Theorem ply1dg3rt0irred 33586

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 12369	. . . . . 6 ⊢ 3 ≠ 0
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 3 ≠ 0)
5	2, 4	eqnetrd 3005	. . . 4 ⊢ (𝜑 → (𝐷‘𝑄) ≠ 0)
6		ply1dg3rt0irred.p	. . . . . 6 ⊢ 𝑃 = (Poly1‘𝐹)
7		eqid 2734	. . . . . 6 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
8		eqid 2734	. . . . . 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 2848	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝑃))
14	6, 7, 8, 9, 10, 11, 13	ply1unit 33579	. . . . 5 ⊢ (𝜑 → (𝑄 ∈ (Unit‘𝑃) ↔ (𝐷‘𝑄) = 0))
15	14	necon3bbid 2975	. . . 4 ⊢ (𝜑 → (¬ 𝑄 ∈ (Unit‘𝑃) ↔ (𝐷‘𝑄) ≠ 0))
16	5, 15	mpbird 257	. . 3 ⊢ (𝜑 → ¬ 𝑄 ∈ (Unit‘𝑃))
17	1, 16	eldifd 3973	. 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 776	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃)))
20	19	eldifad 3974	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑝 ∈ 𝐵)
21	20, 12	eleqtrdi 2848	. . . . . . . . . . . 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 33579	. . . . . . . . . . 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 3975	. . . . . . . . . . 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 1558	. . . . . . . . 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 20758	. . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃)))
32	31	eldifad 3974	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑞 ∈ 𝐵)
33		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑃) = (.r‘𝑃)
34	6, 12, 28, 11, 9, 30, 20, 32, 33	ply1mulrtss 33585	. . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . . . 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 5888	. . . . . . . . . . . . . . 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 6078	. . . . . . . . . . . . . 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 4035	. . . . . . . . . . . . 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 4035	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑝) “ { 0 }) ⊆ ∅)
43		ss0 4407	. . . . . . . . . . . 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 33584	. . . . . . . . . 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 3023	. . . . . . . . 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 4653	. . . . . . . . 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 33585	. . . . . . . . . . . . 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 20787	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 ∈ Field → 𝐹 ∈ IDomn)
57	10, 56	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 ∈ IDomn)
58	6	ply1idom 26178	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ IDomn → 𝑃 ∈ IDomn)
59	57, 58	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑃 ∈ IDomn)
60	59	idomcringd 20743	. . . . . . . . . . . . . . . . . . . 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 20272	. . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝑞(.r‘𝑃)𝑝) = 𝑄)
64	63	fveq2d 6910	. . . . . . . . . . . . . . . 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 5888	. . . . . . . . . . . . . . 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 6078	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . 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 4035	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (◡(𝑂‘𝑞) “ { 0 }) ⊆ ∅)
69		ss0 4407	. . . . . . . . . . . 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 20263	. . . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘𝑃) = (0g‘𝑃)
77	59	idomdomd 20742	. . . . . . . . . . . . . . . . . . 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 12541	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 3 ∈ ℕ0
80	2, 79	eqeltrdi 2846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐷‘𝑄) ∈ ℕ0)
81	11, 6, 76, 12	deg1nn0clb 26143	. . . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → 𝑄 ≠ (0g‘𝑃))
85	35, 84	eqnetrd 3005	. . . . . . . . . . . . . . . . . 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 33260	. . . . . . . . . . . . . . . . 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 26141	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑝 ∈ 𝐵 ∧ 𝑝 ≠ (0g‘𝑃)) → (𝐷‘𝑝) ∈ ℕ0)
89	75, 20, 87, 88	syl3anc 1370	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ ℕ0)
90	89	nn0cnd 12586	. . . . . . . . . . . . . 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 26141	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Ring ∧ 𝑞 ∈ 𝐵 ∧ 𝑞 ≠ (0g‘𝑃)) → (𝐷‘𝑞) ∈ ℕ0)
93	75, 32, 91, 92	syl3anc 1370	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑞) ∈ ℕ0)
