Theorem rmfsupp2 33791

Description: A mapping of a multiplication of a constant with a function into a ring is finitely supported if the function is finitely supported. (Contributed by Thierry Arnoux, 3-Jun-2023.)

Hypotheses

Ref	Expression
rmfsuppf2.r	⊢ 𝑅 = (Base‘𝑀)
rmfsupp2.m	⊢ (𝜑 → 𝑀 ∈ Ring)
rmfsupp2.v	⊢ (𝜑 → 𝑉 ∈ 𝑋)
rmfsupp2.c	⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐶 ∈ 𝑅)
rmfsupp2.a	⊢ (𝜑 → 𝐴:𝑉⟶𝑅)
rmfsupp2.1	⊢ (𝜑 → 𝐴 finSupp (0g‘𝑀))

Assertion

Ref	Expression
rmfsupp2	⊢ (𝜑 → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) finSupp (0g‘𝑀))

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

Allowed substitution hints:   𝐶(𝑣)   𝑋(𝑣)

Proof of Theorem rmfsupp2

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funmpt 6576	. . 3 ⊢ Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))
2	1	a1i 11	. 2 ⊢ (𝜑 → Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)))
3		rmfsupp2.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝑋)
4	3	mptexd 7228	. . . 4 ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) ∈ V)
5		rmfsupp2.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Ring)
6		ringgrp 20457	. . . . 5 ⊢ (𝑀 ∈ Ring → 𝑀 ∈ Grp)
7		rmfsuppf2.r	. . . . . 6 ⊢ 𝑅 = (Base‘𝑀)
8		eqid 2761	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
9	7, 8	grpidcl 19169	. . . . 5 ⊢ (𝑀 ∈ Grp → (0g‘𝑀) ∈ 𝑅)
10	5, 6, 9	3syl 19	. . . 4 ⊢ (𝜑 → (0g‘𝑀) ∈ 𝑅)
11		suppval1 8176	. . . 4 ⊢ ((Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) ∧ (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) ∈ V ∧ (0g‘𝑀) ∈ 𝑅) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) supp (0g‘𝑀)) = {𝑢 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))‘𝑢) ≠ (0g‘𝑀)})
12	2, 4, 10, 11	syl3anc 1398	. . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) supp (0g‘𝑀)) = {𝑢 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))‘𝑢) ≠ (0g‘𝑀)})
13		ovex 7451	. . . . . . 7 ⊢ ((𝐴‘𝑣)(.r‘𝑀)𝐶) ∈ V
14		eqid 2761	. . . . . . 7 ⊢ (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) = (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))
15	13, 14	dmmpti 6681	. . . . . 6 ⊢ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) = 𝑉
16	15	a1i 11	. . . . 5 ⊢ (𝜑 → dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) = 𝑉)
17		ovex 7451	. . . . . . . . 9 ⊢ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ∈ V
18		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑣𝑢
19		nfcv 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝐴‘𝑢)
20		nfcv 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(.r‘𝑀)
21		nfcsb1v 3871	. . . . . . . . . . 11 ⊢ Ⅎ𝑣⦋𝑢 / 𝑣⦌𝐶
22	19, 20, 21	nfov 7448	. . . . . . . . . 10 ⊢ Ⅎ𝑣((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶)
23		fveq2 6883	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑢 → (𝐴‘𝑣) = (𝐴‘𝑢))
24		csbeq1a 3861	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑢 → 𝐶 = ⦋𝑢 / 𝑣⦌𝐶)
25	23, 24	oveq12d 7436	. . . . . . . . . 10 ⊢ (𝑣 = 𝑢 → ((𝐴‘𝑣)(.r‘𝑀)𝐶) = ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶))
26	18, 22, 25, 14	fvmptf 7013	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑉 ∧ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ∈ V) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))‘𝑢) = ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶))
27	17, 26	mpan2 704	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑉 → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))‘𝑢) = ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶))
28	27, 15	eleq2s 2879	. . . . . . 7 ⊢ (𝑢 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))‘𝑢) = ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶))
29	28	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))‘𝑢) = ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶))
30	29	neeq1d 3015	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))) → (((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))‘𝑢) ≠ (0g‘𝑀) ↔ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀)))
31	16, 30	rabeqbidva 3429	. . . 4 ⊢ (𝜑 → {𝑢 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))‘𝑢) ≠ (0g‘𝑀)} = {𝑢 ∈ 𝑉 ∣ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀)})
32		rmfsupp2.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴:𝑉⟶𝑅)
33	32	fdmd 6718	. . . . . . 7 ⊢ (𝜑 → dom 𝐴 = 𝑉)
34	33	rabeqdv 3428	. . . . . 6 ⊢ (𝜑 → {𝑢 ∈ dom 𝐴 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)} = {𝑢 ∈ 𝑉 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)})
35	32	ffund 6712	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐴)
36	7	fvexi 6897	. . . . . . . . . . 11 ⊢ 𝑅 ∈ V
37	36	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ V)
