Theorem minplyirred 33704

Description: A nonzero minimal polynomial is irreducible. (Contributed by Thierry Arnoux, 22-Mar-2025.)

Hypotheses

Ref	Expression
ply1annig1p.o	⊢ 𝑂 = (𝐸 evalSub1 𝐹)
ply1annig1p.p	⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
ply1annig1p.b	⊢ 𝐵 = (Base‘𝐸)
ply1annig1p.e	⊢ (𝜑 → 𝐸 ∈ Field)
ply1annig1p.f	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
ply1annig1p.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)
minplyirred.1	⊢ 𝑀 = (𝐸 minPoly 𝐹)
minplyirred.2	⊢ 𝑍 = (0g‘𝑃)
minplyirred.3	⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)

Assertion

Ref	Expression
minplyirred	⊢ (𝜑 → (𝑀‘𝐴) ∈ (Irred‘𝑃))

Proof of Theorem minplyirred

Dummy variables 𝑞 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ply1annig1p.o	. . 3 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
2		ply1annig1p.p	. . 3 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
3		ply1annig1p.b	. . 3 ⊢ 𝐵 = (Base‘𝐸)
4		ply1annig1p.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ Field)
5		ply1annig1p.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
6		ply1annig1p.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
7		eqid 2740	. . 3 ⊢ (0g‘𝐸) = (0g‘𝐸)
8		eqid 2740	. . 3 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}
9		eqid 2740	. . 3 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
10		eqid 2740	. . 3 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
11		minplyirred.1	. . 3 ⊢ 𝑀 = (𝐸 minPoly 𝐹)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	minplycl 33699	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	minplyval 33698	. . 3 ⊢ (𝜑 → (𝑀‘𝐴) = ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}))
14		eqid 2740	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
15		eqid 2740	. . . . . 6 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
16	15	sdrgdrng 20813	. . . . 5 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
17	5, 16	syl 17	. . . 4 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
18	4	fldcrngd 20764	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ CRing)
19		sdrgsubrg 20814	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
20	5, 19	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
21	1, 2, 3, 18, 20, 6, 7, 8	ply1annidl 33695	. . . 4 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} ∈ (LIdeal‘𝑃))
22	4	flddrngd 20763	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ DivRing)
23		drngnzr 20770	. . . . . 6 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ NzRing)
24	22, 23	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ NzRing)
25	1, 2, 3, 18, 20, 6, 7, 8, 14, 24	ply1annnr 33696	. . . 4 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} ≠ (Base‘𝑃))
26	2, 10, 14, 17, 21, 25	ig1pnunit 33586	. . 3 ⊢ (𝜑 → ¬ ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}) ∈ (Unit‘𝑃))
27	13, 26	eqneltrd 2864	. 2 ⊢ (𝜑 → ¬ (𝑀‘𝐴) ∈ (Unit‘𝑃))
28		fldidom 20793	. . . . . . . . . . 11 ⊢ (𝐸 ∈ Field → 𝐸 ∈ IDomn)
29	4, 28	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ IDomn)
30	29	idomdomd 20748	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Domn)
31	30	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝐸 ∈ Domn)
32	18	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝐸 ∈ CRing)
33	20	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝐹 ∈ (SubRing‘𝐸))
34	6	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝐴 ∈ 𝐵)
35		simpllr 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝑓 ∈ (Base‘𝑃))
36	1, 2, 3, 14, 32, 33, 34, 35	evls1fvcl 22400	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → ((𝑂‘𝑓)‘𝐴) ∈ 𝐵)
37		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝑔 ∈ (Base‘𝑃))
38	1, 2, 3, 14, 32, 33, 34, 37	evls1fvcl 22400	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → ((𝑂‘𝑔)‘𝐴) ∈ 𝐵)
39		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴))
40	39	fveq2d 6924	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (𝑂‘(𝑓(.r‘𝑃)𝑔)) = (𝑂‘(𝑀‘𝐴)))
41	40	fveq1d 6922	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → ((𝑂‘(𝑓(.r‘𝑃)𝑔))‘𝐴) = ((𝑂‘(𝑀‘𝐴))‘𝐴))
42		eqid 2740	. . . . . . . . . 10 ⊢ (.r‘𝑃) = (.r‘𝑃)
43		eqid 2740	. . . . . . . . . 10 ⊢ (.r‘𝐸) = (.r‘𝐸)
