Theorem rmfsupp2 33320

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 6530	. . 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 7170	. . . 4 ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) ∈ V)
5		rmfsupp2.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Ring)
6		ringgrp 20173	. . . . 5 ⊢ (𝑀 ∈ Ring → 𝑀 ∈ Grp)
7		rmfsuppf2.r	. . . . . 6 ⊢ 𝑅 = (Base‘𝑀)
8		eqid 2736	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
9	7, 8	grpidcl 18895	. . . . 5 ⊢ (𝑀 ∈ Grp → (0g‘𝑀) ∈ 𝑅)
10	5, 6, 9	3syl 18	. . . 4 ⊢ (𝜑 → (0g‘𝑀) ∈ 𝑅)
11		suppval1 8108	. . . 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 7391	. . . . . . 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 6636	. . . . . 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 7391	. . . . . . . . 9 ⊢ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ∈ V
18		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑣𝑢
19		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(𝐴‘𝑢)
20		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑣(.r‘𝑀)
21		nfcsb1v 3873	. . . . . . . . . . 11 ⊢ Ⅎ𝑣⦋𝑢 / 𝑣⦌𝐶
22	19, 20, 21	nfov 7388	. . . . . . . . . 10 ⊢ Ⅎ𝑣((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶)
23		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑢 → (𝐴‘𝑣) = (𝐴‘𝑢))
24		csbeq1a 3863	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑢 → 𝐶 = ⦋𝑢 / 𝑣⦌𝐶)
25	23, 24	oveq12d 7376	. . . . . . . . . 10 ⊢ (𝑣 = 𝑢 → ((𝐴‘𝑣)(.r‘𝑀)𝐶) = ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶))
26	18, 22, 25, 14	fvmptf 6962	. . . . . . . . 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 2854	. . . . . . 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 2991	. . . . 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 3415	. . . 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 6672	. . . . . . 7 ⊢ (𝜑 → dom 𝐴 = 𝑉)
34	33	rabeqdv 3414	. . . . . 6 ⊢ (𝜑 → {𝑢 ∈ dom 𝐴 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)} = {𝑢 ∈ 𝑉 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)})
35	32	ffund 6666	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐴)
36	7	fvexi 6848	. . . . . . . . . . 11 ⊢ 𝑅 ∈ V
37	36	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ V)
38	37, 3	elmapd 8777	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ (𝑅 ↑m 𝑉) ↔ 𝐴:𝑉⟶𝑅))
39	32, 38	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝑅 ↑m 𝑉))
40		suppval1 8108	. . . . . . . 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 9272	. . . . . . 7 ⊢ (𝜑 → (𝐴 supp (0g‘𝑀)) ∈ Fin)
44	41, 43	eqeltrrd 2837	. . . . . 6 ⊢ (𝜑 → {𝑢 ∈ dom 𝐴 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)} ∈ Fin)
45	34, 44	eqeltrrd 2837	. . . . 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 7373	. . . . . . . . 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 3128	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 𝐶 ∈ 𝑅)
52	51	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑉) ∧ (𝐴‘𝑢) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑉 𝐶 ∈ 𝑅)
53		rspcsbela 4390	. . . . . . . . . . 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 20228	. . . . . . . . . 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 2953	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑉) → (((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀) → (𝐴‘𝑢) ≠ (0g‘𝑀)))
61	60	ss2rabdv 4027	. . . . 5 ⊢ (𝜑 → {𝑢 ∈ 𝑉 ∣ ((𝐴‘𝑢)(.r‘𝑀)⦋𝑢 / 𝑣⦌𝐶) ≠ (0g‘𝑀)} ⊆ {𝑢 ∈ 𝑉 ∣ (𝐴‘𝑢) ≠ (0g‘𝑀)})
62		ssfi 9097	. . . . 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 2836	. . 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 2836	. 2 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) supp (0g‘𝑀)) ∈ Fin)
66		isfsupp 9268	. . 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 2932  ∀wral 3051  {crab 3399  Vcvv 3440  ⦋csb 3849   ⊆ wss 3901   class class class wbr 5098   ↦ cmpt 5179  dom cdm 5624  Fun wfun 6486  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   supp csupp 8102   ↑_m cmap 8763  Fincfn 8883   finSupp cfsupp 9264  Basecbs 17136  ._rcmulr 17178  0_gc0g 17359  Grpcgrp 18863  Ringcrg 20168

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-plusg 17190  df-0g 17361  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866  df-minusg 18867  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-ring 20170

This theorem is referenced by:  evlextv  33707  fedgmullem1  33786