Theorem irredminply 33912

Description: An irreducible, monic, annihilating polynomial is the minimal polynomial. (Contributed by Thierry Arnoux, 27-Apr-2025.)

Hypotheses

Ref	Expression
irredminply.o	⊢ 𝑂 = (𝐸 evalSub1 𝐹)
irredminply.p	⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
irredminply.b	⊢ 𝐵 = (Base‘𝐸)
irredminply.e	⊢ (𝜑 → 𝐸 ∈ Field)
irredminply.f	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
irredminply.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)
irredminply.0	⊢ 0 = (0g‘𝐸)
irredminply.m	⊢ 𝑀 = (𝐸 minPoly 𝐹)
irredminply.z	⊢ 𝑍 = (0g‘𝑃)
irredminply.1	⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) = 0 )
irredminply.2	⊢ (𝜑 → 𝐺 ∈ (Irred‘𝑃))
irredminply.3	⊢ (𝜑 → 𝐺 ∈ (Monic1p‘(𝐸 ↾s 𝐹)))

Assertion

Ref	Expression
irredminply	⊢ (𝜑 → 𝐺 = (𝑀‘𝐴))

Proof of Theorem irredminply

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

Step	Hyp	Ref	Expression
1		irredminply.p	. 2 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
2		eqid 2741	. 2 ⊢ (Monic1p‘(𝐸 ↾s 𝐹)) = (Monic1p‘(𝐸 ↾s 𝐹))
3		eqid 2741	. 2 ⊢ (Unit‘𝑃) = (Unit‘𝑃)
4		eqid 2741	. 2 ⊢ (.r‘𝑃) = (.r‘𝑃)
5		irredminply.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ Field)
6		irredminply.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
7		fldsdrgfld 20774	. . 3 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
8	5, 6, 7	syl2anc 591	. 2 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
9		irredminply.3	. 2 ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘(𝐸 ↾s 𝐹)))
10		eqid 2741	. . 3 ⊢ (0g‘(Poly1‘𝐸)) = (0g‘(Poly1‘𝐸))
11		irredminply.m	. . 3 ⊢ 𝑀 = (𝐸 minPoly 𝐹)
12		irredminply.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
13		fveq2 6831	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑂‘𝑔) = (𝑂‘𝐺))
14	13	fveq1d 6833	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑂‘𝑔)‘𝐴) = ((𝑂‘𝐺)‘𝐴))
15	14	eqeq1d 2743	. . . . 5 ⊢ (𝑔 = 𝐺 → (((𝑂‘𝑔)‘𝐴) = 0 ↔ ((𝑂‘𝐺)‘𝐴) = 0 ))
16		irredminply.1	. . . . 5 ⊢ (𝜑 → ((𝑂‘𝐺)‘𝐴) = 0 )
17	15, 9, 16	rspcedvdw 3565	. . . 4 ⊢ (𝜑 → ∃𝑔 ∈ (Monic1p‘(𝐸 ↾s 𝐹))((𝑂‘𝑔)‘𝐴) = 0 )
18		irredminply.o	. . . . 5 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
19		eqid 2741	. . . . 5 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
20		irredminply.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐸)
21		irredminply.0	. . . . 5 ⊢ 0 = (0g‘𝐸)
22	5	fldcrngd 20718	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ CRing)
23		sdrgsubrg 20767	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
24	6, 23	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
25	18, 19, 20, 21, 22, 24	elirng 33882	. . . 4 ⊢ (𝜑 → (𝐴 ∈ (𝐸 IntgRing 𝐹) ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑔 ∈ (Monic1p‘(𝐸 ↾s 𝐹))((𝑂‘𝑔)‘𝐴) = 0 )))
26	12, 17, 25	mpbir2and 720	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
27	10, 5, 6, 11, 26, 2	minplym1p 33909	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Monic1p‘(𝐸 ↾s 𝐹)))
28	19	sdrgdrng 20766	. . . . . . 7 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
29	6, 28	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
30	29	drngringd 20713	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
31		irredminply.2	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (Irred‘𝑃))
32		eqid 2741	. . . . . . 7 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
33		eqid 2741	. . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃)
34	32, 33	irredcl 20399	. . . . . 6 ⊢ (𝐺 ∈ (Irred‘𝑃) → 𝐺 ∈ (Base‘𝑃))
35	31, 34	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝑃))
36	1, 33, 2	mon1pcl 26132	. . . . . . 7 ⊢ ((𝑀‘𝐴) ∈ (Monic1p‘(𝐸 ↾s 𝐹)) → (𝑀‘𝐴) ∈ (Base‘𝑃))
37	27, 36	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃))
38	10, 5, 6, 11, 26	irngnminplynz 33908	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸)))
39		irredminply.z	. . . . . . . 8 ⊢ 𝑍 = (0g‘𝑃)
40		eqid 2741	. . . . . . . . 9 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
41	40, 19, 1, 33, 24, 10	ressply10g 33662	. . . . . . . 8 ⊢ (𝜑 → (0g‘(Poly1‘𝐸)) = (0g‘𝑃))
42	39, 41	eqtr4id 2795	. . . . . . 7 ⊢ (𝜑 → 𝑍 = (0g‘(Poly1‘𝐸)))
43	38, 42	neeqtrrd 3010	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)
44		eqid 2741	. . . . . . 7 ⊢ (Unic1p‘(𝐸 ↾s 𝐹)) = (Unic1p‘(𝐸 ↾s 𝐹))
45	1, 33, 39, 44	drnguc1p 26161	. . . . . 6 ⊢ (((𝐸 ↾s 𝐹) ∈ DivRing ∧ (𝑀‘𝐴) ∈ (Base‘𝑃) ∧ (𝑀‘𝐴) ≠ 𝑍) → (𝑀‘𝐴) ∈ (Unic1p‘(𝐸 ↾s 𝐹)))
