Theorem rngunsnply1chr2 34246

Description: When adjoining an element 𝑋 with characteristic polynomial of degree 2 to a subring 𝑆, the elements 𝑌 of the generated subring can be written as a linear form. (Contributed by Thierry Arnoux, 8-Nov-2026.)

Hypotheses

Ref	Expression
rngunsnply1chr2.v	⊢ 𝑉 = (Base‘𝑅)
rngunsnply1chr2.p	⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))
rngunsnply1chr2.u	⊢ 𝑈 = (Base‘𝑃)
rngunsnply1chr2.q	⊢ 𝑄 = (𝑅 evalSub1 𝑆)
rngunsnply1chr2.r	⊢ (𝜑 → 𝑅 ∈ CRing)
rngunsnply1chr2.1	⊢ (𝜑 → 𝑅 ∈ NzRing)
rngunsnply1chr2.s	⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))
rngunsnply1chr2.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
rngunsnply1chr2.y	⊢ (𝜑 → 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑋})))
rngunsnply1chr2.n	⊢ 𝑁 = (RingSpan‘𝑅)
rngunsnply1chr2.2	⊢ + = (+g‘𝑅)
rngunsnply1chr2.3	⊢ · = (.r‘𝑅)
rngunsnply1chr2.4	⊢ ↑ = (.g‘(mulGrp‘𝑅))
rngunsnply1chr2.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
rngunsnply1chr2.b	⊢ (𝜑 → 𝐵 ∈ 𝑆)
rngunsnply1chr2.5	⊢ (𝜑 → (2 ↑ 𝑋) = ((𝐴 · 𝑋) + 𝐵))

Assertion

Ref	Expression
rngunsnply1chr2	⊢ (𝜑 → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 𝑌 = ((𝑎 · 𝑋) + 𝑏))

Distinct variable groups:   + ,𝑎,𝑏   · ,𝑎,𝑏   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏   𝑃,𝑎,𝑏   𝑅,𝑎,𝑏   𝑆,𝑎,𝑏   𝑋,𝑎,𝑏   𝑌,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑎, 𝑏)   𝑄(𝑎, 𝑏)   𝑈(𝑎, 𝑏)   ↑ (𝑎, 𝑏)   𝑁(𝑎, 𝑏)   𝑉(𝑎, 𝑏)

