Theorem ply1dg3rt0irred 33819

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 12350	. . . . . 6 ⊢ 3 ≠ 0
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 3 ≠ 0)
5	2, 4	eqnetrd 3031	. . . 4 ⊢ (𝜑 → (𝐷‘𝑄) ≠ 0)
6		ply1dg3rt0irred.p	. . . . . 6 ⊢ 𝑃 = (Poly1‘𝐹)
7		eqid 2769	. . . . . 6 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
8		eqid 2769	. . . . . 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 2879	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝑃))
14	6, 7, 8, 9, 10, 11, 13	ply1unit 33810	. . . . 5 ⊢ (𝜑 → (𝑄 ∈ (Unit‘𝑃) ↔ (𝐷‘𝑄) = 0))
15	14	necon3bbid 3001	. . . 4 ⊢ (𝜑 → (¬ 𝑄 ∈ (Unit‘𝑃) ↔ (𝐷‘𝑄) ≠ 0))
16	5, 15	mpbird 260	. . 3 ⊢ (𝜑 → ¬ 𝑄 ∈ (Unit‘𝑃))
17	1, 16	eldifd 3924	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝐵 ∖ (Unit‘𝑃)))
18	10	ad3antrrr 742	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝐹 ∈ Field)
19		simpllr 787	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃)))
20	19	eldifad 3925	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑝 ∈ 𝐵)
21	20, 12	eleqtrdi 2879	. . . . . . . . . . . 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 33810	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝑝 ∈ (Unit‘𝑃) ↔ (𝐷‘𝑝) = 0))
23	22	biimpar 482	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 0) → 𝑝 ∈ (Unit‘𝑃))
24	19	eldifbd 3926	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → ¬ 𝑝 ∈ (Unit‘𝑃))
25	24	adantr 485	. . . . . . . . . 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 1589	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 0) → ⊥)
27	26	adantlr 727	. . . . . . . 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 20826	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 ∈ CRing)
30	29	ad3antrrr 742	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝐹 ∈ CRing)
31		simplr 780	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃)))
32	31	eldifad 3925	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑞 ∈ 𝐵)
33		eqid 2769	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑃) = (.r‘𝑃)
34	6, 12, 28, 11, 9, 30, 20, 32, 33	ply1mulrtss 33817	. . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝑝(.r‘𝑃)𝑞) = 𝑄)
36	35	fveq2d 6886	. . . . . . . . . . . . . . . 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 5862	. . . . . . . . . . . . . . 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 6062	. . . . . . . . . . . . . 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 3981	. . . . . . . . . . . . 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 742	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑄) “ { 0 }) = ∅)
42	39, 41	sseqtrd 3981	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑝) “ { 0 }) ⊆ ∅)
43		ss0 4366	. . . . . . . . . . . 12 ⊢ ((◡(𝑂‘𝑝) “ { 0 }) ⊆ ∅ → (◡(𝑂‘𝑝) “ { 0 }) = ∅)
44	42, 43	syl 18	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑝) “ { 0 }) = ∅)
45	44	adantr 485	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 1) → (◡(𝑂‘𝑝) “ { 0 }) = ∅)
46	18	adantr 485	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 1) → 𝐹 ∈ Field)
47	20	adantr 485	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 1) → 𝑝 ∈ 𝐵)
48		simpr 489	. . . . . . . . . . 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 33816	. . . . . . . . . 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 3048	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 1) → ⊥)
51	50	adantlr 727	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) ∈ {0, 1}) ∧ (𝐷‘𝑝) = 1) → ⊥)
52		elpri 4618	. . . . . . . . 9 ⊢ ((𝐷‘𝑝) ∈ {0, 1} → ((𝐷‘𝑝) = 0 ∨ (𝐷‘𝑝) = 1))
