Theorem ply1dg3rt0irred 33662

Description: If a cubic polynomial over a field has no roots, it is irreducible. (Proposed by Saveliy Skresanov, 5-Jun-2025.) (Contributed by Thierry Arnoux, 8-Jun-2025.)

Hypotheses

Ref	Expression
ply1dg3rt0irred.z	⊢ 0 = (0g‘𝐹)
ply1dg3rt0irred.o	⊢ 𝑂 = (eval1‘𝐹)
ply1dg3rt0irred.d	⊢ 𝐷 = (deg1‘𝐹)
ply1dg3rt0irred.p	⊢ 𝑃 = (Poly1‘𝐹)
ply1dg3rt0irred.b	⊢ 𝐵 = (Base‘𝑃)
ply1dg3rt0irred.f	⊢ (𝜑 → 𝐹 ∈ Field)
ply1dg3rt0irred.q	⊢ (𝜑 → 𝑄 ∈ 𝐵)
ply1dg3rt0irred.1	⊢ (𝜑 → (◡(𝑂‘𝑄) “ { 0 }) = ∅)
ply1dg3rt0irred.2	⊢ (𝜑 → (𝐷‘𝑄) = 3)

Assertion

Ref	Expression
ply1dg3rt0irred	⊢ (𝜑 → 𝑄 ∈ (Irred‘𝑃))

Proof of Theorem ply1dg3rt0irred

Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ply1dg3rt0irred.q	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
2		ply1dg3rt0irred.2	. . . . 5 ⊢ (𝜑 → (𝐷‘𝑄) = 3)
3		3ne0 12281	. . . . . 6 ⊢ 3 ≠ 0
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 3 ≠ 0)
5	2, 4	eqnetrd 3000	. . . 4 ⊢ (𝜑 → (𝐷‘𝑄) ≠ 0)
6		ply1dg3rt0irred.p	. . . . . 6 ⊢ 𝑃 = (Poly1‘𝐹)
7		eqid 2737	. . . . . 6 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
8		eqid 2737	. . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹)
9		ply1dg3rt0irred.z	. . . . . 6 ⊢ 0 = (0g‘𝐹)
10		ply1dg3rt0irred.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ Field)
11		ply1dg3rt0irred.d	. . . . . 6 ⊢ 𝐷 = (deg1‘𝐹)
12		ply1dg3rt0irred.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑃)
13	1, 12	eleqtrdi 2847	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝑃))
14	6, 7, 8, 9, 10, 11, 13	ply1unit 33653	. . . . 5 ⊢ (𝜑 → (𝑄 ∈ (Unit‘𝑃) ↔ (𝐷‘𝑄) = 0))
15	14	necon3bbid 2970	. . . 4 ⊢ (𝜑 → (¬ 𝑄 ∈ (Unit‘𝑃) ↔ (𝐷‘𝑄) ≠ 0))
16	5, 15	mpbird 257	. . 3 ⊢ (𝜑 → ¬ 𝑄 ∈ (Unit‘𝑃))
17	1, 16	eldifd 3901	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝐵 ∖ (Unit‘𝑃)))
18	10	ad3antrrr 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝐹 ∈ Field)
19		simpllr 776	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃)))
20	19	eldifad 3902	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑝 ∈ 𝐵)
21	20, 12	eleqtrdi 2847	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑝 ∈ (Base‘𝑃))
22	6, 7, 8, 9, 18, 11, 21	ply1unit 33653	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝑝 ∈ (Unit‘𝑃) ↔ (𝐷‘𝑝) = 0))
23	22	biimpar 477	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 0) → 𝑝 ∈ (Unit‘𝑃))
24	19	eldifbd 3903	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → ¬ 𝑝 ∈ (Unit‘𝑃))
25	24	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 0) → ¬ 𝑝 ∈ (Unit‘𝑃))
26	23, 25	pm2.21fal 1564	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 0) → ⊥)
27	26	adantlr 716	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) ∈ {0, 1}) ∧ (𝐷‘𝑝) = 0) → ⊥)
28		ply1dg3rt0irred.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = (eval1‘𝐹)
29	10	fldcrngd 20713	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 ∈ CRing)
30	29	ad3antrrr 731	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝐹 ∈ CRing)
31		simplr 769	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃)))
32	31	eldifad 3902	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑞 ∈ 𝐵)
33		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑃) = (.r‘𝑃)
