Theorem evl1deg1 34158

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 7432	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑘(.g‘(mulGrp‘𝑅))𝑥) = (𝑘(.g‘(mulGrp‘𝑅))𝑋))
2	1	oveq2d 7440	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑥)) = ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))
3	2	mpteq2dv 5198	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑥))) = (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))))
4	3	oveq2d 7440	. . 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 2760	. . . 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 34146	. . 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 7459	. . 3 ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))) ∈ V)
17	4, 14, 15, 16	fvmptd4 7023	. 2 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑋) = (𝑅 Σg (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))))
18		eqid 2760	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
19		evl1deg1.6	. . 3 ⊢ + = (+g‘𝑅)
20	9	crngringd 20482	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
21	20	ringcmnd 20522	. . 3 ⊢ (𝜑 → 𝑅 ∈ CMnd)
22		nn0ex 12619	. . . 4 ⊢ ℕ0 ∈ V
23	22	a1i 11	. . 3 ⊢ (𝜑 → ℕ0 ∈ V)
24	20	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
25	13, 8, 6, 7	coe1fvalcl 22539	. . . . 5 ⊢ ((𝑀 ∈ 𝑈 ∧ 𝑘 ∈ ℕ0) → (𝐶‘𝑘) ∈ 𝐾)
26	10, 25	sylan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐶‘𝑘) ∈ 𝐾)
27		eqid 2760	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
28	27, 7	mgpbas 20374	. . . . 5 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
29	27	ringmgp 20474	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
30	20, 29	syl 18	. . . . . 6 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd)
31	30	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (mulGrp‘𝑅) ∈ Mnd)
32		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
33	15	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑋 ∈ 𝐾)
34	28, 12, 31, 32, 33	mulgnn0cld 19314	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
35	7, 11, 24, 26, 34	ringcld 20493	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) ∈ 𝐾)
36		fvexd 6905	. . . 4 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
37		fveq2 6890	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐶‘𝑘) = (𝐶‘𝑗))
38		oveq1 7431	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘(.g‘(mulGrp‘𝑅))𝑋) = (𝑗(.g‘(mulGrp‘𝑅))𝑋))
39	37, 38	oveq12d 7442	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)))
40		breq1 5105	. . . . . . 7 ⊢ (𝑖 = (𝐷‘𝑀) → (𝑖 < 𝑗 ↔ (𝐷‘𝑀) < 𝑗))
41	40	imbi1d 344	. . . . . 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 3185	. . . . 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 12629	. . . . . 6 ⊢ 1 ∈ ℕ0
45	43, 44	eqeltrdi 2868	. . . . 5 ⊢ (𝜑 → (𝐷‘𝑀) ∈ ℕ0)
46	10	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → 𝑀 ∈ 𝑈)
47		simplr 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → 𝑗 ∈ ℕ0)
48		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → (𝐷‘𝑀) < 𝑗)
49		evl1deg1.8	. . . . . . . . . . 11 ⊢ 𝐷 = (deg1‘𝑅)
50	49, 6, 8, 18, 13	deg1lt 26424	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑈 ∧ 𝑗 ∈ ℕ0 ∧ (𝐷‘𝑀) < 𝑗) → (𝐶‘𝑗) = (0g‘𝑅))
51	46, 47, 48, 50	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → (𝐶‘𝑗) = (0g‘𝑅))
52	51	oveq1d 7439	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = ((0g‘𝑅) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)))
53	20	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → 𝑅 ∈ Ring)
