Theorem evl1deg1 33511

Description: Evaluation of a univariate polynomial of degree 1. (Contributed by Thierry Arnoux, 8-Jun-2025.)

Hypotheses

Ref	Expression
evl1deg1.1	⊢ 𝑃 = (Poly1‘𝑅)
evl1deg1.2	⊢ 𝑂 = (eval1‘𝑅)
evl1deg1.3	⊢ 𝐾 = (Base‘𝑅)
evl1deg1.4	⊢ 𝑈 = (Base‘𝑃)
evl1deg1.5	⊢ · = (.r‘𝑅)
evl1deg1.6	⊢ + = (+g‘𝑅)
evl1deg1.7	⊢ 𝐶 = (coe1‘𝑀)
evl1deg1.8	⊢ 𝐷 = (deg1‘𝑅)
evl1deg1.9	⊢ 𝐴 = (𝐶‘1)
evl1deg1.10	⊢ 𝐵 = (𝐶‘0)
evl1deg1.11	⊢ (𝜑 → 𝑅 ∈ CRing)
evl1deg1.12	⊢ (𝜑 → 𝑀 ∈ 𝑈)
evl1deg1.13	⊢ (𝜑 → (𝐷‘𝑀) = 1)
evl1deg1.14	⊢ (𝜑 → 𝑋 ∈ 𝐾)

Assertion

Ref	Expression
evl1deg1	⊢ (𝜑 → ((𝑂‘𝑀)‘𝑋) = ((𝐴 · 𝑋) + 𝐵))