46	29, 37, 43, 45	syl3anc 1380	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Unic1p‘(𝐸 ↾s 𝐹)))
47		eqidd 2742	. . . . 5 ⊢ (𝜑 → (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) = (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))
48		eqid 2741	. . . . . . 7 ⊢ (quot1p‘(𝐸 ↾s 𝐹)) = (quot1p‘(𝐸 ↾s 𝐹))
49		eqid 2741	. . . . . . 7 ⊢ (deg1‘(𝐸 ↾s 𝐹)) = (deg1‘(𝐸 ↾s 𝐹))
50		eqid 2741	. . . . . . 7 ⊢ (-g‘𝑃) = (-g‘𝑃)
51	48, 1, 33, 49, 50, 4, 44	q1peqb 26143	. . . . . 6 ⊢ (((𝐸 ↾s 𝐹) ∈ Ring ∧ 𝐺 ∈ (Base‘𝑃) ∧ (𝑀‘𝐴) ∈ (Unic1p‘(𝐸 ↾s 𝐹))) → (((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃) ∧ ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(-g‘𝑃)((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))) < ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴))) ↔ (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) = (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))))
52	51	biimpar 479	. . . . 5 ⊢ ((((𝐸 ↾s 𝐹) ∈ Ring ∧ 𝐺 ∈ (Base‘𝑃) ∧ (𝑀‘𝐴) ∈ (Unic1p‘(𝐸 ↾s 𝐹))) ∧ (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) = (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) → ((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃) ∧ ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(-g‘𝑃)((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))) < ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴))))
53	30, 35, 46, 47, 52	syl31anc 1382	. . . 4 ⊢ (𝜑 → ((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃) ∧ ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(-g‘𝑃)((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))) < ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴))))
54	53	simpld 496	. . 3 ⊢ (𝜑 → (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃))
55		eqid 2741	. . . . . . 7 ⊢ (rem1p‘(𝐸 ↾s 𝐹)) = (rem1p‘(𝐸 ↾s 𝐹))
56		eqid 2741	. . . . . . 7 ⊢ (+g‘𝑃) = (+g‘𝑃)
57	1, 33, 44, 48, 55, 4, 56	r1pid 26148	. . . . . 6 ⊢ (((𝐸 ↾s 𝐹) ∈ Ring ∧ 𝐺 ∈ (Base‘𝑃) ∧ (𝑀‘𝐴) ∈ (Unic1p‘(𝐸 ↾s 𝐹))) → 𝐺 = (((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴))(+g‘𝑃)(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))))
58	30, 35, 46, 57	syl3anc 1380	. . . . 5 ⊢ (𝜑 → 𝐺 = (((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴))(+g‘𝑃)(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))))
59	55, 1, 33, 44, 49	r1pdeglt 26147	. . . . . . . . . 10 ⊢ (((𝐸 ↾s 𝐹) ∈ Ring ∧ 𝐺 ∈ (Base‘𝑃) ∧ (𝑀‘𝐴) ∈ (Unic1p‘(𝐸 ↾s 𝐹))) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) < ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)))
60	30, 35, 46, 59	syl3anc 1380	. . . . . . . . 9 ⊢ (𝜑 → ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) < ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)))
61	60	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) < ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)))
62	30	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝐸 ↾s 𝐹) ∈ Ring)
63	37	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝑀‘𝐴) ∈ (Base‘𝑃))
64	43	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝑀‘𝐴) ≠ 𝑍)
65	49, 1, 39, 33	deg1nn0cl 26075	. . . . . . . . . . 11 ⊢ (((𝐸 ↾s 𝐹) ∈ Ring ∧ (𝑀‘𝐴) ∈ (Base‘𝑃) ∧ (𝑀‘𝐴) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)) ∈ ℕ0)
66	62, 63, 64, 65	syl3anc 1380	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)) ∈ ℕ0)
67	66	nn0red 12494	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)) ∈ ℝ)
68	55, 1, 33, 44	r1pcl 26146	. . . . . . . . . . . . 13 ⊢ (((𝐸 ↾s 𝐹) ∈ Ring ∧ 𝐺 ∈ (Base‘𝑃) ∧ (𝑀‘𝐴) ∈ (Unic1p‘(𝐸 ↾s 𝐹))) → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃))
69	30, 35, 46, 68	syl3anc 1380	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃))
70	69	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃))
71		simpr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍)
72	49, 1, 39, 33	deg1nn0cl 26075	. . . . . . . . . . 11 ⊢ (((𝐸 ↾s 𝐹) ∈ Ring ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃) ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) ∈ ℕ0)
73	62, 70, 71, 72	syl3anc 1380	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) ∈ ℕ0)
74	73	nn0red 12494	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) ∈ ℝ)
75		eqid 2741	. . . . . . . . . . . . 13 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }
76		eqid 2741	. . . . . . . . . . . . 13 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
