Theorem evls1fpws 22498

Description: Evaluation of a univariate subring polynomial as a function in a power series. (Contributed by Thierry Arnoux, 23-Jan-2025.)

Hypotheses

Ref	Expression
ressply1evl2.q	⊢ 𝑄 = (𝑆 evalSub1 𝑅)
ressply1evl2.k	⊢ 𝐾 = (Base‘𝑆)
ressply1evl2.w	⊢ 𝑊 = (Poly1‘𝑈)
ressply1evl2.u	⊢ 𝑈 = (𝑆 ↾s 𝑅)
ressply1evl2.b	⊢ 𝐵 = (Base‘𝑊)
evls1fpws.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evls1fpws.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evls1fpws.y	⊢ (𝜑 → 𝑀 ∈ 𝐵)
evls1fpws.1	⊢ · = (.r‘𝑆)
evls1fpws.2	⊢ ↑ = (.g‘(mulGrp‘𝑆))
evls1fpws.a	⊢ 𝐴 = (coe1‘𝑀)

Assertion

Ref	Expression
evls1fpws	⊢ (𝜑 → (𝑄‘𝑀) = (𝑥 ∈ 𝐾 ↦ (𝑆 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))))

Distinct variable groups:   · ,𝑘,𝑥   𝐴,𝑘,𝑥   𝐵,𝑘   𝑘,𝐾,𝑥   𝑘,𝑀   𝑄,𝑘,𝑥   𝑆,𝑘,𝑥   𝑈,𝑘,𝑥   𝑘,𝑊,𝑥   𝜑,𝑘,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝑅(𝑥,𝑘)   ↑ (𝑥,𝑘)   𝑀(𝑥)

Proof of Theorem evls1fpws

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evls1fpws.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
2		ressply1evl2.u	. . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅)
3	2	subrgring 20659	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
4	1, 3	syl 18	. . . 4 ⊢ (𝜑 → 𝑈 ∈ Ring)
5		evls1fpws.y	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
6		ressply1evl2.w	. . . . 5 ⊢ 𝑊 = (Poly1‘𝑈)
7		eqid 2769	. . . . 5 ⊢ (var1‘𝑈) = (var1‘𝑈)
8		ressply1evl2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
9		eqid 2769	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
10		eqid 2769	. . . . 5 ⊢ (mulGrp‘𝑊) = (mulGrp‘𝑊)
11		eqid 2769	. . . . 5 ⊢ (.g‘(mulGrp‘𝑊)) = (.g‘(mulGrp‘𝑊))
12		evls1fpws.a	. . . . 5 ⊢ 𝐴 = (coe1‘𝑀)
13	6, 7, 8, 9, 10, 11, 12	ply1coe 22427	. . . 4 ⊢ ((𝑈 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝑊 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))))))
14	4, 5, 13	syl2anc 595	. . 3 ⊢ (𝜑 → 𝑀 = (𝑊 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))))))
15	14	fveq2d 6886	. 2 ⊢ (𝜑 → (𝑄‘𝑀) = (𝑄‘(𝑊 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))))))
16		ressply1evl2.q	. . . 4 ⊢ 𝑄 = (𝑆 evalSub1 𝑅)
17		ressply1evl2.k	. . . 4 ⊢ 𝐾 = (Base‘𝑆)
18		eqid 2769	. . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊)
19		eqid 2769	. . . 4 ⊢ (𝑆 ↑s 𝐾) = (𝑆 ↑s 𝐾)
20		evls1fpws.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ CRing)
21	6	ply1lmod 22380	. . . . . . 7 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ LMod)
22	4, 21	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod)
23	22	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑊 ∈ LMod)
24		eqid 2769	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
25	12, 8, 6, 24	coe1fvalcl 22341	. . . . . . 7 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ (Base‘𝑈))
26	5, 25	sylan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ (Base‘𝑈))
27	6	ply1sca 22381	. . . . . . . . 9 ⊢ (𝑈 ∈ Ring → 𝑈 = (Scalar‘𝑊))
28	4, 27	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑈 = (Scalar‘𝑊))
29	28	fveq2d 6886	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑈) = (Base‘(Scalar‘𝑊)))
30	29	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (Base‘𝑈) = (Base‘(Scalar‘𝑊)))
31	26, 30	eleqtrd 2871	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
32	10, 8	mgpbas 20221	. . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑊))
33	6	ply1ring 22376	. . . . . . . . 9 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
34	4, 33	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Ring)
35	34	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑊 ∈ Ring)
36	10	ringmgp 20321	. . . . . . 7 ⊢ (𝑊 ∈ Ring → (mulGrp‘𝑊) ∈ Mnd)
37	35, 36	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (mulGrp‘𝑊) ∈ Mnd)
38		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
39	4	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑈 ∈ Ring)
40	7, 6, 8	vr1cl 22346	. . . . . . 7 ⊢ (𝑈 ∈ Ring → (var1‘𝑈) ∈ 𝐵)
41	39, 40	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (var1‘𝑈) ∈ 𝐵)
42	32, 11, 37, 38, 41	mulgnn0cld 19161	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)) ∈ 𝐵)
43		eqid 2769	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
44		eqid 2769	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
45	8, 43, 9, 44	lmodvscl 20977	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝐴‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)) ∈ 𝐵) → ((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))) ∈ 𝐵)
46	23, 31, 42, 45	syl3anc 1396	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))) ∈ 𝐵)
47		ssidd 3968	. . . 4 ⊢ (𝜑 → ℕ0 ⊆ ℕ0)
48		fvexd 6897	. . . . 5 ⊢ (𝜑 → (0g‘𝑊) ∈ V)
49		fveq2 6882	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
50		oveq1 7418	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)) = (𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈)))
51	49, 50	oveq12d 7429	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))) = ((𝐴‘𝑗)( ·𝑠 ‘𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈))))
52		eqid 2769	. . . . . . . 8 ⊢ (0g‘𝑈) = (0g‘𝑈)
53	12, 8, 6, 52	coe1ae0 22345	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → ∃𝑖 ∈ ℕ0 ∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → (𝐴‘𝑗) = (0g‘𝑈)))
54	5, 53	syl 18	. . . . . 6 ⊢ (𝜑 → ∃𝑖 ∈ ℕ0 ∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → (𝐴‘𝑗) = (0g‘𝑈)))
55		simpr 489	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → (𝐴‘𝑗) = (0g‘𝑈))
56	28	ad3antrrr 742	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → 𝑈 = (Scalar‘𝑊))
57	56	fveq2d 6886	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → (0g‘𝑈) = (0g‘(Scalar‘𝑊)))
58	55, 57	eqtrd 2804	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → (𝐴‘𝑗) = (0g‘(Scalar‘𝑊)))
59	58	oveq1d 7426	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → ((𝐴‘𝑗)( ·𝑠 ‘𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈))) = ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈))))
60	22	ad3antrrr 742	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → 𝑊 ∈ LMod)
61	34, 36	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (mulGrp‘𝑊) ∈ Mnd)
62	61	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (mulGrp‘𝑊) ∈ Mnd)
63		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝑗 ∈ ℕ0)
64	4, 40	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (var1‘𝑈) ∈ 𝐵)
65	64	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (var1‘𝑈) ∈ 𝐵)
66	32, 11, 62, 63, 65	mulgnn0cld 19161	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈)) ∈ 𝐵)
67	66	ad4ant13 763	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → (𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈)) ∈ 𝐵)
68		eqid 2769	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
69	8, 43, 9, 68, 18	lmod0vs 20994	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ (𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈)) ∈ 𝐵) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈))) = (0g‘𝑊))
70	60, 67, 69	syl2anc 595	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈))) = (0g‘𝑊))
71	59, 70	eqtrd 2804	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → ((𝐴‘𝑗)( ·𝑠 ‘𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈))) = (0g‘𝑊))
72	71	ex 417	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) → ((𝐴‘𝑗) = (0g‘𝑈) → ((𝐴‘𝑗)( ·𝑠 ‘𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈))) = (0g‘𝑊)))
73	72	imim2d 58	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) → ((𝑖 < 𝑗 → (𝐴‘𝑗) = (0g‘𝑈)) → (𝑖 < 𝑗 → ((𝐴‘𝑗)( ·𝑠 ‘𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈))) = (0g‘𝑊))))
74	73	ralimdva 3183	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → (𝐴‘𝑗) = (0g‘𝑈)) → ∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → ((𝐴‘𝑗)( ·𝑠 ‘𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈))) = (0g‘𝑊))))
