Theorem evls1fpws 22313

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 20507	. . . . 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 2736	. . . . 5 ⊢ (var1‘𝑈) = (var1‘𝑈)
8		ressply1evl2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
9		eqid 2736	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
10		eqid 2736	. . . . 5 ⊢ (mulGrp‘𝑊) = (mulGrp‘𝑊)
11		eqid 2736	. . . . 5 ⊢ (.g‘(mulGrp‘𝑊)) = (.g‘(mulGrp‘𝑊))
12		evls1fpws.a	. . . . 5 ⊢ 𝐴 = (coe1‘𝑀)
13	6, 7, 8, 9, 10, 11, 12	ply1coe 22242	. . . 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 6838	. 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 2736	. . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊)
19		eqid 2736	. . . 4 ⊢ (𝑆 ↑s 𝐾) = (𝑆 ↑s 𝐾)
20		evls1fpws.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ CRing)
21	6	ply1lmod 22192	. . . . . . 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 2736	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
25	12, 8, 6, 24	coe1fvalcl 22153	. . . . . . 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 22193	. . . . . . . . 9 ⊢ (𝑈 ∈ Ring → 𝑈 = (Scalar‘𝑊))
28	4, 27	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 = (Scalar‘𝑊))
29	28	fveq2d 6838	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑈) = (Base‘(Scalar‘𝑊)))
30	29	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (Base‘𝑈) = (Base‘(Scalar‘𝑊)))
31	26, 30	eleqtrd 2838	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
32	10, 8	mgpbas 20080	. . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑊))
33	6	ply1ring 22188	. . . . . . . . 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 20174	. . . . . . 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 22158	. . . . . . 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 19025	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)) ∈ 𝐵)
43		eqid 2736	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
44		eqid 2736	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
45	8, 43, 9, 44	lmodvscl 20829	. . . . 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 3957	. . . 4 ⊢ (𝜑 → ℕ0 ⊆ ℕ0)
48		fvexd 6849	. . . . 5 ⊢ (𝜑 → (0g‘𝑊) ∈ V)
49		fveq2 6834	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
50		oveq1 7365	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)) = (𝑗(.g‘(mulGrp‘𝑊))(var1‘𝑈)))
51	49, 50	oveq12d 7376	. . . . 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 2736	. . . . . . . 8 ⊢ (0g‘𝑈) = (0g‘𝑈)
53	12, 8, 6, 52	coe1ae0 22157	. . . . . . 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 6838	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → (0g‘𝑈) = (0g‘(Scalar‘𝑊)))
58	55, 57	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) ∧ (𝐴‘𝑗) = (0g‘𝑈)) → (𝐴‘𝑗) = (0g‘(Scalar‘𝑊)))
59	58	oveq1d 7373	. . . . . . . . . . 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 19025	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
69	8, 43, 9, 68, 18	lmod0vs 20846	. . . . . . . . . . . 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 2771	. . . . . . . . . 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 3148	. . . . . . 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 3149	. . . . . 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 13921	. . . 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 22268	. . 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 22266	. . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (algSc‘𝑊) = (algSc‘𝑊)
83	82, 43, 34, 22, 44, 8	asclf 21837	. . . . . . . . 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 7030	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((algSc‘𝑊)‘(𝐴‘𝑘)) ∈ 𝐵)
86		eqid 2736	. . . . . . . 8 ⊢ (.r‘𝑊) = (.r‘𝑊)
87		eqid 2736	. . . . . . . 8 ⊢ (.r‘(𝑆 ↑s 𝐾)) = (.r‘(𝑆 ↑s 𝐾))
88	8, 86, 87	rhmmul 20421	. . . . . . 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 20508	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
91	20, 1, 90	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ CRing)
92	6	ply1assa 22140	. . . . . . . . . 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 21842	. . . . . . . 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 6838	. . . . . 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 2736	. . . . . . . 8 ⊢ (Base‘(𝑆 ↑s 𝐾)) = (Base‘(𝑆 ↑s 𝐾))
99	20	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑆 ∈ CRing)
100	17	fvexi 6848	. . . . . . . . 9 ⊢ 𝐾 ∈ V
101	100	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐾 ∈ V)
102	8, 98	rhmf 20420	. . . . . . . . . 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 7030	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘))) ∈ (Base‘(𝑆 ↑s 𝐾)))
105	103, 42	ffvelcdmd 7030	. . . . . . . 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 17412	. . . . . . 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 17409	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘))):𝐾⟶𝐾)
109	108	ffnd 6663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘((algSc‘𝑊)‘(𝐴‘𝑘))) Fn 𝐾)
110	19, 17, 98, 99, 101, 105	pwselbas 17409	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))):𝐾⟶𝐾)
111	110	ffnd 6663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈))) Fn 𝐾)
112		inidm 4179	. . . . . . . 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 20505	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐾)
116	1, 115	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ⊆ 𝐾)
117	2, 17	ressbas2 17165	. . . . . . . . . . . . 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 2839	. . . . . . . . . 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 22310	. . . . . . . 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 22312	. . . . . . . 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 7631	. . . . . . 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 2771	. . . . . 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 2779	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑄‘((𝐴‘𝑘)( ·𝑠 ‘𝑊)(𝑘(.g‘(mulGrp‘𝑊))(var1‘𝑈)))) = (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))
