Theorem minplyirred 33868

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 2736	. . 3 ⊢ (0g‘𝐸) = (0g‘𝐸)
8		eqid 2736	. . 3 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}
9		eqid 2736	. . 3 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
10		eqid 2736	. . 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 33863	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	minplyval 33862	. . 3 ⊢ (𝜑 → (𝑀‘𝐴) = ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}))
14		eqid 2736	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
15		eqid 2736	. . . . . 6 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
16	15	sdrgdrng 20723	. . . . 5 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
17	5, 16	syl 17	. . . 4 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
18	4	fldcrngd 20675	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ CRing)
19		sdrgsubrg 20724	. . . . . 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 33859	. . . 4 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} ∈ (LIdeal‘𝑃))
22	4	flddrngd 20674	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ DivRing)
23		drngnzr 20681	. . . . . 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 33860	. . . 4 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} ≠ (Base‘𝑃))
26	2, 10, 14, 17, 21, 25	ig1pnunit 33682	. . 3 ⊢ (𝜑 → ¬ ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}) ∈ (Unit‘𝑃))
27	13, 26	eqneltrd 2856	. 2 ⊢ (𝜑 → ¬ (𝑀‘𝐴) ∈ (Unit‘𝑃))
28		fldidom 20704	. . . . . . . . . . 11 ⊢ (𝐸 ∈ Field → 𝐸 ∈ IDomn)
29	4, 28	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ IDomn)
30	29	idomdomd 20659	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Domn)
31	30	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝐸 ∈ Domn)
32	18	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝐸 ∈ CRing)
33	20	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝐹 ∈ (SubRing‘𝐸))
34	6	ad3antrrr 730	. . . . . . . . 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 22319	. . . . . . . 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 22319	. . . . . . . 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 6838	. . . . . . . . . 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 6836	. . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (.r‘𝑃) = (.r‘𝑃)
43		eqid 2736	. . . . . . . . . 10 ⊢ (.r‘𝐸) = (.r‘𝐸)
44	1, 3, 2, 15, 14, 42, 43, 32, 33, 35, 37, 34	evls1muld 22316	. . . . . . . . 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 2736	. . . . . . . . . . . . . . 15 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
46	2, 10, 45	ig1pcl 26140	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 2836	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀‘𝐴) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})
49		fveq2 6834	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = (𝑀‘𝐴) → (𝑂‘𝑞) = (𝑂‘(𝑀‘𝐴)))
50	49	fveq1d 6836	. . . . . . . . . . . . . 14 ⊢ (𝑞 = (𝑀‘𝐴) → ((𝑂‘𝑞)‘𝐴) = ((𝑂‘(𝑀‘𝐴))‘𝐴))
51	50	eqeq1d 2738	. . . . . . . . . . . . 13 ⊢ (𝑞 = (𝑀‘𝐴) → (((𝑂‘𝑞)‘𝐴) = (0g‘𝐸) ↔ ((𝑂‘(𝑀‘𝐴))‘𝐴) = (0g‘𝐸)))
52	51	elrab 3646	. . . . . . . . . . . 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 730	. . . . . . . . 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 2779	. . . . . . . 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 20641	. . . . . . . . 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 1375	. . . . . . 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 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑓)‘𝐴) = (0g‘𝐸)) → 𝐸 ∈ Field)
61	5	ad4antr 732	. . . . . . . . . 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 730	. . . . . . . . . . 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 20731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
72	4, 5, 71	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
73		fldidom 20704	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ↾s 𝐹) ∈ Field → (𝐸 ↾s 𝐹) ∈ IDomn)
74	72, 73	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ IDomn)
75	74	idomdomd 20659	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Domn)
76	2	ply1domn 26085	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸 ↾s 𝐹) ∈ Domn → 𝑃 ∈ Domn)
77	75, 76	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ Domn)
78	77	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → 𝑃 ∈ Domn)
79	39, 65	eqnetrd 2999	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → (𝑓(.r‘𝑃)𝑔) ≠ 𝑍)
80	14, 42, 63	domneq0 20641	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ Domn ∧ 𝑓 ∈ (Base‘𝑃) ∧ 𝑔 ∈ (Base‘𝑃)) → ((𝑓(.r‘𝑃)𝑔) = 𝑍 ↔ (𝑓 = 𝑍 ∨ 𝑔 = 𝑍)))
81	80	necon3abid 2968	. . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) → ¬ (𝑓 = 𝑍 ∨ 𝑔 = 𝑍))
84		neanior 3025	. . . . . . . . . . . . 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 33867	. . . . . . . . 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 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ (𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴)) ∧ ((𝑂‘𝑔)‘𝐴) = (0g‘𝐸)) → 𝐸 ∈ Field)
93	5	ad4antr 732	. . . . . . . . . 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 20675	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ CRing)
99	2	ply1crng 22139	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ↾s 𝐹) ∈ CRing → 𝑃 ∈ CRing)
100	98, 99	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ CRing)
101	100	ad4antr 732	. . . . . . . . . . . 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 20186	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ CRing ∧ 𝑔 ∈ (Base‘𝑃) ∧ 𝑓 ∈ (Base‘𝑃)) → (𝑔(.r‘𝑃)𝑓) = (𝑓(.r‘𝑃)𝑔))
103	101, 96, 97, 102	syl3anc 1373	. . . . . . . . . . 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 2771	. . . . . . . . . 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 33867	. . . . . . . . 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 966	. . . . . . 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 871	. . . . 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 3179	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (Base‘𝑃)∀𝑔 ∈ (Base‘𝑃)((𝑓(.r‘𝑃)𝑔) = (𝑀‘𝐴) → (𝑓 ∈ (Unit‘𝑃) ∨ 𝑔 ∈ (Unit‘𝑃))))
117		eqid 2736	. . 3 ⊢ (Unit‘𝑃) = (Unit‘𝑃)
118		eqid 2736	. . 3 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
119	14, 117, 118, 42	isirred2 20357	. 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 1344	1 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Irred‘𝑃))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  {crab 3399  dom cdm 5624  ‘cfv 6492  (class class class)co 7358  Basecbs 17136   ↾_s cress 17157  ._rcmulr 17178  0_gc0g 17359  CRingccrg 20169  Unitcui 20291  Irredcir 20292  NzRingcnzr 20445  SubRingcsubrg 20502  Domncdomn 20625  IDomncidom 20626  DivRingcdr 20662  Fieldcfield 20663  SubDRingcsdrg 20719  LIdealclidl 21161  RSpancrsp 21162  Poly₁cpl1 22117   evalSub₁ ces1 22257  idlGen_1pcig1p 26091   minPoly cminply 33856

