Theorem evls1fpws 22256

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 20483	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
4	1, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ∈ Ring)
5		evls1fpws.y	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
6		ressply1evl2.w	. . . . 5 ⊢ 𝑊 = (Poly1‘𝑈)
7		eqid 2729	. . . . 5 ⊢ (var1‘𝑈) = (var1‘𝑈)
8		ressply1evl2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
9		eqid 2729	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
10		eqid 2729	. . . . 5 ⊢ (mulGrp‘𝑊) = (mulGrp‘𝑊)
11		eqid 2729	. . . . 5 ⊢ (.g‘(mulGrp‘𝑊)) = (.g‘(mulGrp‘𝑊))
12		evls1fpws.a	. . . . 5 ⊢ 𝐴 = (coe1‘𝑀)
13	6, 7, 8, 9, 10, 11, 12	ply1coe 22185	. . . 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 584	. . 3 ⊢ (𝜑 → 𝑀 = (𝑊 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))))))
15	14	fveq2d 6862	. 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 2729	. . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊)
19		eqid 2729	. . . 4 ⊢ (𝑆 ↑s 𝐾) = (𝑆 ↑s 𝐾)
20		evls1fpws.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ CRing)
21	6	ply1lmod 22136	. . . . . . 7 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ LMod)
22	4, 21	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod)
23	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑊 ∈ LMod)
24		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
25	12, 8, 6, 24	coe1fvalcl 22097	. . . . . . 7 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ (Base‘𝑈))
26	5, 25	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ (Base‘𝑈))
27	6	ply1sca 22137	. . . . . . . . 9 ⊢ (𝑈 ∈ Ring → 𝑈 = (Scalar‘𝑊))
28	4, 27	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 = (Scalar‘𝑊))
29	28	fveq2d 6862	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑈) = (Base‘(Scalar‘𝑊)))
30	29	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (Base‘𝑈) = (Base‘(Scalar‘𝑊)))
31	26, 30	eleqtrd 2830	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
32	10, 8	mgpbas 20054	. . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑊))
33	6	ply1ring 22132	. . . . . . . . 9 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
34	4, 33	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ Ring)
35	34	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑊 ∈ Ring)
36	10	ringmgp 20148	. . . . . . 7 ⊢ (𝑊 ∈ Ring → (mulGrp‘𝑊) ∈ Mnd)
37	35, 36	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (mulGrp‘𝑊) ∈ Mnd)
38		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
39	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑈 ∈ Ring)
40	7, 6, 8	vr1cl 22102	. . . . . . 7 ⊢ (𝑈 ∈ Ring → (var1‘𝑈) ∈ 𝐵)
41	39, 40	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (var1‘𝑈) ∈ 𝐵)
42	32, 11, 37, 38, 41	mulgnn0cld 19027	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)) ∈ 𝐵)
43		eqid 2729	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
44		eqid 2729	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
45	8, 43, 9, 44	lmodvscl 20784	. . . . 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 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))) ∈ 𝐵)
47		ssidd 3970	. . . 4 ⊢ (𝜑 → ℕ0 ⊆ ℕ0)
48		fvexd 6873	. . . . 5 ⊢ (𝜑 → (0g‘𝑊) ∈ V)
49		fveq2 6858	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
50		oveq1 7394	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)) = (𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈)))
51	49, 50	oveq12d 7405	. . . . 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 2729	. . . . . . . 8 ⊢ (0g‘𝑈) = (0g‘𝑈)
53	12, 8, 6, 52	coe1ae0 22101	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → ∃𝑖 ∈ ℕ0 ∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → (𝐴‘𝑗) = (0g‘𝑈)))
54	5, 53	syl 17	. . . . . 6 ⊢ (𝜑 → ∃𝑖 ∈ ℕ0 ∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → (𝐴‘𝑗) = (0g‘𝑈)))
55		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → (𝐴‘𝑗) = (0g‘𝑈))
56	28	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → 𝑈 = (Scalar‘𝑊))
57	56	fveq2d 6862	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → (0g‘𝑈) = (0g‘(Scalar‘𝑊)))
58	55, 57	eqtrd 2764	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → (𝐴‘𝑗) = (0g‘(Scalar‘𝑊)))
