Theorem minplyirred 34454

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 2763	. . 3 ⊢ (0g‘𝐸) = (0g‘𝐸)
8		eqid 2763	. . 3 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}
9		eqid 2763	. . 3 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
10		eqid 2763	. . 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 34449	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	minplyval 34448	. . 3 ⊢ (𝜑 → (𝑀‘𝐴) = ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}))
14		eqid 2763	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
15		eqid 2763	. . . . . 6 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
16	15	sdrgdrng 21098	. . . . 5 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
17	5, 16	syl 18	. . . 4 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
18	4	fldcrngd 21046	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ CRing)
19		sdrgsubrg 21099	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
20	5, 19	syl 18	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
21	1, 2, 3, 18, 20, 6, 7, 8	ply1annidl 34445	. . . 4 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} ∈ (LIdeal‘𝑃))
22	4	flddrngd 21045	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ DivRing)
23		drngnzr 21053	. . . . . 6 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ NzRing)
24	22, 23	syl 18	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ NzRing)
25	1, 2, 3, 18, 20, 6, 7, 8, 14, 24	ply1annnr 34446	. . . 4 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} ≠ (Base‘𝑃))
26	2, 10, 14, 17, 21, 25	ig1pnunit 34236	. . 3 ⊢ (𝜑 → ¬ ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}) ∈ (Unit‘𝑃))
27	13, 26	eqneltrd 2884	. 2 ⊢ (𝜑 → ¬ (𝑀‘𝐴) ∈ (Unit‘𝑃))
28		fldidom 21080	. . . . . . . . . . 11 ⊢ (𝐸 ∈ Field → 𝐸 ∈ IDomn)
29	4, 28	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ IDomn)
30	29	idomdomd 21028	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Domn)
31	30	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝐸 ∈ Domn)
32	18	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝐸 ∈ CRing)
33	20	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝐹 ∈ (SubRing‘𝐸))
34	6	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝐴 ∈ 𝐵)
35		simpllr 788	. . . . . . . . 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 22748	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → ((𝑂‘𝑓)‘𝐴) ∈ 𝐵)
37		simplr 781	. . . . . . . . 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 22748	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → ((𝑂‘𝑔)‘𝐴) ∈ 𝐵)
39		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴))
40	39	fveq2d 6920	. . . . . . . . . 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 6918	. . . . . . . . 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 2763	. . . . . . . . . 10 ⊢ (.r‘𝑃) = (.r‘𝑃)
43		eqid 2763	. . . . . . . . . 10 ⊢ (.r‘𝐸) = (.r‘𝐸)
44	1, 3, 2, 15, 14, 42, 43, 32, 33, 35, 37, 34	evls1muld 22745	. . . . . . . . 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 2763	. . . . . . . . . . . . . . 15 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
46	2, 10, 45	ig1pcl 26552	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 2864	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀‘𝐴) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})
49		fveq2 6916	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = (𝑀‘𝐴) → (𝑂‘𝑞) = (𝑂‘(𝑀‘𝐴)))
50	49	fveq1d 6918	. . . . . . . . . . . . . 14 ⊢ (𝑞 = (𝑀‘𝐴) → ((𝑂‘𝑞)‘𝐴) = ((𝑂‘(𝑀‘𝐴))‘𝐴))
51	50	eqeq1d 2765	. . . . . . . . . . . . 13 ⊢ (𝑞 = (𝑀‘𝐴) → (((𝑂‘𝑞)‘𝐴) = (0g‘𝐸) ↔ ((𝑂‘(𝑀‘𝐴))‘𝐴) = (0g‘𝐸)))
52	51	elrab 3648	. . . . . . . . . . . 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 221	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ dom 𝑂 ∧ ((𝑂‘(𝑀‘𝐴))‘𝐴) = (0g‘𝐸)))
54	53	simprd 501	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑂‘(𝑀‘𝐴))‘𝐴) = (0g‘𝐸))
