Theorem ply1dg3rt0irred 33667

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 12253	. . . . . 6 ⊢ 3 ≠ 0
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 3 ≠ 0)
5	2, 4	eqnetrd 2999	. . . 4 ⊢ (𝜑 → (𝐷‘𝑄) ≠ 0)
6		ply1dg3rt0irred.p	. . . . . 6 ⊢ 𝑃 = (Poly1‘𝐹)
7		eqid 2736	. . . . . 6 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
8		eqid 2736	. . . . . 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 2846	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝑃))
14	6, 7, 8, 9, 10, 11, 13	ply1unit 33658	. . . . 5 ⊢ (𝜑 → (𝑄 ∈ (Unit‘𝑃) ↔ (𝐷‘𝑄) = 0))
15	14	necon3bbid 2969	. . . 4 ⊢ (𝜑 → (¬ 𝑄 ∈ (Unit‘𝑃) ↔ (𝐷‘𝑄) ≠ 0))
16	5, 15	mpbird 257	. . 3 ⊢ (𝜑 → ¬ 𝑄 ∈ (Unit‘𝑃))
17	1, 16	eldifd 3912	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝐵 ∖ (Unit‘𝑃)))
18	10	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝐹 ∈ Field)
19		simpllr 775	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃)))
20	19	eldifad 3913	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑝 ∈ 𝐵)
21	20, 12	eleqtrdi 2846	. . . . . . . . . . . 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 33658	. . . . . . . . . . 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 3914	. . . . . . . . . . 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 1563	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 0) → ⊥)
27	26	adantlr 715	. . . . . . . 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 20677	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 ∈ CRing)
30	29	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝐹 ∈ CRing)
31		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃)))
32	31	eldifad 3913	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑞 ∈ 𝐵)
33		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑃) = (.r‘𝑃)
34	6, 12, 28, 11, 9, 30, 20, 32, 33	ply1mulrtss 33665	. . . . . . . . . . . . . 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 6838	. . . . . . . . . . . . . . . 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 5824	. . . . . . . . . . . . . . 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 6018	. . . . . . . . . . . . . 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 3970	. . . . . . . . . . . . 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 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑄) “ { 0 }) = ∅)
42	39, 41	sseqtrd 3970	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑝) “ { 0 }) ⊆ ∅)
43		ss0 4354	. . . . . . . . . . . 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 33664	. . . . . . . . . 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 3016	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 1) → ⊥)
51	50	adantlr 715	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) ∈ {0, 1}) ∧ (𝐷‘𝑝) = 1) → ⊥)
52		elpri 4604	. . . . . . . . 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 960	. . . . . . 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 33665	. . . . . . . . . . . . 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 20706	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ Field → 𝐹 ∈ IDomn)
57	10, 56	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 ∈ IDomn)
58	6	ply1idom 26088	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ IDomn → 𝑃 ∈ IDomn)
59	57, 58	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑃 ∈ IDomn)
60	59	idomcringd 20662	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑃 ∈ CRing)
61	60	ad3antrrr 730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑃 ∈ CRing)
62	12, 33, 61, 32, 20	crngcomd 20192	. . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝑞(.r‘𝑃)𝑝) = 𝑄)
64	63	fveq2d 6838	. . . . . . . . . . . . . . . 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 5824	. . . . . . . . . . . . . . 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 6018	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 3970	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑞) “ { 0 }) ⊆ ∅)
69		ss0 4354	. . . . . . . . . . . 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 20183	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ Ring)
75	74	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝐹 ∈ Ring)
76		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘𝑃) = (0g‘𝑃)
77	59	idomdomd 20661	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ Domn)
78	77	ad3antrrr 730	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑃 ∈ Domn)
79		3nn0 12421	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 3 ∈ ℕ0
80	2, 79	eqeltrdi 2844	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐷‘𝑄) ∈ ℕ0)
81	11, 6, 76, 12	deg1nn0clb 26053	. . . . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑄 ≠ (0g‘𝑃))
84	83	ad3antrrr 730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑄 ≠ (0g‘𝑃))
85	35, 84	eqnetrd 2999	. . . . . . . . . . . . . . . . . 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 33358	. . . . . . . . . . . . . . . . 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 26051	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑝 ∈ 𝐵 ∧ 𝑝 ≠ (0g‘𝑃)) → (𝐷‘𝑝) ∈ ℕ0)
89	75, 20, 87, 88	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ ℕ0)
90	89	nn0cnd 12466	. . . . . . . . . . . . . 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 26051	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑞 ∈ 𝐵 ∧ 𝑞 ≠ (0g‘𝑃)) → (𝐷‘𝑞) ∈ ℕ0)
93	75, 32, 91, 92	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑞) ∈ ℕ0)
94	93	nn0cnd 12466	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑞) ∈ ℂ)
95	35	fveq2d 6838	. . . . . . . . . . . . . . 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 20661	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ Domn)
