Theorem irredminply 33719

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 2730	. 2 ⊢ (Monic1p‘(𝐸 ↾s 𝐹)) = (Monic1p‘(𝐸 ↾s 𝐹))
3		eqid 2730	. 2 ⊢ (Unit‘𝑃) = (Unit‘𝑃)
4		eqid 2730	. 2 ⊢ (.r‘𝑃) = (.r‘𝑃)
5		irredminply.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ Field)
6		irredminply.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
7		fldsdrgfld 20706	. . 3 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
8	5, 6, 7	syl2anc 584	. 2 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
9		irredminply.3	. 2 ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘(𝐸 ↾s 𝐹)))
10		eqid 2730	. . 3 ⊢ (0g‘(Poly1‘𝐸)) = (0g‘(Poly1‘𝐸))
11		irredminply.m	. . 3 ⊢ 𝑀 = (𝐸 minPoly 𝐹)
12		irredminply.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
13		fveq2 6817	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑂‘𝑔) = (𝑂‘𝐺))
14	13	fveq1d 6819	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑂‘𝑔)‘𝐴) = ((𝑂‘𝐺)‘𝐴))
15	14	eqeq1d 2732	. . . . 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 3578	. . . 4 ⊢ (𝜑 → ∃𝑔 ∈ (Monic1p‘(𝐸 ↾s 𝐹))((𝑂‘𝑔)‘𝐴) = 0 )
18		irredminply.o	. . . . 5 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
19		eqid 2730	. . . . 5 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
20		irredminply.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐸)
21		irredminply.0	. . . . 5 ⊢ 0 = (0g‘𝐸)
22	5	fldcrngd 20650	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ CRing)
23		sdrgsubrg 20699	. . . . . 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 33689	. . . 4 ⊢ (𝜑 → (𝐴 ∈ (𝐸 IntgRing 𝐹) ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑔 ∈ (Monic1p‘(𝐸 ↾s 𝐹))((𝑂‘𝑔)‘𝐴) = 0 )))
26	12, 17, 25	mpbir2and 713	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
27	10, 5, 6, 11, 26, 2	minplym1p 33716	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Monic1p‘(𝐸 ↾s 𝐹)))
28	19	sdrgdrng 20698	. . . . . . 7 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
29	6, 28	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
30	29	drngringd 20645	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
31		irredminply.2	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (Irred‘𝑃))
32		eqid 2730	. . . . . . 7 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
33		eqid 2730	. . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃)
34	32, 33	irredcl 20335	. . . . . 6 ⊢ (𝐺 ∈ (Irred‘𝑃) → 𝐺 ∈ (Base‘𝑃))
35	31, 34	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝑃))
36	1, 33, 2	mon1pcl 26070	. . . . . . 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 33715	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸)))
39		irredminply.z	. . . . . . . 8 ⊢ 𝑍 = (0g‘𝑃)
40		eqid 2730	. . . . . . . . 9 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
41	40, 19, 1, 33, 24, 10	ressply10g 33520	. . . . . . . 8 ⊢ (𝜑 → (0g‘(Poly1‘𝐸)) = (0g‘𝑃))
42	39, 41	eqtr4id 2784	. . . . . . 7 ⊢ (𝜑 → 𝑍 = (0g‘(Poly1‘𝐸)))
43	38, 42	neeqtrrd 3000	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)
44		eqid 2730	. . . . . . 7 ⊢ (Unic1p‘(𝐸 ↾s 𝐹)) = (Unic1p‘(𝐸 ↾s 𝐹))
45	1, 33, 39, 44	drnguc1p 26099	. . . . . 6 ⊢ (((𝐸 ↾s 𝐹) ∈ DivRing ∧ (𝑀‘𝐴) ∈ (Base‘𝑃) ∧ (𝑀‘𝐴) ≠ 𝑍) → (𝑀‘𝐴) ∈ (Unic1p‘(𝐸 ↾s 𝐹)))
46	29, 37, 43, 45	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Unic1p‘(𝐸 ↾s 𝐹)))
47		eqidd 2731	. . . . 5 ⊢ (𝜑 → (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) = (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))
48		eqid 2730	. . . . . . 7 ⊢ (quot1p‘(𝐸 ↾s 𝐹)) = (quot1p‘(𝐸 ↾s 𝐹))
49		eqid 2730	. . . . . . 7 ⊢ (deg1‘(𝐸 ↾s 𝐹)) = (deg1‘(𝐸 ↾s 𝐹))
50		eqid 2730	. . . . . . 7 ⊢ (-g‘𝑃) = (-g‘𝑃)
51	48, 1, 33, 49, 50, 4, 44	q1peqb 26081	. . . . . 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 477	. . . . 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 1375	. . . 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 494	. . 3 ⊢ (𝜑 → (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃))
55		eqid 2730	. . . . . . 7 ⊢ (rem1p‘(𝐸 ↾s 𝐹)) = (rem1p‘(𝐸 ↾s 𝐹))
56		eqid 2730	. . . . . . 7 ⊢ (+g‘𝑃) = (+g‘𝑃)
57	1, 33, 44, 48, 55, 4, 56	r1pid 26086	. . . . . 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 1373	. . . . 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 26085	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ (𝜑 → ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) < ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)))
61	60	adantr 480	. . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝐸 ↾s 𝐹) ∈ Ring)
63	37	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝑀‘𝐴) ∈ (Base‘𝑃))