Proof of Theorem evl1deg1

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

Step	Hyp	Ref	Expression
1		oveq2 7357	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑘(.g‘(mulGrp‘𝑅))𝑥) = (𝑘(.g‘(mulGrp‘𝑅))𝑋))
2	1	oveq2d 7365	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑥)) = ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))
3	2	mpteq2dv 5186	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑥))) = (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))))
4	3	oveq2d 7365	. . 3 ⊢ (𝑥 = 𝑋 → (𝑅 Σg (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑥)))) = (𝑅 Σg (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))))
5		evl1deg1.2	. . . 4 ⊢ 𝑂 = (eval1‘𝑅)
6		evl1deg1.1	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
7		evl1deg1.3	. . . 4 ⊢ 𝐾 = (Base‘𝑅)
8		evl1deg1.4	. . . 4 ⊢ 𝑈 = (Base‘𝑃)
9		evl1deg1.11	. . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing)
10		evl1deg1.12	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
11		evl1deg1.5	. . . 4 ⊢ · = (.r‘𝑅)
12		eqid 2729	. . . 4 ⊢ (.g‘(mulGrp‘𝑅)) = (.g‘(mulGrp‘𝑅))
13		evl1deg1.7	. . . 4 ⊢ 𝐶 = (coe1‘𝑀)
14	5, 6, 7, 8, 9, 10, 11, 12, 13	evl1fpws 33499	. . 3 ⊢ (𝜑 → (𝑂‘𝑀) = (𝑥 ∈ 𝐾 ↦ (𝑅 Σg (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑥))))))
15		evl1deg1.14	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐾)
16		ovexd 7384	. . 3 ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))) ∈ V)
17	4, 14, 15, 16	fvmptd4 6954	. 2 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑋) = (𝑅 Σg (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))))
18		eqid 2729	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
19		evl1deg1.6	. . 3 ⊢ + = (+g‘𝑅)
20	9	crngringd 20131	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
21	20	ringcmnd 20169	. . 3 ⊢ (𝜑 → 𝑅 ∈ CMnd)
22		nn0ex 12390	. . . 4 ⊢ ℕ0 ∈ V
23	22	a1i 11	. . 3 ⊢ (𝜑 → ℕ0 ∈ V)
24	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
25	13, 8, 6, 7	coe1fvalcl 22095	. . . . 5 ⊢ ((𝑀 ∈ 𝑈 ∧ 𝑘 ∈ ℕ0) → (𝐶‘𝑘) ∈ 𝐾)
26	10, 25	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐶‘𝑘) ∈ 𝐾)
27		eqid 2729	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
28	27, 7	mgpbas 20030	. . . . 5 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
29	27	ringmgp 20124	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
30	20, 29	syl 17	. . . . . 6 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd)
31	30	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (mulGrp‘𝑅) ∈ Mnd)
32		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
33	15	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑋 ∈ 𝐾)
34	28, 12, 31, 32, 33	mulgnn0cld 18974	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
35	7, 11, 24, 26, 34	ringcld 20145	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) ∈ 𝐾)
36		fvexd 6837	. . . 4 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
37		fveq2 6822	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐶‘𝑘) = (𝐶‘𝑗))
38		oveq1 7356	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘(.g‘(mulGrp‘𝑅))𝑋) = (𝑗(.g‘(mulGrp‘𝑅))𝑋))
39	37, 38	oveq12d 7367	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)))
40		breq1 5095	. . . . . . 7 ⊢ (𝑖 = (𝐷‘𝑀) → (𝑖 < 𝑗 ↔ (𝐷‘𝑀) < 𝑗))
41	40	imbi1d 341	. . . . . 6 ⊢ (𝑖 = (𝐷‘𝑀) → ((𝑖 < 𝑗 → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅)) ↔ ((𝐷‘𝑀) < 𝑗 → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))))
42	41	ralbidv 3152	. . . . 5 ⊢ (𝑖 = (𝐷‘𝑀) → (∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅)) ↔ ∀𝑗 ∈ ℕ0 ((𝐷‘𝑀) < 𝑗 → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))))
43		evl1deg1.13	. . . . . 6 ⊢ (𝜑 → (𝐷‘𝑀) = 1)
44		1nn0 12400	. . . . . 6 ⊢ 1 ∈ ℕ0
45	43, 44	eqeltrdi 2836	. . . . 5 ⊢ (𝜑 → (𝐷‘𝑀) ∈ ℕ0)
46	10	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → 𝑀 ∈ 𝑈)
47		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → 𝑗 ∈ ℕ0)
48		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → (𝐷‘𝑀) < 𝑗)
49		evl1deg1.8	. . . . . . . . . . 11 ⊢ 𝐷 = (deg1‘𝑅)