130	129	mpteq2dva 5191	. . . 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 7374	. . 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 2736	. . . 4 ⊢ (0g‘(𝑆 ↑s 𝐾)) = (0g‘(𝑆 ↑s 𝐾))
133	100	a1i 11	. . . 4 ⊢ (𝜑 → 𝐾 ∈ V)
134		nn0ex 12407	. . . . 5 ⊢ ℕ0 ∈ V
135	134	a1i 11	. . . 4 ⊢ (𝜑 → ℕ0 ∈ V)
136	20	crngringd 20181	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
137	136	ringcmnd 20219	. . . 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 3934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ 𝐾)
142	141	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → (𝐴‘𝑘) ∈ 𝐾)
143		eqid 2736	. . . . . . . . . 10 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
144	143, 17	mgpbas 20080	. . . . . . . . 9 ⊢ 𝐾 = (Base‘(mulGrp‘𝑆))
145	143	ringmgp 20174	. . . . . . . . . . 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 19025	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ 𝐾) → (𝑘 ↑ 𝑥) ∈ 𝐾)
149	17, 106, 138, 142, 148	ringcld 20195	. . . . . . 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 7170	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)))) ∈ V)
154		funmpt 6530	. . . . . 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 6849	. . . . 5 ⊢ (𝜑 → (0g‘(𝑆 ↑s 𝐾)) ∈ V)
157	12, 8, 6, 52	coe1sfi 22154	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → 𝐴 finSupp (0g‘𝑈))
158	5, 157	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑈))
159	158	fsuppimpd 9272	. . . . 5 ⊢ (𝜑 → (𝐴 supp (0g‘𝑈)) ∈ Fin)
160	12, 8, 6, 24	coe1f 22152	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ 𝐵 → 𝐴:ℕ0⟶(Base‘𝑈))
161	5, 160	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴:ℕ0⟶(Base‘𝑈))
162	161	ffnd 6663	. . . . . . . . . . . . . . 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 6849	. . . . . . . . . . . . . 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 8113	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝐴‘𝑘) = (0g‘𝑈))
168		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑆) = (0g‘𝑆)
169	2, 168	subrg0 20512	. . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 7373	. . . . . . . . . 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 3913	. . . . . . . . . . . 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 19025	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → (𝑘 ↑ 𝑥) ∈ 𝐾)
181	17, 106, 168	ringlz 20228	. . . . . . . . . . 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 2771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) ∧ 𝑥 ∈ 𝐾) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)) = (0g‘𝑆))
184	183	mpteq2dva 5191	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))) = (𝑥 ∈ 𝐾 ↦ (0g‘𝑆)))
185		fconstmpt 5686	. . . . . . . 8 ⊢ (𝐾 × {(0g‘𝑆)}) = (𝑥 ∈ 𝐾 ↦ (0g‘𝑆))
186	184, 185	eqtr4di 2789	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))) = (𝐾 × {(0g‘𝑆)}))
187	137	cmnmndd 19733	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Mnd)
188	19, 168	pws0g 18698	. . . . . . . . 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 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℕ0 ∖ (𝐴 supp (0g‘𝑈)))) → (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))) = (0g‘(𝑆 ↑s 𝐾)))
192	191, 135	suppss2 8142	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦ (𝑥 ∈ 𝐾 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥)))) supp (0g‘(𝑆 ↑s 𝐾))) ⊆ (𝐴 supp (0g‘𝑈)))
193		suppssfifsupp 9283	. . . . 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 19911	. . 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 2775	. 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 2771	1 ⊢ (𝜑 → (𝑄‘𝑀) = (𝑥 ∈ 𝐾 ↦ (𝑆 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ∖ cdif 3898   ⊆ wss 3901  {csn 4580   class class class wbr 5098   ↦ cmpt 5179   × cxp 5622  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∘_f cof 7620   supp csupp 8102  Fincfn 8883   finSupp cfsupp 9264   < clt 11166  ℕ₀cn0 12401  Basecbs 17136   ↾_s cress 17157  ._rcmulr 17178  Scalarcsca 17180   ·_𝑠 cvsca 17181  0_gc0g 17359   Σ_g cgsu 17360   ↑_s cpws 17366  Mndcmnd 18659  ._gcmg 18997  mulGrpcmgp 20075  Ringcrg 20168  CRingccrg 20169   RingHom crh 20405  SubRingcsubrg 20502  LModclmod 20811  AssAlgcasa 21805  algSccascl 21807  var₁cv1 22116  Poly₁cpl1 22117  coe₁cco1 22118   evalSub₁ ces1 22257

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-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  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-se 5578  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-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ofr 7623  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-sup 9345  df-oi 9415  df-card 9851  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-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-hom 17201  df-cco 17202  df-0g 17361  df-gsum 17362  df-prds 17367  df-pws 17369  df-mre 17505  df-mrc 17506  df-acs 17508  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-subg 19053  df-ghm 19142  df-cntz 19246  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-srg 20122  df-ring 20170  df-cring 20171  df-rhm 20408  df-subrng 20479  df-subrg 20503  df-lmod 20813  df-lss 20883  df-lsp 20923  df-assa 21808  df-asp 21809  df-ascl 21810  df-psr 21865  df-mvr 21866  df-mpl 21867  df-opsr 21869  df-evls 22029  df-evl 22030  df-psr1 22120  df-vr1 22121  df-ply1 22122  df-coe1 22123  df-evls1 22259  df-evl1 22260

This theorem is referenced by:  ressply1evl  22314  evl1fpws  33645  ressply1evls1  33646  evls1fldgencl  33827