64	43	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝑀‘𝐴) ≠ 𝑍)
65	49, 1, 39, 33	deg1nn0cl 26013	. . . . . . . . . . 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 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)) ∈ ℕ0)
67	66	nn0red 12435	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)) ∈ ℝ)
68	55, 1, 33, 44	r1pcl 26084	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃))
70	69	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃))
71		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍)
72	49, 1, 39, 33	deg1nn0cl 26013	. . . . . . . . . . 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 1373	. . . . . . . . . 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 12435	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) ∈ ℝ)
75		eqid 2730	. . . . . . . . . . . . 13 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }
76		eqid 2730	. . . . . . . . . . . . 13 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
77		eqid 2730	. . . . . . . . . . . . 13 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
78	18, 1, 20, 5, 6, 12, 21, 75, 76, 77, 11	minplyval 33708	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀‘𝐴) = ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }))
79	78	fveq2d 6821	. . . . . . . . . . 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 480	. . . . . . . . . 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 480	. . . . . . . . . . . 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 33705	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } ∈ (LIdeal‘𝑃))
84	83	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } ∈ (LIdeal‘𝑃))
85		fveq2 6817	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) → (𝑂‘𝑞) = (𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))))
86	85	fveq1d 6819	. . . . . . . . . . . . . 14 ⊢ (𝑞 = (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) → ((𝑂‘𝑞)‘𝐴) = ((𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴))
87	86	eqeq1d 2732	. . . . . . . . . . . . 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 33514	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑂 = (Base‘𝑃))
89	69, 88	eleqtrrd 2832	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ dom 𝑂)
90	55, 1, 33, 48, 4, 50	r1pval 26083	. . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . 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 6821	. . . . . . . . . . . . . . 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 6819	. . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . 16 ⊢ (-g‘𝐸) = (-g‘𝐸)
95	1	ply1ring 22153	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ↾s 𝐹) ∈ Ring → 𝑃 ∈ Ring)
96	30, 95	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ Ring)
97	33, 4, 96, 54, 37	ringcld 20171	. . . . . . . . . . . . . . . 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 33525	. . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (.r‘𝐸) = (.r‘𝐸)
100	18, 20, 1, 19, 33, 4, 99, 22, 24, 54, 37, 12	evls1muld 22280	. . . . . . . . . . . . . . . . 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 33712	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑂‘(𝑀‘𝐴))‘𝐴) = 0 )
102	101	oveq2d 7357	. . . . . . . . . . . . . . . . 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 20157	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ Ring)
104	18, 1, 20, 33, 22, 24, 12, 54	evls1fvcl 22283	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑂‘(𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴) ∈ 𝐵)
105	20, 99, 21, 103, 104	ringrzd 20207	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝑂‘(𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴)(.r‘𝐸) 0 ) = 0 )
106	100, 102, 105	3eqtrd 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑂‘((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))‘𝐴) = 0 )
107	16, 106	oveq12d 7359	. . . . . . . . . . . . . . 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 20158	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 ∈ Grp)
109	20, 21	grpidcl 18870	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∈ Grp → 0 ∈ 𝐵)
110	20, 21, 94	grpsubid1 18930	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 (-g‘𝐸) 0 ) = 0 )
111	108, 109, 110	syl2anc2 585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ( 0 (-g‘𝐸) 0 ) = 0 )
112	98, 107, 111	3eqtrd 2769	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑂‘(𝐺(-g‘𝑃)((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴))))‘𝐴) = 0 )
113	93, 112	eqtrd 2765	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴) = 0 )
114	87, 89, 113	elrabd 3647	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 })
115	114	adantr 480	. . . . . . . . . . 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 33552	. . . . . . . . . 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 5111	. . . . . . . . 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 11256	. . . . . . . 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 816	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍)
120		nne 2930	. . . . . . 7 ⊢ (¬ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍 ↔ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) = 𝑍)
121	119, 120	sylib 218	. . . . . 6 ⊢ (𝜑 → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) = 𝑍)
122	121	oveq2d 7357	. . . . 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 20153	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Grp)
124	33, 56, 39, 123, 97	grpridd 18875	. . . . 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 2769	. . . 4 ⊢ (𝜑 → 𝐺 = ((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))
126	125, 31	eqeltrrd 2830	. . 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 33714	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Irred‘𝑃))
128	32, 3	irrednu 20336	. . . 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 20340	. . . . 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 871	. . . 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 1004	. . 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 1375	. 2 ⊢ (𝜑 → (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Unit‘𝑃))
134	1, 2, 3, 4, 8, 9, 27, 133, 125	m1pmeq 33537	1 ⊢ (𝜑 → 𝐺 = (𝑀‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110   ≠ wne 2926  ∃wrex 3054  {crab 3393   class class class wbr 5089  dom cdm 5614  ‘cfv 6477  (class class class)co 7341   < clt 11138   ≤ cle 11139  ℕ₀cn0 12373  Basecbs 17112   ↾_s cress 17133  +_gcplusg 17153  ._rcmulr 17154  0_gc0g 17335  Grpcgrp 18838  -_gcsg 18840  Ringcrg 20144  Unitcui 20266  Irredcir 20267  SubRingcsubrg 20477  DivRingcdr 20637  Fieldcfield 20638  SubDRingcsdrg 20694  LIdealclidl 21136  RSpancrsp 21137  Poly₁cpl1 22082   evalSub₁ ces1 22221  deg₁cdg1 25979  Monic_1pcmn1 26051  Unic_1pcuc1p 26052  quot_1pcq1p 26053  rem_1pcr1p 26054  idlGen_1pcig1p 26055   IntgRing cirng 33686   minPoly cminply 33702

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075  ax-pre-sup 11076  ax-addf 11077

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-isom 6486  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-ofr 7606  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-tpos 8151  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-pm 8748  df-ixp 8817  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fsupp 9241  df-sup 9321  df-inf 9322  df-oi 9391  df-card 9824  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-3 12181  df-4 12182  df-5 12183  df-6 12184  df-7 12185  df-8 12186  df-9 12187  df-n0 12374  df-z 12461  df-dec 12581  df-uz 12725  df-fz 13400  df-fzo 13547  df-seq 13901  df-hash 14230  df-struct 17050  df-sets 17067  df-slot 17085  df-ndx 17097  df-base 17113  df-ress 17134  df-plusg 17166  df-mulr 17167  df-starv 17168  df-sca 17169  df-vsca 17170  df-ip 17171  df-tset 17172  df-ple 17173  df-ds 17175  df-unif 17176  df-hom 17177  df-cco 17178  df-0g 17337  df-gsum 17338  df-prds 17343  df-pws 17345  df-mre 17480  df-mrc 17481  df-acs 17483  df-mgm 18540  df-sgrp 18619  df-mnd 18635  df-mhm 18683  df-submnd 18684  df-grp 18841  df-minusg 18842  df-sbg 18843  df-mulg 18973  df-subg 19028  df-ghm 19118  df-cntz 19222  df-cmn 19687  df-abl 19688  df-mgp 20052  df-rng 20064  df-ur 20093  df-srg 20098  df-ring 20146  df-cring 20147  df-oppr 20248  df-dvdsr 20268  df-unit 20269  df-irred 20270  df-invr 20299  df-rhm 20383  df-nzr 20421  df-subrng 20454  df-subrg 20478  df-rlreg 20602  df-domn 20603  df-idom 20604  df-drng 20639  df-field 20640  df-sdrg 20695  df-lmod 20788  df-lss 20858  df-lsp 20898  df-sra 21100  df-rgmod 21101  df-lidl 21138  df-rsp 21139  df-cnfld 21285  df-assa 21783  df-asp 21784  df-ascl 21785  df-psr 21839  df-mvr 21840  df-mpl 21841  df-opsr 21843  df-evls 22002  df-evl 22003  df-psr1 22085  df-vr1 22086  df-ply1 22087  df-coe1 22088  df-evls1 22223  df-evl1 22224  df-mdeg 25980  df-deg1 25981  df-mon1 26056  df-uc1p 26057  df-q1p 26058  df-r1p 26059  df-ig1p 26060  df-irng 33687  df-minply 33703

This theorem is referenced by:  2sqr3minply  33783  cos9thpiminply  33791