Proof of Theorem rngunsnply1chr2

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7456	. . . . 5 ⊢ (𝑎 = ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) → (𝑎 · 𝑋) = (((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) · 𝑋))
2	1	oveq1d 7464	. . . 4 ⊢ (𝑎 = ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) → ((𝑎 · 𝑋) + 𝑏) = ((((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) · 𝑋) + 𝑏))
3	2	eqeq2d 2774	. . 3 ⊢ (𝑎 = ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) → (𝑌 = ((𝑎 · 𝑋) + 𝑏) ↔ 𝑌 = ((((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) · 𝑋) + 𝑏)))
4		oveq2 7457	. . . 4 ⊢ (𝑏 = ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘0) → ((((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) · 𝑋) + 𝑏) = ((((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) · 𝑋) + ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘0)))
5	4	eqeq2d 2774	. . 3 ⊢ (𝑏 = ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘0) → (𝑌 = ((((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) · 𝑋) + 𝑏) ↔ 𝑌 = ((((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) · 𝑋) + ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘0))))
6		rngunsnply1chr2.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))
7		eqid 2763	. . . . . . . . . 10 ⊢ (𝑅 ↾s 𝑆) = (𝑅 ↾s 𝑆)
8	7	subrgring 20868	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝑆) ∈ Ring)
9	6, 8	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ↾s 𝑆) ∈ Ring)
10	9	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑅 ↾s 𝑆) ∈ Ring)
11		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝑝 ∈ 𝑈)
12		rngunsnply1chr2.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑅)
13		rngunsnply1chr2.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆))
14		rngunsnply1chr2.u	. . . . . . . 8 ⊢ 𝑈 = (Base‘𝑃)
15		rngunsnply1chr2.q	. . . . . . . 8 ⊢ 𝑄 = (𝑅 evalSub1 𝑆)
16		rngunsnply1chr2.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CRing)
17	16	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝑅 ∈ CRing)
18		rngunsnply1chr2.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ NzRing)
19	18	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝑅 ∈ NzRing)
20	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝑆 ∈ (SubRing‘𝑅))
21		rngunsnply1chr2.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
22	21	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝑋 ∈ 𝑉)
23		rngunsnply1chr2.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑋})))
24	23	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑋})))
25		rngunsnply1chr2.n	. . . . . . . 8 ⊢ 𝑁 = (RingSpan‘𝑅)
26		rngunsnply1chr2.2	. . . . . . . 8 ⊢ + = (+g‘𝑅)
27		rngunsnply1chr2.3	. . . . . . . 8 ⊢ · = (.r‘𝑅)
28		rngunsnply1chr2.4	. . . . . . . 8 ⊢ ↑ = (.g‘(mulGrp‘𝑅))
29		rngunsnply1chr2.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
30	29	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐴 ∈ 𝑆)
31		rngunsnply1chr2.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
32	31	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐵 ∈ 𝑆)
33		rngunsnply1chr2.5	. . . . . . . . 9 ⊢ (𝜑 → (2 ↑ 𝑋) = ((𝐴 · 𝑋) + 𝐵))
34	33	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (2 ↑ 𝑋) = ((𝐴 · 𝑋) + 𝐵))
35		eqid 2763	. . . . . . . 8 ⊢ (deg1‘𝑅) = (deg1‘𝑅)
36		eqid 2763	. . . . . . . 8 ⊢ (rem1p‘(𝑅 ↾s 𝑆)) = (rem1p‘(𝑅 ↾s 𝑆))
37		eqid 2763	. . . . . . . 8 ⊢ ((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵))) = ((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))
38		eqid 2763	. . . . . . . 8 ⊢ (𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))) = (𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵))))
39	12, 13, 14, 15, 17, 19, 20, 22, 24, 25, 26, 27, 28, 30, 32, 34, 35, 36, 37, 38, 11	rngunsnply1chr2lem2 34244	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵))) ∈ (Unic1p‘(𝑅 ↾s 𝑆)))
40		eqid 2763	. . . . . . . 8 ⊢ (Unic1p‘(𝑅 ↾s 𝑆)) = (Unic1p‘(𝑅 ↾s 𝑆))
41	36, 13, 14, 40	r1pcl 26521	. . . . . . 7 ⊢ (((𝑅 ↾s 𝑆) ∈ Ring ∧ 𝑝 ∈ 𝑈 ∧ ((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵))) ∈ (Unic1p‘(𝑅 ↾s 𝑆))) → (𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))) ∈ 𝑈)
42	10, 11, 39, 41	syl3anc 1399	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))) ∈ 𝑈)
43		1nn0 12660	. . . . . 6 ⊢ 1 ∈ ℕ0
44		eqid 2763	. . . . . . 7 ⊢ (coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵))))) = (coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))
45		eqid 2763	. . . . . . 7 ⊢ (Base‘(𝑅 ↾s 𝑆)) = (Base‘(𝑅 ↾s 𝑆))