55	54	ad3antrrr 743	. . . . . . . . 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 2806	. . . . . . . 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 21011	. . . . . . . . 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 482	. . . . . . . 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 1401	. . . . . . 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 745	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → 𝐸 ∈ Field)
61	5	ad4antr 745	. . . . . . . . . 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 486	. . . . . . . . . 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 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (𝑀‘𝐴) ≠ 𝑍)
66	65	adantr 486	. . . . . . . . . 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 486	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → 𝑓 ∈ (Base‘𝑃))
68		simpllr 788	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → 𝑔 ∈ (Base‘𝑃))
69		simplr 781	. . . . . . . . . 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 490	. . . . . . . . . 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 21106	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
72	4, 5, 71	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
73		fldidom 21080	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ↾s 𝐹) ∈ Field → (𝐸 ↾s 𝐹) ∈ IDomn)
74	72, 73	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ IDomn)
75	74	idomdomd 21028	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Domn)
76	2	ply1domn 26497	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸 ↾s 𝐹) ∈ Domn → 𝑃 ∈ Domn)
77	75, 76	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ Domn)
78	77	ad3antrrr 743	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝑃 ∈ Domn)
79	39, 65	eqnetrd 3026	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (𝑓(.r‘𝑃)𝑔) ≠ 𝑍)
80	14, 42, 63	domneq0 21011	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ Domn ∧ 𝑓 ∈ (Base‘𝑃) ∧ 𝑔 ∈ (Base‘𝑃)) → ((𝑓(.r‘𝑃)𝑔) = 𝑍 ↔ (𝑓 = 𝑍 ∨ 𝑔 = 𝑍)))
81	80	necon3abid 2995	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Domn ∧ 𝑓 ∈ (Base‘𝑃) ∧ 𝑔 ∈ (Base‘𝑃)) → ((𝑓(.r‘𝑃)𝑔) ≠ 𝑍 ↔ ¬ (𝑓 = 𝑍 ∨ 𝑔 = 𝑍)))
82	81	biimpa 482	. . . . . . . . . . . . . 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 1401	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → ¬ (𝑓 = 𝑍 ∨ 𝑔 = 𝑍))
84		neanior 3052	. . . . . . . . . . . . 13 ⊢ ((𝑓 ≠ 𝑍 ∧ 𝑔 ≠ 𝑍) ↔ ¬ (𝑓 = 𝑍 ∨ 𝑔 = 𝑍))
85	83, 84	sylibr 237	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (𝑓 ≠ 𝑍 ∧ 𝑔 ≠ 𝑍))
86	85	simpld 500	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝑓 ≠ 𝑍)
87	86	adantr 486	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → 𝑓 ≠ 𝑍)
88	85	simprd 501	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝑔 ≠ 𝑍)
89	88	adantr 486	. . . . . . . . . 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 34453	. . . . . . . . 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 418	. . . . . . . 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 745	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → 𝐸 ∈ Field)
93	5	ad4antr 745	. . . . . . . . . 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 486	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → 𝐴 ∈ 𝐵)
95	65	adantr 486	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → (𝑀‘𝐴) ≠ 𝑍)
96		simpllr 788	. . . . . . . . . 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 486	. . . . . . . . . 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 21046	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ CRing)
99	2	ply1crng 22571	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ↾s 𝐹) ∈ CRing → 𝑃 ∈ CRing)
100	98, 99	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ CRing)
101	100	ad4antr 745	. . . . . . . . . . . 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 20529	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ CRing ∧ 𝑔 ∈ (Base‘𝑃) ∧ 𝑓 ∈ (Base‘𝑃)) → (𝑔(.r‘𝑃)𝑓) = (𝑓(.r‘𝑃)𝑔))
