Theorem scmfsupp 48698

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 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀))

Distinct variable groups:   𝑣,𝐴   𝑣,𝑀   𝑣,𝑅   𝑣,𝑉

Allowed substitution hint:   𝑆(𝑣)

Proof of Theorem scmfsupp

Step	Hyp	Ref	Expression
1		funmpt 6531	. . 3 ⊢ Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))
2	1	a1i 11	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)))
3		id 22	. . . 4 ⊢ (𝐴 finSupp (0g‘𝑆) → 𝐴 finSupp (0g‘𝑆))
4	3	fsuppimpd 9277	. . 3 ⊢ (𝐴 finSupp (0g‘𝑆) → (𝐴 supp (0g‘𝑆)) ∈ Fin)
5		scmsuppfi.s	. . . 4 ⊢ 𝑆 = (Scalar‘𝑀)
6		scmsuppfi.r	. . . 4 ⊢ 𝑅 = (Base‘𝑆)
7	5, 6	scmsuppfi 48697	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (𝐴 supp (0g‘𝑆)) ∈ Fin) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin)
8	4, 7	syl3an3 1166	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin)
9		mptexg 7170	. . . . 5 ⊢ (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V)
10	9	adantl 481	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V)
11	10	3ad2ant1 1134	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V)
12		fvex 6848	. . 3 ⊢ (0g‘𝑀) ∈ V
13		isfsupp 9273	. . 3 ⊢ (((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V ∧ (0g‘𝑀) ∈ V) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀) ↔ (Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∧ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin)))
14	11, 12, 13	sylancl 587	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀) ↔ (Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∧ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin)))
15	2, 8, 14	mpbir2and 714	1 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3441  𝒫 cpw 4555   class class class wbr 5099   ↦ cmpt 5180  Fun wfun 6487  ‘cfv 6493  (class class class)co 7361   supp csupp 8105   ↑_m cmap 8768  Fincfn 8888   finSupp cfsupp 9269  Basecbs 17141  Scalarcsca 17185   ·_𝑠 cvsca 17186  0_gc0g 17364  LModclmod 20816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-supp 8106  df-1o 8400  df-map 8770  df-en 8889  df-fin 8892  df-fsupp 9270  df-0g 17366  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-grp 18871  df-ring 20175  df-lmod 20818

This theorem is referenced by:  gsumlsscl  48703  lincfsuppcl  48736  linccl  48737  lincdifsn  48747  lincsum  48752  lincscm  48753  lincresunit3lem2  48803  lincresunit3  48804