54	53, 29	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → (mulGrp‘𝑅) ∈ Mnd)
55	15	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → 𝑋 ∈ 𝐾)
56	28, 12, 54, 47, 55	mulgnn0cld 19314	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → (𝑗(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
57	7, 11, 18, 53, 56	ringlzd 20535	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → ((0g‘𝑅) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))
58	52, 57	eqtrd 2795	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ (𝐷‘𝑀) < 𝑗) → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))
59	58	ex 418	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → ((𝐷‘𝑀) < 𝑗 → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅)))
60	59	ralrimiva 3154	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ ℕ0 ((𝐷‘𝑀) < 𝑗 → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅)))
61	42, 45, 60	rspcedvdw 3579	. . . 4 ⊢ (𝜑 → ∃𝑖 ∈ ℕ0 ∀𝑗 ∈ ℕ0 (𝑖 < 𝑗 → ((𝐶‘𝑗) · (𝑗(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅)))
62	36, 35, 39, 61	mptnn0fsuppd 14148	. . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))) finSupp (0g‘𝑅))
63		nn0disj01 33460	. . . 4 ⊢ ({0, 1} ∩ (ℤ≥‘2)) = ∅
64	63	a1i 11	. . 3 ⊢ (𝜑 → ({0, 1} ∩ (ℤ≥‘2)) = ∅)
65		nn0split01 33459	. . . 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 20152	. 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 12628	. . . . . 6 ⊢ 0 ∈ ℕ0
69	68	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ ℕ0)
70	44	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ0)
71		0ne1 12421	. . . . . 6 ⊢ 0 ≠ 1
72	71	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ≠ 1)
73	13, 8, 6, 7	coe1fvalcl 22539	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑈 ∧ 0 ∈ ℕ0) → (𝐶‘0) ∈ 𝐾)
74	10, 68, 73	sylancl 598	. . . . . 6 ⊢ (𝜑 → (𝐶‘0) ∈ 𝐾)
75	28, 12, 30, 69, 15	mulgnn0cld 19314	. . . . . 6 ⊢ (𝜑 → (0(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
76	7, 11, 20, 74, 75	ringcld 20493	. . . . 5 ⊢ (𝜑 → ((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)) ∈ 𝐾)
77	13, 8, 6, 7	coe1fvalcl 22539	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑈 ∧ 1 ∈ ℕ0) → (𝐶‘1) ∈ 𝐾)
78	10, 44, 77	sylancl 598	. . . . . 6 ⊢ (𝜑 → (𝐶‘1) ∈ 𝐾)
79	28, 12, 30, 70, 15	mulgnn0cld 19314	. . . . . 6 ⊢ (𝜑 → (1(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
80	7, 11, 20, 78, 79	ringcld 20493	. . . . 5 ⊢ (𝜑 → ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋)) ∈ 𝐾)
81		fveq2 6890	. . . . . . 7 ⊢ (𝑘 = 0 → (𝐶‘𝑘) = (𝐶‘0))
82		oveq1 7431	. . . . . . 7 ⊢ (𝑘 = 0 → (𝑘(.g‘(mulGrp‘𝑅))𝑋) = (0(.g‘(mulGrp‘𝑅))𝑋))
83	81, 82	oveq12d 7442	. . . . . 6 ⊢ (𝑘 = 0 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = ((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)))
84		fveq2 6890	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐶‘𝑘) = (𝐶‘1))
85		oveq1 7431	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑘(.g‘(mulGrp‘𝑅))𝑋) = (1(.g‘(mulGrp‘𝑅))𝑋))
86	84, 85	oveq12d 7442	. . . . . 6 ⊢ (𝑘 = 1 → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋)))
87	7, 19, 83, 86	gsumpr 20178	. . . . 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 1415	. . . 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 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → 𝑀 ∈ 𝑈)
90		2eluzge0 13015	. . . . . . . . . . . . 13 ⊢ 2 ∈ (ℤ≥‘0)
91		uzss 12995	. . . . . . . . . . . . 13 ⊢ (2 ∈ (ℤ≥‘0) → (ℤ≥‘2) ⊆ (ℤ≥‘0))