75	74	reximdva 3184	. . . . . 6 ⊢ (𝜑 → (∃𝑖 ∈ ℕ0 ∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → (𝐴‘𝑗) = (0g‘𝑈)) → ∃𝑖 ∈ ℕ0 ∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → ((𝐴‘𝑗)( ·𝑠 ‘𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈))) = (0g‘𝑊))))
76	54, 75	mpd 16	. . . . 5 ⊢ (𝜑 → ∃𝑖 ∈ ℕ0 ∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → ((𝐴‘𝑗)( ·𝑠 ‘𝑊)(𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈))) = (0g‘𝑊)))
77	48, 46, 51, 76	mptnn0fsuppd 14034	. . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))) finSupp (0g‘𝑊))
78	16, 17, 6, 18, 2, 19, 8, 20, 1, 46, 47, 77	evls1gsumadd 22453	. . 3 ⊢ (𝜑 → (𝑄‘(𝑊 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))))) = ((𝑆 ↑s 𝐾) Σg (𝑘 ∈ ℕ0 ↦ (𝑄‘((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))))))
79	16, 17, 19, 2, 6	evls1rhm 22451	. . . . . . . . 9 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom (𝑆 ↑s 𝐾)))
80	20, 1, 79	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom (𝑆 ↑s 𝐾)))
81	80	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑄 ∈ (𝑊 RingHom (𝑆 ↑s 𝐾)))
82		eqid 2769	. . . . . . . . . 10 ⊢ (algSc‘𝑊) = (algSc‘𝑊)
83	82, 43, 34, 22, 44, 8	asclf 22000	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘𝑊):(Base‘(Scalar‘𝑊))⟶𝐵)
84	83	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (algSc‘𝑊):(Base‘(Scalar‘𝑊))⟶𝐵)
85	84, 31	ffvelcdmd 7081	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((algSc‘𝑊)‘(𝐴‘𝑘)) ∈ 𝐵)
86		eqid 2769	. . . . . . . 8 ⊢ (.r‘𝑊) = (.r‘𝑊)
87		eqid 2769	. . . . . . . 8 ⊢ (.r‘(𝑆 ↑s 𝐾)) = (.r‘(𝑆 ↑s 𝐾))
88	8, 86, 87	rhmmul 20568	. . . . . . 7 ⊢ ((𝑄 ∈ (𝑊 RingHom (𝑆 ↑s 𝐾)) ∧ ((algSc‘𝑊)‘(𝐴‘𝑘)) ∈ 𝐵 ∧ (𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)) ∈ 𝐵) → (𝑄‘(((algSc‘𝑊)‘(𝐴‘𝑘))(.r‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))) = ((𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘)))(.r‘(𝑆 ↑s 𝐾))(𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))))
89	81, 85, 42, 88	syl3anc 1396	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘(((algSc‘𝑊)‘(𝐴‘𝑘))(.r‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))) = ((𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘)))(.r‘(𝑆 ↑s 𝐾))(𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))))
90	2	subrgcrng 20660	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
91	20, 1, 90	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ CRing)
92	6	ply1assa 22328	. . . . . . . . . 10 ⊢ (𝑈 ∈ CRing → 𝑊 ∈ AssAlg)
93	91, 92	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ AssAlg)
94	93	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑊 ∈ AssAlg)
95	82, 43, 44, 8, 86, 9	asclmul1 22005	. . . . . . . 8 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)) ∈ 𝐵) → (((algSc‘𝑊)‘(𝐴‘𝑘))(.r‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))) = ((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))))
96	94, 31, 42, 95	syl3anc 1396	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((algSc‘𝑊)‘(𝐴‘𝑘))(.r‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))) = ((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))))
97	96	fveq2d 6886	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘(((algSc‘𝑊)‘(𝐴‘𝑘))(.r‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))) = (𝑄‘((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))))
98		eqid 2769	. . . . . . . 8 ⊢ (Base‘(𝑆 ↑s 𝐾)) = (Base‘(𝑆 ↑s 𝐾))
99	20	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑆 ∈ CRing)
100	17	fvexi 6896	. . . . . . . . 9 ⊢ 𝐾 ∈ V
101	100	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐾 ∈ V)
102	8, 98	rhmf 20566	. . . . . . . . . 10 ⊢ (𝑄 ∈ (𝑊 RingHom (𝑆 ↑s 𝐾)) → 𝑄:𝐵⟶(Base‘(𝑆 ↑s 𝐾)))
103	81, 102	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑄:𝐵⟶(Base‘(𝑆 ↑s 𝐾)))
