Theorem cos9thpiminply 33972

Description: The polynomial ((𝑋↑3) + ((-3 · 𝑋) + 1)) is the minimal polynomial for 𝐴 over ℚ, and its degree is 3. (Contributed by Thierry Arnoux, 14-Nov-2025.)

Hypotheses

Ref	Expression
cos9thpiminplylem3.1	⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))
cos9thpiminplylem4.2	⊢ 𝑍 = (𝑂↑𝑐(1 / 3))
cos9thpiminplylem5.3	⊢ 𝐴 = (𝑍 + (1 / 𝑍))
cos9thpiminply.q	⊢ 𝑄 = (ℂfld ↾s ℚ)
cos9thpiminply.4	⊢ + = (+g‘𝑃)
cos9thpiminply.5	⊢ · = (.r‘𝑃)
cos9thpiminply.6	⊢ ↑ = (.g‘(mulGrp‘𝑃))
cos9thpiminply.p	⊢ 𝑃 = (Poly1‘𝑄)
cos9thpiminply.k	⊢ 𝐾 = (algSc‘𝑃)
cos9thpiminply.x	⊢ 𝑋 = (var1‘𝑄)
cos9thpiminply.d	⊢ 𝐷 = (deg1‘𝑄)
cos9thpiminply.f	⊢ 𝐹 = ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))
cos9thpiminply.m	⊢ 𝑀 = (ℂfld minPoly ℚ)

Assertion

Ref	Expression
cos9thpiminply	⊢ (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3)

Proof of Theorem cos9thpiminply

Dummy variables 𝑖 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (ℂfld evalSub1 ℚ) = (ℂfld evalSub1 ℚ)
2		cos9thpiminply.p	. . . . 5 ⊢ 𝑃 = (Poly1‘𝑄)
3		cos9thpiminply.q	. . . . . 6 ⊢ 𝑄 = (ℂfld ↾s ℚ)
4	3	fveq2i 6847	. . . . 5 ⊢ (Poly1‘𝑄) = (Poly1‘(ℂfld ↾s ℚ))
5	2, 4	eqtri 2760	. . . 4 ⊢ 𝑃 = (Poly1‘(ℂfld ↾s ℚ))
6		cnfldbas 21330	. . . 4 ⊢ ℂ = (Base‘ℂfld)
7		cnfldfld 33441	. . . . 5 ⊢ ℂfld ∈ Field
8	7	a1i 11	. . . 4 ⊢ (⊤ → ℂfld ∈ Field)
9		cndrng 21370	. . . . . 6 ⊢ ℂfld ∈ DivRing
10		qsubdrg 21391	. . . . . . 7 ⊢ (ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing)
11	10	simpli 483	. . . . . 6 ⊢ ℚ ∈ (SubRing‘ℂfld)
12	10	simpri 485	. . . . . 6 ⊢ (ℂfld ↾s ℚ) ∈ DivRing
13		issdrg 20738	. . . . . 6 ⊢ (ℚ ∈ (SubDRing‘ℂfld) ↔ (ℂfld ∈ DivRing ∧ ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing))
14	9, 11, 12, 13	mpbir3an 1343	. . . . 5 ⊢ ℚ ∈ (SubDRing‘ℂfld)
15	14	a1i 11	. . . 4 ⊢ (⊤ → ℚ ∈ (SubDRing‘ℂfld))
16		cos9thpiminplylem5.3	. . . . 5 ⊢ 𝐴 = (𝑍 + (1 / 𝑍))
17		cos9thpiminplylem4.2	. . . . . . 7 ⊢ 𝑍 = (𝑂↑𝑐(1 / 3))
18		cos9thpiminplylem3.1	. . . . . . . . 9 ⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))
19		ax-icn 11099	. . . . . . . . . . . . 13 ⊢ i ∈ ℂ
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → i ∈ ℂ)
21		2cnd 12237	. . . . . . . . . . . . 13 ⊢ (⊤ → 2 ∈ ℂ)
22		picn 26440	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
23	22	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → π ∈ ℂ)
24	21, 23	mulcld 11166	. . . . . . . . . . . 12 ⊢ (⊤ → (2 · π) ∈ ℂ)
25	20, 24	mulcld 11166	. . . . . . . . . . 11 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
26		3cn 12240	. . . . . . . . . . . 12 ⊢ 3 ∈ ℂ
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℂ)
28		3ne0 12265	. . . . . . . . . . . 12 ⊢ 3 ≠ 0
29	28	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ≠ 0)
30	25, 27, 29	divcld 11931	. . . . . . . . . 10 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
31	30	efcld 16020	. . . . . . . . 9 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
32	18, 31	eqeltrid 2841	. . . . . . . 8 ⊢ (⊤ → 𝑂 ∈ ℂ)
33	27, 29	reccld 11924	. . . . . . . 8 ⊢ (⊤ → (1 / 3) ∈ ℂ)
34	32, 33	cxpcld 26690	. . . . . . 7 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ∈ ℂ)
