Theorem rmfsupp2 33238

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 6579	. . 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 7221	. . . 4 ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) ∈ V)
5		rmfsupp2.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Ring)
6		ringgrp 20203	. . . . 5 ⊢ (𝑀 ∈ Ring → 𝑀 ∈ Grp)
7		rmfsuppf2.r	. . . . . 6 ⊢ 𝑅 = (Base‘𝑀)
8		eqid 2736	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
9	7, 8	grpidcl 18953	. . . . 5 ⊢ (𝑀 ∈ Grp → (0g‘𝑀) ∈ 𝑅)
10	5, 6, 9	3syl 18	. . . 4 ⊢ (𝜑 → (0g‘𝑀) ∈ 𝑅)
11		suppval1 8170	. . . 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 1373	. . 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 7443	. . . . . . 7 ⊢ ((𝐴‘𝑣)(.r‘𝑀)𝐶) ∈ V
14		eqid 2736	. . . . . . 7 ⊢ (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) = (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))
15	13, 14	dmmpti 6687	. . . . . 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 7443	. . . . . . . . 9 ⊢ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ∈ V
18		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑣𝑢
19		nfcv 2899	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝐴‘𝑢)
20		nfcv 2899	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(.r‘𝑀)
21		nfcsb1v 3903	. . . . . . . . . . 11 ⊢ Ⅎ𝑣⦋𝑢 / 𝑣⦌𝐶
22	19, 20, 21	nfov 7440	. . . . . . . . . 10 ⊢ Ⅎ𝑣((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶)
23		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑢 → (𝐴‘𝑣) = (𝐴‘𝑢))
24		csbeq1a 3893	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑢 → 𝐶 = ⦋𝑢 / 𝑣⦌𝐶)
25	23, 24	oveq12d 7428	. . . . . . . . . 10 ⊢ (𝑣 = 𝑢 → ((𝐴‘𝑣)(.r‘𝑀)𝐶) = ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶))
26	18, 22, 25, 14	fvmptf 7012	. . . . . . . . 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 691	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑉 → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))‘𝑢) = ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶))
28	27, 15	eleq2s 2853	. . . . . . 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 2992	. . . . 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 3437	. . . 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 6721	. . . . . . 7 ⊢ (𝜑 → dom 𝐴 = 𝑉)
34	33	rabeqdv 3436	. . . . . 6 ⊢ (𝜑 → {𝑢 ∈ dom 𝐴 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)} = {𝑢 ∈ 𝑉 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)})
35	32	ffund 6715	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐴)
36	7	fvexi 6895	. . . . . . . . . . 11 ⊢ 𝑅 ∈ V
37	36	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ V)
38	37, 3	elmapd 8859	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ (𝑅 ↑m 𝑉) ↔ 𝐴:𝑉⟶𝑅))
39	32, 38	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝑅 ↑m 𝑉))
40		suppval1 8170	. . . . . . . 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 1373	. . . . . . 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 9386	. . . . . . 7 ⊢ (𝜑 → (𝐴 supp (0g‘𝑀)) ∈ Fin)
44	41, 43	eqeltrrd 2836	. . . . . 6 ⊢ (𝜑 → {𝑢 ∈ dom 𝐴 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)} ∈ Fin)
45	34, 44	eqeltrrd 2836	. . . . 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 7425	. . . . . . . . 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 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → 𝑀 ∈ Ring)
49		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → 𝑢 ∈ 𝑉)
50		rmfsupp2.c	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐶 ∈ 𝑅)
51	50	ralrimiva 3133	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 𝐶 ∈ 𝑅)
52	51	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑉 𝐶 ∈ 𝑅)
53		rspcsbela 4418	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ 𝑉 ∧ ∀𝑣 ∈ 𝑉 𝐶 ∈ 𝑅) → ⦋𝑢 / 𝑣⦌𝐶 ∈ 𝑅)
54	49, 52, 53	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → ⦋𝑢 / 𝑣⦌𝐶 ∈ 𝑅)
55		eqid 2736	. . . . . . . . . . 11 ⊢ (.r‘𝑀) = (.r‘𝑀)
56	7, 55, 8	ringlz 20258	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Ring ∧ ⦋𝑢 / 𝑣⦌𝐶 ∈ 𝑅) → ((0g‘𝑀)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) = (0g‘𝑀))
57	48, 54, 56	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → ((0g‘𝑀)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) = (0g‘𝑀))
58	47, 57	eqtrd 2771	. . . . . . . 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 2954	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → (((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀) → (𝐴‘𝑢) ≠ (0g‘𝑀)))
61	60	ss2rabdv 4056	. . . . 5 ⊢ (𝜑 → {𝑢 ∈ 𝑉 ∣ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀)} ⊆ {𝑢 ∈ 𝑉 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)})
62		ssfi 9192	. . . . 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 584	. . . 4 ⊢ (𝜑 → {𝑢 ∈ 𝑉 ∣ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀)} ∈ Fin)
64	31, 63	eqeltrd 2835	. . 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 2835	. 2 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) supp (0g‘𝑀)) ∈ Fin)
66		isfsupp 9382	. . 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 584	. 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 713	1 ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) finSupp (0g‘𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  {crab 3420  Vcvv 3464  ⦋csb 3879   ⊆ wss 3931   class class class wbr 5124   ↦ cmpt 5206  dom cdm 5659  Fun wfun 6530  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   supp csupp 8164   ↑_m cmap 8845  Fincfn 8964   finSupp cfsupp 9378  Basecbs 17233  ._rcmulr 17277  0_gc0g 17458  Grpcgrp 18921  Ringcrg 20198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-2nd 7994  df-supp 8165  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9379  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-plusg 17289  df-0g 17460  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-grp 18924  df-minusg 18925  df-cmn 19768  df-abl 19769  df-mgp 20106  df-rng 20118  df-ur 20147  df-ring 20200

This theorem is referenced by:  fedgmullem1  33674