Theorem evl1deg1 33601

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 7439	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑘(.g‘(mulGrp‘𝑅))𝑥) = (𝑘(.g‘(mulGrp‘𝑅))𝑋))
2	1	oveq2d 7447	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑥)) = ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))
3	2	mpteq2dv 5244	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑥))) = (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))))
4	3	oveq2d 7447	. . 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 33590	. . 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 7466	. . 3 ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))) ∈ V)
17	4, 14, 15, 16	fvmptd4 7040	. 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 20243	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
21	20	ringcmnd 20281	. . 3 ⊢ (𝜑 → 𝑅 ∈ CMnd)
22		nn0ex 12532	. . . 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 22214	. . . . 5 ⊢ ((𝑀 ∈ 𝑈 ∧ 𝑘 ∈ ℕ0) → (𝐶‘𝑘) ∈ 𝐾)
26	10, 25	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐶‘𝑘) ∈ 𝐾)
27		eqid 2737	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
28	27, 7	mgpbas 20142	. . . . 5 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
29	27	ringmgp 20236	. . . . . . 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 19113	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
35	7, 11, 24, 26, 34	ringcld 20257	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) ∈ 𝐾)
36		fvexd 6921	. . . 4 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
37		fveq2 6906	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐶‘𝑘) = (𝐶‘𝑗))
38		oveq1 7438	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘(.g‘(mulGrp‘𝑅))𝑋) = (𝑗(.g‘(mulGrp‘𝑅))𝑋))
39	37, 38	oveq12d 7449	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)))
40		breq1 5146	. . . . . . 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 3178	. . . . 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 12542	. . . . . 6 ⊢ 1 ∈ ℕ0
45	43, 44	eqeltrdi 2849	. . . . 5 ⊢ (𝜑 → (𝐷‘𝑀) ∈ ℕ0)
46	10	ad2antrr 726	. . . . . . . . . 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 26136	. . . . . . . . . 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 7446	. . . . . . . 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 19113	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → (𝑗(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
57	7, 11, 18, 53, 56	ringlzd 20292	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → ((0g‘𝑅) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))
58	52, 57	eqtrd 2777	. . . . . . 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 3146	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ ℕ0 ((𝐷‘𝑀) < 𝑗 → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅)))
61	42, 45, 60	rspcedvdw 3625	. . . 4 ⊢ (𝜑 → ∃𝑖 ∈ ℕ0 ∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅)))
62	36, 35, 39, 61	mptnn0fsuppd 14039	. . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))) finSupp (0g‘𝑅))
63		nn0disj01 32820	. . . 4 ⊢ ({0, 1} ∩ (ℤ≥‘2)) = ∅
64	63	a1i 11	. . 3 ⊢ (𝜑 → ({0, 1} ∩ (ℤ≥‘2)) = ∅)
65		nn0split01 32819	. . . 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 19947	. 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 12541	. . . . . 6 ⊢ 0 ∈ ℕ0
69	68	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ ℕ0)
70	44	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ0)
71		0ne1 12337	. . . . . 6 ⊢ 0 ≠ 1
72	71	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ≠ 1)
73	13, 8, 6, 7	coe1fvalcl 22214	. . . . . . 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 19113	. . . . . 6 ⊢ (𝜑 → (0(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
76	7, 11, 20, 74, 75	ringcld 20257	. . . . 5 ⊢ (𝜑 → ((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)) ∈ 𝐾)
77	13, 8, 6, 7	coe1fvalcl 22214	. . . . . . 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 19113	. . . . . 6 ⊢ (𝜑 → (1(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
80	7, 11, 20, 78, 79	ringcld 20257	. . . . 5 ⊢ (𝜑 → ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋)) ∈ 𝐾)
81		fveq2 6906	. . . . . . 7 ⊢ (𝑘 = 0 → (𝐶‘𝑘) = (𝐶‘0))
82		oveq1 7438	. . . . . . 7 ⊢ (𝑘 = 0 → (𝑘(.g‘(mulGrp‘𝑅))𝑋) = (0(.g‘(mulGrp‘𝑅))𝑋))
83	81, 82	oveq12d 7449	. . . . . 6 ⊢ (𝑘 = 0 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = ((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)))
84		fveq2 6906	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐶‘𝑘) = (𝐶‘1))
85		oveq1 7438	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑘(.g‘(mulGrp‘𝑅))𝑋) = (1(.g‘(mulGrp‘𝑅))𝑋))
