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Theorem lincfsuppcl 31054
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 988 . . 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 5792 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  (Scalar `  M
) )
52, 4eqtri 2480 . . . . . . . 8  |-  S  =  ( Base `  (Scalar `  M ) )
65oveq1i 6200 . . . . . . 7  |-  ( S  ^m  V )  =  ( ( Base `  (Scalar `  M ) )  ^m  V )
76eleq2i 2529 . . . . . 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 465 . . . 4  |-  ( ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  )  ->  F  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) )
1093ad2ant3 1011 . . 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 3966 . . . . . 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 2459 . . . . . . 7  |-  ( V  e.  W  ->  ( Base `  M )  =  B )
1514sseq2d 3482 . . . . . 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 484 . . . 4  |-  ( ( V  e.  W  /\  V  C_  B )  ->  V  e.  ~P ( Base `  M ) )
18173ad2ant2 1010 . . 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 31050 . . 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 1219 . 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 2451 . . 3  |-  ( 0g
`  M )  =  ( 0g `  M
)
22 lmodcmn 17099 . . . 4  |-  ( M  e.  LMod  ->  M  e. CMnd
)
23223ad2ant1 1009 . . 3  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  ->  M  e. CMnd )
24 simpl 457 . . . 4  |-  ( ( V  e.  W  /\  V  C_  B )  ->  V  e.  W )
25243ad2ant2 1010 . . 3  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  ->  V  e.  W )
261adantr 465 . . . . 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 7334 . . . . . . . . 9  |-  ( F  e.  ( S  ^m  V )  ->  F : V --> S )
28 ffvelrn 5940 . . . . . . . . . 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 465 . . . . . . 7  |-  ( ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  )  ->  (
v  e.  V  -> 
( F `  v
)  e.  S ) )
32313ad2ant3 1011 . . . . . 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 3448 . . . . . . . 8  |-  ( V 
C_  B  ->  (
v  e.  V  -> 
v  e.  B ) )
3534adantl 466 . . . . . . 7  |-  ( ( V  e.  W  /\  V  C_  B )  -> 
( v  e.  V  ->  v  e.  B ) )
36353ad2ant2 1010 . . . . . 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 2451 . . . . . 6  |-  ( .s
`  M )  =  ( .s `  M
)
3912, 3, 38, 2lmodvscl 17071 . . . . 5  |-  ( ( M  e.  LMod  /\  ( F `  v )  e.  S  /\  v  e.  B )  ->  (
( F `  v
) ( .s `  M ) v )  e.  B )
4026, 33, 37, 39syl3anc 1219 . . . 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 2451 . . . 4  |-  ( v  e.  V  |->  ( ( F `  v ) ( .s `  M
) v ) )  =  ( v  e.  V  |->  ( ( F `
 v ) ( .s `  M ) v ) )
4240, 41fmptd 5966 . . 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 457 . . . . 5  |-  ( ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  )  ->  F  e.  ( S  ^m  V
) )
44433ad2ant3 1011 . . . 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 1017 . . . . 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 4429 . . . 4  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  ->  F finSupp  ( 0g `  R
) )
483, 2scmfsupp 30930 . . . 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 1225 . . 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 16501 . 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 2539 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 965    = wceq 1370    e. wcel 1758    C_ wss 3426   ~Pcpw 3958   class class class wbr 4390    |-> cmpt 4448   -->wf 5512   ` cfv 5516  (class class class)co 6190    ^m cmap 7314   finSupp cfsupp 7721   Basecbs 14276  Scalarcsca 14343   .scvsca 14344   0gc0g 14480    gsumg cgsu 14481  CMndccmn 16381   LModclmod 17054   linC clinc 31045
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 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-fzo 11650  df-seq 11908  df-hash 12205  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-plusg 14353  df-0g 14482  df-gsum 14483  df-mnd 15517  df-grp 15647  df-minusg 15648  df-cntz 15937  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-rng 16753  df-lmod 17056  df-linc 31047
This theorem is referenced by:  lindslinindimp2lem4  31102
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