104	103, 85	ffvelcdmd 7081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘))) ∈ (Base‘(𝑆 ↑s 𝐾)))
105	103, 42	ffvelcdmd 7081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))) ∈ (Base‘(𝑆 ↑s 𝐾)))
106		evls1fpws.1	. . . . . . . 8 ⊢ · = (.r‘𝑆)
107	19, 98, 99, 101, 104, 105, 106, 87	pwsmulrval 17545	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘)))(.r‘(𝑆 ↑s 𝐾))(𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))) = ((𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘))) ∘f · (𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))))
108	19, 17, 98, 99, 101, 104	pwselbas 17542	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘))):𝐾⟶𝐾)
109	108	ffnd 6707	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘))) Fn 𝐾)
110	19, 17, 98, 99, 101, 105	pwselbas 17542	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))):𝐾⟶𝐾)
111	110	ffnd 6707	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))) Fn 𝐾)
112		inidm 4187	. . . . . . . 8 ⊢ (𝐾 ∩ 𝐾) = 𝐾
113	20	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → 𝑆 ∈ CRing)
114	1	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → 𝑅 ∈ (SubRing‘𝑆))
115	17	subrgss 20657	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐾)
116	1, 115	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ⊆ 𝐾)
117	2, 17	ressbas2 17298	. . . . . . . . . . . . 13 ⊢ (𝑅 ⊆ 𝐾 → 𝑅 = (Base‘𝑈))
118	116, 117	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 = (Base‘𝑈))
119	118	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑅 = (Base‘𝑈))
120	26, 119	eleqtrrd 2872	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ 𝑅)
121	120	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → (𝐴‘𝑘) ∈ 𝑅)
122		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝐾)
123	16, 6, 2, 17, 82, 113, 114, 121, 122	evls1scafv 22495	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → ((𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘)))‘𝑥) = (𝐴‘𝑘))
124		evls1fpws.2	. . . . . . . . 9 ⊢ ↑ = (.g‘(mulGrp‘𝑆))
125		simplr 780	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → 𝑘 ∈ ℕ0)
126	16, 2, 6, 7, 17, 11, 124, 113, 114, 125, 122	evls1varpwval 22497	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → ((𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))‘𝑥) = (𝑘 ↑ 𝑥))
127	109, 111, 101, 101, 112, 123, 126	offval 7684	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘))) ∘f · (𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))) = (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))
128	107, 127	eqtrd 2804	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘)))(.r‘(𝑆 ↑s 𝐾))(𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))) = (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))
129	89, 97, 128	3eqtr3d 2812	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))) = (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))
130	129	mpteq2dva 5208	. . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ (𝑄‘((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))))) = (𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)))))
131	130	oveq2d 7427	. . 3 ⊢ (𝜑 → ((𝑆 ↑s 𝐾) Σg (𝑘 ∈ ℕ0 ↦ (𝑄‘((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))))) = ((𝑆 ↑s 𝐾) Σg (𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))))
132		eqid 2769	. . . 4 ⊢ (0g‘(𝑆 ↑s 𝐾)) = (0g‘(𝑆 ↑s 𝐾))
133	100	a1i 11	. . . 4 ⊢ (𝜑 → 𝐾 ∈ V)
134		nn0ex 12510	. . . . 5 ⊢ ℕ0 ∈ V
135	134	a1i 11	. . . 4 ⊢ (𝜑 → ℕ0 ∈ V)
136	20	crngringd 20328	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
137	136	ringcmnd 20367	. . . 4 ⊢ (𝜑 → 𝑆 ∈ CMnd)
138	136	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → 𝑆 ∈ Ring)
139	1	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ (SubRing‘𝑆))
140	139, 115	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑅 ⊆ 𝐾)