77		eqid 2741	. . . . . . . . . . . . 13 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
78	18, 1, 20, 5, 6, 12, 21, 75, 76, 77, 11	minplyval 33901	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀‘𝐴) = ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }))
79	78	fveq2d 6835	. . . . . . . . . . 11 ⊢ (𝜑 → ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)) = ((deg1‘(𝐸 ↾s 𝐹))‘((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 })))
80	79	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)) = ((deg1‘(𝐸 ↾s 𝐹))‘((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 })))
81	6	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → 𝐹 ∈ (SubDRing‘𝐸))
82	81, 28	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝐸 ↾s 𝐹) ∈ DivRing)
83	18, 1, 20, 22, 24, 12, 21, 75	ply1annidl 33898	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } ∈ (LIdeal‘𝑃))
84	83	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } ∈ (LIdeal‘𝑃))
85		fveq2 6831	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) → (𝑂‘𝑞) = (𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))))
86	85	fveq1d 6833	. . . . . . . . . . . . . 14 ⊢ (𝑞 = (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) → ((𝑂‘𝑞)‘𝐴) = ((𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴))
87	86	eqeq1d 2743	. . . . . . . . . . . . 13 ⊢ (𝑞 = (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) → (((𝑂‘𝑞)‘𝐴) = 0 ↔ ((𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴) = 0 ))
88	18, 1, 33, 22, 24	evls1dm 33656	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑂 = (Base‘𝑃))
89	69, 88	eleqtrrd 2844	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ dom 𝑂)
90	55, 1, 33, 48, 4, 50	r1pval 26145	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ (Base‘𝑃) ∧ (𝑀‘𝐴) ∈ (Base‘𝑃)) → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) = (𝐺(-g‘𝑃)((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴))))
91	35, 37, 90	syl2anc 591	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) = (𝐺(-g‘𝑃)((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴))))
92	91	fveq2d 6835	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) = (𝑂‘(𝐺(-g‘𝑃)((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))))
93	92	fveq1d 6833	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴) = ((𝑂‘(𝐺(-g‘𝑃)((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴))))‘𝐴))
94		eqid 2741	. . . . . . . . . . . . . . . 16 ⊢ (-g‘𝐸) = (-g‘𝐸)
95	1	ply1ring 22236	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ↾s 𝐹) ∈ Ring → 𝑃 ∈ Ring)
96	30, 95	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ Ring)
97	33, 4, 96, 54, 37	ringcld 20236	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)) ∈ (Base‘𝑃))
98	18, 20, 1, 19, 33, 50, 94, 22, 24, 35, 97, 12	evls1subd 33667	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑂‘(𝐺(-g‘𝑃)((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴))))‘𝐴) = (((𝑂‘𝐺)‘𝐴)(-g‘𝐸)((𝑂‘((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))‘𝐴)))
99		eqid 2741	. . . . . . . . . . . . . . . . . 18 ⊢ (.r‘𝐸) = (.r‘𝐸)
100	18, 20, 1, 19, 33, 4, 99, 22, 24, 54, 37, 12	evls1muld 22362	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑂‘((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))‘𝐴) = (((𝑂‘(𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴)(.r‘𝐸)((𝑂‘(𝑀‘𝐴))‘𝐴)))
101	18, 1, 20, 5, 6, 12, 21, 11	minplyann 33905	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑂‘(𝑀‘𝐴))‘𝐴) = 0 )
102	101	oveq2d 7376	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝑂‘(𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴)(.r‘𝐸)((𝑂‘(𝑀‘𝐴))‘𝐴)) = (((𝑂‘(𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴)(.r‘𝐸) 0 ))
103	22	crngringd 20222	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ Ring)
104	18, 1, 20, 33, 22, 24, 12, 54	evls1fvcl 22365	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑂‘(𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴) ∈ 𝐵)
105	20, 99, 21, 103, 104	ringrzd 20272	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝑂‘(𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴)(.r‘𝐸) 0 ) = 0 )
106	100, 102, 105	3eqtrd 2780	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑂‘((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))‘𝐴) = 0 )
107	16, 106	oveq12d 7378	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝑂‘𝐺)‘𝐴)(-g‘𝐸)((𝑂‘((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))‘𝐴)) = ( 0 (-g‘𝐸) 0 ))
