Theorem irredminply 33822

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 2734	. 2 ⊢ (Monic1p‘(𝐸 ↾s 𝐹)) = (Monic1p‘(𝐸 ↾s 𝐹))
3		eqid 2734	. 2 ⊢ (Unit‘𝑃) = (Unit‘𝑃)
4		eqid 2734	. 2 ⊢ (.r‘𝑃) = (.r‘𝑃)
5		irredminply.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ Field)
6		irredminply.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
7		fldsdrgfld 20729	. . 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 2734	. . 3 ⊢ (0g‘(Poly1‘𝐸)) = (0g‘(Poly1‘𝐸))
11		irredminply.m	. . 3 ⊢ 𝑀 = (𝐸 minPoly 𝐹)
12		irredminply.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
13		fveq2 6832	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑂‘𝑔) = (𝑂‘𝐺))
14	13	fveq1d 6834	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑂‘𝑔)‘𝐴) = ((𝑂‘𝐺)‘𝐴))
15	14	eqeq1d 2736	. . . . 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 3577	. . . 4 ⊢ (𝜑 → ∃𝑔 ∈ (Monic1p‘(𝐸 ↾s 𝐹))((𝑂‘𝑔)‘𝐴) = 0 )
18		irredminply.o	. . . . 5 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
19		eqid 2734	. . . . 5 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
20		irredminply.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐸)
21		irredminply.0	. . . . 5 ⊢ 0 = (0g‘𝐸)
22	5	fldcrngd 20673	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ CRing)
23		sdrgsubrg 20722	. . . . . 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 33792	. . . 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 33819	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Monic1p‘(𝐸 ↾s 𝐹)))
28	19	sdrgdrng 20721	. . . . . . 7 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
29	6, 28	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
30	29	drngringd 20668	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
31		irredminply.2	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (Irred‘𝑃))
32		eqid 2734	. . . . . . 7 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
33		eqid 2734	. . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃)
34	32, 33	irredcl 20358	. . . . . 6 ⊢ (𝐺 ∈ (Irred‘𝑃) → 𝐺 ∈ (Base‘𝑃))
35	31, 34	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝑃))
36	1, 33, 2	mon1pcl 26104	. . . . . . 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 33818	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸)))
39		irredminply.z	. . . . . . . 8 ⊢ 𝑍 = (0g‘𝑃)
40		eqid 2734	. . . . . . . . 9 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
41	40, 19, 1, 33, 24, 10	ressply10g 33597	. . . . . . . 8 ⊢ (𝜑 → (0g‘(Poly1‘𝐸)) = (0g‘𝑃))
42	39, 41	eqtr4id 2788	. . . . . . 7 ⊢ (𝜑 → 𝑍 = (0g‘(Poly1‘𝐸)))
43	38, 42	neeqtrrd 3004	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)
44		eqid 2734	. . . . . . 7 ⊢ (Unic1p‘(𝐸 ↾s 𝐹)) = (Unic1p‘(𝐸 ↾s 𝐹))
45	1, 33, 39, 44	drnguc1p 26133	. . . . . 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 2735	. . . . 5 ⊢ (𝜑 → (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) = (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))
48		eqid 2734	. . . . . . 7 ⊢ (quot1p‘(𝐸 ↾s 𝐹)) = (quot1p‘(𝐸 ↾s 𝐹))
49		eqid 2734	. . . . . . 7 ⊢ (deg1‘(𝐸 ↾s 𝐹)) = (deg1‘(𝐸 ↾s 𝐹))
50		eqid 2734	. . . . . . 7 ⊢ (-g‘𝑃) = (-g‘𝑃)
51	48, 1, 33, 49, 50, 4, 44	q1peqb 26115	. . . . . 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 2734	. . . . . . 7 ⊢ (rem1p‘(𝐸 ↾s 𝐹)) = (rem1p‘(𝐸 ↾s 𝐹))
56		eqid 2734	. . . . . . 7 ⊢ (+g‘𝑃) = (+g‘𝑃)
57	1, 33, 44, 48, 55, 4, 56	r1pid 26120	. . . . . 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 26119	. . . . . . . . . 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 26047	. . . . . . . . . . 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 12461	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)) ∈ ℝ)
68	55, 1, 33, 44	r1pcl 26118	. . . . . . . . . . . . 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 26047	. . . . . . . . . . 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 12461	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) ∈ ℝ)
75		eqid 2734	. . . . . . . . . . . . 13 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }
76		eqid 2734	. . . . . . . . . . . . 13 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
77		eqid 2734	. . . . . . . . . . . . 13 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
78	18, 1, 20, 5, 6, 12, 21, 75, 76, 77, 11	minplyval 33811	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀‘𝐴) = ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }))
79	78	fveq2d 6836	. . . . . . . . . . 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 33808	. . . . . . . . . . . 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 6832	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) → (𝑂‘𝑞) = (𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))))
86	85	fveq1d 6834	. . . . . . . . . . . . . 14 ⊢ (𝑞 = (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) → ((𝑂‘𝑞)‘𝐴) = ((𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴))
87	86	eqeq1d 2736	. . . . . . . . . . . . 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 33591	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑂 = (Base‘𝑃))
89	69, 88	eleqtrrd 2837	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ dom 𝑂)
90	55, 1, 33, 48, 4, 50	r1pval 26117	. . . . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . 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 6834	. . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . 16 ⊢ (-g‘𝐸) = (-g‘𝐸)
95	1	ply1ring 22186	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ↾s 𝐹) ∈ Ring → 𝑃 ∈ Ring)
96	30, 95	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ Ring)
97	33, 4, 96, 54, 37	ringcld 20193	. . . . . . . . . . . . . . . 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 33602	. . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (.r‘𝐸) = (.r‘𝐸)
100	18, 20, 1, 19, 33, 4, 99, 22, 24, 54, 37, 12	evls1muld 22314	. . . . . . . . . . . . . . . . 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 33815	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑂‘(𝑀‘𝐴))‘𝐴) = 0 )
102	101	oveq2d 7372	. . . . . . . . . . . . . . . . 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 20179	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ Ring)
104	18, 1, 20, 33, 22, 24, 12, 54	evls1fvcl 22317	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑂‘(𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴) ∈ 𝐵)
105	20, 99, 21, 103, 104	ringrzd 20229	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝑂‘(𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴)(.r‘𝐸) 0 ) = 0 )
106	100, 102, 105	3eqtrd 2773	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑂‘((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))‘𝐴) = 0 )
107	16, 106	oveq12d 7374	. . . . . . . . . . . . . . 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 20180	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 ∈ Grp)
109	20, 21	grpidcl 18893	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∈ Grp → 0 ∈ 𝐵)
110	20, 21, 94	grpsubid1 18953	. . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑂‘(𝐺(-g‘𝑃)((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴))))‘𝐴) = 0 )
113	93, 112	eqtrd 2769	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴) = 0 )
114	87, 89, 113	elrabd 3646	. . . . . . . . . . . 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 33632	. . . . . . . . . 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 5118	. . . . . . . . 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 11282	. . . . . . . 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 2934	. . . . . . 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 7372	. . . . 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 20175	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Grp)
124	33, 56, 39, 123, 97	grpridd 18898	. . . . 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 2773	. . . 4 ⊢ (𝜑 → 𝐺 = ((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))
126	125, 31	eqeltrrd 2835	. . 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 33817	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Irred‘𝑃))
128	32, 3	irrednu 20359	. . . 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 20363	. . . . 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 33615	1 ⊢ (𝜑 → 𝐺 = (𝑀‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∃wrex 3058  {crab 3397   class class class wbr 5096  dom cdm 5622  ‘cfv 6490  (class class class)co 7356   < clt 11164   ≤ cle 11165  ℕ₀cn0 12399  Basecbs 17134   ↾_s cress 17155  +_gcplusg 17175  ._rcmulr 17176  0_gc0g 17357  Grpcgrp 18861  -_gcsg 18863  Ringcrg 20166  Unitcui 20289  Irredcir 20290  SubRingcsubrg 20500  DivRingcdr 20660  Fieldcfield 20661  SubDRingcsdrg 20717  LIdealclidl 21159  RSpancrsp 21160  Poly₁cpl1 22115   evalSub₁ ces1 22255  deg₁cdg1 26013  Monic_1pcmn1 26085  Unic_1pcuc1p 26086  quot_1pcq1p 26087  rem_1pcr1p 26088  idlGen_1pcig1p 26089   IntgRing cirng 33789   minPoly cminply 33805