38	37, 3	elmapd 8853	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ (𝑅 ↑m 𝑉) ↔ 𝐴:𝑉⟶𝑅))
39	32, 38	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝑅 ↑m 𝑉))
40		suppval1 8176	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (0g‘𝑀) ∈ 𝑅) → (𝐴 supp (0g‘𝑀)) = {𝑢 ∈ dom 𝐴 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)})
41	35, 39, 10, 40	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → (𝐴 supp (0g‘𝑀)) = {𝑢 ∈ dom 𝐴 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)})
42		rmfsupp2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑀))
43	42	fsuppimpd 9354	. . . . . . 7 ⊢ (𝜑 → (𝐴 supp (0g‘𝑀)) ∈ Fin)
44	41, 43	eqeltrrd 2862	. . . . . 6 ⊢ (𝜑 → {𝑢 ∈ dom 𝐴 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)} ∈ Fin)
45	34, 44	eqeltrrd 2862	. . . . 5 ⊢ (𝜑 → {𝑢 ∈ 𝑉 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)} ∈ Fin)
46		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → (𝐴‘𝑢) = (0g‘𝑀))
47	46	oveq1d 7433	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) = ((0g‘𝑀)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶))
48	5	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → 𝑀 ∈ Ring)
49		simplr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → 𝑢 ∈ 𝑉)
50		rmfsupp2.c	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐶 ∈ 𝑅)
51	50	ralrimiva 3155	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 𝐶 ∈ 𝑅)
52	51	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑉 𝐶 ∈ 𝑅)
53		rspcsbela 4396	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ 𝑉 ∧ ∀𝑣 ∈ 𝑉 𝐶 ∈ 𝑅) → ⦋𝑢 / 𝑣⦌𝐶 ∈ 𝑅)
54	49, 52, 53	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → ⦋𝑢 / 𝑣⦌𝐶 ∈ 𝑅)
55		eqid 2761	. . . . . . . . . . 11 ⊢ (.r‘𝑀) = (.r‘𝑀)
56	7, 55, 8	ringlz 20517	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Ring ∧ ⦋𝑢 / 𝑣⦌𝐶 ∈ 𝑅) → ((0g‘𝑀)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) = (0g‘𝑀))
57	48, 54, 56	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → ((0g‘𝑀)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) = (0g‘𝑀))
58	47, 57	eqtrd 2796	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) = (0g‘𝑀))
59	58	ex 418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → ((𝐴‘𝑢) = (0g‘𝑀) → ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) = (0g‘𝑀)))
60	59	necon3d 2977	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → (((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀) → (𝐴‘𝑢) ≠ (0g‘𝑀)))
61	60	ss2rabdv 4023	. . . . 5 ⊢ (𝜑 → {𝑢 ∈ 𝑉 ∣ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀)} ⊆ {𝑢 ∈ 𝑉 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)})
62		ssfi 9181	. . . . 5 ⊢ (({𝑢 ∈ 𝑉 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)} ∈ Fin ∧ {𝑢 ∈ 𝑉 ∣ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀)} ⊆ {𝑢 ∈ 𝑉 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)}) → {𝑢 ∈ 𝑉 ∣ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀)} ∈ Fin)
63	45, 61, 62	syl2anc 596	. . . 4 ⊢ (𝜑 → {𝑢 ∈ 𝑉 ∣ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀)} ∈ Fin)
64	31, 63	eqeltrd 2861	. . 3 ⊢ (𝜑 → {𝑢 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))‘𝑢) ≠ (0g‘𝑀)} ∈ Fin)
65	12, 64	eqeltrd 2861	. 2 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) supp (0g‘𝑀)) ∈ Fin)
66		isfsupp 9350	. . 3 ⊢ (((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) ∈ V ∧ (0g‘𝑀) ∈ 𝑅) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) finSupp (0g‘𝑀) ↔ (Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) ∧ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) supp (0g‘𝑀)) ∈ Fin)))
67	4, 10, 66	syl2anc 596	. 2 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) finSupp (0g‘𝑀) ↔ (Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) ∧ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) supp (0g‘𝑀)) ∈ Fin)))
68	2, 65, 67	mpbir2and 726	1 ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) finSupp (0g‘𝑀))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  {crab 3413  Vcvv 3451  ⦋csb 3847   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5651  Fun wfun 6531  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418   supp csupp 8170   ↑_m cmap 8840  Fincfn 8966   finSupp cfsupp 9346  Basecbs 17380  ._rcmulr 17422  0_gc0g 17603  Grpcgrp 19137  Ringcrg 20452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-2nd 8000  df-supp 8171  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9347  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-plusg 17434  df-0g 17605  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-grp 19140  df-minusg 19141  df-cmn 19989  df-abl 19990  df-mgp 20354  df-rng 20368  df-ur 20401  df-ring 20454

This theorem is used by:  evlextv  34167  fedgmullem1  34254