Theorem evl1deg1 33654

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 7369	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑘(.g‘(mulGrp‘𝑅))𝑥) = (𝑘(.g‘(mulGrp‘𝑅))𝑋))
2	1	oveq2d 7377	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑥)) = ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))
3	2	mpteq2dv 5180	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑥))) = (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))))
4	3	oveq2d 7377	. . 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 2737	. . . 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 33642	. . 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 7396	. . 3 ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))) ∈ V)
17	4, 14, 15, 16	fvmptd4 6967	. 2 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑋) = (𝑅 Σg (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))))
18		eqid 2737	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
19		evl1deg1.6	. . 3 ⊢ + = (+g‘𝑅)
20	9	crngringd 20221	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
21	20	ringcmnd 20259	. . 3 ⊢ (𝜑 → 𝑅 ∈ CMnd)
22		nn0ex 12437	. . . 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 22189	. . . . 5 ⊢ ((𝑀 ∈ 𝑈 ∧ 𝑘 ∈ ℕ0) → (𝐶‘𝑘) ∈ 𝐾)
26	10, 25	sylan 581	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐶‘𝑘) ∈ 𝐾)
27		eqid 2737	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
28	27, 7	mgpbas 20120	. . . . 5 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
29	27	ringmgp 20214	. . . . . . 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 19065	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
35	7, 11, 24, 26, 34	ringcld 20235	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) ∈ 𝐾)
36		fvexd 6850	. . . 4 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
37		fveq2 6835	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐶‘𝑘) = (𝐶‘𝑗))
38		oveq1 7368	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘(.g‘(mulGrp‘𝑅))𝑋) = (𝑗(.g‘(mulGrp‘𝑅))𝑋))
39	37, 38	oveq12d 7379	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)))
40		breq1 5089	. . . . . . 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 3161	. . . . 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 12447	. . . . . 6 ⊢ 1 ∈ ℕ0
45	43, 44	eqeltrdi 2845	. . . . 5 ⊢ (𝜑 → (𝐷‘𝑀) ∈ ℕ0)
46	10	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → 𝑀 ∈ 𝑈)
47		simplr 769	. . . . . . . . . 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 26075	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑈 ∧ 𝑗 ∈ ℕ0 ∧ (𝐷‘𝑀) < 𝑗) → (𝐶‘𝑗) = (0g‘𝑅))
51	46, 47, 48, 50	syl3anc 1374	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → (𝐶‘𝑗) = (0g‘𝑅))
52	51	oveq1d 7376	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = ((0g‘𝑅) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)))
53	20	ad2antrr 727	. . . . . . . . 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 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → 𝑋 ∈ 𝐾)
56	28, 12, 54, 47, 55	mulgnn0cld 19065	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → (𝑗(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
57	7, 11, 18, 53, 56	ringlzd 20270	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → ((0g‘𝑅) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))
58	52, 57	eqtrd 2772	. . . . . . 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 3130	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ ℕ0 ((𝐷‘𝑀) < 𝑗 → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅)))
61	42, 45, 60	rspcedvdw 3568	. . . 4 ⊢ (𝜑 → ∃𝑖 ∈ ℕ0 ∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅)))
62	36, 35, 39, 61	mptnn0fsuppd 13954	. . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))) finSupp (0g‘𝑅))
63		nn0disj01 32910	. . . 4 ⊢ ({0, 1} ∩ (ℤ≥‘2)) = ∅
64	63	a1i 11	. . 3 ⊢ (𝜑 → ({0, 1} ∩ (ℤ≥‘2)) = ∅)
65		nn0split01 32909	. . . 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 19898	. 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 12446	. . . . . 6 ⊢ 0 ∈ ℕ0