50	49, 6, 8, 18, 13	deg1lt 26000	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑈 ∧ 𝑗 ∈ ℕ0 ∧ (𝐷‘𝑀) < 𝑗) → (𝐶‘𝑗) = (0g‘𝑅))
51	46, 47, 48, 50	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → (𝐶‘𝑗) = (0g‘𝑅))
52	51	oveq1d 7364	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = ((0g‘𝑅) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)))
53	20	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → 𝑅 ∈ Ring)
54	53, 29	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → (mulGrp‘𝑅) ∈ Mnd)
55	15	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → 𝑋 ∈ 𝐾)
56	28, 12, 54, 47, 55	mulgnn0cld 18974	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → (𝑗(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
57	7, 11, 18, 53, 56	ringlzd 20180	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → ((0g‘𝑅) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))
58	52, 57	eqtrd 2764	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))
59	58	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → ((𝐷‘𝑀) < 𝑗 → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅)))
60	59	ralrimiva 3121	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ ℕ0 ((𝐷‘𝑀) < 𝑗 → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅)))
61	42, 45, 60	rspcedvdw 3580	. . . 4 ⊢ (𝜑 → ∃𝑖 ∈ ℕ0 ∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅)))
62	36, 35, 39, 61	mptnn0fsuppd 13905	. . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))) finSupp (0g‘𝑅))
63		nn0disj01 32763	. . . 4 ⊢ ({0, 1} ∩ (ℤ≥‘2)) = ∅
64	63	a1i 11	. . 3 ⊢ (𝜑 → ({0, 1} ∩ (ℤ≥‘2)) = ∅)
65		nn0split01 32762	. . . 4 ⊢ ℕ0 = ({0, 1} ∪ (ℤ≥‘2))
66	65	a1i 11	. . 3 ⊢ (𝜑 → ℕ0 = ({0, 1} ∪ (ℤ≥‘2)))
67	7, 18, 19, 21, 23, 35, 62, 64, 66	gsumsplit2 19808	. 2 ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))) = ((𝑅 Σg (𝑘 ∈ {0, 1} ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))) + (𝑅 Σg (𝑘 ∈ (ℤ≥‘2) ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))))))
68		0nn0 12399	. . . . . 6 ⊢ 0 ∈ ℕ0
69	68	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ ℕ0)
70	44	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ0)
71		0ne1 12199	. . . . . 6 ⊢ 0 ≠ 1
72	71	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ≠ 1)
73	13, 8, 6, 7	coe1fvalcl 22095	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑈 ∧ 0 ∈ ℕ0) → (𝐶‘0) ∈ 𝐾)
74	10, 68, 73	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝐶‘0) ∈ 𝐾)
75	28, 12, 30, 69, 15	mulgnn0cld 18974	. . . . . 6 ⊢ (𝜑 → (0(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
76	7, 11, 20, 74, 75	ringcld 20145	. . . . 5 ⊢ (𝜑 → ((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)) ∈ 𝐾)
77	13, 8, 6, 7	coe1fvalcl 22095	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑈 ∧ 1 ∈ ℕ0) → (𝐶‘1) ∈ 𝐾)
78	10, 44, 77	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝐶‘1) ∈ 𝐾)
79	28, 12, 30, 70, 15	mulgnn0cld 18974	. . . . . 6 ⊢ (𝜑 → (1(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
80	7, 11, 20, 78, 79	ringcld 20145	. . . . 5 ⊢ (𝜑 → ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋)) ∈ 𝐾)
81		fveq2 6822	. . . . . . 7 ⊢ (𝑘 = 0 → (𝐶‘𝑘) = (𝐶‘0))
82		oveq1 7356	. . . . . . 7 ⊢ (𝑘 = 0 → (𝑘(.g‘(mulGrp‘𝑅))𝑋) = (0(.g‘(mulGrp‘𝑅))𝑋))
83	81, 82	oveq12d 7367	. . . . . 6 ⊢ (𝑘 = 0 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = ((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)))
84		fveq2 6822	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐶‘𝑘) = (𝐶‘1))
85		oveq1 7356	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑘(.g‘(mulGrp‘𝑅))𝑋) = (1(.g‘(mulGrp‘𝑅))𝑋))
86	84, 85	oveq12d 7367	. . . . . 6 ⊢ (𝑘 = 1 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋)))
87	7, 19, 83, 86	gsumpr 19834	. . . . 5 ⊢ ((𝑅 ∈ CMnd ∧ (0 ∈ ℕ0 ∧ 1 ∈ ℕ0 ∧ 0 ≠ 1) ∧ (((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)) ∈ 𝐾 ∧ ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋)) ∈ 𝐾)) → (𝑅 Σg (𝑘 ∈ {0, 1} ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))) = (((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)) + ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋))))
