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Theorem scmfsupp 41477
Description: A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
Hypotheses
Ref Expression
scmsuppfi.s 𝑆 = (Scalar‘𝑀)
scmsuppfi.r 𝑅 = (Base‘𝑆)
Assertion
Ref Expression
scmfsupp (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐴 finSupp (0g𝑆)) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) finSupp (0g𝑀))
Distinct variable groups:   𝑣,𝐴   𝑣,𝑀   𝑣,𝑅   𝑣,𝑉
Allowed substitution hint:   𝑆(𝑣)

Proof of Theorem scmfsupp
StepHypRef Expression
1 funmpt 5894 . . 3 Fun (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))
21a1i 11 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐴 finSupp (0g𝑆)) → Fun (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)))
3 id 22 . . . 4 (𝐴 finSupp (0g𝑆) → 𝐴 finSupp (0g𝑆))
43fsuppimpd 8242 . . 3 (𝐴 finSupp (0g𝑆) → (𝐴 supp (0g𝑆)) ∈ Fin)
5 scmsuppfi.s . . . 4 𝑆 = (Scalar‘𝑀)
6 scmsuppfi.r . . . 4 𝑅 = (Base‘𝑆)
75, 6scmsuppfi 41476 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ (𝐴 supp (0g𝑆)) ∈ Fin) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) supp (0g𝑀)) ∈ Fin)
84, 7syl3an3 1358 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐴 finSupp (0g𝑆)) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) supp (0g𝑀)) ∈ Fin)
9 mptexg 6449 . . . . 5 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V)
109adantl 482 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V)
11103ad2ant1 1080 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐴 finSupp (0g𝑆)) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V)
12 fvex 6168 . . 3 (0g𝑀) ∈ V
13 isfsupp 8239 . . 3 (((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V ∧ (0g𝑀) ∈ V) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) finSupp (0g𝑀) ↔ (Fun (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∧ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) supp (0g𝑀)) ∈ Fin)))
1411, 12, 13sylancl 693 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐴 finSupp (0g𝑆)) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) finSupp (0g𝑀) ↔ (Fun (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∧ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) supp (0g𝑀)) ∈ Fin)))
152, 8, 14mpbir2and 956 1 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐴 finSupp (0g𝑆)) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) finSupp (0g𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3190  𝒫 cpw 4136   class class class wbr 4623  cmpt 4683  Fun wfun 5851  cfv 5857  (class class class)co 6615   supp csupp 7255  𝑚 cmap 7817  Fincfn 7915   finSupp cfsupp 8235  Basecbs 15800  Scalarcsca 15884   ·𝑠 cvsca 15885  0gc0g 16040  LModclmod 18803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-er 7702  df-map 7819  df-en 7916  df-fin 7919  df-fsupp 8236  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-ring 18489  df-lmod 18805
This theorem is referenced by:  gsumlsscl  41482  lincfsuppcl  41520  linccl  41521  lincdifsn  41531  lincsum  41536  lincscm  41537  lincresunit3lem2  41587  lincresunit3  41588
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