69	68	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ ℕ0)
70	44	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ0)
71		0ne1 12246	. . . . . 6 ⊢ 0 ≠ 1
72	71	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ≠ 1)
73	13, 8, 6, 7	coe1fvalcl 22189	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑈 ∧ 0 ∈ ℕ0) → (𝐶‘0) ∈ 𝐾)
74	10, 68, 73	sylancl 587	. . . . . 6 ⊢ (𝜑 → (𝐶‘0) ∈ 𝐾)
75	28, 12, 30, 69, 15	mulgnn0cld 19065	. . . . . 6 ⊢ (𝜑 → (0(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
76	7, 11, 20, 74, 75	ringcld 20235	. . . . 5 ⊢ (𝜑 → ((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)) ∈ 𝐾)
77	13, 8, 6, 7	coe1fvalcl 22189	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑈 ∧ 1 ∈ ℕ0) → (𝐶‘1) ∈ 𝐾)
78	10, 44, 77	sylancl 587	. . . . . 6 ⊢ (𝜑 → (𝐶‘1) ∈ 𝐾)
79	28, 12, 30, 70, 15	mulgnn0cld 19065	. . . . . 6 ⊢ (𝜑 → (1(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
80	7, 11, 20, 78, 79	ringcld 20235	. . . . 5 ⊢ (𝜑 → ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋)) ∈ 𝐾)
81		fveq2 6835	. . . . . . 7 ⊢ (𝑘 = 0 → (𝐶‘𝑘) = (𝐶‘0))
82		oveq1 7368	. . . . . . 7 ⊢ (𝑘 = 0 → (𝑘(.g‘(mulGrp‘𝑅))𝑋) = (0(.g‘(mulGrp‘𝑅))𝑋))
83	81, 82	oveq12d 7379	. . . . . 6 ⊢ (𝑘 = 0 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = ((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)))
84		fveq2 6835	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐶‘𝑘) = (𝐶‘1))
85		oveq1 7368	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑘(.g‘(mulGrp‘𝑅))𝑋) = (1(.g‘(mulGrp‘𝑅))𝑋))
86	84, 85	oveq12d 7379	. . . . . 6 ⊢ (𝑘 = 1 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋)))
87	7, 19, 83, 86	gsumpr 19924	. . . . 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 1391	. . . 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 12825	. . . . . . . . . . . . 13 ⊢ 2 ∈ (ℤ≥‘0)
91		uzss 12805	. . . . . . . . . . . . 13 ⊢ (2 ∈ (ℤ≥‘0) → (ℤ≥‘2) ⊆ (ℤ≥‘0))
92	90, 91	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℤ≥‘2) ⊆ (ℤ≥‘0)
93		nn0uz 12820	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
94	92, 93	sseqtrri 3972	. . . . . . . . . . 11 ⊢ (ℤ≥‘2) ⊆ ℕ0
95	94	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ≥‘2) ⊆ ℕ0)
96	95	sselda 3922	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → 𝑘 ∈ ℕ0)
97	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝐷‘𝑀) = 1)
98		eluz2gt1 12864	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘2) → 1 < 𝑘)
99	98	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → 1 < 𝑘)
100	97, 99	eqbrtrd 5108	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝐷‘𝑀) < 𝑘)
101	49, 6, 8, 18, 13	deg1lt 26075	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑈 ∧ 𝑘 ∈ ℕ0 ∧ (𝐷‘𝑀) < 𝑘) → (𝐶‘𝑘) = (0g‘𝑅))
102	89, 96, 100, 101	syl3anc 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝐶‘𝑘) = (0g‘𝑅))
103	102	oveq1d 7376	. . . . . . 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 19065	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝑘(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
108	7, 11, 18, 104, 107	ringlzd 20270	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → ((0g‘𝑅) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))
109	103, 108	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))
110	109	mpteq2dva 5179	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ (ℤ≥‘2) ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))) = (𝑘 ∈ (ℤ≥‘2) ↦ (0g‘𝑅)))
111	110	oveq2d 7377	. . . 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 7379	. . 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 2737	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
114		evl1deg1.10	. . . . . . . 8 ⊢ 𝐵 = (𝐶‘0)
115	114, 74	eqeltrid 2841	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝐾)
116	7, 11, 113, 20, 115	ringridmd 20248	. . . . . 6 ⊢ (𝜑 → (𝐵 · (1r‘𝑅)) = 𝐵)
117	116	oveq1d 7376	. . . . 5 ⊢ (𝜑 → ((𝐵 · (1r‘𝑅)) + (𝐴 · 𝑋)) = (𝐵 + (𝐴 · 𝑋)))
118	114	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝐶‘0))