97	96	ad3antrrr 730	. . . . . . . . . . . . . . . 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 26078	. . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑄) = 3)
100	95, 98, 99	3eqtr3d 2779	. . . . . . . . . . . . . 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 11550	. . . . . . . . . . . . 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 7374	. . . . . . . . . . . 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 12228	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
106		2cn 12222	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
107		ax-1cn 11086	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
108		2p1e3 12284	. . . . . . . . . . . . . 14 ⊢ (2 + 1) = 3
109	105, 106, 107, 108	subaddrii 11472	. . . . . . . . . . . . 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 2775	. . . . . . . . . . 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 33664	. . . . . . . . . 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 3016	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → ⊥)
114	113	adantlr 715	. . . . . . . 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 7374	. . . . . . . . . . . 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 11454	. . . . . . . . . . . . 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 2775	. . . . . . . . . . 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 2846	. . . . . . . . . . . . 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 33658	. . . . . . . . . . 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 3914	. . . . . . . . . . 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 1563	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → ⊥)
129	128	adantlr 715	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) ∈ {2, 3}) ∧ (𝐷‘𝑝) = 3) → ⊥)
130		elpri 4604	. . . . . . . . 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 960	. . . . . . 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 12465	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ ℝ)
135		nn0addge1 12449	. . . . . . . . . . . 12 ⊢ (((𝐷‘𝑝) ∈ ℝ ∧ (𝐷‘𝑞) ∈ ℕ0) → (𝐷‘𝑝) ≤ ((𝐷‘𝑝) + (𝐷‘𝑞)))
136	134, 93, 135	syl2anc 584	. . . . . . . . . . 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 5124	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ≤ 3)
138		fznn0 13537	. . . . . . . . . . 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 836	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ (0...3))
141		fz0to3un2pr 13547	. . . . . . . . 9 ⊢ (0...3) = ({0, 1} ∪ {2, 3})
142	140, 141	eleqtrdi 2846	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ ({0, 1} ∪ {2, 3}))
143		elun 4105	. . . . . . . 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 960	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → ⊥)
146	145	r19.29ffa 32547	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∃𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))(𝑝(.r‘𝑃)𝑞) = 𝑄) → ⊥)
147	146	inegd 1561	. . . 4 ⊢ (𝜑 → ¬ ∃𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∃𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))(𝑝(.r‘𝑃)𝑞) = 𝑄)
148		ralnex2 3116	. . . 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 2933	. . . 4 ⊢ ((𝑝(.r‘𝑃)𝑞) ≠ 𝑄 ↔ ¬ (𝑝(.r‘𝑃)𝑞) = 𝑄)
151	150	2ralbii 3111	. . 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 2736	. . 3 ⊢ (Unit‘𝑃) = (Unit‘𝑃)
154		eqid 2736	. . 3 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
155		eqid 2736	. . 3 ⊢ (𝐵 ∖ (Unit‘𝑃)) = (𝐵 ∖ (Unit‘𝑃))
156	12, 153, 154, 155, 33	isirred 20357	. 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 583	1 ⊢ (𝜑 → 𝑄 ∈ (Irred‘𝑃))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541  ⊥wfal 1553   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060   ∖ cdif 3898   ∪ cun 3899   ⊆ wss 3901  ∅c0 4285  {csn 4580  {cpr 4582   class class class wbr 5098  ^◡ccnv 5623   “ cima 5627  ‘cfv 6492  (class class class)co 7358  ℝcr 11027  0cc0 11028  1c1 11029   + caddc 11031   ≤ cle 11169   − cmin 11366  2c2 12202  3c3 12203  ℕ₀cn0 12403  ...cfz 13425  Basecbs 17138  ._rcmulr 17180  0_gc0g 17361  Ringcrg 20170  CRingccrg 20171  Unitcui 20293  Irredcir 20294  Domncdomn 20627  IDomncidom 20628  Fieldcfield 20665  algSccascl 21809  Poly₁cpl1 22119  eval₁ce1 22260  deg₁cdg1 26017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106  ax-addf 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ofr 7623  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8767  df-pm 8768  df-ixp 8838  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-fsupp 9267  df-sup 9347  df-oi 9417  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12610  df-uz 12754  df-fz 13426  df-fzo 13573  df-seq 13927  df-hash 14256  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-mulr 17193  df-starv 17194  df-sca 17195  df-vsca 17196  df-ip 17197  df-tset 17198  df-ple 17199  df-ds 17201  df-unif 17202  df-hom 17203  df-cco 17204  df-0g 17363  df-gsum 17364  df-prds 17369  df-pws 17371  df-mre 17507  df-mrc 17508  df-acs 17510  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-ghm 19144  df-cntz 19248  df-cmn 19713  df-abl 19714  df-mgp 20078  df-rng 20090  df-ur 20119  df-srg 20124  df-ring 20172  df-cring 20173  df-oppr 20275  df-dvdsr 20295  df-unit 20296  df-irred 20297  df-invr 20326  df-dvr 20339  df-rhm 20410  df-nzr 20448  df-subrng 20481  df-subrg 20505  df-rlreg 20629  df-domn 20630  df-idom 20631  df-drng 20666  df-field 20667  df-lmod 20815  df-lss 20885  df-lsp 20925  df-cnfld 21312  df-assa 21810  df-asp 21811  df-ascl 21812  df-psr 21867  df-mvr 21868  df-mpl 21869  df-opsr 21871  df-evls 22031  df-evl 22032  df-psr1 22122  df-vr1 22123  df-ply1 22124  df-coe1 22125  df-evls1 22261  df-evl1 22262  df-mdeg 26018  df-deg1 26019  df-mon1 26094

This theorem is referenced by:  2sqr3minply  33939  cos9thpiminply  33947