88	21, 69, 70, 72, 76, 80, 87	syl132anc 1390	. . . 4 ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ {0, 1} ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))) = (((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)) + ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋))))
89	10	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → 𝑀 ∈ 𝑈)
90		2eluzge0 12782	. . . . . . . . . . . . 13 ⊢ 2 ∈ (ℤ≥‘0)
91		uzss 12758	. . . . . . . . . . . . 13 ⊢ (2 ∈ (ℤ≥‘0) → (ℤ≥‘2) ⊆ (ℤ≥‘0))
92	90, 91	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℤ≥‘2) ⊆ (ℤ≥‘0)
93		nn0uz 12777	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
94	92, 93	sseqtrri 3985	. . . . . . . . . . 11 ⊢ (ℤ≥‘2) ⊆ ℕ0
95	94	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ≥‘2) ⊆ ℕ0)
96	95	sselda 3935	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → 𝑘 ∈ ℕ0)
97	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝐷‘𝑀) = 1)
98		eluz2gt1 12821	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘2) → 1 < 𝑘)
99	98	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → 1 < 𝑘)
100	97, 99	eqbrtrd 5114	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝐷‘𝑀) < 𝑘)
101	49, 6, 8, 18, 13	deg1lt 26000	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑈 ∧ 𝑘 ∈ ℕ0 ∧ (𝐷‘𝑀) < 𝑘) → (𝐶‘𝑘) = (0g‘𝑅))
102	89, 96, 100, 101	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝐶‘𝑘) = (0g‘𝑅))
103	102	oveq1d 7364	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = ((0g‘𝑅) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))
104	20	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → 𝑅 ∈ Ring)
105	104, 29	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (mulGrp‘𝑅) ∈ Mnd)
106	15	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → 𝑋 ∈ 𝐾)
107	28, 12, 105, 96, 106	mulgnn0cld 18974	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝑘(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
108	7, 11, 18, 104, 107	ringlzd 20180	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → ((0g‘𝑅) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))
109	103, 108	eqtrd 2764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))
110	109	mpteq2dva 5185	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ (ℤ≥‘2) ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))) = (𝑘 ∈ (ℤ≥‘2) ↦ (0g‘𝑅)))
111	110	oveq2d 7365	. . . 4 ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ (ℤ≥‘2) ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))) = (𝑅 Σg (𝑘 ∈ (ℤ≥‘2) ↦ (0g‘𝑅))))
112	88, 111	oveq12d 7367	. . 3 ⊢ (𝜑 → ((𝑅 Σg (𝑘 ∈ {0, 1} ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))) + (𝑅 Σg (𝑘 ∈ (ℤ≥‘2) ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))))) = ((((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)) + ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋))) + (𝑅 Σg (𝑘 ∈ (ℤ≥‘2) ↦ (0g‘𝑅)))))
113		eqid 2729	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
114		evl1deg1.10	. . . . . . . 8 ⊢ 𝐵 = (𝐶‘0)
115	114, 74	eqeltrid 2832	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝐾)
116	7, 11, 113, 20, 115	ringridmd 20158	. . . . . 6 ⊢ (𝜑 → (𝐵 · (1r‘𝑅)) = 𝐵)
117	116	oveq1d 7364	. . . . 5 ⊢ (𝜑 → ((𝐵 · (1r‘𝑅)) + (𝐴 · 𝑋)) = (𝐵 + (𝐴 · 𝑋)))
118	114	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝐶‘0))
119	27, 113	ringidval 20068	. . . . . . . . . 10 ⊢ (1r‘𝑅) = (0g‘(mulGrp‘𝑅))
120	28, 119, 12	mulg0 18953	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐾 → (0(.g‘(mulGrp‘𝑅))𝑋) = (1r‘𝑅))
121	15, 120	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0(.g‘(mulGrp‘𝑅))𝑋) = (1r‘𝑅))
122	121	eqcomd 2735	. . . . . . 7 ⊢ (𝜑 → (1r‘𝑅) = (0(.g‘(mulGrp‘𝑅))𝑋))
123	118, 122	oveq12d 7367	. . . . . 6 ⊢ (𝜑 → (𝐵 · (1r‘𝑅)) = ((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)))
124		evl1deg1.9	. . . . . . . 8 ⊢ 𝐴 = (𝐶‘1)
125	124	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐴 = (𝐶‘1))