35	17, 34	eqeltrid 2841	. . . . . 6 ⊢ (⊤ → 𝑍 ∈ ℂ)
36	17	a1i 11	. . . . . . . 8 ⊢ (⊤ → 𝑍 = (𝑂↑𝑐(1 / 3)))
37	18	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 𝑂 = (exp‘((i · (2 · π)) / 3)))
38	30	efne0d 16034	. . . . . . . . . 10 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ≠ 0)
39	37, 38	eqnetrd 3000	. . . . . . . . 9 ⊢ (⊤ → 𝑂 ≠ 0)
40	32, 39, 33	cxpne0d 26695	. . . . . . . 8 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ≠ 0)
41	36, 40	eqnetrd 3000	. . . . . . 7 ⊢ (⊤ → 𝑍 ≠ 0)
42	35, 41	reccld 11924	. . . . . 6 ⊢ (⊤ → (1 / 𝑍) ∈ ℂ)
43	35, 42	addcld 11165	. . . . 5 ⊢ (⊤ → (𝑍 + (1 / 𝑍)) ∈ ℂ)
44	16, 43	eqeltrid 2841	. . . 4 ⊢ (⊤ → 𝐴 ∈ ℂ)
45		cnfld0 21364	. . . 4 ⊢ 0 = (0g‘ℂfld)
46		cos9thpiminply.m	. . . 4 ⊢ 𝑀 = (ℂfld minPoly ℚ)
47		eqid 2737	. . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃)
48		cos9thpiminply.4	. . . . . 6 ⊢ + = (+g‘𝑃)
49		cos9thpiminply.5	. . . . . 6 ⊢ · = (.r‘𝑃)
50		cos9thpiminply.6	. . . . . 6 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
51		cos9thpiminply.k	. . . . . 6 ⊢ 𝐾 = (algSc‘𝑃)
52		cos9thpiminply.x	. . . . . 6 ⊢ 𝑋 = (var1‘𝑄)
53		cos9thpiminply.d	. . . . . 6 ⊢ 𝐷 = (deg1‘𝑄)
54		cos9thpiminply.f	. . . . . 6 ⊢ 𝐹 = ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))
55	18, 17, 16, 3, 48, 49, 50, 2, 51, 52, 53, 54, 44	cos9thpiminplylem6 33971	. . . . 5 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝐴) = ((𝐴↑3) + ((-3 · 𝐴) + 1)))
56	18, 17, 16	cos9thpiminplylem5 33970	. . . . 5 ⊢ ((𝐴↑3) + ((-3 · 𝐴) + 1)) = 0
57	55, 56	eqtrdi 2788	. . . 4 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝐴) = 0)
58	3	qrng0 27605	. . . . 5 ⊢ 0 = (0g‘𝑄)
59		eqid 2737	. . . . 5 ⊢ (eval1‘𝑄) = (eval1‘𝑄)
60		eqid 2737	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
61	3	qfld 33397	. . . . . 6 ⊢ 𝑄 ∈ Field
62	61	a1i 11	. . . . 5 ⊢ (⊤ → 𝑄 ∈ Field)
63	3	qdrng 27604	. . . . . . . . . . 11 ⊢ 𝑄 ∈ DivRing
64	63	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 𝑄 ∈ DivRing)
65	64	drngringd 20687	. . . . . . . . 9 ⊢ (⊤ → 𝑄 ∈ Ring)
66	2	ply1ring 22205	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → 𝑃 ∈ Ring)
67	65, 66	syl 17	. . . . . . . 8 ⊢ (⊤ → 𝑃 ∈ Ring)
68	67	ringgrpd 20194	. . . . . . 7 ⊢ (⊤ → 𝑃 ∈ Grp)
69		eqid 2737	. . . . . . . . 9 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
70	69, 60	mgpbas 20097	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
71	69	ringmgp 20191	. . . . . . . . 9 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
72	67, 71	syl 17	. . . . . . . 8 ⊢ (⊤ → (mulGrp‘𝑃) ∈ Mnd)
73		3nn0 12433	. . . . . . . . 9 ⊢ 3 ∈ ℕ0
74	73	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℕ0)
75	52, 2, 60	vr1cl 22175	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
76	65, 75	syl 17	. . . . . . . 8 ⊢ (⊤ → 𝑋 ∈ (Base‘𝑃))
77	70, 50, 72, 74, 76	mulgnn0cld 19042	. . . . . . 7 ⊢ (⊤ → (3 ↑ 𝑋) ∈ (Base‘𝑃))
78	2	ply1sca 22210	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ DivRing → 𝑄 = (Scalar‘𝑃))
79	63, 78	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝑄 = (Scalar‘𝑃)
80	2	ply1lmod 22209	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ Ring → 𝑃 ∈ LMod)
