Theorem irredminply 34415

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 2763	. 2 ⊢ (Monic1p‘(𝐸 ↾s 𝐹)) = (Monic1p‘(𝐸 ↾s 𝐹))
3		eqid 2763	. 2 ⊢ (Unit‘𝑃) = (Unit‘𝑃)
4		eqid 2763	. 2 ⊢ (.r‘𝑃) = (.r‘𝑃)
5		irredminply.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ Field)
6		irredminply.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
7		fldsdrgfld 21082	. . 3 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
8	5, 6, 7	syl2anc 596	. 2 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
9		irredminply.3	. 2 ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘(𝐸 ↾s 𝐹)))
10		eqid 2763	. . 3 ⊢ (0g‘(Poly1‘𝐸)) = (0g‘(Poly1‘𝐸))
11		irredminply.m	. . 3 ⊢ 𝑀 = (𝐸 minPoly 𝐹)
12		irredminply.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
13		fveq2 6905	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑂‘𝑔) = (𝑂‘𝐺))
14	13	fveq1d 6907	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑂‘𝑔)‘𝐴) = ((𝑂‘𝐺)‘𝐴))
15	14	eqeq1d 2765	. . . . 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 3583	. . . 4 ⊢ (𝜑 → ∃𝑔 ∈ (Monic1p‘(𝐸 ↾s 𝐹))((𝑂‘𝑔)‘𝐴) = 0 )
18		irredminply.o	. . . . 5 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
19		eqid 2763	. . . . 5 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
20		irredminply.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐸)
21		irredminply.0	. . . . 5 ⊢ 0 = (0g‘𝐸)
22	5	fldcrngd 21022	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ CRing)
23		sdrgsubrg 21075	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
24	6, 23	syl 18	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
25	18, 19, 20, 21, 22, 24	elirng 34385	. . . 4 ⊢ (𝜑 → (𝐴 ∈ (𝐸 IntgRing 𝐹) ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑔 ∈ (Monic1p‘(𝐸 ↾s 𝐹))((𝑂‘𝑔)‘𝐴) = 0 )))
26	12, 17, 25	mpbir2and 726	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
27	10, 5, 6, 11, 26, 2	minplym1p 34412	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Monic1p‘(𝐸 ↾s 𝐹)))
28	19	sdrgdrng 21074	. . . . . . 7 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
29	6, 28	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
30	29	drngringd 21015	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
31		irredminply.2	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (Irred‘𝑃))
32		eqid 2763	. . . . . . 7 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
33		eqid 2763	. . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃)
34	32, 33	irredcl 20681	. . . . . 6 ⊢ (𝐺 ∈ (Irred‘𝑃) → 𝐺 ∈ (Base‘𝑃))
35	31, 34	syl 18	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝑃))
36	1, 33, 2	mon1pcl 26490	. . . . . . 7 ⊢ ((𝑀‘𝐴) ∈ (Monic1p‘(𝐸 ↾s 𝐹)) → (𝑀‘𝐴) ∈ (Base‘𝑃))
37	27, 36	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃))
38	10, 5, 6, 11, 26	irngnminplynz 34411	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸)))
39		irredminply.z	. . . . . . . 8 ⊢ 𝑍 = (0g‘𝑃)
40		eqid 2763	. . . . . . . . 9 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
41	40, 19, 1, 33, 24, 10	ressply10g 34166	. . . . . . . 8 ⊢ (𝜑 → (0g‘(Poly1‘𝐸)) = (0g‘𝑃))
42	39, 41	eqtr4id 2817	. . . . . . 7 ⊢ (𝜑 → 𝑍 = (0g‘(Poly1‘𝐸)))
43	38, 42	neeqtrrd 3033	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ≠ 𝑍)
44		eqid 2763	. . . . . . 7 ⊢ (Unic1p‘(𝐸 ↾s 𝐹)) = (Unic1p‘(𝐸 ↾s 𝐹))
45	1, 33, 39, 44	drnguc1p 26519	. . . . . 6 ⊢ (((𝐸 ↾s 𝐹) ∈ DivRing ∧ (𝑀‘𝐴) ∈ (Base‘𝑃) ∧ (𝑀‘𝐴) ≠ 𝑍) → (𝑀‘𝐴) ∈ (Unic1p‘(𝐸 ↾s 𝐹)))
46	29, 37, 43, 45	syl3anc 1399	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Unic1p‘(𝐸 ↾s 𝐹)))
47		eqidd 2764	. . . . 5 ⊢ (𝜑 → (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) = (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))
48		eqid 2763	. . . . . . 7 ⊢ (quot1p‘(𝐸 ↾s 𝐹)) = (quot1p‘(𝐸 ↾s 𝐹))
49		eqid 2763	. . . . . . 7 ⊢ (deg1‘(𝐸 ↾s 𝐹)) = (deg1‘(𝐸 ↾s 𝐹))
50		eqid 2763	. . . . . . 7 ⊢ (-g‘𝑃) = (-g‘𝑃)
51	48, 1, 33, 49, 50, 4, 44	q1peqb 26501	. . . . . 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 483	. . . . 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 1401	. . . 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 500	. . 3 ⊢ (𝜑 → (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃))
55		eqid 2763	. . . . . . 7 ⊢ (rem1p‘(𝐸 ↾s 𝐹)) = (rem1p‘(𝐸 ↾s 𝐹))
56		eqid 2763	. . . . . . 7 ⊢ (+g‘𝑃) = (+g‘𝑃)
57	1, 33, 44, 48, 55, 4, 56	r1pid 26506	. . . . . 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 1399	. . . . 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 26505	. . . . . . . . . 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 1399	. . . . . . . . 9 ⊢ (𝜑 → ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) < ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)))