126	28, 12	mulg1 18960	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐾 → (1(.g‘(mulGrp‘𝑅))𝑋) = 𝑋)
127	15, 126	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1(.g‘(mulGrp‘𝑅))𝑋) = 𝑋)
128	127	eqcomd 2735	. . . . . . 7 ⊢ (𝜑 → 𝑋 = (1(.g‘(mulGrp‘𝑅))𝑋))
129	125, 128	oveq12d 7367	. . . . . 6 ⊢ (𝜑 → (𝐴 · 𝑋) = ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋)))
130	123, 129	oveq12d 7367	. . . . 5 ⊢ (𝜑 → ((𝐵 · (1r‘𝑅)) + (𝐴 · 𝑋)) = (((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)) + ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋))))
131	124, 78	eqeltrid 2832	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐾)
132	7, 11, 20, 131, 15	ringcld 20145	. . . . . 6 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ 𝐾)
133	7, 19	ringcom 20165	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐵 ∈ 𝐾 ∧ (𝐴 · 𝑋) ∈ 𝐾) → (𝐵 + (𝐴 · 𝑋)) = ((𝐴 · 𝑋) + 𝐵))
134	20, 115, 132, 133	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝐵 + (𝐴 · 𝑋)) = ((𝐴 · 𝑋) + 𝐵))
135	117, 130, 134	3eqtr3d 2772	. . . 4 ⊢ (𝜑 → (((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)) + ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋))) = ((𝐴 · 𝑋) + 𝐵))
136	9	crnggrpd 20132	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
137	136	grpmndd 18825	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Mnd)
138		fvexd 6837	. . . . 5 ⊢ (𝜑 → (ℤ≥‘2) ∈ V)
139	18	gsumz 18710	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (ℤ≥‘2) ∈ V) → (𝑅 Σg (𝑘 ∈ (ℤ≥‘2) ↦ (0g‘𝑅))) = (0g‘𝑅))
140	137, 138, 139	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ (ℤ≥‘2) ↦ (0g‘𝑅))) = (0g‘𝑅))
141	135, 140	oveq12d 7367	. . 3 ⊢ (𝜑 → ((((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)) + ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋))) + (𝑅 Σg (𝑘 ∈ (ℤ≥‘2) ↦ (0g‘𝑅)))) = (((𝐴 · 𝑋) + 𝐵) + (0g‘𝑅)))
142	7, 19, 136, 132, 115	grpcld 18826	. . . 4 ⊢ (𝜑 → ((𝐴 · 𝑋) + 𝐵) ∈ 𝐾)
143	7, 19, 18, 136, 142	grpridd 18849	. . 3 ⊢ (𝜑 → (((𝐴 · 𝑋) + 𝐵) + (0g‘𝑅)) = ((𝐴 · 𝑋) + 𝐵))
144	112, 141, 143	3eqtrd 2768	. 2 ⊢ (𝜑 → ((𝑅 Σg (𝑘 ∈ {0, 1} ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))) + (𝑅 Σg (𝑘 ∈ (ℤ≥‘2) ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))))) = ((𝐴 · 𝑋) + 𝐵))
145	17, 67, 144	3eqtrd 2768	1 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑋) = ((𝐴 · 𝑋) + 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3436   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  {cpr 4579   class class class wbr 5092   ↦ cmpt 5173  ‘cfv 6482  (class class class)co 7349  0cc0 11009  1c1 11010   < clt 11149  2c2 12183  ℕ₀cn0 12384  ℤ_≥cuz 12735  Basecbs 17120  +_gcplusg 17161  ._rcmulr 17162  0_gc0g 17343   Σ_g cgsu 17344  Mndcmnd 18608  ._gcmg 18946  CMndccmn 19659  mulGrpcmgp 20025  1_rcur 20066  Ringcrg 20118  CRingccrg 20119  Poly₁cpl1 22059  coe₁cco1 22060  eval₁ce1 22199  deg₁cdg1 25957

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087  ax-addf 11088

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-ofr 7614  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-pm 8756  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-sup 9332  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-fzo 13558  df-seq 13909  df-hash 14238  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-hom 17185  df-cco 17186  df-0g 17345  df-gsum 17346  df-prds 17351  df-pws 17353  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-mhm 18657  df-submnd 18658  df-grp 18815  df-minusg 18816  df-sbg 18817  df-mulg 18947  df-subg 19002  df-ghm 19092  df-cntz 19196  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-srg 20072  df-ring 20120  df-cring 20121  df-rhm 20357  df-subrng 20431  df-subrg 20455  df-lmod 20765  df-lss 20835  df-lsp 20875  df-cnfld 21262  df-assa 21760  df-asp 21761  df-ascl 21762  df-psr 21816  df-mvr 21817  df-mpl 21818  df-opsr 21820  df-evls 21979  df-evl 21980  df-psr1 22062  df-vr1 22063  df-ply1 22064  df-coe1 22065  df-evls1 22200  df-evl1 22201  df-mdeg 25958  df-deg1 25959

This theorem is referenced by:  ply1dg1rt  33515