81	65, 80	syl 17	. . . . . . . . . . 11 ⊢ (⊤ → 𝑃 ∈ LMod)
82	3	qrngbas 27603	. . . . . . . . . . 11 ⊢ ℚ = (Base‘𝑄)
83	51, 79, 67, 81, 82, 60	asclf 21854	. . . . . . . . . 10 ⊢ (⊤ → 𝐾:ℚ⟶(Base‘𝑃))
84	74	nn0zd 12527	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℤ)
85		zq 12881	. . . . . . . . . . 11 ⊢ (3 ∈ ℤ → 3 ∈ ℚ)
86		qnegcl 12893	. . . . . . . . . . 11 ⊢ (3 ∈ ℚ → -3 ∈ ℚ)
87	84, 85, 86	3syl 18	. . . . . . . . . 10 ⊢ (⊤ → -3 ∈ ℚ)
88	83, 87	ffvelcdmd 7041	. . . . . . . . 9 ⊢ (⊤ → (𝐾‘-3) ∈ (Base‘𝑃))
89	60, 49, 67, 88, 76	ringcld 20212	. . . . . . . 8 ⊢ (⊤ → ((𝐾‘-3) · 𝑋) ∈ (Base‘𝑃))
90		1zzd 12536	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℤ)
91		zq 12881	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
92	90, 91	syl 17	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℚ)
93	83, 92	ffvelcdmd 7041	. . . . . . . 8 ⊢ (⊤ → (𝐾‘1) ∈ (Base‘𝑃))
94	60, 48, 68, 89, 93	grpcld 18894	. . . . . . 7 ⊢ (⊤ → (((𝐾‘-3) · 𝑋) + (𝐾‘1)) ∈ (Base‘𝑃))
95	60, 48, 68, 77, 94	grpcld 18894	. . . . . 6 ⊢ (⊤ → ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))) ∈ (Base‘𝑃))
96	54, 95	eqeltrid 2841	. . . . 5 ⊢ (⊤ → 𝐹 ∈ (Base‘𝑃))
97	62	fldcrngd 20692	. . . . . . . . 9 ⊢ (⊤ → 𝑄 ∈ CRing)
98	59, 2, 60, 97, 82, 96	evl1fvf 33662	. . . . . . . 8 ⊢ (⊤ → ((eval1‘𝑄)‘𝐹):ℚ⟶ℚ)
99	98	ffnd 6673	. . . . . . 7 ⊢ (⊤ → ((eval1‘𝑄)‘𝐹) Fn ℚ)
100		fniniseg2 7018	. . . . . . 7 ⊢ (((eval1‘𝑄)‘𝐹) Fn ℚ → (◡((eval1‘𝑄)‘𝐹) “ {0}) = {𝑥 ∈ ℚ ∣ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0})
101	99, 100	syl 17	. . . . . 6 ⊢ (⊤ → (◡((eval1‘𝑄)‘𝐹) “ {0}) = {𝑥 ∈ ℚ ∣ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0})
102	59, 82	evl1fval1 22292	. . . . . . . . . . . . . . 15 ⊢ (eval1‘𝑄) = (𝑄 evalSub1 ℚ)
103	102	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → (eval1‘𝑄) = (𝑄 evalSub1 ℚ))
104	103	fveq1d 6846	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → ((eval1‘𝑄)‘𝐹) = ((𝑄 evalSub1 ℚ)‘𝐹))
105	104	fveq1d 6846	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) = (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥))
106		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑄 evalSub1 ℚ) = (𝑄 evalSub1 ℚ)
107		cncrng 21360	. . . . . . . . . . . . . . . 16 ⊢ ℂfld ∈ CRing
108	107	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℚ → ℂfld ∈ CRing)
109	11	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℚ → ℚ ∈ (SubRing‘ℂfld))
110	97	mptru 1549	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑄 ∈ CRing
111	110	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℚ → 𝑄 ∈ CRing)
112	111	crngringd 20198	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℚ → 𝑄 ∈ Ring)
113	82	subrgid 20523	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 ∈ Ring → ℚ ∈ (SubRing‘𝑄))
114	112, 113	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℚ → ℚ ∈ (SubRing‘𝑄))
115	96	mptru 1549	. . . . . . . . . . . . . . . 16 ⊢ 𝐹 ∈ (Base‘𝑃)
116	115	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℚ → 𝐹 ∈ (Base‘𝑃))
117	3, 1, 106, 2, 3, 60, 108, 109, 114, 116	ressply1evls1 33664	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → ((𝑄 evalSub1 ℚ)‘𝐹) = (((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ))
