Theorem rmfsupp2 31394

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 6456	. . 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 7082	. . . 4 ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) ∈ V)
5		rmfsupp2.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Ring)
6		ringgrp 19703	. . . . 5 ⊢ (𝑀 ∈ Ring → 𝑀 ∈ Grp)
7		rmfsuppf2.r	. . . . . 6 ⊢ 𝑅 = (Base‘𝑀)
8		eqid 2738	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
9	7, 8	grpidcl 18522	. . . . 5 ⊢ (𝑀 ∈ Grp → (0g‘𝑀) ∈ 𝑅)
10	5, 6, 9	3syl 18	. . . 4 ⊢ (𝜑 → (0g‘𝑀) ∈ 𝑅)
11		suppval1 7954	. . . 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 1369	. . 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 7288	. . . . . . 7 ⊢ ((𝐴‘𝑣)(.r‘𝑀)𝐶) ∈ V
14		eqid 2738	. . . . . . 7 ⊢ (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) = (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))
15	13, 14	dmmpti 6561	. . . . . 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 7288	. . . . . . . . 9 ⊢ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ∈ V
18		nfcv 2906	. . . . . . . . . 10 ⊢ Ⅎ𝑣𝑢
19		nfcv 2906	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝐴‘𝑢)
20		nfcv 2906	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(.r‘𝑀)
21		nfcsb1v 3853	. . . . . . . . . . 11 ⊢ Ⅎ𝑣⦋𝑢 / 𝑣⦌𝐶
22	19, 20, 21	nfov 7285	. . . . . . . . . 10 ⊢ Ⅎ𝑣((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶)
23		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑢 → (𝐴‘𝑣) = (𝐴‘𝑢))
24		csbeq1a 3842	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑢 → 𝐶 = ⦋𝑢 / 𝑣⦌𝐶)
25	23, 24	oveq12d 7273	. . . . . . . . . 10 ⊢ (𝑣 = 𝑢 → ((𝐴‘𝑣)(.r‘𝑀)𝐶) = ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶))
26	18, 22, 25, 14	fvmptf 6878	. . . . . . . . 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 687	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑉 → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))‘𝑢) = ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶))
28	27, 15	eleq2s 2857	. . . . . . 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 481	. . . . . 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 3002	. . . . 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 3411	. . . 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 6595	. . . . . . 7 ⊢ (𝜑 → dom 𝐴 = 𝑉)
34	33	rabeqdv 3409	. . . . . 6 ⊢ (𝜑 → {𝑢 ∈ dom 𝐴 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)} = {𝑢 ∈ 𝑉 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)})
35	32	ffund 6588	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐴)
36	7	fvexi 6770	. . . . . . . . . . 11 ⊢ 𝑅 ∈ V
37	36	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ V)
38	37, 3	elmapd 8587	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ (𝑅 ↑m 𝑉) ↔ 𝐴:𝑉⟶𝑅))
39	32, 38	mpbird 256	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝑅 ↑m 𝑉))
40		suppval1 7954	. . . . . . . 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 1369	. . . . . . 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 9065	. . . . . . 7 ⊢ (𝜑 → (𝐴 supp (0g‘𝑀)) ∈ Fin)
44	41, 43	eqeltrrd 2840	. . . . . 6 ⊢ (𝜑 → {𝑢 ∈ dom 𝐴 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)} ∈ Fin)
45	34, 44	eqeltrrd 2840	. . . . 5 ⊢ (𝜑 → {𝑢 ∈ 𝑉 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)} ∈ Fin)
46		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → (𝐴‘𝑢) = (0g‘𝑀))
47	46	oveq1d 7270	. . . . . . . . 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 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → 𝑀 ∈ Ring)
49		simplr 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → 𝑢 ∈ 𝑉)
50		rmfsupp2.c	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐶 ∈ 𝑅)
51	50	ralrimiva 3107	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 𝐶 ∈ 𝑅)
52	51	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑉 𝐶 ∈ 𝑅)
53		rspcsbela 4366	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ 𝑉 ∧ ∀𝑣 ∈ 𝑉 𝐶 ∈ 𝑅) → ⦋𝑢 / 𝑣⦌𝐶 ∈ 𝑅)
54	49, 52, 53	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → ⦋𝑢 / 𝑣⦌𝐶 ∈ 𝑅)
55		eqid 2738	. . . . . . . . . . 11 ⊢ (.r‘𝑀) = (.r‘𝑀)
56	7, 55, 8	ringlz 19741	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Ring ∧ ⦋𝑢 / 𝑣⦌𝐶 ∈ 𝑅) → ((0g‘𝑀)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) = (0g‘𝑀))
57	48, 54, 56	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → ((0g‘𝑀)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) = (0g‘𝑀))
58	47, 57	eqtrd 2778	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) = (0g‘𝑀))
59	58	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → ((𝐴‘𝑢) = (0g‘𝑀) → ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) = (0g‘𝑀)))
60	59	necon3d 2963	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → (((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀) → (𝐴‘𝑢) ≠ (0g‘𝑀)))
61	60	ss2rabdv 4005	. . . . 5 ⊢ (𝜑 → {𝑢 ∈ 𝑉 ∣ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀)} ⊆ {𝑢 ∈ 𝑉 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)})
62		ssfi 8918	. . . . 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 583	. . . 4 ⊢ (𝜑 → {𝑢 ∈ 𝑉 ∣ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀)} ∈ Fin)
64	31, 63	eqeltrd 2839	. . 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 2839	. 2 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) supp (0g‘𝑀)) ∈ Fin)
66		isfsupp 9062	. . 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 583	. 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 709	1 ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) finSupp (0g‘𝑀))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  {crab 3067  Vcvv 3422  ⦋csb 3828   ⊆ wss 3883   class class class wbr 5070   ↦ cmpt 5153  dom cdm 5580  Fun wfun 6412  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   supp csupp 7948   ↑_m cmap 8573  Fincfn 8691   finSupp cfsupp 9058  Basecbs 16840  ._rcmulr 16889  0_gc0g 17067  Grpcgrp 18492  Ringcrg 19698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-plusg 16901  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-mgp 19636  df-ring 19700

This theorem is referenced by:  fedgmullem1  31612