34	6, 12, 28, 11, 9, 30, 20, 32, 33	ply1mulrtss 33660	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑝) “ { 0 }) ⊆ (◡(𝑂‘(𝑝(.r‘𝑃)𝑞)) “ { 0 }))
35		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝑝(.r‘𝑃)𝑞) = 𝑄)
36	35	fveq2d 6839	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝑂‘(𝑝(.r‘𝑃)𝑞)) = (𝑂‘𝑄))
37	36	cnveqd 5825	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → ◡(𝑂‘(𝑝(.r‘𝑃)𝑞)) = ◡(𝑂‘𝑄))
38	37	imaeq1d 6019	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘(𝑝(.r‘𝑃)𝑞)) “ { 0 }) = (◡(𝑂‘𝑄) “ { 0 }))
39	34, 38	sseqtrd 3959	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑝) “ { 0 }) ⊆ (◡(𝑂‘𝑄) “ { 0 }))
40		ply1dg3rt0irred.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (◡(𝑂‘𝑄) “ { 0 }) = ∅)
41	40	ad3antrrr 731	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑄) “ { 0 }) = ∅)
42	39, 41	sseqtrd 3959	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑝) “ { 0 }) ⊆ ∅)
43		ss0 4343	. . . . . . . . . . . 12 ⊢ ((◡(𝑂‘𝑝) “ { 0 }) ⊆ ∅ → (◡(𝑂‘𝑝) “ { 0 }) = ∅)
44	42, 43	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑝) “ { 0 }) = ∅)
45	44	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 1) → (◡(𝑂‘𝑝) “ { 0 }) = ∅)
46	18	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 1) → 𝐹 ∈ Field)
47	20	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 1) → 𝑝 ∈ 𝐵)
48		simpr 484	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 1) → (𝐷‘𝑝) = 1)
49	6, 12, 28, 11, 9, 46, 47, 48	ply1dg1rtn0 33659	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 1) → (◡(𝑂‘𝑝) “ { 0 }) ≠ ∅)
50	45, 49	pm2.21ddne 3017	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 1) → ⊥)
51	50	adantlr 716	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) ∈ {0, 1}) ∧ (𝐷‘𝑝) = 1) → ⊥)
52		elpri 4592	. . . . . . . . 9 ⊢ ((𝐷‘𝑝) ∈ {0, 1} → ((𝐷‘𝑝) = 0 ∨ (𝐷‘𝑝) = 1))
53	52	adantl 481	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) ∈ {0, 1}) → ((𝐷‘𝑝) = 0 ∨ (𝐷‘𝑝) = 1))
54	27, 51, 53	mpjaodan 961	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) ∈ {0, 1}) → ⊥)
55	6, 12, 28, 11, 9, 30, 32, 20, 33	ply1mulrtss 33660	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑞) “ { 0 }) ⊆ (◡(𝑂‘(𝑞(.r‘𝑃)𝑝)) “ { 0 }))
56		fldidom 20742	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ Field → 𝐹 ∈ IDomn)
57	10, 56	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 ∈ IDomn)
58	6	ply1idom 26103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ IDomn → 𝑃 ∈ IDomn)
59	57, 58	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑃 ∈ IDomn)
60	59	idomcringd 20698	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑃 ∈ CRing)
61	60	ad3antrrr 731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑃 ∈ CRing)
62	12, 33, 61, 32, 20	crngcomd 20230	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝑞(.r‘𝑃)𝑝) = (𝑝(.r‘𝑃)𝑞))
63	62, 35	eqtrd 2772	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝑞(.r‘𝑃)𝑝) = 𝑄)
64	63	fveq2d 6839	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝑂‘(𝑞(.r‘𝑃)𝑝)) = (𝑂‘𝑄))
65	64	cnveqd 5825	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → ◡(𝑂‘(𝑞(.r‘𝑃)𝑝)) = ◡(𝑂‘𝑄))
66	65	imaeq1d 6019	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘(𝑞(.r‘𝑃)𝑝)) “ { 0 }) = (◡(𝑂‘𝑄) “ { 0 }))
67	66, 41	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘(𝑞(.r‘𝑃)𝑝)) “ { 0 }) = ∅)