46	44, 14, 13, 45	coe1fvalcl 22574	. . . . . 6 ⊢ (((𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))) ∈ 𝑈 ∧ 1 ∈ ℕ0) → ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) ∈ (Base‘(𝑅 ↾s 𝑆)))
47	42, 43, 46	sylancl 598	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) ∈ (Base‘(𝑅 ↾s 𝑆)))
48	7	subrgbas 20875	. . . . . 6 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 = (Base‘(𝑅 ↾s 𝑆)))
49	20, 48	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝑆 = (Base‘(𝑅 ↾s 𝑆)))
50	47, 49	eleqtrrd 2867	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) ∈ 𝑆)
51	50	adantr 486	. . 3 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑈) ∧ 𝑌 = ((𝑄‘𝑝)‘𝑋)) → ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) ∈ 𝑆)
52		0nn0 12659	. . . . . 6 ⊢ 0 ∈ ℕ0
53	44, 14, 13, 45	coe1fvalcl 22574	. . . . . 6 ⊢ (((𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))) ∈ 𝑈 ∧ 0 ∈ ℕ0) → ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘0) ∈ (Base‘(𝑅 ↾s 𝑆)))
54	42, 52, 53	sylancl 598	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘0) ∈ (Base‘(𝑅 ↾s 𝑆)))
55	54, 49	eleqtrrd 2867	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘0) ∈ 𝑆)
56	55	adantr 486	. . 3 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑈) ∧ 𝑌 = ((𝑄‘𝑝)‘𝑋)) → ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘0) ∈ 𝑆)
57		simpr 490	. . . 4 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑈) ∧ 𝑌 = ((𝑄‘𝑝)‘𝑋)) → 𝑌 = ((𝑄‘𝑝)‘𝑋))
58	12, 13, 14, 15, 17, 19, 20, 22, 24, 25, 26, 27, 28, 30, 32, 34, 35, 36, 37, 38, 11	rngunsnply1chr2lem3 34245	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (((deg1‘𝑅)‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵))))) ≤ 1 ∧ ((𝑄‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘𝑋) = ((𝑄‘𝑝)‘𝑋)))
59	58	simprd 501	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((𝑄‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘𝑋) = ((𝑄‘𝑝)‘𝑋))
60	59	adantr 486	. . . 4 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑈) ∧ 𝑌 = ((𝑄‘𝑝)‘𝑋)) → ((𝑄‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘𝑋) = ((𝑄‘𝑝)‘𝑋))
61		eqid 2763	. . . . . 6 ⊢ ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) = ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1)
62		eqid 2763	. . . . . 6 ⊢ ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘0) = ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘0)
63	58	simpld 500	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((deg1‘𝑅)‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵))))) ≤ 1)
64	13, 15, 12, 14, 27, 26, 44, 35, 61, 62, 17, 20, 42, 63, 22	evls1degle1 34232	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((𝑄‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘𝑋) = ((((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) · 𝑋) + ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘0)))
65	64	adantr 486	. . . 4 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑈) ∧ 𝑌 = ((𝑄‘𝑝)‘𝑋)) → ((𝑄‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘𝑋) = ((((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) · 𝑋) + ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘0)))
66	57, 60, 65	3eqtr2d 2804	. . 3 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑈) ∧ 𝑌 = ((𝑄‘𝑝)‘𝑋)) → 𝑌 = ((((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘1) · 𝑋) + ((coe1‘(𝑝(rem1p‘(𝑅 ↾s 𝑆))((2(.g‘(mulGrp‘𝑃))(var1‘𝑅))(-g‘𝑃)((𝐴( ·𝑠 ‘𝑃)(var1‘𝑅))(+g‘𝑃)((algSc‘𝑃)‘𝐵)))))‘0)))
67	3, 5, 51, 56, 66	2rspcedvdw 3593	. 2 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑈) ∧ 𝑌 = ((𝑄‘𝑝)‘𝑋)) → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 𝑌 = ((𝑎 · 𝑋) + 𝑏))
68	12, 13, 14, 15, 16, 6, 21, 25	rngunsnply1 34242	. . 3 ⊢ (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑋})) ↔ ∃𝑝 ∈ 𝑈 𝑌 = ((𝑄‘𝑝)‘𝑋)))
69	23, 68	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑝 ∈ 𝑈 𝑌 = ((𝑄‘𝑝)‘𝑋))
70	67, 69	r19.29a 3174	1 ⊢ (𝜑 → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 𝑌 = ((𝑎 · 𝑋) + 𝑏))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ∪ cun 4152  {csn 4588   class class class wbr 5106  ‘cfv 6564  (class class class)co 7449  0cc0 11237  1c1 11238   ≤ cle 11381  2c2 12435  ℕ₀cn0 12644  Basecbs 17428   ↾_s cress 17449  +_gcplusg 17469  ._rcmulr 17470   ·_𝑠 cvsca 17473  -_gcsg 19187  ._gcmg 19318  mulGrpcmgp 20402  Ringcrg 20501  CRingccrg 20502  NzRingcnzr 20804  SubRingcsubrg 20863  RingSpancrgspn 20904  algSccascl 22204  var₁cv1 22538  Poly₁cpl1 22539  coe₁cco1 22540   evalSub₁ ces1 22675  deg₁cdg1 26416  Unic_1pcuc1p 26489  rem_1pcr1p 26491