86	84, 85	oveq12d 7449	. . . . . 6 ⊢ (𝑘 = 1 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋)))
87	7, 19, 83, 86	gsumpr 19973	. . . . 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 12935	. . . . . . . . . . . . 13 ⊢ 2 ∈ (ℤ≥‘0)
91		uzss 12901	. . . . . . . . . . . . 13 ⊢ (2 ∈ (ℤ≥‘0) → (ℤ≥‘2) ⊆ (ℤ≥‘0))
92	90, 91	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℤ≥‘2) ⊆ (ℤ≥‘0)
93		nn0uz 12920	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
94	92, 93	sseqtrri 4033	. . . . . . . . . . 11 ⊢ (ℤ≥‘2) ⊆ ℕ0
95	94	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ≥‘2) ⊆ ℕ0)
96	95	sselda 3983	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → 𝑘 ∈ ℕ0)
97	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝐷‘𝑀) = 1)
98		eluz2gt1 12962	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘2) → 1 < 𝑘)
99	98	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → 1 < 𝑘)
100	97, 99	eqbrtrd 5165	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝐷‘𝑀) < 𝑘)
101	49, 6, 8, 18, 13	deg1lt 26136	. . . . . . . . 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 7446	. . . . . . 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 19113	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝑘(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
108	7, 11, 18, 104, 107	ringlzd 20292	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → ((0g‘𝑅) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))
109	103, 108	eqtrd 2777	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))
110	109	mpteq2dva 5242	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ (ℤ≥‘2) ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))) = (𝑘 ∈ (ℤ≥‘2) ↦ (0g‘𝑅)))
111	110	oveq2d 7447	. . . 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 7449	. . 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 2845	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝐾)
116	7, 11, 113, 20, 115	ringridmd 20270	. . . . . 6 ⊢ (𝜑 → (𝐵 · (1r‘𝑅)) = 𝐵)
117	116	oveq1d 7446	. . . . 5 ⊢ (𝜑 → ((𝐵 · (1r‘𝑅)) + (𝐴 · 𝑋)) = (𝐵 + (𝐴 · 𝑋)))
118	114	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝐶‘0))
119	27, 113	ringidval 20180	. . . . . . . . . 10 ⊢ (1r‘𝑅) = (0g‘(mulGrp‘𝑅))
120	28, 119, 12	mulg0 19092	. . . . . . . . 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 7449	. . . . . 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 19099	. . . . . . . . 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 7449	. . . . . 6 ⊢ (𝜑 → (𝐴 · 𝑋) = ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋)))
130	123, 129	oveq12d 7449	. . . . 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 2845	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐾)
132	7, 11, 20, 131, 15	ringcld 20257	. . . . . 6 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ 𝐾)
133	7, 19	ringcom 20277	. . . . . 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 2785	. . . 4 ⊢ (𝜑 → (((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)) + ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋))) = ((𝐴 · 𝑋) + 𝐵))
136	9	crnggrpd 20244	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
137	136	grpmndd 18964	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Mnd)
138		fvexd 6921	. . . . 5 ⊢ (𝜑 → (ℤ≥‘2) ∈ V)
139	18	gsumz 18849	. . . . 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 7449	. . 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 18965	. . . 4 ⊢ (𝜑 → ((𝐴 · 𝑋) + 𝐵) ∈ 𝐾)
143	7, 19, 18, 136, 142	grpridd 18988	. . 3 ⊢ (𝜑 → (((𝐴 · 𝑋) + 𝐵) + (0g‘𝑅)) = ((𝐴 · 𝑋) + 𝐵))
144	112, 141, 143	3eqtrd 2781	. 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 2781	1 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑋) = ((𝐴 · 𝑋) + 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  Vcvv 3480   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  {cpr 4628   class class class wbr 5143   ↦ cmpt 5225  ‘cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156   < clt 11295  2c2 12321  ℕ₀cn0 12526  ℤ_≥cuz 12878  Basecbs 17247  +_gcplusg 17297  ._rcmulr 17298  0_gc0g 17484   Σ_g cgsu 17485  Mndcmnd 18747  ._gcmg 19085  CMndccmn 19798  mulGrpcmgp 20137  1_rcur 20178  Ringcrg 20230  CRingccrg 20231  Poly₁cpl1 22178  coe₁cco1 22179  eval₁ce1 22318  deg₁cdg1 26093

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-rhm 20472  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-lsp 20970  df-cnfld 21365  df-assa 21873  df-asp 21874  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-evls 22098  df-evl 22099  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-evls1 22319  df-evl1 22320  df-mdeg 26094  df-deg1 26095

This theorem is referenced by:  ply1dg1rt  33604