118	117	fveq1d 6846	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥) = ((((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ)‘𝑥))
119		fvres 6863	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → ((((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ)‘𝑥) = (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥))
120	118, 119	eqtr2d 2773	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥) = (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥))
121		qcn 12890	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℂ)
122	18, 17, 16, 3, 48, 49, 50, 2, 51, 52, 53, 54, 121	cos9thpiminplylem6 33971	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥) = ((𝑥↑3) + ((-3 · 𝑥) + 1)))
123	105, 120, 122	3eqtr2d 2778	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) = ((𝑥↑3) + ((-3 · 𝑥) + 1)))
124		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℚ)
125	124	cos9thpiminplylem2 33967	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℚ → ((𝑥↑3) + ((-3 · 𝑥) + 1)) ≠ 0)
126	123, 125	eqnetrd 3000	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) ≠ 0)
127	126	neneqd 2938	. . . . . . . . 9 ⊢ (𝑥 ∈ ℚ → ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
128	127	rgen 3054	. . . . . . . 8 ⊢ ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0
129	128	a1i 11	. . . . . . 7 ⊢ (⊤ → ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
130		rabeq0 4342	. . . . . . 7 ⊢ ({𝑥 ∈ ℚ ∣ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0} = ∅ ↔ ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
131	129, 130	sylibr 234	. . . . . 6 ⊢ (⊤ → {𝑥 ∈ ℚ ∣ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0} = ∅)
132	101, 131	eqtrd 2772	. . . . 5 ⊢ (⊤ → (◡((eval1‘𝑄)‘𝐹) “ {0}) = ∅)
133	54	a1i 11	. . . . . . 7 ⊢ (⊤ → 𝐹 = ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))))
134	133	fveq2d 6848	. . . . . 6 ⊢ (⊤ → (𝐷‘𝐹) = (𝐷‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))))
135		1lt3 12327	. . . . . . . . 9 ⊢ 1 < 3
136	135	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1 < 3)
137		0lt1 11673	. . . . . . . . . . . 12 ⊢ 0 < 1
138	137	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 0 < 1)
139	138	gt0ne0d 11715	. . . . . . . . . . . 12 ⊢ (⊤ → 1 ≠ 0)
140	53, 2, 82, 51, 58	deg1scl 26091	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ Ring ∧ 1 ∈ ℚ ∧ 1 ≠ 0) → (𝐷‘(𝐾‘1)) = 0)
141	65, 92, 139, 140	syl3anc 1374	. . . . . . . . . . 11 ⊢ (⊤ → (𝐷‘(𝐾‘1)) = 0)
142		drngdomn 20699	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ Domn)
143	63, 142	mp1i 13	. . . . . . . . . . . . 13 ⊢ (⊤ → 𝑄 ∈ Domn)
144	27, 29	negne0d 11504	. . . . . . . . . . . . . 14 ⊢ (⊤ → -3 ≠ 0)
145	2, 51, 58, 47, 82	ply1scln0 22251	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ Ring ∧ -3 ∈ ℚ ∧ -3 ≠ 0) → (𝐾‘-3) ≠ (0g‘𝑃))
146	65, 87, 144, 145	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (⊤ → (𝐾‘-3) ≠ (0g‘𝑃))
147	107	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → ℂfld ∈ CRing)
148		drngnzr 20698	. . . . . . . . . . . . . . 15 ⊢ (ℂfld ∈ DivRing → ℂfld ∈ NzRing)