68	55, 67	sseqtrd 3959	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑞) “ { 0 }) ⊆ ∅)
69		ss0 4343	. . . . . . . . . . . 12 ⊢ ((◡(𝑂‘𝑞) “ { 0 }) ⊆ ∅ → (◡(𝑂‘𝑞) “ { 0 }) = ∅)
70	68, 69	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑞) “ { 0 }) = ∅)
71	70	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → (◡(𝑂‘𝑞) “ { 0 }) = ∅)
72	18	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → 𝐹 ∈ Field)
73	32	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → 𝑞 ∈ 𝐵)
74	29	crngringd 20221	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ Ring)
75	74	ad3antrrr 731	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝐹 ∈ Ring)
76		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘𝑃) = (0g‘𝑃)
77	59	idomdomd 20697	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ Domn)
78	77	ad3antrrr 731	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑃 ∈ Domn)
79		3nn0 12449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 3 ∈ ℕ0
80	2, 79	eqeltrdi 2845	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐷‘𝑄) ∈ ℕ0)
81	11, 6, 76, 12	deg1nn0clb 26068	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 ∈ Ring ∧ 𝑄 ∈ 𝐵) → (𝑄 ≠ (0g‘𝑃) ↔ (𝐷‘𝑄) ∈ ℕ0))
82	81	biimpar 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ Ring ∧ 𝑄 ∈ 𝐵) ∧ (𝐷‘𝑄) ∈ ℕ0) → 𝑄 ≠ (0g‘𝑃))
83	74, 1, 80, 82	syl21anc 838	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑄 ≠ (0g‘𝑃))
84	83	ad3antrrr 731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑄 ≠ (0g‘𝑃))
85	35, 84	eqnetrd 3000	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝑝(.r‘𝑃)𝑞) ≠ (0g‘𝑃))
86	12, 33, 76, 78, 20, 32, 85	domnmuln0rd 33353	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝑝 ≠ (0g‘𝑃) ∧ 𝑞 ≠ (0g‘𝑃)))
87	86	simpld 494	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑝 ≠ (0g‘𝑃))
88	11, 6, 76, 12	deg1nn0cl 26066	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑝 ∈ 𝐵 ∧ 𝑝 ≠ (0g‘𝑃)) → (𝐷‘𝑝) ∈ ℕ0)
89	75, 20, 87, 88	syl3anc 1374	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ ℕ0)
90	89	nn0cnd 12494	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ ℂ)
91	86	simprd 495	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑞 ≠ (0g‘𝑃))
92	11, 6, 76, 12	deg1nn0cl 26066	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑞 ∈ 𝐵 ∧ 𝑞 ≠ (0g‘𝑃)) → (𝐷‘𝑞) ∈ ℕ0)
93	75, 32, 91, 92	syl3anc 1374	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑞) ∈ ℕ0)
94	93	nn0cnd 12494	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑞) ∈ ℂ)
95	35	fveq2d 6839	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘(𝑝(.r‘𝑃)𝑞)) = (𝐷‘𝑄))
96	57	idomdomd 20697	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ Domn)
97	96	ad3antrrr 731	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝐹 ∈ Domn)
98	11, 6, 12, 33, 76, 97, 20, 87, 32, 91	deg1mul 26093	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘(𝑝(.r‘𝑃)𝑞)) = ((𝐷‘𝑝) + (𝐷‘𝑞)))
99	2	ad3antrrr 731	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑄) = 3)
100	95, 98, 99	3eqtr3d 2780	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → ((𝐷‘𝑝) + (𝐷‘𝑞)) = 3)
101	90, 94, 100	mvlladdd 11555	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑞) = (3 − (𝐷‘𝑝)))
102	101	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → (𝐷‘𝑞) = (3 − (𝐷‘𝑝)))
103		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → (𝐷‘𝑝) = 2)
