Theorem rmfsupp2 33248

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 6527	. . 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 7167	. . . 4 ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) ∈ V)
5		rmfsupp2.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Ring)
6		ringgrp 20164	. . . . 5 ⊢ (𝑀 ∈ Ring → 𝑀 ∈ Grp)
7		rmfsuppf2.r	. . . . . 6 ⊢ 𝑅 = (Base‘𝑀)
8		eqid 2733	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
9	7, 8	grpidcl 18886	. . . . 5 ⊢ (𝑀 ∈ Grp → (0g‘𝑀) ∈ 𝑅)
10	5, 6, 9	3syl 18	. . . 4 ⊢ (𝜑 → (0g‘𝑀) ∈ 𝑅)
11		suppval1 8105	. . . 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 7388	. . . . . . 7 ⊢ ((𝐴‘𝑣)(.r‘𝑀)𝐶) ∈ V
14		eqid 2733	. . . . . . 7 ⊢ (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) = (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶))
15	13, 14	dmmpti 6633	. . . . . 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 7388	. . . . . . . . 9 ⊢ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ∈ V
18		nfcv 2895	. . . . . . . . . 10 ⊢ Ⅎ𝑣𝑢
19		nfcv 2895	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝐴‘𝑢)
20		nfcv 2895	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(.r‘𝑀)
21		nfcsb1v 3870	. . . . . . . . . . 11 ⊢ Ⅎ𝑣⦋𝑢 / 𝑣⦌𝐶
22	19, 20, 21	nfov 7385	. . . . . . . . . 10 ⊢ Ⅎ𝑣((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶)
23		fveq2 6831	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑢 → (𝐴‘𝑣) = (𝐴‘𝑢))
24		csbeq1a 3860	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑢 → 𝐶 = ⦋𝑢 / 𝑣⦌𝐶)
25	23, 24	oveq12d 7373	. . . . . . . . . 10 ⊢ (𝑣 = 𝑢 → ((𝐴‘𝑣)(.r‘𝑀)𝐶) = ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶))
26	18, 22, 25, 14	fvmptf 6959	. . . . . . . . 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 2851	. . . . . . 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 2988	. . . . 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 3412	. . . 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 6669	. . . . . . 7 ⊢ (𝜑 → dom 𝐴 = 𝑉)
34	33	rabeqdv 3411	. . . . . 6 ⊢ (𝜑 → {𝑢 ∈ dom 𝐴 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)} = {𝑢 ∈ 𝑉 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)})
35	32	ffund 6663	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐴)
36	7	fvexi 6845	. . . . . . . . . . 11 ⊢ 𝑅 ∈ V
37	36	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ V)
38	37, 3	elmapd 8773	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ (𝑅 ↑m 𝑉) ↔ 𝐴:𝑉⟶𝑅))
39	32, 38	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝑅 ↑m 𝑉))
40		suppval1 8105	. . . . . . . 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 9264	. . . . . . 7 ⊢ (𝜑 → (𝐴 supp (0g‘𝑀)) ∈ Fin)
44	41, 43	eqeltrrd 2834	. . . . . 6 ⊢ (𝜑 → {𝑢 ∈ dom 𝐴 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)} ∈ Fin)
45	34, 44	eqeltrrd 2834	. . . . 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 7370	. . . . . . . . 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 3125	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 𝐶 ∈ 𝑅)
52	51	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑉 𝐶 ∈ 𝑅)
53		rspcsbela 4387	. . . . . . . . . . 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 2733	. . . . . . . . . . 11 ⊢ (.r‘𝑀) = (.r‘𝑀)
56	7, 55, 8	ringlz 20219	. . . . . . . . . 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 2768	. . . . . . . 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 2950	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → (((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀) → (𝐴‘𝑢) ≠ (0g‘𝑀)))
61	60	ss2rabdv 4024	. . . . 5 ⊢ (𝜑 → {𝑢 ∈ 𝑉 ∣ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀)} ⊆ {𝑢 ∈ 𝑉 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)})
62		ssfi 9093	. . . . 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 2833	. . 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 2833	. 2 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) supp (0g‘𝑀)) ∈ Fin)
66		isfsupp 9260	. . 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 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  {crab 3396  Vcvv 3437  ⦋csb 3846   ⊆ wss 3898   class class class wbr 5095   ↦ cmpt 5176  dom cdm 5621  Fun wfun 6483  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355   supp csupp 8099   ↑_m cmap 8759  Fincfn 8879   finSupp cfsupp 9256  Basecbs 17127  ._rcmulr 17169  0_gc0g 17350  Grpcgrp 18854  Ringcrg 20159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-2nd 7931  df-supp 8100  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-map 8761  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-fsupp 9257  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-2 12199  df-sets 17082  df-slot 17100  df-ndx 17112  df-base 17128  df-plusg 17181  df-0g 17352  df-mgm 18556  df-sgrp 18635  df-mnd 18651  df-grp 18857  df-minusg 18858  df-cmn 19702  df-abl 19703  df-mgp 20067  df-rng 20079  df-ur 20108  df-ring 20161

This theorem is referenced by:  evlextv  33635  fedgmullem1  33714