  e.  W  ->  ( Base `  M )  =  B )
1514sseq2d 3381 . . . . . 6  |-  ( V  e.  W  ->  ( V  C_  ( Base `  M
)  <->  V  C_  B ) )
1611, 15bitr2d 254 . . . . 5  |-  ( V  e.  W  ->  ( V  C_  B  <->  V  e.  ~P ( Base `  M
) ) )
1716biimpa 481 . . . 4  |-  ( ( V  e.  W  /\  V  C_  B )  ->  V  e.  ~P ( Base `  M ) )
18173ad2ant2 1005 . . 3  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  ->  V  e.  ~P ( Base `  M ) )
19 lincval 30784 . . 3  |-  ( ( 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 ) ) ) )
201, 10, 18, 19syl3anc 1213 . 2  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  -> 
( F ( linC  `  M ) V )  =  ( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s `  M
) v ) ) ) )
21 eqid 2441 . . 3  |-  ( 0g
`  M )  =  ( 0g `  M
)
22 lmodcmn 16973 . . . 4  |-  ( M  e.  LMod  ->  M  e. CMnd
)
23223ad2ant1 1004 . . 3  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  ->  M  e. CMnd )
24 simpl 454 . . . 4  |-  ( ( V  e.  W  /\  V  C_  B )  ->  V  e.  W )
25243ad2ant2 1005 . . 3  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  ->  V  e.  W )
261adantr 462 . . . . 5  |-  ( ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V
)  /\  F finSupp  .0.  )
)  /\  v  e.  V )  ->  M  e.  LMod )
27 elmapi 7230 . . . . . . . . 9  |-  ( F  e.  ( S  ^m  V )  ->  F : V --> S )
28 ffvelrn 5838 . . . . . . . . . 10  |-  ( ( F : V --> S  /\  v  e.  V )  ->  ( F `  v
)  e.  S )
2928ex 434 . . . . . . . . 9  |-  ( F : V --> S  -> 
( v  e.  V  ->  ( F `  v
)  e.  S ) )
3027, 29syl 16 . . . . . . . 8  |-  ( F  e.  ( S  ^m  V )  ->  (
v  e.  V  -> 
( F `  v
)  e.  S ) )
3130adantr 462 . . . . . . 7  |-  ( ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  )  ->  (
v  e.  V  -> 
( F `  v
)  e.  S ) )
32313ad2ant3 1006 . . . . . 6  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  -> 
( v  e.  V  ->  ( F `  v
)  e.  S ) )
3332imp 429 . . . . 5  |-  ( ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V
)  /\  F finSupp  .0.  )
)  /\  v  e.  V )  ->  ( F `  v )  e.  S )
34 ssel 3347 . . . . . . . 8  |-  ( V 
C_  B  ->  (
v  e.  V  -> 
v  e.  B ) )
3534adantl 463 . . . . . . 7  |-  ( ( V  e.  W  /\  V  C_  B )  -> 
( v  e.  V  ->  v  e.  B ) )
36353ad2ant2 1005 . . . . . 6  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  -> 
( v  e.  V  ->  v  e.  B ) )
3736imp 429 . . . . 5  |-  ( ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V
)  /\  F finSupp  .0.  )
)  /\  v  e.  V )  ->  v  e.  B )
38 eqid 2441 . . . . . 6  |-  ( .s
`  M )  =  ( .s `  M
)
3912, 3, 38, 2lmodvscl 16945 . . . . 5  |-  ( ( M  e.  LMod  /\  ( F `  v )  e.  S  /\  v  e.  B )  ->  (
( F `  v
) ( .s `  M ) v )  e.  B )
4026, 33, 37, 39syl3anc 1213 . . . 4  |-  ( ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V
)  /\  F finSupp  .0.  )
)  /\  v  e.  V )  ->  (
( F `  v
) ( .s `  M ) v )  e.  B )
41 eqid 2441 . . . 4  |-  ( v  e.  V  |->  ( ( F `  v ) ( .s `  M
) v ) )  =  ( v  e.  V  |->  ( ( F `
 v ) ( .s `  M ) v ) )
4240, 41fmptd 5864 . . 3  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  -> 
( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) ) : V --> B )
43 simpl 454 . . . . 5  |-  ( ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  )  ->  F  e.  ( S  ^m  V
) )
44433ad2ant3 1006 . . . 4  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  ->  F  e.  ( S  ^m  V ) )
45 simp3r 1012 . . . . 5  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  ->  F finSupp  .0.  )
46 lincfsuppcl.0 . . . . 5  |-  .0.  =  ( 0g `  R )
4745, 46syl6breq 4328 . . . 4  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  ->  F finSupp  ( 0g `  R
) )
483, 2scmfsupp 30708 . . . 4  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  F  e.  ( S  ^m  V )  /\  F finSupp  ( 0g `  R ) )  -> 
( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) ) finSupp  ( 0g
`  M ) )
491, 18, 44, 47, 48syl211anc 1219 . . 3  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  -> 
( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) ) finSupp  ( 0g
`  M ) )
5012, 21, 23, 25, 42, 49gsumcl 16390 . 2  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  -> 
( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s
`  M ) v ) ) )  e.  B )
5120, 50eqeltrd 2515 1  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  -> 
( F ( linC  `  M ) V )  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    C_ wss 3325   ~Pcpw 3857   class class class wbr 4289    e. cmpt 4347   -->wf 5411   ` cfv 5415  (class class class)co 6090    ^m cmap 7210   finSupp cfsupp 7616   Basecbs 14170  Scalarcsca 14237   .scvsca 14238   0gc0g 14374    gsumg cgsu 14375  CMndccmn 16270   LModclmod 16928   linC clinc 30779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-plusg 14247  df-0g 14376  df-gsum 14377  df-mnd 15411  df-grp 15538  df-minusg 15539  df-cntz 15828  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-lmod 16930  df-linc 30781
This theorem is referenced by:  lindslinindimp2lem4  30836
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