104	103	oveq2d 7377	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → (3 − (𝐷‘𝑝)) = (3 − 2))
105		3cn 12256	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
106		2cn 12250	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
107		ax-1cn 11090	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
108		2p1e3 12312	. . . . . . . . . . . . . 14 ⊢ (2 + 1) = 3
109	105, 106, 107, 108	subaddrii 11477	. . . . . . . . . . . . 13 ⊢ (3 − 2) = 1
110	109	a1i 11	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → (3 − 2) = 1)
111	102, 104, 110	3eqtrd 2776	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → (𝐷‘𝑞) = 1)
112	6, 12, 28, 11, 9, 72, 73, 111	ply1dg1rtn0 33659	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → (◡(𝑂‘𝑞) “ { 0 }) ≠ ∅)
113	71, 112	pm2.21ddne 3017	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → ⊥)
114	113	adantlr 716	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) ∈ {2, 3}) ∧ (𝐷‘𝑝) = 2) → ⊥)
115	101	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → (𝐷‘𝑞) = (3 − (𝐷‘𝑝)))
116		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → (𝐷‘𝑝) = 3)
117	116	oveq2d 7377	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → (3 − (𝐷‘𝑝)) = (3 − 3))
118	105	subidi 11459	. . . . . . . . . . . . 13 ⊢ (3 − 3) = 0
119	118	a1i 11	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → (3 − 3) = 0)
120	115, 117, 119	3eqtrd 2776	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → (𝐷‘𝑞) = 0)
121	18	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → 𝐹 ∈ Field)
122	32, 12	eleqtrdi 2847	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑞 ∈ (Base‘𝑃))
123	122	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → 𝑞 ∈ (Base‘𝑃))
124	6, 7, 8, 9, 121, 11, 123	ply1unit 33653	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → (𝑞 ∈ (Unit‘𝑃) ↔ (𝐷‘𝑞) = 0))
125	120, 124	mpbird 257	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → 𝑞 ∈ (Unit‘𝑃))
126	31	eldifbd 3903	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → ¬ 𝑞 ∈ (Unit‘𝑃))
127	126	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → ¬ 𝑞 ∈ (Unit‘𝑃))
128	125, 127	pm2.21fal 1564	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → ⊥)
129	128	adantlr 716	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) ∈ {2, 3}) ∧ (𝐷‘𝑝) = 3) → ⊥)
130		elpri 4592	. . . . . . . . 9 ⊢ ((𝐷‘𝑝) ∈ {2, 3} → ((𝐷‘𝑝) = 2 ∨ (𝐷‘𝑝) = 3))
131	130	adantl 481	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) ∈ {2, 3}) → ((𝐷‘𝑝) = 2 ∨ (𝐷‘𝑝) = 3))
132	114, 129, 131	mpjaodan 961	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) ∈ {2, 3}) → ⊥)
133	79	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 3 ∈ ℕ0)
134	89	nn0red 12493	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ ℝ)
135		nn0addge1 12477	. . . . . . . . . . . 12 ⊢ (((𝐷‘𝑝) ∈ ℝ ∧ (𝐷‘𝑞) ∈ ℕ0) → (𝐷‘𝑝) ≤ ((𝐷‘𝑝) + (𝐷‘𝑞)))
136	134, 93, 135	syl2anc 585	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ≤ ((𝐷‘𝑝) + (𝐷‘𝑞)))
137	136, 100	breqtrd 5112	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ≤ 3)
138		fznn0 13567	. . . . . . . . . . 11 ⊢ (3 ∈ ℕ0 → ((𝐷‘𝑝) ∈ (0...3) ↔ ((𝐷‘𝑝) ∈ ℕ0 ∧ (𝐷‘𝑝) ≤ 3)))
139	138	biimpar 477	. . . . . . . . . 10 ⊢ ((3 ∈ ℕ0 ∧ ((𝐷‘𝑝) ∈ ℕ0 ∧ (𝐷‘𝑝) ≤ 3)) → (𝐷‘𝑝) ∈ (0...3))
140	133, 89, 137, 139	syl12anc 837	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ (0...3))