44	1, 3, 2, 15, 14, 42, 43, 32, 33, 35, 37, 34	evls1muld 22397	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → ((𝑂‘(𝑓(.r‘𝑃)𝑔))‘𝐴) = (((𝑂‘𝑓)‘𝐴)(.r‘𝐸)((𝑂‘𝑔)‘𝐴)))
45		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
46	2, 10, 45	ig1pcl 26238	. . . . . . . . . . . . . 14 ⊢ (((𝐸 ↾s 𝐹) ∈ DivRing ∧ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} ∈ (LIdeal‘𝑃)) → ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})
47	17, 21, 46	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})
48	13, 47	eqeltrd 2844	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀‘𝐴) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})
49		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = (𝑀‘𝐴) → (𝑂‘𝑞) = (𝑂‘(𝑀‘𝐴)))
50	49	fveq1d 6922	. . . . . . . . . . . . . 14 ⊢ (𝑞 = (𝑀‘𝐴) → ((𝑂‘𝑞)‘𝐴) = ((𝑂‘(𝑀‘𝐴))‘𝐴))
51	50	eqeq1d 2742	. . . . . . . . . . . . 13 ⊢ (𝑞 = (𝑀‘𝐴) → (((𝑂‘𝑞)‘𝐴) = (0g‘𝐸) ↔ ((𝑂‘(𝑀‘𝐴))‘𝐴) = (0g‘𝐸)))
52	51	elrab 3708	. . . . . . . . . . . 12 ⊢ ((𝑀‘𝐴) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} ↔ ((𝑀‘𝐴) ∈ dom 𝑂 ∧ ((𝑂‘(𝑀‘𝐴))‘𝐴) = (0g‘𝐸)))
53	48, 52	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ dom 𝑂 ∧ ((𝑂‘(𝑀‘𝐴))‘𝐴) = (0g‘𝐸)))
54	53	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑂‘(𝑀‘𝐴))‘𝐴) = (0g‘𝐸))
55	54	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → ((𝑂‘(𝑀‘𝐴))‘𝐴) = (0g‘𝐸))
56	41, 44, 55	3eqtr3d 2788	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (((𝑂‘𝑓)‘𝐴)(.r‘𝐸)((𝑂‘𝑔)‘𝐴)) = (0g‘𝐸))
57	3, 43, 7	domneq0 20730	. . . . . . . . 9 ⊢ ((𝐸 ∈ Domn ∧ ((𝑂‘𝑓)‘𝐴) ∈ 𝐵 ∧ ((𝑂‘𝑔)‘𝐴) ∈ 𝐵) → ((((𝑂‘𝑓)‘𝐴)(.r‘𝐸)((𝑂‘𝑔)‘𝐴)) = (0g‘𝐸) ↔ (((𝑂‘𝑓)‘𝐴) = (0g‘𝐸) ∨ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸))))
58	57	biimpa 476	. . . . . . . 8 ⊢ (((𝐸 ∈ Domn ∧ ((𝑂‘𝑓)‘𝐴) ∈ 𝐵 ∧ ((𝑂‘𝑔)‘𝐴) ∈ 𝐵) ∧ (((𝑂‘𝑓)‘𝐴)(.r‘𝐸)((𝑂‘𝑔)‘𝐴)) = (0g‘𝐸)) → (((𝑂‘𝑓)‘𝐴) = (0g‘𝐸) ∨ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)))
59	31, 36, 38, 56, 58	syl31anc 1373	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (((𝑂‘𝑓)‘𝐴) = (0g‘𝐸) ∨ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)))
60	4	ad4antr 731	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → 𝐸 ∈ Field)
61	5	ad4antr 731	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → 𝐹 ∈ (SubDRing‘𝐸))
62	34	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → 𝐴 ∈ 𝐵)
63		minplyirred.2	. . . . . . . . . 10 ⊢ 𝑍 = (0g‘𝑃)
64		minplyirred.3	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)
65	64	ad3antrrr 729	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (𝑀‘𝐴) ≠ 𝑍)
66	65	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → (𝑀‘𝐴) ≠ 𝑍)
67	35	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → 𝑓 ∈ (Base‘𝑃))
68		simpllr 775	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → 𝑔 ∈ (Base‘𝑃))
69		simplr 768	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴))
70		simpr 484	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸))
71		fldsdrgfld 20821	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
72	4, 5, 71	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
73		fldidom 20793	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ↾s 𝐹) ∈ Field → (𝐸 ↾s 𝐹) ∈ IDomn)
74	72, 73	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ IDomn)
75	74	idomdomd 20748	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Domn)
76	2	ply1domn 26183	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸 ↾s 𝐹) ∈ Domn → 𝑃 ∈ Domn)
77	75, 76	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ Domn)
78	77	ad3antrrr 729	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝑃 ∈ Domn)
79	39, 65	eqnetrd 3014	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (𝑓(.r‘𝑃)𝑔) ≠ 𝑍)
80	14, 42, 63	domneq0 20730	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ Domn ∧ 𝑓 ∈ (Base‘𝑃) ∧ 𝑔 ∈ (Base‘𝑃)) → ((𝑓(.r‘𝑃)𝑔) = 𝑍 ↔ (𝑓 = 𝑍 ∨ 𝑔 = 𝑍)))
81	80	necon3abid 2983	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Domn ∧ 𝑓 ∈ (Base‘𝑃) ∧ 𝑔 ∈ (Base‘𝑃)) → ((𝑓(.r‘𝑃)𝑔) ≠ 𝑍 ↔ ¬ (𝑓 = 𝑍 ∨ 𝑔 = 𝑍)))
82	81	biimpa 476	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ Domn ∧ 𝑓 ∈ (Base‘𝑃) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) ≠ 𝑍) → ¬ (𝑓 = 𝑍 ∨ 𝑔 = 𝑍))
83	78, 35, 37, 79, 82	syl31anc 1373	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → ¬ (𝑓 = 𝑍 ∨ 𝑔 = 𝑍))
84		neanior 3041	. . . . . . . . . . . . 13 ⊢ ((𝑓 ≠ 𝑍 ∧ 𝑔 ≠ 𝑍) ↔ ¬ (𝑓 = 𝑍 ∨ 𝑔 = 𝑍))