92	90, 91	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℤ≥‘2) ⊆ (ℤ≥‘0)
93		nn0uz 13010	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
94	92, 93	sseqtrri 3958	. . . . . . . . . . 11 ⊢ (ℤ≥‘2) ⊆ ℕ0
95	94	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ≥‘2) ⊆ ℕ0)
96	95	sselda 3909	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → 𝑘 ∈ ℕ0)
97	43	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝐷‘𝑀) = 1)
98		eluz2gt1 13054	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘2) → 1 < 𝑘)
99	98	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → 1 < 𝑘)
100	97, 99	eqbrtrd 5126	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝐷‘𝑀) < 𝑘)
101	49, 6, 8, 18, 13	deg1lt 26424	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑈 ∧ 𝑘 ∈ ℕ0 ∧ (𝐷‘𝑀) < 𝑘) → (𝐶‘𝑘) = (0g‘𝑅))
102	89, 96, 100, 101	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝐶‘𝑘) = (0g‘𝑅))
103	102	oveq1d 7439	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = ((0g‘𝑅) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)))
104	20	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → 𝑅 ∈ Ring)
105	104, 29	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (mulGrp‘𝑅) ∈ Mnd)
106	15	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → 𝑋 ∈ 𝐾)
107	28, 12, 105, 96, 106	mulgnn0cld 19314	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → (𝑘(.g‘(mulGrp‘𝑅))𝑋) ∈ 𝐾)
108	7, 11, 18, 104, 107	ringlzd 20535	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → ((0g‘𝑅) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))
109	103, 108	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘2)) → ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋)) = (0g‘𝑅))
110	109	mpteq2dva 5197	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ (ℤ≥‘2) ↦ ((𝐶‘𝑘) · (𝑘(.g‘(mulGrp‘𝑅))𝑋))) = (𝑘 ∈ (ℤ≥‘2) ↦ (0g‘𝑅)))
111	110	oveq2d 7440	. . . 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 7442	. . 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 2760	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
114		evl1deg1.10	. . . . . . . 8 ⊢ 𝐵 = (𝐶‘0)
115	114, 74	eqeltrid 2864	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝐾)
116	7, 11, 113, 20, 115	ringridmd 20511	. . . . . 6 ⊢ (𝜑 → (𝐵 · (1r‘𝑅)) = 𝐵)
117	116	oveq1d 7439	. . . . 5 ⊢ (𝜑 → ((𝐵 · (1r‘𝑅)) + (𝐴 · 𝑋)) = (𝐵 + (𝐴 · 𝑋)))
118	114	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝐶‘0))
119	27, 113	ringidval 20418	. . . . . . . . . 10 ⊢ (1r‘𝑅) = (0g‘(mulGrp‘𝑅))
120	28, 119, 12	mulg0 19293	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐾 → (0(.g‘(mulGrp‘𝑅))𝑋) = (1r‘𝑅))
121	15, 120	syl 18	. . . . . . . 8 ⊢ (𝜑 → (0(.g‘(mulGrp‘𝑅))𝑋) = (1r‘𝑅))
122	121	eqcomd 2766	. . . . . . 7 ⊢ (𝜑 → (1r‘𝑅) = (0(.g‘(mulGrp‘𝑅))𝑋))
123	118, 122	oveq12d 7442	. . . . . 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 19300	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐾 → (1(.g‘(mulGrp‘𝑅))𝑋) = 𝑋)
127	15, 126	syl 18	. . . . . . . 8 ⊢ (𝜑 → (1(.g‘(mulGrp‘𝑅))𝑋) = 𝑋)
128	127	eqcomd 2766	. . . . . . 7 ⊢ (𝜑 → 𝑋 = (1(.g‘(mulGrp‘𝑅))𝑋))
129	125, 128	oveq12d 7442	. . . . . 6 ⊢ (𝜑 → (𝐴 · 𝑋) = ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋)))
130	123, 129	oveq12d 7442	. . . . 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 2864	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐾)
132	7, 11, 20, 131, 15	ringcld 20493	. . . . . 6 ⊢ (𝜑 → (𝐴 · 𝑋) ∈ 𝐾)
