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Theorem lincfsuppcl 33287
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 994 . . 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 5851 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  (Scalar `  M
) )
52, 4eqtri 2483 . . . . . . . 8  |-  S  =  ( Base `  (Scalar `  M ) )
65oveq1i 6280 . . . . . . 7  |-  ( S  ^m  V )  =  ( ( Base `  (Scalar `  M ) )  ^m  V )
76eleq2i 2532 . . . . . 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 463 . . . 4  |-  ( ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  )  ->  F  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) )
1093ad2ant3 1017 . . 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 4007 . . . . . 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 2462 . . . . . . 7  |-  ( V  e.  W  ->  ( Base `  M )  =  B )
1514sseq2d 3517 . . . . . 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 482 . . . 4  |-  ( ( V  e.  W  /\  V  C_  B )  ->  V  e.  ~P ( Base `  M ) )
18173ad2ant2 1016 . . 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 33283 . . 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 1226 . 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 2454 . . 3  |-  ( 0g
`  M )  =  ( 0g `  M
)
22 lmodcmn 17756 . . . 4  |-  ( M  e.  LMod  ->  M  e. CMnd
)
23223ad2ant1 1015 . . 3  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  ->  M  e. CMnd )
24 simpl 455 . . . 4  |-  ( ( V  e.  W  /\  V  C_  B )  ->  V  e.  W )
25243ad2ant2 1016 . . 3  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  ->  V  e.  W )
261adantr 463 . . . . 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 7433 . . . . . . . . 9  |-  ( F  e.  ( S  ^m  V )  ->  F : V --> S )
28 ffvelrn 6005 . . . . . . . . . 10  |-  ( ( F : V --> S  /\  v  e.  V )  ->  ( F `  v
)  e.  S )
2928ex 432 . . . . . . . . 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 463 . . . . . . 7  |-  ( ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  )  ->  (
v  e.  V  -> 
( F `  v
)  e.  S ) )
32313ad2ant3 1017 . . . . . 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 427 . . . . 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 3483 . . . . . . . 8  |-  ( V 
C_  B  ->  (
v  e.  V  -> 
v  e.  B ) )
3534adantl 464 . . . . . . 7  |-  ( ( V  e.  W  /\  V  C_  B )  -> 
( v  e.  V  ->  v  e.  B ) )
36353ad2ant2 1016 . . . . . 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 427 . . . . 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 2454 . . . . . 6  |-  ( .s
`  M )  =  ( .s `  M
)
3912, 3, 38, 2lmodvscl 17727 . . . . 5  |-  ( ( M  e.  LMod  /\  ( F `  v )  e.  S  /\  v  e.  B )  ->  (
( F `  v
) ( .s `  M ) v )  e.  B )
4026, 33, 37, 39syl3anc 1226 . . . 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 2454 . . . 4  |-  ( v  e.  V  |->  ( ( F `  v ) ( .s `  M
) v ) )  =  ( v  e.  V  |->  ( ( F `
 v ) ( .s `  M ) v ) )
4240, 41fmptd 6031 . . 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 455 . . . . 5  |-  ( ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  )  ->  F  e.  ( S  ^m  V
) )
44433ad2ant3 1017 . . . 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 1023 . . . . 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 4478 . . . 4  |-  ( ( M  e.  LMod  /\  ( V  e.  W  /\  V  C_  B )  /\  ( F  e.  ( S  ^m  V )  /\  F finSupp  .0.  ) )  ->  F finSupp  ( 0g `  R
) )
483, 2scmfsupp 33244 . . . 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 1232 . . 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 17125 . 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 2542 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    C_ wss 3461   ~Pcpw 3999   class class class wbr 4439    |-> cmpt 4497   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   finSupp cfsupp 7821   Basecbs 14719  Scalarcsca 14790   .scvsca 14791   0gc0g 14932    gsumg cgsu 14933  CMndccmn 17000   LModclmod 17710   linC clinc 33278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12093  df-hash 12391  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-plusg 14800  df-0g 14934  df-gsum 14935  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-grp 16259  df-minusg 16260  df-cntz 16557  df-cmn 17002  df-abl 17003  df-mgp 17340  df-ur 17352  df-ring 17398  df-lmod 17712  df-linc 33280
This theorem is referenced by:  lindslinindimp2lem4  33335
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