85	83, 84	sylibr 234	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (𝑓 ≠ 𝑍 ∧ 𝑔 ≠ 𝑍))
86	85	simpld 494	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝑓 ≠ 𝑍)
87	86	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → 𝑓 ≠ 𝑍)
88	85	simprd 495	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝑔 ≠ 𝑍)
89	88	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → 𝑔 ≠ 𝑍)
90	1, 2, 3, 60, 61, 62, 11, 63, 66, 67, 68, 69, 70, 87, 89	minplyirredlem 33703	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → 𝑔 ∈ (Unit‘𝑃))
91	90	ex 412	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (((𝑂‘𝑓)‘𝐴) = (0g‘𝐸) → 𝑔 ∈ (Unit‘𝑃)))
92	4	ad4antr 731	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → 𝐸 ∈ Field)
93	5	ad4antr 731	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → 𝐹 ∈ (SubDRing‘𝐸))
94	34	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → 𝐴 ∈ 𝐵)
95	65	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → (𝑀‘𝐴) ≠ 𝑍)
96		simpllr 775	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → 𝑔 ∈ (Base‘𝑃))
97	35	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → 𝑓 ∈ (Base‘𝑃))
98	72	fldcrngd 20764	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ CRing)
99	2	ply1crng 22221	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ↾s 𝐹) ∈ CRing → 𝑃 ∈ CRing)
100	98, 99	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ CRing)
101	100	ad4antr 731	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → 𝑃 ∈ CRing)
102	14, 42	crngcom 20278	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ CRing ∧ 𝑔 ∈ (Base‘𝑃) ∧ 𝑓 ∈ (Base‘𝑃)) → (𝑔(.r‘𝑃)𝑓) = (𝑓(.r‘𝑃)𝑔))
103	101, 96, 97, 102	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → (𝑔(.r‘𝑃)𝑓) = (𝑓(.r‘𝑃)𝑔))
104		simplr 768	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴))
105	103, 104	eqtrd 2780	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → (𝑔(.r‘𝑃)𝑓) = (𝑀‘𝐴))
106		simpr 484	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸))
107	88	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → 𝑔 ≠ 𝑍)
108	86	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → 𝑓 ≠ 𝑍)
109	1, 2, 3, 92, 93, 94, 11, 63, 95, 96, 97, 105, 106, 107, 108	minplyirredlem 33703	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → 𝑓 ∈ (Unit‘𝑃))
110	109	ex 412	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (((𝑂‘𝑔)‘𝐴) = (0g‘𝐸) → 𝑓 ∈ (Unit‘𝑃)))
111	91, 110	orim12d 965	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → ((((𝑂‘𝑓)‘𝐴) = (0g‘𝐸) ∨ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → (𝑔 ∈ (Unit‘𝑃) ∨ 𝑓 ∈ (Unit‘𝑃))))
112	59, 111	mpd 15	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (𝑔 ∈ (Unit‘𝑃) ∨ 𝑓 ∈ (Unit‘𝑃)))
113	112	orcomd 870	. . . . 5 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (𝑓 ∈ (Unit‘𝑃) ∨ 𝑔 ∈ (Unit‘𝑃)))
114	113	ex 412	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) → ((𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴) → (𝑓 ∈ (Unit‘𝑃) ∨ 𝑔 ∈ (Unit‘𝑃))))
115	114	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘𝑃) ∧ 𝑔 ∈ (Base‘𝑃))) → ((𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴) → (𝑓 ∈ (Unit‘𝑃) ∨ 𝑔 ∈ (Unit‘𝑃))))
116	115	ralrimivva 3208	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (Base‘𝑃)∀𝑔 ∈ (Base‘𝑃)((𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴) → (𝑓 ∈ (Unit‘𝑃) ∨ 𝑔 ∈ (Unit‘𝑃))))
117		eqid 2740	. . 3 ⊢ (Unit‘𝑃) = (Unit‘𝑃)
118		eqid 2740	. . 3 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
119	14, 117, 118, 42	isirred2 20447	. 2 ⊢ ((𝑀‘𝐴) ∈ (Irred‘𝑃) ↔ ((𝑀‘𝐴) ∈ (Base‘𝑃) ∧ ¬ (𝑀‘𝐴) ∈ (Unit‘𝑃) ∧ ∀𝑓 ∈ (Base‘𝑃)∀𝑔 ∈ (Base‘𝑃)((𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴) → (𝑓 ∈ (Unit‘𝑃) ∨ 𝑔 ∈ (Unit‘𝑃)))))
120	12, 27, 116, 119	syl3anbrc 1343	1 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Irred‘𝑃))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  {crab 3443  dom cdm 5700  ‘cfv 6573  (class class class)co 7448  Basecbs 17258   ↾_s cress 17287  ._rcmulr 17312  0_gc0g 17499  CRingccrg 20261  Unitcui 20381  Irredcir 20382  NzRingcnzr 20538  SubRingcsubrg 20595  Domncdomn 20714  IDomncidom 20715  DivRingcdr 20751  Fieldcfield 20752  SubDRingcsdrg 20809  LIdealclidl 21239  RSpancrsp 21240  Poly₁cpl1 22199   evalSub₁ ces1 22338  idlGen_1pcig1p 26189   minPoly cminply 33692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262  ax-addf 11263