119	27, 113	ringidval 20158	. . . . . . . . . 10 ⊢ (1r‘𝑅) = (0g‘(mulGrp‘𝑅))
120	28, 119, 12	mulg0 19044	. . . . . . . . 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 2743	. . . . . . 7 ⊢ (𝜑 → (1r‘𝑅) = (0(.g‘(mulGrp‘𝑅))𝑋))
123	118, 122	oveq12d 7379	. . . . . 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 19051	. . . . . . . . 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 2743	. . . . . . 7 ⊢ (𝜑 → 𝑋 = (1(.g‘(mulGrp‘𝑅))𝑋))
129	125, 128	oveq12d 7379	. . . . . 6 ⊢ (𝜑 → (𝐴 · 𝑋) = ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋)))
130	123, 129	oveq12d 7379	. . . . 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 2841	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐾)
132	7, 11, 20, 131, 15	ringcld 20235	. . . . . 6 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ 𝐾)
133	7, 19	ringcom 20255	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐵 ∈ 𝐾 ∧ (𝐴 · 𝑋) ∈ 𝐾) → (𝐵 + (𝐴 · 𝑋)) = ((𝐴 · 𝑋) + 𝐵))
134	20, 115, 132, 133	syl3anc 1374	. . . . 5 ⊢ (𝜑 → (𝐵 + (𝐴 · 𝑋)) = ((𝐴 · 𝑋) + 𝐵))
135	117, 130, 134	3eqtr3d 2780	. . . 4 ⊢ (𝜑 → (((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)) + ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋))) = ((𝐴 · 𝑋) + 𝐵))
136	9	crnggrpd 20222	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
137	136	grpmndd 18916	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Mnd)
138		fvexd 6850	. . . . 5 ⊢ (𝜑 → (ℤ≥‘2) ∈ V)
139	18	gsumz 18798	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (ℤ≥‘2) ∈ V) → (𝑅 Σg (𝑘 ∈ (ℤ≥‘2) ↦ (0g‘𝑅))) = (0g‘𝑅))
140	137, 138, 139	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ (ℤ≥‘2) ↦ (0g‘𝑅))) = (0g‘𝑅))
141	135, 140	oveq12d 7379	. . 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 18917	. . . 4 ⊢ (𝜑 → ((𝐴 · 𝑋) + 𝐵) ∈ 𝐾)
143	7, 19, 18, 136, 142	grpridd 18940	. . 3 ⊢ (𝜑 → (((𝐴 · 𝑋) + 𝐵) + (0g‘𝑅)) = ((𝐴 · 𝑋) + 𝐵))
144	112, 141, 143	3eqtrd 2776	. 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 2776	1 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑋) = ((𝐴 · 𝑋) + 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3430   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  {cpr 4570   class class class wbr 5086   ↦ cmpt 5167  ‘cfv 6493  (class class class)co 7361  0cc0 11032  1c1 11033   < clt 11173  2c2 12230  ℕ₀cn0 12431  ℤ_≥cuz 12782  Basecbs 17173  +_gcplusg 17214  ._rcmulr 17215  0_gc0g 17396   Σ_g cgsu 17397  Mndcmnd 18696  ._gcmg 19037  CMndccmn 19749  mulGrpcmgp 20115  1_rcur 20156  Ringcrg 20208  CRingccrg 20209  Poly₁cpl1 22153  coe₁cco1 22154  eval₁ce1 22292  deg₁cdg1 26032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110  ax-addf 11111

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7625  df-ofr 7626  df-om 7812  df-1st 7936  df-2nd 7937  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-sup 9349  df-oi 9419  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-uz 12783  df-fz 13456  df-fzo 13603  df-seq 13958  df-hash 14287  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-starv 17229  df-sca 17230  df-vsca 17231  df-ip 17232  df-tset 17233  df-ple 17234  df-ds 17236  df-unif 17237  df-hom 17238  df-cco 17239  df-0g 17398  df-gsum 17399  df-prds 17404  df-pws 17406  df-mre 17542  df-mrc 17543  df-acs 17545  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-submnd 18746  df-grp 18906  df-minusg 18907  df-sbg 18908  df-mulg 19038  df-subg 19093  df-ghm 19182  df-cntz 19286  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-srg 20162  df-ring 20210  df-cring 20211  df-rhm 20446  df-subrng 20517  df-subrg 20541  df-lmod 20851  df-lss 20921  df-lsp 20961  df-cnfld 21348  df-assa 21846  df-asp 21847  df-ascl 21848  df-psr 21902  df-mvr 21903  df-mpl 21904  df-opsr 21906  df-evls 22065  df-evl 22066  df-psr1 22156  df-vr1 22157  df-ply1 22158  df-coe1 22159  df-evls1 22293  df-evl1 22294  df-mdeg 26033  df-deg1 26034

This theorem is referenced by:  ply1dg1rt  33658