61	60	adantr 486	. . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝐸 ↾s 𝐹) ∈ Ring)
63	37	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝑀‘𝐴) ∈ (Base‘𝑃))
64	43	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝑀‘𝐴) ≠ 𝑍)
65	49, 1, 39, 33	deg1nn0cl 26433	. . . . . . . . . . 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 1399	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)) ∈ ℕ0)
67	66	nn0red 12692	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝑀‘𝐴)) ∈ ℝ)
68	55, 1, 33, 44	r1pcl 26504	. . . . . . . . . . . . 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 1399	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃))
70	69	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Base‘𝑃))
71		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍)
72	49, 1, 39, 33	deg1nn0cl 26433	. . . . . . . . . . 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 1399	. . . . . . . . . 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 12692	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → ((deg1‘(𝐸 ↾s 𝐹))‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))) ∈ ℝ)
75		eqid 2763	. . . . . . . . . . . . 13 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }
76		eqid 2763	. . . . . . . . . . . . 13 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
77		eqid 2763	. . . . . . . . . . . . 13 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
78	18, 1, 20, 5, 6, 12, 21, 75, 76, 77, 11	minplyval 34404	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀‘𝐴) = ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }))
79	78	fveq2d 6909	. . . . . . . . . . 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 486	. . . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → 𝐹 ∈ (SubDRing‘𝐸))
82	81, 28	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → (𝐸 ↾s 𝐹) ∈ DivRing)
83	18, 1, 20, 22, 24, 12, 21, 75	ply1annidl 34401	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } ∈ (LIdeal‘𝑃))
84	83	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍) → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } ∈ (LIdeal‘𝑃))
85		fveq2 6905	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) → (𝑂‘𝑞) = (𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))))
86	85	fveq1d 6907	. . . . . . . . . . . . . 14 ⊢ (𝑞 = (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) → ((𝑂‘𝑞)‘𝐴) = ((𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴))
87	86	eqeq1d 2765	. . . . . . . . . . . . 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 34160	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑂 = (Base‘𝑃))
89	69, 88	eleqtrrd 2867	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ dom 𝑂)
90	55, 1, 33, 48, 4, 50	r1pval 26503	. . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . 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 6907	. . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . 16 ⊢ (-g‘𝐸) = (-g‘𝐸)
95	1	ply1ring 22592	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ↾s 𝐹) ∈ Ring → 𝑃 ∈ Ring)
96	30, 95	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ Ring)
97	33, 4, 96, 54, 37	ringcld 20511	. . . . . . . . . . . . . . . 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 34171	. . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (.r‘𝐸) = (.r‘𝐸)
100	18, 20, 1, 19, 33, 4, 99, 22, 24, 54, 37, 12	evls1muld 22717	. . . . . . . . . . . . . . . . 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 34408	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑂‘(𝑀‘𝐴))‘𝐴) = 0 )
102	101	oveq2d 7456	. . . . . . . . . . . . . . . . 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 20500	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ Ring)
104	18, 1, 20, 33, 22, 24, 12, 54	evls1fvcl 22720	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑂‘(𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴) ∈ 𝐵)
105	20, 99, 21, 103, 104	ringrzd 20554	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝑂‘(𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴)(.r‘𝐸) 0 ) = 0 )
106	100, 102, 105	3eqtrd 2802	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑂‘((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))‘𝐴) = 0 )
107	16, 106	oveq12d 7458	. . . . . . . . . . . . . . 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 20501	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 ∈ Grp)
109	20, 21	grpidcl 19202	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∈ Grp → 0 ∈ 𝐵)
110	20, 21, 94	grpsubid1 19261	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 (-g‘𝐸) 0 ) = 0 )
111	108, 109, 110	syl2anc2 597	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ( 0 (-g‘𝐸) 0 ) = 0 )