149	9, 148	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (⊤ → ℂfld ∈ NzRing)
150	11	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → ℚ ∈ (SubRing‘ℂfld))
151	52, 47, 3, 2, 147, 149, 150	vr1nz 33692	. . . . . . . . . . . . 13 ⊢ (⊤ → 𝑋 ≠ (0g‘𝑃))
152	53, 2, 60, 49, 47, 143, 88, 146, 76, 151	deg1mul 26093	. . . . . . . . . . . 12 ⊢ (⊤ → (𝐷‘((𝐾‘-3) · 𝑋)) = ((𝐷‘(𝐾‘-3)) + (𝐷‘𝑋)))
153	53, 2, 82, 51, 58	deg1scl 26091	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ Ring ∧ -3 ∈ ℚ ∧ -3 ≠ 0) → (𝐷‘(𝐾‘-3)) = 0)
154	65, 87, 144, 153	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (⊤ → (𝐷‘(𝐾‘-3)) = 0)
155		drngnzr 20698	. . . . . . . . . . . . . . 15 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ NzRing)
156	63, 155	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (⊤ → 𝑄 ∈ NzRing)
157	53, 2, 52, 156	deg1vr 33691	. . . . . . . . . . . . 13 ⊢ (⊤ → (𝐷‘𝑋) = 1)
158	154, 157	oveq12d 7388	. . . . . . . . . . . 12 ⊢ (⊤ → ((𝐷‘(𝐾‘-3)) + (𝐷‘𝑋)) = (0 + 1))
159		1cnd 11141	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℂ)
160	159	addlidd 11348	. . . . . . . . . . . 12 ⊢ (⊤ → (0 + 1) = 1)
161	152, 158, 160	3eqtrd 2776	. . . . . . . . . . 11 ⊢ (⊤ → (𝐷‘((𝐾‘-3) · 𝑋)) = 1)
162	138, 141, 161	3brtr4d 5132	. . . . . . . . . 10 ⊢ (⊤ → (𝐷‘(𝐾‘1)) < (𝐷‘((𝐾‘-3) · 𝑋)))
163	2, 53, 65, 60, 48, 89, 93, 162	deg1add 26081	. . . . . . . . 9 ⊢ (⊤ → (𝐷‘(((𝐾‘-3) · 𝑋) + (𝐾‘1))) = (𝐷‘((𝐾‘-3) · 𝑋)))
164	163, 161	eqtrd 2772	. . . . . . . 8 ⊢ (⊤ → (𝐷‘(((𝐾‘-3) · 𝑋) + (𝐾‘1))) = 1)
165	53, 2, 52, 69, 50	deg1pw 26099	. . . . . . . . 9 ⊢ ((𝑄 ∈ NzRing ∧ 3 ∈ ℕ0) → (𝐷‘(3 ↑ 𝑋)) = 3)
166	156, 74, 165	syl2anc 585	. . . . . . . 8 ⊢ (⊤ → (𝐷‘(3 ↑ 𝑋)) = 3)
167	136, 164, 166	3brtr4d 5132	. . . . . . 7 ⊢ (⊤ → (𝐷‘(((𝐾‘-3) · 𝑋) + (𝐾‘1))) < (𝐷‘(3 ↑ 𝑋)))
168	2, 53, 65, 60, 48, 77, 94, 167	deg1add 26081	. . . . . 6 ⊢ (⊤ → (𝐷‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))) = (𝐷‘(3 ↑ 𝑋)))
169	134, 168, 166	3eqtrd 2776	. . . . 5 ⊢ (⊤ → (𝐷‘𝐹) = 3)
170	58, 59, 53, 2, 60, 62, 96, 132, 169	ply1dg3rt0irred 33683	. . . 4 ⊢ (⊤ → 𝐹 ∈ (Irred‘𝑃))
171		eqid 2737	. . . . . . 7 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
172	171, 47	irredn0 20376	. . . . . 6 ⊢ ((𝑃 ∈ Ring ∧ 𝐹 ∈ (Irred‘𝑃)) → 𝐹 ≠ (0g‘𝑃))
173	67, 170, 172	syl2anc 585	. . . . 5 ⊢ (⊤ → 𝐹 ≠ (0g‘𝑃))
174	169	fveq2d 6848	. . . . . 6 ⊢ (⊤ → ((coe1‘𝐹)‘(𝐷‘𝐹)) = ((coe1‘𝐹)‘3))
175	133	fveq2d 6848	. . . . . . . 8 ⊢ (⊤ → (coe1‘𝐹) = (coe1‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))))
176	175	fveq1d 6846	. . . . . . 7 ⊢ (⊤ → ((coe1‘𝐹)‘3) = ((coe1‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))))‘3))
177		cnfldadd 21332	. . . . . . . . . . 11 ⊢ + = (+g‘ℂfld)
178	3, 177	ressplusg 17225	. . . . . . . . . 10 ⊢ (ℚ ∈ (SubRing‘ℂfld) → + = (+g‘𝑄))
179	11, 178	ax-mp 5	. . . . . . . . 9 ⊢ + = (+g‘𝑄)
180	2, 60, 48, 179	coe1addfv 22224	. . . . . . . 8 ⊢ (((𝑄 ∈ Ring ∧ (3 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (((𝐾‘-3) · 𝑋) + (𝐾‘1)) ∈ (Base‘𝑃)) ∧ 3 ∈ ℕ0) → ((coe1‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))))‘3) = (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3)))