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 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104  ax-addf 11105

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 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-sup 9345  df-inf 9346  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-starv 17192  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-unif 17200  df-hom 17201  df-cco 17202  df-0g 17361  df-gsum 17362  df-prds 17367  df-pws 17369  df-mre 17505  df-mrc 17506  df-acs 17508  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-subg 19053  df-ghm 19142  df-cntz 19246  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-srg 20122  df-ring 20170  df-cring 20171  df-oppr 20273  df-dvdsr 20293  df-unit 20294  df-irred 20295  df-invr 20324  df-rhm 20408  df-nzr 20446  df-subrng 20479  df-subrg 20503  df-rlreg 20627  df-domn 20628  df-idom 20629  df-drng 20664  df-field 20665  df-sdrg 20720  df-lmod 20813  df-lss 20883  df-lsp 20923  df-sra 21125  df-rgmod 21126  df-lidl 21163  df-cnfld 21310  df-assa 21808  df-asp 21809  df-ascl 21810  df-psr 21865  df-mvr 21866  df-mpl 21867  df-opsr 21869  df-evls 22029  df-evl 22030  df-psr1 22120  df-vr1 22121  df-ply1 22122  df-coe1 22123  df-evls1 22259  df-evl1 22260  df-mdeg 26016  df-deg1 26017  df-mon1 26092  df-uc1p 26093  df-ig1p 26096  df-minply 33857

This theorem is referenced by:  irredminply  33873  algextdeglem4  33877