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-ofr 7715  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-starv 17326  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-unif 17334  df-hom 17335  df-cco 17336  df-0g 17501  df-gsum 17502  df-prds 17507  df-pws 17509  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-mulg 19108  df-subg 19163  df-ghm 19253  df-cntz 19357  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-srg 20214  df-ring 20262  df-cring 20263  df-oppr 20360  df-dvdsr 20383  df-unit 20384  df-irred 20385  df-invr 20414  df-rhm 20498  df-nzr 20539  df-subrng 20572  df-subrg 20597  df-rlreg 20716  df-domn 20717  df-idom 20718  df-drng 20753  df-field 20754  df-sdrg 20810  df-lmod 20882  df-lss 20953  df-lsp 20993  df-sra 21195  df-rgmod 21196  df-lidl 21241  df-cnfld 21388  df-assa 21896  df-asp 21897  df-ascl 21898  df-psr 21952  df-mvr 21953  df-mpl 21954  df-opsr 21956  df-evls 22121  df-evl 22122  df-psr1 22202  df-vr1 22203  df-ply1 22204  df-coe1 22205  df-evls1 22340  df-evl1 22341  df-mdeg 26114  df-deg1 26115  df-mon1 26190  df-uc1p 26191  df-ig1p 26194  df-minply 33693

This theorem is referenced by:  irredminply  33707  algextdeglem4  33711