181	65, 77, 94, 74, 180	syl31anc 1376	. . . . . . 7 ⊢ (⊤ → ((coe1‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))))‘3) = (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3)))
182		iftrue 4487	. . . . . . . . . 10 ⊢ (𝑖 = 3 → if(𝑖 = 3, 1, 0) = 1)
183	3	qrng1 27606	. . . . . . . . . . 11 ⊢ 1 = (1r‘𝑄)
184	2, 52, 50, 65, 74, 58, 183	coe1mon 33686	. . . . . . . . . 10 ⊢ (⊤ → (coe1‘(3 ↑ 𝑋)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 3, 1, 0)))
185	182, 184, 74, 159	fvmptd4 6976	. . . . . . . . 9 ⊢ (⊤ → ((coe1‘(3 ↑ 𝑋))‘3) = 1)
186	2, 60, 48, 179	coe1addfv 22224	. . . . . . . . . . 11 ⊢ (((𝑄 ∈ Ring ∧ ((𝐾‘-3) · 𝑋) ∈ (Base‘𝑃) ∧ (𝐾‘1) ∈ (Base‘𝑃)) ∧ 3 ∈ ℕ0) → ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3) = (((coe1‘((𝐾‘-3) · 𝑋))‘3) + ((coe1‘(𝐾‘1))‘3)))
187	65, 89, 93, 74, 186	syl31anc 1376	. . . . . . . . . 10 ⊢ (⊤ → ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3) = (((coe1‘((𝐾‘-3) · 𝑋))‘3) + ((coe1‘(𝐾‘1))‘3)))
188	2	ply1assa 22157	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 ∈ CRing → 𝑃 ∈ AssAlg)
189	97, 188	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → 𝑃 ∈ AssAlg)
190		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
191	51, 79, 82, 60, 49, 190	asclmul1 21859	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ AssAlg ∧ -3 ∈ ℚ ∧ 𝑋 ∈ (Base‘𝑃)) → ((𝐾‘-3) · 𝑋) = (-3( ·𝑠 ‘𝑃)𝑋))
192	189, 87, 76, 191	syl3anc 1374	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ((𝐾‘-3) · 𝑋) = (-3( ·𝑠 ‘𝑃)𝑋))
193	70, 50	mulg1 19028	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ (Base‘𝑃) → (1 ↑ 𝑋) = 𝑋)
194	76, 193	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (1 ↑ 𝑋) = 𝑋)
195	194	oveq2d 7386	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)) = (-3( ·𝑠 ‘𝑃)𝑋))
196	192, 195	eqtr4d 2775	. . . . . . . . . . . . . 14 ⊢ (⊤ → ((𝐾‘-3) · 𝑋) = (-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))
197	196	fveq2d 6848	. . . . . . . . . . . . 13 ⊢ (⊤ → (coe1‘((𝐾‘-3) · 𝑋)) = (coe1‘(-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋))))
198	197	fveq1d 6846	. . . . . . . . . . . 12 ⊢ (⊤ → ((coe1‘((𝐾‘-3) · 𝑋))‘3) = ((coe1‘(-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))‘3))
199		1nn0 12431	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ0
200	199	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℕ0)
201		1red 11147	. . . . . . . . . . . . . 14 ⊢ (⊤ → 1 ∈ ℝ)
202	201, 136	ltned 11283	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ≠ 3)
203	58, 82, 2, 52, 190, 69, 50, 65, 87, 200, 74, 202	coe1tmfv2 22234	. . . . . . . . . . . 12 ⊢ (⊤ → ((coe1‘(-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))‘3) = 0)
204	198, 203	eqtrd 2772	. . . . . . . . . . 11 ⊢ (⊤ → ((coe1‘((𝐾‘-3) · 𝑋))‘3) = 0)
205	2, 51, 82, 58	coe1scl 22246	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ Ring ∧ 1 ∈ ℚ) → (coe1‘(𝐾‘1)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, 1, 0)))
206	65, 92, 205	syl2anc 585	. . . . . . . . . . . 12 ⊢ (⊤ → (coe1‘(𝐾‘1)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, 1, 0)))
207		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑖 = 3) → 𝑖 = 3)