141	140, 120	sseldd 3946	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ 𝐾)
142	141	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → (𝐴‘𝑘) ∈ 𝐾)
143		eqid 2769	. . . . . . . . . 10 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
144	143, 17	mgpbas 20221	. . . . . . . . 9 ⊢ 𝐾 = (Base‘(mulGrp‘𝑆))
145	143	ringmgp 20321	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd)
146	136, 145	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd)
147	146	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → (mulGrp‘𝑆) ∈ Mnd)
148	144, 124, 147, 125, 122	mulgnn0cld 19161	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → (𝑘 ↑ 𝑥) ∈ 𝐾)
149	17, 106, 138, 142, 148	ringcld 20342	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)) ∈ 𝐾)
150	149	3impa 1125	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝑥 ∈ 𝐾) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)) ∈ 𝐾)
151	150	3com23 1142	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)) ∈ 𝐾)
152	151	3expb 1136	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑘 ∈ ℕ0)) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)) ∈ 𝐾)
153	135	mptexd 7223	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)))) ∈ V)
154		funmpt 6575	. . . . . 6 ⊢ Fun (𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))
155	154	a1i 11	. . . . 5 ⊢ (𝜑 → Fun (𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)))))
156		fvexd 6897	. . . . 5 ⊢ (𝜑 → (0g‘(𝑆 ↑s 𝐾)) ∈ V)
157	12, 8, 6, 52	coe1sfi 22342	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → 𝐴 finSupp (0g‘𝑈))
158	5, 157	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑈))
159	158	fsuppimpd 9329	. . . . 5 ⊢ (𝜑 → (𝐴 supp (0g‘𝑈)) ∈ Fin)
160	12, 8, 6, 24	coe1f 22340	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ 𝐵 → 𝐴:ℕ0⟶(Base‘𝑈))
161	5, 160	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴:ℕ0⟶(Base‘𝑈))
162	161	ffnd 6707	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 Fn ℕ0)
163	162	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → 𝐴 Fn ℕ0)
164	134	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → ℕ0 ∈ V)
165		fvexd 6897	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (0g‘𝑈) ∈ V)
166		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈))))
167	163, 164, 165, 166	fvdifsupp 8167	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝐴‘𝑘) = (0g‘𝑈))
168		eqid 2769	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑆) = (0g‘𝑆)
169	2, 168	subrg0 20664	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (0g‘𝑆) = (0g‘𝑈))
170	1, 169	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝑆) = (0g‘𝑈))
171	170	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (0g‘𝑆) = (0g‘𝑈))
172	167, 171	eqtr4d 2807	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝐴‘𝑘) = (0g‘𝑆))
173	172	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → (𝐴‘𝑘) = (0g‘𝑆))
174	173	oveq1d 7426	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)) = ((0g‘𝑆) · (𝑘 ↑ 𝑥)))
175	136	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → 𝑆 ∈ Ring)
176	175, 145	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → (mulGrp‘𝑆) ∈ Mnd)
177		simplr 780	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈))))
178	177	eldifad 3925	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → 𝑘 ∈ ℕ0)
179		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝐾)
180	144, 124, 176, 178, 179	mulgnn0cld 19161	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → (𝑘 ↑ 𝑥) ∈ 𝐾)
181	17, 106, 168	ringlz 20376	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Ring ∧ (𝑘 ↑ 𝑥) ∈ 𝐾) → ((0g‘𝑆) · (𝑘 ↑ 𝑥)) = (0g‘𝑆))
182	175, 180, 181	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → ((0g‘𝑆) · (𝑘 ↑ 𝑥)) = (0g‘𝑆))
183	174, 182	eqtrd 2804	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)) = (0g‘𝑆))
184	183	mpteq2dva 5208	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))) = (𝑥 ∈ 𝐾 ↦ (0g‘𝑆)))