103	101, 96, 97, 102	syl3anc 1399	. . . . . . . . . . 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 781	. . . . . . . . . . 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 2798	. . . . . . . . . 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 490	. . . . . . . . . 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 486	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → 𝑔 ≠ 𝑍)
108	86	adantr 486	. . . . . . . . . 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 34453	. . . . . . . . 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 418	. . . . . . . 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 979	. . . . . . 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 16	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (𝑔 ∈ (Unit‘𝑃) ∨ 𝑓 ∈ (Unit‘𝑃)))
113	112	orcomd 885	. . . . 5 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (𝑓 ∈ (Unit‘𝑃) ∨ 𝑔 ∈ (Unit‘𝑃)))
114	113	ex 418	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) → ((𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴) → (𝑓 ∈ (Unit‘𝑃) ∨ 𝑔 ∈ (Unit‘𝑃))))
115	114	anasss 472	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (Base‘𝑃) ∧ 𝑔 ∈ (Base‘𝑃))) → ((𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴) → (𝑓 ∈ (Unit‘𝑃) ∨ 𝑔 ∈ (Unit‘𝑃))))
116	115	ralrimivva 3209	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (Base‘𝑃)∀𝑔 ∈ (Base‘𝑃)((𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴) → (𝑓 ∈ (Unit‘𝑃) ∨ 𝑔 ∈ (Unit‘𝑃))))
117		eqid 2763	. . 3 ⊢ (Unit‘𝑃) = (Unit‘𝑃)
118		eqid 2763	. . 3 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
119	14, 117, 118, 42	isirred2 20702	. 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 1362	1 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Irred‘𝑃))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  {crab 3416  dom cdm 5658  ‘cfv 6566  (class class class)co 7451  Basecbs 17431   ↾_s cress 17452  ._rcmulr 17473  0_gc0g 17660  CRingccrg 20511  Unitcui 20636  Irredcir 20637  NzRingcnzr 20813  SubRingcsubrg 20872  Domncdomn 20995  IDomncidom 20996  DivRingcdr 21031  Fieldcfield 21032  SubDRingcsdrg 21094  LIdealclidl 21535  RSpancrsp 21536  Poly₁cpl1 22550   evalSub₁ ces1 22686  idlGen_1pcig1p 26503   minPoly cminply 34442

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317  ax-addf 11318

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725  df-ofr 7726  df-om 7912  df-1st 8035  df-2nd 8036  df-supp 8215  df-tpos 8281  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-pm 8888  df-ixp 8964  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-fsupp 9392  df-sup 9471  df-inf 9472  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-fz 13680  df-fzo 13829  df-seq 14185  df-hash 14516  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-mulr 17487  df-starv 17488  df-sca 17489  df-vsca 17490  df-ip 17491  df-tset 17492  df-ple 17493  df-ds 17495  df-unif 17496  df-hom 17497  df-cco 17498  df-euf 17499  df-0g 17662  df-gsum 17663  df-prds 17668  df-pws 17670  df-mre 17806  df-mrc 17807  df-acs 17809  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-mhm 19028  df-submnd 19029  df-grp 19197  df-minusg 19198  df-sbg 19199  df-mulg 19328  df-subg 19383  df-ghm 19478  df-cntz 19582  df-cmn 20047  df-abl 20048  df-mgp 20412  df-rng 20426  df-ur 20459  df-srg 20464  df-ring 20512  df-cring 20513  df-oppr 20618  df-dvdsr 20638  df-unit 20639  df-irred 20640  df-invr 20669  df-rhm 20753  df-nzr 20814  df-subrng 20849  df-subrg 20873  df-rlreg 20997  df-domn 20998  df-idom 20999  df-drng 21033  df-field 21034  df-sdrg 21095  df-lmod 21188  df-lss 21258  df-lsp 21298  df-sra 21499  df-rgmod 21500  df-lidl 21537  df-cnfld 21730  df-assa 22216  df-asp 22217  df-ascl 22218  df-psr 22272  df-mvr 22273  df-mpl 22274  df-opsr 22276  df-evls 22438  df-evl 22439  df-psr1 22553  df-vr1 22554  df-ply1 22555  df-coe1 22556  df-evls1 22688  df-evl1 22689  df-mdeg 26428  df-deg1 26429  df-mon1 26504  df-uc1p 26505  df-ig1p 26508  df-minply 34443

This theorem is used by:  irredminply  34459  algextdeglem4  34463