94	93	nn0cnd 12586	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑞) ∈ ℂ)
95	35	fveq2d 6910	. . . . . . . . . . . . . . 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 20742	. . . . . . . . . . . . . . . . 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 26168	. . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . 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 11671	. . . . . . . . . . . . 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 7446	. . . . . . . . . . . 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 12344	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
106		2cn 12338	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
107		ax-1cn 11210	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
108		2p1e3 12405	. . . . . . . . . . . . . 14 ⊢ (2 + 1) = 3
109	105, 106, 107, 108	subaddrii 11595	. . . . . . . . . . . . 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 2778	. . . . . . . . . . 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 33584	. . . . . . . . . 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 3023	. . . . . . . . 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 7446	. . . . . . . . . . . 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 11577	. . . . . . . . . . . . 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 2778	. . . . . . . . . . 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 2848	. . . . . . . . . . . . 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 33579	. . . . . . . . . . 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 3975	. . . . . . . . . . 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 1558	. . . . . . . . 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 4653	. . . . . . . . 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 12585	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ ℝ)
135		nn0addge1 12569	. . . . . . . . . . . 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 5173	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ≤ 3)
138		fznn0 13655	. . . . . . . . . . 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 13665	. . . . . . . . 9 ⊢ (0...3) = ({0, 1} ∪ {2, 3})
142	140, 141	eleqtrdi 2848	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ 𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))) ∧ (𝑝(.r‘𝑃)𝑞) = 𝑄) → (𝐷‘𝑝) ∈ ({0, 1} ∪ {2, 3}))
143		elun 4162	. . . . . . . 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 32499	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∃𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))(𝑝(.r‘𝑃)𝑞) = 𝑄) → ⊥)
147	146	inegd 1556	. . . 4 ⊢ (𝜑 → ¬ ∃𝑝 ∈ (𝐵 ∖ (Unit‘𝑃))∃𝑞 ∈ (𝐵 ∖ (Unit‘𝑃))(𝑝(.r‘𝑃)𝑞) = 𝑄)
148		ralnex2 3130	. . . 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 2938	. . . 4 ⊢ ((𝑝(.r‘𝑃)𝑞) ≠ 𝑄 ↔ ¬ (𝑝(.r‘𝑃)𝑞) = 𝑄)
151	150	2ralbii 3125	. . 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 2734	. . 3 ⊢ (Unit‘𝑃) = (Unit‘𝑃)
154		eqid 2734	. . 3 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
155		eqid 2734	. . 3 ⊢ (𝐵 ∖ (Unit‘𝑃)) = (𝐵 ∖ (Unit‘𝑃))
156	12, 153, 154, 155, 33	isirred 20435	. 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 1536  ⊥wfal 1548   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  ∃wrex 3067   ∖ cdif 3959   ∪ cun 3960   ⊆ wss 3962  ∅c0 4338  {csn 4630  {cpr 4632   class class class wbr 5147  ^◡ccnv 5687   “ cima 5691  ‘cfv 6562  (class class class)co 7430  ℝcr 11151  0cc0 11152  1c1 11153   + caddc 11155   ≤ cle 11293   − cmin 11489  2c2 12318  3c3 12319  ℕ₀cn0 12523  ...cfz 13543  Basecbs 17244  ._rcmulr 17298  0_gc0g 17485  Ringcrg 20250  CRingccrg 20251  Unitcui 20371  Irredcir 20372  Domncdomn 20708  IDomncidom 20709  Fieldcfield 20746  algSccascl 21889  Poly₁cpl1 22193  eval₁ce1 22333  deg₁cdg1 26107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-srg 20204  df-ring 20252  df-cring 20253  df-oppr 20350  df-dvdsr 20373  df-unit 20374  df-irred 20375  df-invr 20404  df-dvr 20417  df-rhm 20488  df-nzr 20529  df-subrng 20562  df-subrg 20586  df-rlreg 20710  df-domn 20711  df-idom 20712  df-drng 20747  df-field 20748  df-lmod 20876  df-lss 20947  df-lsp 20987  df-cnfld 21382  df-assa 21890  df-asp 21891  df-ascl 21892  df-psr 21946  df-mvr 21947  df-mpl 21948  df-opsr 21950  df-evls 22115  df-evl 22116  df-psr1 22196  df-vr1 22197  df-ply1 22198  df-coe1 22199  df-evls1 22334  df-evl1 22335  df-mdeg 26108  df-deg1 26109  df-mon1 26184

This theorem is referenced by:  2sqr3minply  33752