133	7, 19	ringcom 20518	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐵 ∈ 𝐾 ∧ (𝐴 · 𝑋) ∈ 𝐾) → (𝐵 + (𝐴 · 𝑋)) = ((𝐴 · 𝑋) + 𝐵))
134	20, 115, 132, 133	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (𝐵 + (𝐴 · 𝑋)) = ((𝐴 · 𝑋) + 𝐵))
135	117, 130, 134	3eqtr3d 2803	. . . 4 ⊢ (𝜑 → (((𝐶‘0) · (0(.g‘(mulGrp‘𝑅))𝑋)) + ((𝐶‘1) · (1(.g‘(mulGrp‘𝑅))𝑋))) = ((𝐴 · 𝑋) + 𝐵))
136	9	crnggrpd 20483	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
137	136	grpmndd 19166	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Mnd)
138		fvexd 6905	. . . . 5 ⊢ (𝜑 → (ℤ≥‘2) ∈ V)
139	18	gsumz 19041	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ (ℤ≥‘2) ∈ V) → (𝑅 Σg (𝑘 ∈ (ℤ≥‘2) ↦ (0g‘𝑅))) = (0g‘𝑅))
140	137, 138, 139	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ (ℤ≥‘2) ↦ (0g‘𝑅))) = (0g‘𝑅))
141	135, 140	oveq12d 7442	. . 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 19167	. . . 4 ⊢ (𝜑 → ((𝐴 · 𝑋) + 𝐵) ∈ 𝐾)
143	7, 19, 18, 136, 142	grpridd 19190	. . 3 ⊢ (𝜑 → (((𝐴 · 𝑋) + 𝐵) + (0g‘𝑅)) = ((𝐴 · 𝑋) + 𝐵))
144	112, 141, 143	3eqtrd 2799	. 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 2799	1 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑋) = ((𝐴 · 𝑋) + 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088   ∪ cun 4148  ∅c0 4277  {cpr 4586   class class class wbr 5102   ↦ cmpt 5185  ‘cfv 6543  (class class class)co 7424  0cc0 11207  1c1 11208   < clt 11350  2c2 12404  ℕ₀cn0 12613  ℤ_≥cuz 12972  Basecbs 17396  +_gcplusg 17437  ._rcmulr 17438  0_gc0g 17619   Σ_g cgsu 17620  Mndcmnd 18932  ._gcmg 19286  CMndccmn 20003  mulGrpcmgp 20369  1_rcur 20416  Ringcrg 20468  CRingccrg 20469  Poly₁cpl1 22504  coe₁cco1 22505  eval₁ce1 22641  deg₁cdg1 26381

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284  ax-pre-sup 11285  ax-addf 11286

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7697  df-ofr 7698  df-om 7882  df-1st 8005  df-2nd 8006  df-supp 8185  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-2o 8485  df-er 8725  df-map 8857  df-pm 8858  df-ixp 8934  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-fsupp 9362  df-sup 9441  df-oi 9511  df-card 10027  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-2 12412  df-3 12413  df-4 12414  df-5 12415  df-6 12416  df-7 12417  df-8 12418  df-9 12419  df-n0 12614  df-z 12701  df-dec 12822  df-uz 12973  df-fz 13647  df-fzo 13796  df-seq 14152  df-hash 14482  df-struct 17334  df-sets 17351  df-slot 17369  df-ndx 17381  df-base 17397  df-ress 17418  df-plusg 17450  df-mulr 17451  df-starv 17452  df-sca 17453  df-vsca 17454  df-ip 17455  df-tset 17456  df-ple 17457  df-ds 17459  df-unif 17460  df-hom 17461  df-cco 17462  df-0g 17621  df-gsum 17622  df-prds 17627  df-pws 17629  df-mre 17765  df-mrc 17766  df-acs 17768  df-mgm 18825  df-sgrp 18917  df-mnd 18933  df-mhm 18987  df-submnd 18988  df-grp 19156  df-minusg 19157  df-sbg 19158  df-mulg 19287  df-subg 19342  df-ghm 19437  df-cntz 19540  df-cmn 20005  df-abl 20006  df-mgp 20370  df-rng 20384  df-ur 20417  df-srg 20422  df-ring 20470  df-cring 20471  df-rhm 20711  df-subrng 20807  df-subrg 20831  df-lmod 21146  df-lss 21216  df-lsp 21256  df-cnfld 21688  df-assa 22170  df-asp 22171  df-ascl 22172  df-psr 22226  df-mvr 22227  df-mpl 22228  df-opsr 22230  df-evls 22392  df-evl 22393  df-psr1 22507  df-vr1 22508  df-ply1 22509  df-coe1 22510  df-evls1 22642  df-evl1 22643  df-mdeg 26382  df-deg1 26383

This theorem is used by:  ply1dg1rt  34162