59	58	oveq1d 7402	. . . . . . . . . . 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 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → 𝑊 ∈ LMod)
61	34, 36	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (mulGrp‘𝑊) ∈ Mnd)
62	61	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (mulGrp‘𝑊) ∈ Mnd)
63		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝑗 ∈ ℕ0)
64	4, 40	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (var1‘𝑈) ∈ 𝐵)
65	64	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (var1‘𝑈) ∈ 𝐵)
66	32, 11, 62, 63, 65	mulgnn0cld 19027	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈)) ∈ 𝐵)
67	66	ad4ant13 751	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → (𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈)) ∈ 𝐵)
68		eqid 2729	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
69	8, 43, 9, 68, 18	lmod0vs 20801	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 2764	. . . . . . . . . 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 412	. . . . . . . . 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 57	. . . . . . . 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 3145	. . . . . . 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 3146	. . . . . 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 15	. . . . 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 13963	. . . 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 22211	. . 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 22209	. . . . . . . . 9 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom (𝑆 ↑s 𝐾)))
80	20, 1, 79	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom (𝑆 ↑s 𝐾)))
81	80	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑄 ∈ (𝑊 RingHom (𝑆 ↑s 𝐾)))
82		eqid 2729	. . . . . . . . . 10 ⊢ (algSc‘𝑊) = (algSc‘𝑊)
83	82, 43, 34, 22, 44, 8	asclf 21791	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘𝑊):(Base‘(Scalar‘𝑊))⟶𝐵)
84	83	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (algSc‘𝑊):(Base‘(Scalar‘𝑊))⟶𝐵)
85	84, 31	ffvelcdmd 7057	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((algSc‘𝑊)‘(𝐴‘𝑘)) ∈ 𝐵)
86		eqid 2729	. . . . . . . 8 ⊢ (.r‘𝑊) = (.r‘𝑊)
87		eqid 2729	. . . . . . . 8 ⊢ (.r‘(𝑆 ↑s 𝐾)) = (.r‘(𝑆 ↑s 𝐾))
88	8, 86, 87	rhmmul 20395	. . . . . . 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 1373	. . . . . 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 20484	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
91	20, 1, 90	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ CRing)
92	6	ply1assa 22084	. . . . . . . . . 10 ⊢ (𝑈 ∈ CRing → 𝑊 ∈ AssAlg)
93	91, 92	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ AssAlg)
94	93	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑊 ∈ AssAlg)
95	82, 43, 44, 8, 86, 9	asclmul1 21795	. . . . . . . 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 1373	. . . . . . 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 6862	. . . . . 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 2729	. . . . . . . 8 ⊢ (Base‘(𝑆 ↑s 𝐾)) = (Base‘(𝑆 ↑s 𝐾))
99	20	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑆 ∈ CRing)
100	17	fvexi 6872	. . . . . . . . 9 ⊢ 𝐾 ∈ V
101	100	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐾 ∈ V)
102	8, 98	rhmf 20394	. . . . . . . . . 10 ⊢ (𝑄 ∈ (𝑊 RingHom (𝑆 ↑s 𝐾)) → 𝑄:𝐵⟶(Base‘(𝑆 ↑s 𝐾)))
103	81, 102	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑄:𝐵⟶(Base‘(𝑆 ↑s 𝐾)))
104	103, 85	ffvelcdmd 7057	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘))) ∈ (Base‘(𝑆 ↑s 𝐾)))
105	103, 42	ffvelcdmd 7057	. . . . . . . 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 17454	. . . . . . 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 17452	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘))):𝐾⟶𝐾)
109	108	ffnd 6689	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘))) Fn 𝐾)
110	19, 17, 98, 99, 101, 105	pwselbas 17452	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))):𝐾⟶𝐾)
111	110	ffnd 6689	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))) Fn 𝐾)
112		inidm 4190	. . . . . . . 8 ⊢ (𝐾 ∩ 𝐾) = 𝐾
113	20	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → 𝑆 ∈ CRing)