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 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102  ax-addf 11103

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-ofr 7621  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-map 8763  df-pm 8764  df-ixp 8834  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-fsupp 9263  df-sup 9343  df-inf 9344  df-oi 9413  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-z 12487  df-dec 12606  df-uz 12750  df-fz 13422  df-fzo 13569  df-seq 13923  df-hash 14252  df-struct 17072  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-mulr 17189  df-starv 17190  df-sca 17191  df-vsca 17192  df-ip 17193  df-tset 17194  df-ple 17195  df-ds 17197  df-unif 17198  df-hom 17199  df-cco 17200  df-0g 17359  df-gsum 17360  df-prds 17365  df-pws 17367  df-mre 17503  df-mrc 17504  df-acs 17506  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-mhm 18706  df-submnd 18707  df-grp 18864  df-minusg 18865  df-sbg 18866  df-mulg 18996  df-subg 19051  df-ghm 19140  df-cntz 19244  df-cmn 19709  df-abl 19710  df-mgp 20074  df-rng 20086  df-ur 20115  df-srg 20120  df-ring 20168  df-cring 20169  df-oppr 20271  df-dvdsr 20291  df-unit 20292  df-irred 20293  df-invr 20322  df-rhm 20406  df-nzr 20444  df-subrng 20477  df-subrg 20501  df-rlreg 20625  df-domn 20626  df-idom 20627  df-drng 20662  df-field 20663  df-sdrg 20718  df-lmod 20811  df-lss 20881  df-lsp 20921  df-sra 21123  df-rgmod 21124  df-lidl 21161  df-rsp 21162  df-cnfld 21308  df-assa 21806  df-asp 21807  df-ascl 21808  df-psr 21863  df-mvr 21864  df-mpl 21865  df-opsr 21867  df-evls 22027  df-evl 22028  df-psr1 22118  df-vr1 22119  df-ply1 22120  df-coe1 22121  df-evls1 22257  df-evl1 22258  df-mdeg 26014  df-deg1 26015  df-mon1 26090  df-uc1p 26091  df-q1p 26092  df-r1p 26093  df-ig1p 26094  df-irng 33790  df-minply 33806

This theorem is referenced by:  2sqr3minply  33886  cos9thpiminply  33894