108	22	crnggrpd 20223	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 ∈ Grp)
109	20, 21	grpidcl 18936	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∈ Grp → 0 ∈ 𝐵)
110	20, 21, 94	grpsubid1 18996	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 (-g‘𝐸) 0 ) = 0 )
111	108, 109, 110	syl2anc2 592	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ( 0 (-g‘𝐸) 0 ) = 0 )
112	98, 107, 111	3eqtrd 2780	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑂‘(𝐺(-g‘𝑃)((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴))))‘𝐴) = 0 )
113	93, 112	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴) = 0 )
114	87, 89, 113	elrabd 3633	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 })
115	114	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 })
116	1, 77, 33, 82, 84, 49, 39, 115, 71	ig1pmindeg 33697	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 })) ≤ ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))))
117	80, 116	eqbrtrd 5097	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)) ≤ ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))))
118	67, 74, 117	lensymd 11292	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ¬ ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) < ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)))
119	61, 118	pm2.65da 823	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍)
120		nne 2940	. . . . . . 7 ⊢ (¬ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍 ↔ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) = 𝑍)
121	119, 120	sylib 220	. . . . . 6 ⊢ (𝜑 → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) = 𝑍)
122	121	oveq2d 7376	. . . . 5 ⊢ (𝜑 → (((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴))(+g‘𝑃)(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) = (((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴))(+g‘𝑃)𝑍))
123	96	ringgrpd 20218	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Grp)
124	33, 56, 39, 123, 97	grpridd 18941	. . . . 5 ⊢ (𝜑 → (((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴))(+g‘𝑃)𝑍) = ((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))
125	58, 122, 124	3eqtrd 2780	. . . 4 ⊢ (𝜑 → 𝐺 = ((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))
126	125, 31	eqeltrrd 2842	. . 3 ⊢ (𝜑 → ((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)) ∈ (Irred‘𝑃))
127	18, 1, 20, 5, 6, 12, 11, 39, 43	minplyirred 33907	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Irred‘𝑃))
128	32, 3	irrednu 20400	. . . 4 ⊢ ((𝑀‘𝐴) ∈ (Irred‘𝑃) → ¬ (𝑀‘𝐴) ∈ (Unit‘𝑃))
129	127, 128	syl 17	. . 3 ⊢ (𝜑 → ¬ (𝑀‘𝐴) ∈ (Unit‘𝑃))
130	32, 33, 3, 4	irredmul 20404	. . . . 5 ⊢ (((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃) ∧ (𝑀‘𝐴) ∈ (Base‘𝑃) ∧ ((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)) ∈ (Irred‘𝑃)) → ((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Unit‘𝑃) ∨ (𝑀‘𝐴) ∈ (Unit‘𝑃)))
131	130	orcomd 878	. . . 4 ⊢ (((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃) ∧ (𝑀‘𝐴) ∈ (Base‘𝑃) ∧ ((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)) ∈ (Irred‘𝑃)) → ((𝑀‘𝐴) ∈ (Unit‘𝑃) ∨ (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Unit‘𝑃)))
132	131	orcanai 1011	. . 3 ⊢ ((((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃) ∧ (𝑀‘𝐴) ∈ (Base‘𝑃) ∧ ((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)) ∈ (Irred‘𝑃)) ∧ ¬ (𝑀‘𝐴) ∈ (Unit‘𝑃)) → (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Unit‘𝑃))
133	54, 37, 126, 129, 132	syl31anc 1382	. 2 ⊢ (𝜑 → (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Unit‘𝑃))
134	1, 2, 3, 4, 8, 9, 27, 133, 125	m1pmeq 33680	1 ⊢ (𝜑 → 𝐺 = (𝑀‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∃wrex 3065  {crab 3393   class class class wbr 5075  dom cdm 5621  ‘cfv 6489  (class class class)co 7360   < clt 11174   ≤ cle 11175  ℕ₀cn0 12432  Basecbs 17174   ↾_s cress 17195  +_gcplusg 17215  ._rcmulr 17216  0_gc0g 17397  Grpcgrp 18904  -_gcsg 18906  Ringcrg 20209  Unitcui 20330  Irredcir 20331  SubRingcsubrg 20545  DivRingcdr 20705  Fieldcfield 20706  SubDRingcsdrg 20762  LIdealclidl 21203  RSpancrsp 21204  Poly₁cpl1 22166   evalSub₁ ces1 22303  deg₁cdg1 26041  Monic_1pcmn1 26113  Unic_1pcuc1p 26114  quot_1pcq1p 26115  rem_1pcr1p 26116  idlGen_1pcig1p 26117   IntgRing cirng 33879   minPoly cminply 33895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111  ax-addf 11112