53	52	adantl 486	. . . . . . . 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 973	. . . . . . 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 33817	. . . . . . . . . . . . 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 20853	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ Field → 𝐹 ∈ IDomn)
57	10, 56	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 ∈ IDomn)
58	6	ply1idom 26251	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ IDomn → 𝑃 ∈ IDomn)
59	57, 58	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑃 ∈ IDomn)
60	59	idomcringd 20811	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑃 ∈ CRing)
61	60	ad3antrrr 742	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑃 ∈ CRing)
62	12, 33, 61, 32, 20	crngcomd 20337	. . . . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝑞(.r‘𝑃)𝑝) = 𝑄)
64	63	fveq2d 6886	. . . . . . . . . . . . . . . 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 5862	. . . . . . . . . . . . . . 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 6062	. . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . 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 3981	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑞) “ { 0 }) ⊆ ∅)
69		ss0 4366	. . . . . . . . . . . 12 ⊢ ((◡(𝑂‘𝑞) “ { 0 }) ⊆ ∅ → (◡(𝑂‘𝑞) “ { 0 }) = ∅)
70	68, 69	syl 18	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑞) “ { 0 }) = ∅)
71	70	adantr 485	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → (◡(𝑂‘𝑞) “ { 0 }) = ∅)
72	18	adantr 485	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → 𝐹 ∈ Field)
73	32	adantr 485	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → 𝑞 ∈ 𝐵)
74	29	crngringd 20328	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ Ring)
75	74	ad3antrrr 742	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝐹 ∈ Ring)
76		eqid 2769	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘𝑃) = (0g‘𝑃)
77	59	idomdomd 20810	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ Domn)
78	77	ad3antrrr 742	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑃 ∈ Domn)
79		3nn0 12522	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 3 ∈ ℕ0
80	2, 79	eqeltrdi 2877	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐷‘𝑄) ∈ ℕ0)
81	11, 6, 76, 12	deg1nn0clb 26216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 ∈ Ring ∧ 𝑄 ∈ 𝐵) → (𝑄 ≠ (0g‘𝑃) ↔ (𝐷‘𝑄) ∈ ℕ0))
82	81	biimpar 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ Ring ∧ 𝑄 ∈ 𝐵) ∧ (𝐷‘𝑄) ∈ ℕ0) → 𝑄 ≠ (0g‘𝑃))
83	74, 1, 80, 82	syl21anc 850	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑄 ≠ (0g‘𝑃))
84	83	ad3antrrr 742	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑄 ≠ (0g‘𝑃))
85	35, 84	eqnetrd 3031	. . . . . . . . . . . . . . . . . 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 33538	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝑝 ≠ (0g‘𝑃) ∧ 𝑞 ≠ (0g‘𝑃)))
87	86	simpld 499	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑝 ≠ (0g‘𝑃))
88	11, 6, 76, 12	deg1nn0cl 26214	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑝 ∈ 𝐵 ∧ 𝑝 ≠ (0g‘𝑃)) → (𝐷‘𝑝) ∈ ℕ0)
89	75, 20, 87, 88	syl3anc 1396	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ ℕ0)
90	89	nn0cnd 12567	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ ℂ)
91	86	simprd 500	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑞 ≠ (0g‘𝑃))
92	11, 6, 76, 12	deg1nn0cl 26214	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑞 ∈ 𝐵 ∧ 𝑞 ≠ (0g‘𝑃)) → (𝐷‘𝑞) ∈ ℕ0)
93	75, 32, 91, 92	syl3anc 1396	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑞) ∈ ℕ0)
94	93	nn0cnd 12567	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑞) ∈ ℂ)
95	35	fveq2d 6886	. . . . . . . . . . . . . . 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 20810	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ Domn)