114	1	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → 𝑅 ∈ (SubRing‘𝑆))
115	17	subrgss 20481	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐾)
116	1, 115	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ⊆ 𝐾)
117	2, 17	ressbas2 17208	. . . . . . . . . . . . 13 ⊢ (𝑅 ⊆ 𝐾 → 𝑅 = (Base‘𝑈))
118	116, 117	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 = (Base‘𝑈))
119	118	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑅 = (Base‘𝑈))
120	26, 119	eleqtrrd 2831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ 𝑅)
121	120	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → (𝐴‘𝑘) ∈ 𝑅)
122		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝐾)
123	16, 6, 2, 17, 82, 113, 114, 121, 122	evls1scafv 22253	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → ((𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘)))‘𝑥) = (𝐴‘𝑘))
124		evls1fpws.2	. . . . . . . . 9 ⊢ ↑ = (.g‘(mulGrp‘𝑆))
125		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → 𝑘 ∈ ℕ0)
126	16, 2, 6, 7, 17, 11, 124, 113, 114, 125, 122	evls1varpwval 22255	. . . . . . . 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 7662	. . . . . . 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 2764	. . . . . 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 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))) = (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))
130	129	mpteq2dva 5200	. . . 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 7403	. . 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 2729	. . . 4 ⊢ (0g‘(𝑆 ↑s 𝐾)) = (0g‘(𝑆 ↑s 𝐾))
133	100	a1i 11	. . . 4 ⊢ (𝜑 → 𝐾 ∈ V)
134		nn0ex 12448	. . . . 5 ⊢ ℕ0 ∈ V
135	134	a1i 11	. . . 4 ⊢ (𝜑 → ℕ0 ∈ V)
136	20	crngringd 20155	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
137	136	ringcmnd 20193	. . . 4 ⊢ (𝜑 → 𝑆 ∈ CMnd)
138	136	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → 𝑆 ∈ Ring)
139	1	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ (SubRing‘𝑆))
140	139, 115	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑅 ⊆ 𝐾)
141	140, 120	sseldd 3947	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ 𝐾)
142	141	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → (𝐴‘𝑘) ∈ 𝐾)
143		eqid 2729	. . . . . . . . . 10 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
144	143, 17	mgpbas 20054	. . . . . . . . 9 ⊢ 𝐾 = (Base‘(mulGrp‘𝑆))
145	143	ringmgp 20148	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd)
146	136, 145	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd)
147	146	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → (mulGrp‘𝑆) ∈ Mnd)
148	144, 124, 147, 125, 122	mulgnn0cld 19027	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → (𝑘 ↑ 𝑥) ∈ 𝐾)
149	17, 106, 138, 142, 148	ringcld 20169	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)) ∈ 𝐾)
150	149	3impa 1109	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0 ∧ 𝑥 ∈ 𝐾) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)) ∈ 𝐾)
151	150	3com23 1126	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)) ∈ 𝐾)
152	151	3expb 1120	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑘 ∈ ℕ0)) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)) ∈ 𝐾)
153	135	mptexd 7198	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)))) ∈ V)
154		funmpt 6554	. . . . . 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 6873	. . . . 5 ⊢ (𝜑 → (0g‘(𝑆 ↑s 𝐾)) ∈ V)
157	12, 8, 6, 52	coe1sfi 22098	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → 𝐴 finSupp (0g‘𝑈))
158	5, 157	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑈))
159	158	fsuppimpd 9320	. . . . 5 ⊢ (𝜑 → (𝐴 supp (0g‘𝑈)) ∈ Fin)
160	12, 8, 6, 24	coe1f 22096	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ 𝐵 → 𝐴:ℕ0⟶(Base‘𝑈))
161	5, 160	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴:ℕ0⟶(Base‘𝑈))
162	161	ffnd 6689	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 Fn ℕ0)
163	162	adantr 480	. . . . . . . . . . . . . 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 6873	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (0g‘𝑈) ∈ V)
166		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈))))
167	163, 164, 165, 166	fvdifsupp 8150	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝐴‘𝑘) = (0g‘𝑈))