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-ofr 7625  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8105  df-tpos 8170  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-sup 9349  df-inf 9350  df-oi 9419  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-fz 13457  df-fzo 13604  df-seq 13959  df-hash 14288  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-mulr 17229  df-starv 17230  df-sca 17231  df-vsca 17232  df-ip 17233  df-tset 17234  df-ple 17235  df-ds 17237  df-unif 17238  df-hom 17239  df-cco 17240  df-0g 17399  df-gsum 17400  df-prds 17405  df-pws 17407  df-mre 17543  df-mrc 17544  df-acs 17546  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mhm 18746  df-submnd 18747  df-grp 18907  df-minusg 18908  df-sbg 18909  df-mulg 19039  df-subg 19094  df-ghm 19183  df-cntz 19287  df-cmn 19752  df-abl 19753  df-mgp 20117  df-rng 20129  df-ur 20158  df-srg 20163  df-ring 20211  df-cring 20212  df-oppr 20312  df-dvdsr 20332  df-unit 20333  df-irred 20334  df-invr 20363  df-rhm 20447  df-nzr 20489  df-subrng 20522  df-subrg 20546  df-rlreg 20670  df-domn 20671  df-idom 20672  df-drng 20707  df-field 20708  df-sdrg 20763  df-lmod 20856  df-lss 20926  df-lsp 20966  df-sra 21167  df-rgmod 21168  df-lidl 21205  df-rsp 21206  df-cnfld 21352  df-assa 21832  df-asp 21833  df-ascl 21834  df-psr 21888  df-mvr 21889  df-mpl 21890  df-opsr 21892  df-evls 22054  df-evl 22055  df-psr1 22169  df-vr1 22170  df-ply1 22171  df-coe1 22172  df-evls1 22305  df-evl1 22306  df-mdeg 26042  df-deg1 26043  df-mon1 26118  df-uc1p 26119  df-q1p 26120  df-r1p 26121  df-ig1p 26122  df-irng 33880  df-minply 33896

This theorem is referenced by:  2sqr3minply  33976  cos9thpiminply  33984