208	28	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑖 = 3) → 3 ≠ 0)
209	207, 208	eqnetrd 3000	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑖 = 3) → 𝑖 ≠ 0)
210	209	neneqd 2938	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑖 = 3) → ¬ 𝑖 = 0)
211	210	iffalsed 4492	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑖 = 3) → if(𝑖 = 0, 1, 0) = 0)
212		0zd 12514	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ∈ ℤ)
213	206, 211, 74, 212	fvmptd 6959	. . . . . . . . . . 11 ⊢ (⊤ → ((coe1‘(𝐾‘1))‘3) = 0)
214	204, 213	oveq12d 7388	. . . . . . . . . 10 ⊢ (⊤ → (((coe1‘((𝐾‘-3) · 𝑋))‘3) + ((coe1‘(𝐾‘1))‘3)) = (0 + 0))
215		00id 11322	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
216	215	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (0 + 0) = 0)
217	187, 214, 216	3eqtrd 2776	. . . . . . . . 9 ⊢ (⊤ → ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3) = 0)
218	185, 217	oveq12d 7388	. . . . . . . 8 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3)) = (1 + 0))
219	159	addridd 11347	. . . . . . . 8 ⊢ (⊤ → (1 + 0) = 1)
220	218, 219	eqtrd 2772	. . . . . . 7 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3)) = 1)
221	176, 181, 220	3eqtrd 2776	. . . . . 6 ⊢ (⊤ → ((coe1‘𝐹)‘3) = 1)
222	174, 221	eqtrd 2772	. . . . 5 ⊢ (⊤ → ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1)
223	3	fveq2i 6847	. . . . . . 7 ⊢ (Monic1p‘𝑄) = (Monic1p‘(ℂfld ↾s ℚ))
224	223	eqcomi 2746	. . . . . 6 ⊢ (Monic1p‘(ℂfld ↾s ℚ)) = (Monic1p‘𝑄)
225	2, 60, 47, 53, 224, 183	ismon1p 26121	. . . . 5 ⊢ (𝐹 ∈ (Monic1p‘(ℂfld ↾s ℚ)) ↔ (𝐹 ∈ (Base‘𝑃) ∧ 𝐹 ≠ (0g‘𝑃) ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1))
226	96, 173, 222, 225	syl3anbrc 1345	. . . 4 ⊢ (⊤ → 𝐹 ∈ (Monic1p‘(ℂfld ↾s ℚ)))
227	1, 5, 6, 8, 15, 44, 45, 46, 47, 57, 170, 226	irredminply 33900	. . 3 ⊢ (⊤ → 𝐹 = (𝑀‘𝐴))
228	227, 169	jca 511	. 2 ⊢ (⊤ → (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3))
229	228	mptru 1549	1 ⊢ (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {crab 3401  ∅c0 4287  ifcif 4481  {csn 4582   class class class wbr 5100   ↦ cmpt 5181  ^◡ccnv 5633   ↾ cres 5636   “ cima 5637   Fn wfn 6497  ‘cfv 6502  (class class class)co 7370  ℂcc 11038  0cc0 11040  1c1 11041  ici 11042   + caddc 11043   · cmul 11045   < clt 11180  -cneg 11379   / cdiv 11808  2c2 12214  3c3 12215  ℕ₀cn0 12415  ℤcz 12502  ℚcq 12875  ↑cexp 13998  expce 15998  πcpi 16003  Basecbs 17150   ↾_s cress 17171  +_gcplusg 17191  ._rcmulr 17192  Scalarcsca 17194   ·_𝑠 cvsca 17195  0_gc0g 17373  Mndcmnd 18673  ._gcmg 19014  mulGrpcmgp 20092  Ringcrg 20185  CRingccrg 20186  Irredcir 20309  NzRingcnzr 20462  SubRingcsubrg 20519  Domncdomn 20642  DivRingcdr 20679  Fieldcfield 20680  SubDRingcsdrg 20736  LModclmod 20828  ℂ_fldccnfld 21326  AssAlgcasa 21822  algSccascl 21824  var₁cv1 22133  Poly₁cpl1 22134  coe₁cco1 22135   evalSub₁ ces1 22274  eval₁ce1 22275  deg₁cdg1 26032  Monic_1pcmn1 26104  ↑_𝑐ccxp 26537   minPoly cminply 33883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-inf2 9564  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117  ax-pre-sup 11118  ax-addf 11119  ax-mulf 11120