97	96	ad3antrrr 742	. . . . . . . . . . . . . . . 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 26241	. . . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑄) = 3)
100	95, 98, 99	3eqtr3d 2812	. . . . . . . . . . . . . 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 11625	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑞) = (3 − (𝐷‘𝑝)))
102	101	adantr 485	. . . . . . . . . . . 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 489	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → (𝐷‘𝑝) = 2)
104	103	oveq2d 7427	. . . . . . . . . . . 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 12322	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
106		2cn 12316	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
107		ax-1cn 11158	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
108		2p1e3 12382	. . . . . . . . . . . . . 14 ⊢ (2 + 1) = 3
109	105, 106, 107, 108	subaddrii 11547	. . . . . . . . . . . . 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 2808	. . . . . . . . . . 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 33816	. . . . . . . . . 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 3048	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 2) → ⊥)
114	113	adantlr 727	. . . . . . . 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 485	. . . . . . . . . . . 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 489	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → (𝐷‘𝑝) = 3)
117	116	oveq2d 7427	. . . . . . . . . . . 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 11529	. . . . . . . . . . . . 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 2808	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → (𝐷‘𝑞) = 0)
121	18	adantr 485	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → 𝐹 ∈ Field)
122	32, 12	eleqtrdi 2879	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑞 ∈ (Base‘𝑃))
123	122	adantr 485	. . . . . . . . . . . 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 33810	. . . . . . . . . . 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 260	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → 𝑞 ∈ (Unit‘𝑃))
126	31	eldifbd 3926	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → ¬ 𝑞 ∈ (Unit‘𝑃))
127	126	adantr 485	. . . . . . . . . 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 1589	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) = 3) → ⊥)
129	128	adantlr 727	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) ∧ (𝐷‘𝑝) ∈ {2, 3}) ∧ (𝐷‘𝑝) = 3) → ⊥)
130		elpri 4618	. . . . . . . . 9 ⊢ ((𝐷‘𝑝) ∈ {2, 3} → ((𝐷‘𝑝) = 2 ∨ (𝐷‘𝑝) = 3))
131	130	adantl 486	. . . . . . . 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 973	. . . . . . 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 12566	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ ℝ)
135		nn0addge1 12550	. . . . . . . . . . . 12 ⊢ (((𝐷‘𝑝) ∈ ℝ ∧ (𝐷‘𝑞) ∈ ℕ0) → (𝐷‘𝑝) ≤ ((𝐷‘𝑝) + (𝐷‘𝑞)))
136	134, 93, 135	syl2anc 595	. . . . . . . . . . 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 5141	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ≤ 3)
138		fznn0 13647	. . . . . . . . . . 11 ⊢ (3 ∈ ℕ0 → ((𝐷‘𝑝) ∈ (0...3) ↔ ((𝐷‘𝑝) ∈ ℕ0 ∧ (𝐷‘𝑝) ≤ 3)))
139	138	biimpar 482	. . . . . . . . . 10 ⊢ ((3 ∈ ℕ0 ∧ ((𝐷‘𝑝) ∈ ℕ0 ∧ (𝐷‘𝑝) ≤ 3)) → (𝐷‘𝑝) ∈ (0...3))
140	133, 89, 137, 139	syl12anc 849	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ (0...3))
141		fz0to3un2pr 13657	. . . . . . . . 9 ⊢ (0...3) = ({0, 1} ∪ {2, 3})
142	140, 141	eleqtrdi 2879	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ ({0, 1} ∪ {2, 3}))
143		elun 4115	. . . . . . . 8 ⊢ ((𝐷‘𝑝) ∈ ({0, 1} ∪ {2, 3}) ↔ ((𝐷‘𝑝) ∈ {0, 1} ∨ (𝐷‘𝑝) ∈ {2, 3}))