185		fconstmpt 5724	. . . . . . . 8 ⊢ (𝐾 × {(0g‘𝑆)}) = (𝑥 ∈ 𝐾 ↦ (0g‘𝑆))
186	184, 185	eqtr4di 2822	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))) = (𝐾 × {(0g‘𝑆)}))
187	137	cmnmndd 19874	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Mnd)
188	19, 168	pws0g 18831	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ 𝐾 ∈ V) → (𝐾 × {(0g‘𝑆)}) = (0g‘(𝑆 ↑s 𝐾)))
189	187, 133, 188	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (𝐾 × {(0g‘𝑆)}) = (0g‘(𝑆 ↑s 𝐾)))
190	189	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝐾 × {(0g‘𝑆)}) = (0g‘(𝑆 ↑s 𝐾)))
191	186, 190	eqtrd 2804	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))) = (0g‘(𝑆 ↑s 𝐾)))
192	191, 135	suppss2 8196	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)))) supp (0g‘(𝑆 ↑s 𝐾))) ⊆ (𝐴 supp (0g‘𝑈)))
193		suppssfifsupp 9340	. . . . 5 ⊢ ((((𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)))) ∧ (0g‘(𝑆 ↑s 𝐾)) ∈ V) ∧ ((𝐴 supp (0g‘𝑈)) ∈ Fin ∧ ((𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)))) supp (0g‘(𝑆 ↑s 𝐾))) ⊆ (𝐴 supp (0g‘𝑈)))) → (𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)))) finSupp (0g‘(𝑆 ↑s 𝐾)))
194	153, 155, 156, 159, 192, 193	syl32anc 1403	. . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)))) finSupp (0g‘(𝑆 ↑s 𝐾)))
195	19, 17, 132, 133, 135, 137, 152, 194	pwsgsum 20052	. . 3 ⊢ (𝜑 → ((𝑆 ↑s 𝐾) Σg (𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))) = (𝑥 ∈ 𝐾 ↦ (𝑆 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))))
196	78, 131, 195	3eqtrd 2808	. 2 ⊢ (𝜑 → (𝑄‘(𝑊 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))))) = (𝑥 ∈ 𝐾 ↦ (𝑆 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))))
197	15, 196	eqtrd 2804	1 ⊢ (𝜑 → (𝑄‘𝑀) = (𝑥 ∈ 𝐾 ↦ (𝑆 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913  {csn 4594   class class class wbr 5113   ↦ cmpt 5196   × cxp 5660  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∘_f cof 7673   supp csupp 8156  Fincfn 8943   finSupp cfsupp 9321   < clt 11243  ℕ₀cn0 12504  Basecbs 17269   ↾_s cress 17290  ._rcmulr 17311  Scalarcsca 17313   ·_𝑠 cvsca 17314  0_gc0g 17492   Σ_g cgsu 17493   ↑_s cpws 17499  Mndcmnd 18792  ._gcmg 19133  mulGrpcmgp 20216  Ringcrg 20315  CRingccrg 20316   RingHom crh 20551  SubRingcsubrg 20654  LModclmod 20959  AssAlgcasa 21969  algSccascl 21971  var₁cv1 22305  Poly₁cpl1 22306  coe₁cco1 22307   evalSub₁ ces1 22442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-ofr 7676  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8157  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8694  df-map 8826  df-pm 8827  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9322  df-sup 9402  df-oi 9472  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-uz 12863  df-fz 13536  df-fzo 13683  df-seq 14038  df-hash 14367  df-struct 17207  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-mulr 17324  df-sca 17326  df-vsca 17327  df-ip 17328  df-tset 17329  df-ple 17330  df-ds 17332  df-hom 17334  df-cco 17335  df-0g 17494  df-gsum 17495  df-prds 17500  df-pws 17502  df-mre 17638  df-mrc 17639  df-acs 17641  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-mhm 18841  df-submnd 18842  df-grp 19003  df-minusg 19004  df-sbg 19005  df-mulg 19134  df-subg 19189  df-ghm 19284  df-cntz 19387  df-cmn 19852  df-abl 19853  df-mgp 20217  df-rng 20231  df-ur 20264  df-srg 20269  df-ring 20317  df-cring 20318  df-rhm 20554  df-subrng 20631  df-subrg 20655  df-lmod 20961  df-lss 21031  df-lsp 21071  df-assa 21972  df-asp 21973  df-ascl 21974  df-psr 22028  df-mvr 22029  df-mpl 22030  df-opsr 22032  df-evls 22194  df-evl 22195  df-psr1 22309  df-vr1 22310  df-ply1 22311  df-coe1 22312  df-evls1 22444  df-evl1 22445

This theorem is referenced by:  ressply1evl  22499  evl1fpws  33799  ressply1evls1  33800  evls1fldgencl  34005