168		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑆) = (0g‘𝑆)
169	2, 168	subrg0 20488	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (0g‘𝑆) = (0g‘𝑈))
170	1, 169	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝑆) = (0g‘𝑈))
171	170	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (0g‘𝑆) = (0g‘𝑈))
172	167, 171	eqtr4d 2767	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝐴‘𝑘) = (0g‘𝑆))
173	172	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → (𝐴‘𝑘) = (0g‘𝑆))
174	173	oveq1d 7402	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)) = ((0g‘𝑆) · (𝑘 ↑ 𝑥)))
175	136	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → 𝑆 ∈ Ring)
176	175, 145	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → (mulGrp‘𝑆) ∈ Mnd)
177		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈))))
178	177	eldifad 3926	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → 𝑘 ∈ ℕ0)
179		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → 𝑥 ∈ 𝐾)
180	144, 124, 176, 178, 179	mulgnn0cld 19027	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → (𝑘 ↑ 𝑥) ∈ 𝐾)
181	17, 106, 168	ringlz 20202	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Ring ∧ (𝑘 ↑ 𝑥) ∈ 𝐾) → ((0g‘𝑆) · (𝑘 ↑ 𝑥)) = (0g‘𝑆))
182	175, 180, 181	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → ((0g‘𝑆) · (𝑘 ↑ 𝑥)) = (0g‘𝑆))
183	174, 182	eqtrd 2764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)) = (0g‘𝑆))
184	183	mpteq2dva 5200	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))) = (𝑥 ∈ 𝐾 ↦ (0g‘𝑆)))
185		fconstmpt 5700	. . . . . . . 8 ⊢ (𝐾 × {(0g‘𝑆)}) = (𝑥 ∈ 𝐾 ↦ (0g‘𝑆))
186	184, 185	eqtr4di 2782	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))) = (𝐾 × {(0g‘𝑆)}))
187	137	cmnmndd 19734	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Mnd)
188	19, 168	pws0g 18700	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ 𝐾 ∈ V) → (𝐾 × {(0g‘𝑆)}) = (0g‘(𝑆 ↑s 𝐾)))
189	187, 133, 188	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐾 × {(0g‘𝑆)}) = (0g‘(𝑆 ↑s 𝐾)))
190	189	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝐾 × {(0g‘𝑆)}) = (0g‘(𝑆 ↑s 𝐾)))
191	186, 190	eqtrd 2764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))) = (0g‘(𝑆 ↑s 𝐾)))
192	191, 135	suppss2 8179	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)))) supp (0g‘(𝑆 ↑s 𝐾))) ⊆ (𝐴 supp (0g‘𝑈)))
193		suppssfifsupp 9331	. . . . 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 1380	. . . 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 19912	. . 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 2768	. 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 2764	1 ⊢ (𝜑 → (𝑄‘𝑀) = (𝑥 ∈ 𝐾 ↦ (𝑆 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ∖ cdif 3911   ⊆ wss 3914  {csn 4589   class class class wbr 5107   ↦ cmpt 5188   × cxp 5636  Fun wfun 6505   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651   supp csupp 8139  Fincfn 8918   finSupp cfsupp 9312   < clt 11208  ℕ₀cn0 12442  Basecbs 17179   ↾_s cress 17200  ._rcmulr 17221  Scalarcsca 17223   ·_𝑠 cvsca 17224  0_gc0g 17402   Σ_g cgsu 17403   ↑_s cpws 17409  Mndcmnd 18661  ._gcmg 18999  mulGrpcmgp 20049  Ringcrg 20142  CRingccrg 20143   RingHom crh 20378  SubRingcsubrg 20478  LModclmod 20766  AssAlgcasa 21759  algSccascl 21761  var₁cv1 22060  Poly₁cpl1 22061  coe₁cco1 22062   evalSub₁ ces1 22200

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-ofr 7654  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-sup 9393  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17547  df-mrc 17548  df-acs 17550  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-ghm 19145  df-cntz 19249  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-srg 20096  df-ring 20144  df-cring 20145  df-rhm 20381  df-subrng 20455  df-subrg 20479  df-lmod 20768  df-lss 20838  df-lsp 20878  df-assa 21762  df-asp 21763  df-ascl 21764  df-psr 21818  df-mvr 21819  df-mpl 21820  df-opsr 21822  df-evls 21981  df-evl 21982  df-psr1 22064  df-vr1 22065  df-ply1 22066  df-coe1 22067  df-evls1 22202  df-evl1 22203

This theorem is referenced by:  ressply1evl  22257  evl1fpws  33533  ressply1evls1  33534  evls1fldgencl  33665