144	142, 143	sylib 221	. . . . . . 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 973	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → ⊥)
146	145	r19.29ffa 32759	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∃𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))(𝑝(.r‘𝑃)𝑞) = 𝑄) → ⊥)
147	146	inegd 1587	. . . 4 ⊢ (𝜑 → ¬ ∃𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∃𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))(𝑝(.r‘𝑃)𝑞) = 𝑄)
148		ralnex2 3151	. . . 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 237	. . 3 ⊢ (𝜑 → ∀𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∀𝑞 ∈ (𝐵 ∖ (Unit‘𝑃)) ¬ (𝑝(.r‘𝑃)𝑞) = 𝑄)
150		df-ne 2965	. . . 4 ⊢ ((𝑝(.r‘𝑃)𝑞) ≠ 𝑄 ↔ ¬ (𝑝(.r‘𝑃)𝑞) = 𝑄)
151	150	2ralbii 3146	. . 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 237	. 2 ⊢ (𝜑 → ∀𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∀𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))(𝑝(.r‘𝑃)𝑞) ≠ 𝑄)
153		eqid 2769	. . 3 ⊢ (Unit‘𝑃) = (Unit‘𝑃)
154		eqid 2769	. . 3 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
155		eqid 2769	. . 3 ⊢ (𝐵 ∖ (Unit‘𝑃)) = (𝐵 ∖ (Unit‘𝑃))
156	12, 153, 154, 155, 33	isirred 20501	. 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 594	1 ⊢ (𝜑 → 𝑄 ∈ (Irred‘𝑃))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567  ⊥wfal 1579   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∖ cdif 3910   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294  {csn 4594  {cpr 4596   class class class wbr 5113  ^◡ccnv 5661   “ cima 5665  ‘cfv 6537  (class class class)co 7411  ℝcr 11099  0cc0 11100  1c1 11101   + caddc 11103   ≤ cle 11244   − cmin 11441  2c2 12295  3c3 12296  ℕ₀cn0 12504  ...cfz 13535  Basecbs 17269  ._rcmulr 17311  0_gc0g 17492  Ringcrg 20315  CRingccrg 20316  Unitcui 20437  Irredcir 20438  Domncdomn 20777  IDomncidom 20778  Fieldcfield 20814  algSccascl 21971  Poly₁cpl1 22306  eval₁ce1 22443  deg₁cdg1 26180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177  ax-pre-sup 11178  ax-addf 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-ofr 7676  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8157  df-tpos 8222  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8694  df-map 8826  df-pm 8827  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9322  df-sup 9402  df-oi 9472  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-uz 12863  df-fz 13536  df-fzo 13683  df-seq 14038  df-hash 14367  df-struct 17207  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-mulr 17324  df-starv 17325  df-sca 17326  df-vsca 17327  df-ip 17328  df-tset 17329  df-ple 17330  df-ds 17332  df-unif 17333  df-hom 17334  df-cco 17335  df-0g 17494  df-gsum 17495  df-prds 17500  df-pws 17502  df-mre 17638  df-mrc 17639  df-acs 17641  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-mhm 18841  df-submnd 18842  df-grp 19003  df-minusg 19004  df-sbg 19005  df-mulg 19134  df-subg 19189  df-ghm 19284  df-cntz 19387  df-cmn 19852  df-abl 19853  df-mgp 20217  df-rng 20231  df-ur 20264  df-srg 20269  df-ring 20317  df-cring 20318  df-oppr 20419  df-dvdsr 20439  df-unit 20440  df-irred 20441  df-invr 20470  df-dvr 20483  df-rhm 20554  df-nzr 20596  df-subrng 20631  df-subrg 20655  df-rlreg 20779  df-domn 20780  df-idom 20781  df-drng 20815  df-field 20816  df-lmod 20961  df-lss 21031  df-lsp 21071  df-cnfld 21492  df-assa 21972  df-asp 21973  df-ascl 21974  df-psr 22028  df-mvr 22029  df-mpl 22030  df-opsr 22032  df-evls 22194  df-evl 22195  df-psr1 22309  df-vr1 22310  df-ply1 22311  df-coe1 22312  df-evls1 22444  df-evl1 22445  df-mdeg 26181  df-deg1 26182  df-mon1 26257

This theorem is referenced by:  2sqr3minply  34115  cos9thpiminply  34123