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781  ax-cnex 11293  ax-resscn 11294  ax-1cn 11295  ax-icn 11296  ax-addcl 11297  ax-addrcl 11298  ax-mulcl 11299  ax-mulrcl 11300  ax-mulcom 11301  ax-addass 11302  ax-mulass 11303  ax-distr 11304  ax-i2m1 11305  ax-1ne0 11306  ax-1rid 11307  ax-rnegex 11308  ax-rrecex 11309  ax-cnre 11310  ax-pre-lttri 11311  ax-pre-lttrn 11312  ax-pre-ltadd 11313  ax-pre-mulgt0 11314  ax-pre-sup 11315  ax-addf 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-isom 6573  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-of 7723  df-ofr 7724  df-om 7910  df-1st 8033  df-2nd 8034  df-supp 8213  df-tpos 8279  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-1o 8512  df-2o 8513  df-er 8753  df-map 8885  df-pm 8886  df-ixp 8962  df-en 9010  df-dom 9011  df-sdom 9012  df-fin 9013  df-fsupp 9390  df-sup 9469  df-oi 9539  df-card 10055  df-pnf 11382  df-mnf 11383  df-xr 11384  df-ltxr 11385  df-le 11386  df-sub 11580  df-neg 11581  df-nn 12374  df-2 12443  df-3 12444  df-4 12445  df-5 12446  df-6 12447  df-7 12448  df-8 12449  df-9 12450  df-n0 12645  df-z 12732  df-dec 12853  df-uz 13004  df-fz 13678  df-fzo 13827  df-seq 14183  df-hash 14514  df-struct 17366  df-sets 17383  df-slot 17401  df-ndx 17413  df-base 17429  df-ress 17450  df-plusg 17482  df-mulr 17483  df-starv 17484  df-sca 17485  df-vsca 17486  df-ip 17487  df-tset 17488  df-ple 17489  df-ds 17491  df-unif 17492  df-hom 17493  df-cco 17494  df-0g 17653  df-gsum 17654  df-prds 17659  df-pws 17661  df-mre 17797  df-mrc 17798  df-acs 17800  df-mgm 18857  df-sgrp 18949  df-mnd 18965  df-mhm 19019  df-submnd 19020  df-grp 19188  df-minusg 19189  df-sbg 19190  df-mulg 19319  df-subg 19374  df-ghm 19469  df-cntz 19573  df-cmn 20038  df-abl 20039  df-mgp 20403  df-rng 20417  df-ur 20450  df-srg 20455  df-ring 20503  df-cring 20504  df-oppr 20609  df-dvdsr 20629  df-unit 20630  df-invr 20660  df-rhm 20744  df-nzr 20805  df-subrng 20840  df-subrg 20864  df-rgspn 20905  df-rlreg 20988  df-lmod 21179  df-lss 21249  df-lsp 21289  df-cnfld 21721  df-assa 22205  df-asp 22206  df-ascl 22207  df-psr 22261  df-mvr 22262  df-mpl 22263  df-opsr 22265  df-evls 22427  df-evl 22428  df-psr1 22542  df-vr1 22543  df-ply1 22544  df-coe1 22545  df-evls1 22677  df-evl1 22678  df-mdeg 26417  df-deg1 26418  df-mon1 26493  df-uc1p 26494  df-q1p 26495  df-r1p 26496

This theorem is used by: (None)