141		fz0to3un2pr 13577	. . . . . . . . 9 ⊢ (0...3) = ({0, 1} ∪ {2, 3})
142	140, 141	eleqtrdi 2847	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ ({0, 1} ∪ {2, 3}))
143		elun 4094	. . . . . . . 8 ⊢ ((𝐷‘𝑝) ∈ ({0, 1} ∪ {2, 3}) ↔ ((𝐷‘𝑝) ∈ {0, 1} ∨ (𝐷‘𝑝) ∈ {2, 3}))
144	142, 143	sylib 218	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → ((𝐷‘𝑝) ∈ {0, 1} ∨ (𝐷‘𝑝) ∈ {2, 3}))
145	54, 132, 144	mpjaodan 961	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → ⊥)
146	145	r19.29ffa 32558	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∃𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))(𝑝(.r‘𝑃)𝑞) = 𝑄) → ⊥)
147	146	inegd 1562	. . . 4 ⊢ (𝜑 → ¬ ∃𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∃𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))(𝑝(.r‘𝑃)𝑞) = 𝑄)
148		ralnex2 3118	. . . 4 ⊢ (∀𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∀𝑞 ∈ (𝐵 ∖ (Unit‘𝑃)) ¬ (𝑝(.r‘𝑃)𝑞) = 𝑄 ↔ ¬ ∃𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∃𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))(𝑝(.r‘𝑃)𝑞) = 𝑄)
149	147, 148	sylibr 234	. . 3 ⊢ (𝜑 → ∀𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∀𝑞 ∈ (𝐵 ∖ (Unit‘𝑃)) ¬ (𝑝(.r‘𝑃)𝑞) = 𝑄)
150		df-ne 2934	. . . 4 ⊢ ((𝑝(.r‘𝑃)𝑞) ≠ 𝑄 ↔ ¬ (𝑝(.r‘𝑃)𝑞) = 𝑄)
151	150	2ralbii 3113	. . 3 ⊢ (∀𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∀𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))(𝑝(.r‘𝑃)𝑞) ≠ 𝑄 ↔ ∀𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∀𝑞 ∈ (𝐵 ∖ (Unit‘𝑃)) ¬ (𝑝(.r‘𝑃)𝑞) = 𝑄)
152	149, 151	sylibr 234	. 2 ⊢ (𝜑 → ∀𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∀𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))(𝑝(.r‘𝑃)𝑞) ≠ 𝑄)
153		eqid 2737	. . 3 ⊢ (Unit‘𝑃) = (Unit‘𝑃)
154		eqid 2737	. . 3 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
155		eqid 2737	. . 3 ⊢ (𝐵 ∖ (Unit‘𝑃)) = (𝐵 ∖ (Unit‘𝑃))
156	12, 153, 154, 155, 33	isirred 20393	. 2 ⊢ (𝑄 ∈ (Irred‘𝑃) ↔ (𝑄 ∈ (𝐵 ∖ (Unit‘𝑃)) ∧ ∀𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∀𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))(𝑝(.r‘𝑃)𝑞) ≠ 𝑄))
157	17, 152, 156	sylanbrc 584	1 ⊢ (𝜑 → 𝑄 ∈ (Irred‘𝑃))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542  ⊥wfal 1554   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   ∖ cdif 3887   ∪ cun 3888   ⊆ wss 3890  ∅c0 4274  {csn 4568  {cpr 4570   class class class wbr 5086  ^◡ccnv 5624   “ cima 5628  ‘cfv 6493  (class class class)co 7361  ℝcr 11031  0cc0 11032  1c1 11033   + caddc 11035   ≤ cle 11174   − cmin 11371  2c2 12230  3c3 12231  ℕ₀cn0 12431  ...cfz 13455  Basecbs 17173  ._rcmulr 17215  0_gc0g 17396  Ringcrg 20208  CRingccrg 20209  Unitcui 20329  Irredcir 20330  Domncdomn 20663  IDomncidom 20664  Fieldcfield 20701  algSccascl 21845  Poly₁cpl1 22153  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-tpos 8170  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-oppr 20311  df-dvdsr 20331  df-unit 20332  df-irred 20333  df-invr 20362  df-dvr 20375  df-rhm 20446  df-nzr 20484  df-subrng 20517  df-subrg 20541  df-rlreg 20665  df-domn 20666  df-idom 20667  df-drng 20702  df-field 20703  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  df-mon1 26109

This theorem is referenced by:  2sqr3minply  33943  cos9thpiminply  33951