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-of 7634  df-ofr 7635  df-om 7821  df-1st 7945  df-2nd 7946  df-supp 8115  df-tpos 8180  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-er 8647  df-map 8779  df-pm 8780  df-ixp 8850  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-fsupp 9279  df-fi 9328  df-sup 9359  df-inf 9360  df-oi 9429  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-div 11809  df-nn 12160  df-2 12222  df-3 12223  df-4 12224  df-5 12225  df-6 12226  df-7 12227  df-8 12228  df-9 12229  df-n0 12416  df-z 12503  df-dec 12622  df-uz 12766  df-q 12876  df-rp 12920  df-xneg 13040  df-xadd 13041  df-xmul 13042  df-ioo 13279  df-ioc 13280  df-ico 13281  df-icc 13282  df-fz 13438  df-fzo 13585  df-fl 13726  df-mod 13804  df-seq 13939  df-exp 13999  df-fac 14211  df-bc 14240  df-hash 14268  df-shft 15004  df-sgn 15024  df-cj 15036  df-re 15037  df-im 15038  df-sqrt 15172  df-abs 15173  df-limsup 15408  df-clim 15425  df-rlim 15426  df-sum 15624  df-ef 16004  df-sin 16006  df-cos 16007  df-pi 16009  df-dvds 16194  df-gcd 16436  df-prm 16613  df-struct 17088  df-sets 17105  df-slot 17123  df-ndx 17135  df-base 17151  df-ress 17172  df-plusg 17204  df-mulr 17205  df-starv 17206  df-sca 17207  df-vsca 17208  df-ip 17209  df-tset 17210  df-ple 17211  df-ds 17213  df-unif 17214  df-hom 17215  df-cco 17216  df-rest 17356  df-topn 17357  df-0g 17375  df-gsum 17376  df-topgen 17377  df-pt 17378  df-prds 17381  df-pws 17383  df-xrs 17437  df-qtop 17442  df-imas 17443  df-xps 17445  df-mre 17519  df-mrc 17520  df-acs 17522  df-mgm 18579  df-sgrp 18658  df-mnd 18674  df-mhm 18722  df-submnd 18723  df-grp 18883  df-minusg 18884  df-sbg 18885  df-mulg 19015  df-subg 19070  df-ghm 19159  df-cntz 19263  df-cmn 19728  df-abl 19729  df-mgp 20093  df-rng 20105  df-ur 20134  df-srg 20139  df-ring 20187  df-cring 20188  df-oppr 20290  df-dvdsr 20310  df-unit 20311  df-irred 20312  df-invr 20341  df-dvr 20354  df-rhm 20425  df-nzr 20463  df-subrng 20496  df-subrg 20520  df-rlreg 20644  df-domn 20645  df-idom 20646  df-drng 20681  df-field 20682  df-sdrg 20737  df-lmod 20830  df-lss 20900  df-lsp 20940  df-sra 21142  df-rgmod 21143  df-lidl 21180  df-rsp 21181  df-psmet 21318  df-xmet 21319  df-met 21320  df-bl 21321  df-mopn 21322  df-fbas 21323  df-fg 21324  df-cnfld 21327  df-assa 21825  df-asp 21826  df-ascl 21827  df-psr 21882  df-mvr 21883  df-mpl 21884  df-opsr 21886  df-evls 22046  df-evl 22047  df-psr1 22137  df-vr1 22138  df-ply1 22139  df-coe1 22140  df-evls1 22276  df-evl1 22277  df-top 22855  df-topon 22872  df-topsp 22894  df-bases 22907  df-cld 22980  df-ntr 22981  df-cls 22982  df-nei 23059  df-lp 23097  df-perf 23098  df-cn 23188  df-cnp 23189  df-haus 23276  df-tx 23523  df-hmeo 23716  df-fil 23807  df-fm 23899  df-flim 23900  df-flf 23901  df-xms 24281  df-ms 24282  df-tms 24283  df-cncf 24844  df-limc 25840  df-dv 25841  df-mdeg 26033  df-deg1 26034  df-mon1 26109  df-uc1p 26110  df-q1p 26111  df-r1p 26112  df-ig1p 26113  df-log 26538  df-cxp 26539  df-irng 33868  df-minply 33884

This theorem is referenced by:  cos9thpinconstrlem2  33974