112	98, 107, 111	3eqtrd 2802	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑂‘(𝐺(-g‘𝑃)((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴))))‘𝐴) = 0 )
113	93, 112	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑂‘(𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)))‘𝐴) = 0 )
114	87, 89, 113	elrabd 3650	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 })
115	114	adantr 486	. . . . . . . . . . 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 34201	. . . . . . . . . 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 5130	. . . . . . . . 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 11485	. . . . . . . 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 829	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍)
120		nne 2963	. . . . . . 7 ⊢ (¬ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ≠ 𝑍 ↔ (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) = 𝑍)
121	119, 120	sylib 221	. . . . . 6 ⊢ (𝜑 → (𝐺(rem1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) = 𝑍)
122	121	oveq2d 7456	. . . . 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 20496	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Grp)
124	33, 56, 39, 123, 97	grpridd 19207	. . . . 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 2802	. . . 4 ⊢ (𝜑 → 𝐺 = ((𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴))(.r‘𝑃)(𝑀‘𝐴)))
126	125, 31	eqeltrrd 2865	. . 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 34410	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Irred‘𝑃))
128	32, 3	irrednu 20682	. . . 4 ⊢ ((𝑀‘𝐴) ∈ (Irred‘𝑃) → ¬ (𝑀‘𝐴) ∈ (Unit‘𝑃))
129	127, 128	syl 18	. . 3 ⊢ (𝜑 → ¬ (𝑀‘𝐴) ∈ (Unit‘𝑃))
130	32, 33, 3, 4	irredmul 20686	. . . . 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 885	. . . 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 1018	. . 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 1401	. 2 ⊢ (𝜑 → (𝐺(quot1p‘(𝐸 ↾s 𝐹))(𝑀‘𝐴)) ∈ (Unit‘𝑃))
134	1, 2, 3, 4, 8, 9, 27, 133, 125	m1pmeq 34184	1 ⊢ (𝜑 → 𝐺 = (𝑀‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090  {crab 3416   class class class wbr 5106  dom cdm 5655  ‘cfv 6555  (class class class)co 7440   < clt 11367   ≤ cle 11368  ℕ₀cn0 12630  Basecbs 17413   ↾_s cress 17434  +_gcplusg 17454  ._rcmulr 17455  0_gc0g 17636  Grpcgrp 19170  -_gcsg 19172  Ringcrg 20486  Unitcui 20612  Irredcir 20613  SubRingcsubrg 20848  DivRingcdr 21007  Fieldcfield 21008  SubDRingcsdrg 21070  LIdealclidl 21511  RSpancrsp 21512  Poly₁cpl1 22522   evalSub₁ ces1 22658  deg₁cdg1 26399  Monic_1pcmn1 26471  Unic_1pcuc1p 26472  quot_1pcq1p 26473  rem_1pcr1p 26474  idlGen_1pcig1p 26475   IntgRing cirng 34382   minPoly cminply 34398

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 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-pre-sup 11302  ax-addf 11303

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 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-of 7714  df-ofr 7715  df-om 7899  df-1st 8022  df-2nd 8023  df-supp 8202  df-tpos 8268  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-er 8742  df-map 8874  df-pm 8875  df-ixp 8951  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-fsupp 9379  df-sup 9458  df-inf 9459  df-oi 9528  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-uz 12990  df-fz 13664  df-fzo 13813  df-seq 14169  df-hash 14499  df-struct 17351  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-mulr 17468  df-starv 17469  df-sca 17470  df-vsca 17471  df-ip 17472  df-tset 17473  df-ple 17474  df-ds 17476  df-unif 17477  df-hom 17478  df-cco 17479  df-0g 17638  df-gsum 17639  df-prds 17644  df-pws 17646  df-mre 17782  df-mrc 17783  df-acs 17785  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-mhm 19004  df-submnd 19005  df-grp 19173  df-minusg 19174  df-sbg 19175  df-mulg 19304  df-subg 19359  df-ghm 19454  df-cntz 19558  df-cmn 20023  df-abl 20024  df-mgp 20388  df-rng 20402  df-ur 20435  df-srg 20440  df-ring 20488  df-cring 20489  df-oppr 20594  df-dvdsr 20614  df-unit 20615  df-irred 20616  df-invr 20645  df-rhm 20729  df-nzr 20790  df-subrng 20825  df-subrg 20849  df-rlreg 20973  df-domn 20974  df-idom 20975  df-drng 21009  df-field 21010  df-sdrg 21071  df-lmod 21164  df-lss 21234  df-lsp 21274  df-sra 21475  df-rgmod 21476  df-lidl 21513  df-rsp 21514  df-cnfld 21706  df-assa 22188  df-asp 22189  df-ascl 22190  df-psr 22244  df-mvr 22245  df-mpl 22246  df-opsr 22248  df-evls 22410  df-evl 22411  df-psr1 22525  df-vr1 22526  df-ply1 22527  df-coe1 22528  df-evls1 22660  df-evl1 22661  df-mdeg 26400  df-deg1 26401  df-mon1 26476  df-uc1p 26477  df-q1p 26478  df-r1p 26479  df-ig1p 26480  df-irng 34383  df-minply 34399

This theorem is used by:  2sqr3minply  34479  cos9thpiminply  34487