Theorem cos9thpiminply 34123

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 2769	. . . 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 6885	. . . . 5 ⊢ (Poly1‘𝑄) = (Poly1‘(ℂfld ↾s ℚ))
5	2, 4	eqtri 2792	. . . 4 ⊢ 𝑃 = (Poly1‘(ℂfld ↾s ℚ))
6		cnfldbas 21495	. . . 4 ⊢ ℂ = (Base‘ℂfld)
7		cnfldfld 33605	. . . . 5 ⊢ ℂfld ∈ Field
8	7	a1i 11	. . . 4 ⊢ (⊤ → ℂfld ∈ Field)
9		cndrng 21520	. . . . . 6 ⊢ ℂfld ∈ DivRing
10		qsubdrg 21538	. . . . . . 7 ⊢ (ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing)
11	10	simpli 488	. . . . . 6 ⊢ ℚ ∈ (SubRing‘ℂfld)
12	10	simpri 490	. . . . . 6 ⊢ (ℂfld ↾s ℚ) ∈ DivRing
13		issdrg 20869	. . . . . 6 ⊢ (ℚ ∈ (SubDRing‘ℂfld) ↔ (ℂfld ∈ DivRing ∧ ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing))
14	9, 11, 12, 13	mpbir3an 1358	. . . . 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 11159	. . . . . . . . . . . . 13 ⊢ i ∈ ℂ
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → i ∈ ℂ)
21		2cnd 12319	. . . . . . . . . . . . 13 ⊢ (⊤ → 2 ∈ ℂ)
22		picn 26587	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
23	22	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → π ∈ ℂ)
24	21, 23	mulcld 11229	. . . . . . . . . . . 12 ⊢ (⊤ → (2 · π) ∈ ℂ)
25	20, 24	mulcld 11229	. . . . . . . . . . 11 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
26		3cn 12322	. . . . . . . . . . . 12 ⊢ 3 ∈ ℂ
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℂ)
28		3ne0 12350	. . . . . . . . . . . 12 ⊢ 3 ≠ 0
29	28	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ≠ 0)
30	25, 27, 29	divcld 11991	. . . . . . . . . 10 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
31	30	efcld 16137	. . . . . . . . 9 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
32	18, 31	eqeltrid 2873	. . . . . . . 8 ⊢ (⊤ → 𝑂 ∈ ℂ)
33	27, 29	reccld 11984	. . . . . . . 8 ⊢ (⊤ → (1 / 3) ∈ ℂ)
34	32, 33	cxpcld 26839	. . . . . . 7 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ∈ ℂ)
35	17, 34	eqeltrid 2873	. . . . . 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 16151	. . . . . . . . . 10 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ≠ 0)
39	37, 38	eqnetrd 3031	. . . . . . . . 9 ⊢ (⊤ → 𝑂 ≠ 0)
40	32, 39, 33	cxpne0d 26844	. . . . . . . 8 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ≠ 0)
41	36, 40	eqnetrd 3031	. . . . . . 7 ⊢ (⊤ → 𝑍 ≠ 0)
42	35, 41	reccld 11984	. . . . . 6 ⊢ (⊤ → (1 / 𝑍) ∈ ℂ)
43	35, 42	addcld 11228	. . . . 5 ⊢ (⊤ → (𝑍 + (1 / 𝑍)) ∈ ℂ)
44	16, 43	eqeltrid 2873	. . . 4 ⊢ (⊤ → 𝐴 ∈ ℂ)
45		cnfld0 21515	. . . 4 ⊢ 0 = (0g‘ℂfld)
46		cos9thpiminply.m	. . . 4 ⊢ 𝑀 = (ℂfld minPoly ℚ)
47		eqid 2769	. . . 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 34122	. . . . 5 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝐴) = ((𝐴↑3) + ((-3 · 𝐴) + 1)))
56	18, 17, 16	cos9thpiminplylem5 34121	. . . . 5 ⊢ ((𝐴↑3) + ((-3 · 𝐴) + 1)) = 0
57	55, 56	eqtrdi 2820	. . . 4 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝐴) = 0)
58	3	qrng0 27751	. . . . 5 ⊢ 0 = (0g‘𝑄)
59		eqid 2769	. . . . 5 ⊢ (eval1‘𝑄) = (eval1‘𝑄)
60		eqid 2769	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
61	3	qfld 33561	. . . . . 6 ⊢ 𝑄 ∈ Field
62	61	a1i 11	. . . . 5 ⊢ (⊤ → 𝑄 ∈ Field)
63	3	qdrng 27750	. . . . . . . . . . 11 ⊢ 𝑄 ∈ DivRing
64	63	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 𝑄 ∈ DivRing)
65	64	drngringd 20821	. . . . . . . . 9 ⊢ (⊤ → 𝑄 ∈ Ring)
66	2	ply1ring 22376	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → 𝑃 ∈ Ring)
67	65, 66	syl 18	. . . . . . . 8 ⊢ (⊤ → 𝑃 ∈ Ring)
68	67	ringgrpd 20324	. . . . . . 7 ⊢ (⊤ → 𝑃 ∈ Grp)
69		eqid 2769	. . . . . . . . 9 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
70	69, 60	mgpbas 20221	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
71	69	ringmgp 20321	. . . . . . . . 9 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
72	67, 71	syl 18	. . . . . . . 8 ⊢ (⊤ → (mulGrp‘𝑃) ∈ Mnd)
73		3nn0 12522	. . . . . . . . 9 ⊢ 3 ∈ ℕ0
74	73	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℕ0)
75	52, 2, 60	vr1cl 22346	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
76	65, 75	syl 18	. . . . . . . 8 ⊢ (⊤ → 𝑋 ∈ (Base‘𝑃))
77	70, 50, 72, 74, 76	mulgnn0cld 19161	. . . . . . 7 ⊢ (⊤ → (3 ↑ 𝑋) ∈ (Base‘𝑃))
78	2	ply1sca 22381	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ DivRing → 𝑄 = (Scalar‘𝑃))
79	63, 78	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝑄 = (Scalar‘𝑃)
80	2	ply1lmod 22380	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ Ring → 𝑃 ∈ LMod)
81	65, 80	syl 18	. . . . . . . . . . 11 ⊢ (⊤ → 𝑃 ∈ LMod)
82	3	qrngbas 27749	. . . . . . . . . . 11 ⊢ ℚ = (Base‘𝑄)
83	51, 79, 67, 81, 82, 60	asclf 22000	. . . . . . . . . 10 ⊢ (⊤ → 𝐾:ℚ⟶(Base‘𝑃))
84	74	nn0zd 12616	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℤ)
85		zq 12978	. . . . . . . . . . 11 ⊢ (3 ∈ ℤ → 3 ∈ ℚ)
86		qnegcl 12990	. . . . . . . . . . 11 ⊢ (3 ∈ ℚ → -3 ∈ ℚ)
87	84, 85, 86	3syl 19	. . . . . . . . . 10 ⊢ (⊤ → -3 ∈ ℚ)
88	83, 87	ffvelcdmd 7081	. . . . . . . . 9 ⊢ (⊤ → (𝐾‘-3) ∈ (Base‘𝑃))
89	60, 49, 67, 88, 76	ringcld 20342	. . . . . . . 8 ⊢ (⊤ → ((𝐾‘-3) · 𝑋) ∈ (Base‘𝑃))
90		1zzd 12625	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℤ)
91		zq 12978	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
92	90, 91	syl 18	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℚ)
93	83, 92	ffvelcdmd 7081	. . . . . . . 8 ⊢ (⊤ → (𝐾‘1) ∈ (Base‘𝑃))
94	60, 48, 68, 89, 93	grpcld 19014	. . . . . . 7 ⊢ (⊤ → (((𝐾‘-3) · 𝑋) + (𝐾‘1)) ∈ (Base‘𝑃))
95	60, 48, 68, 77, 94	grpcld 19014	. . . . . 6 ⊢ (⊤ → ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))) ∈ (Base‘𝑃))
96	54, 95	eqeltrid 2873	. . . . 5 ⊢ (⊤ → 𝐹 ∈ (Base‘𝑃))
97	62	fldcrngd 20826	. . . . . . . . 9 ⊢ (⊤ → 𝑄 ∈ CRing)
98	59, 2, 60, 97, 82, 96	evl1fvf 33798	. . . . . . . 8 ⊢ (⊤ → ((eval1‘𝑄)‘𝐹):ℚ⟶ℚ)
99	98	ffnd 6707	. . . . . . 7 ⊢ (⊤ → ((eval1‘𝑄)‘𝐹) Fn ℚ)
100		fniniseg2 7058	. . . . . . 7 ⊢ (((eval1‘𝑄)‘𝐹) Fn ℚ → (◡((eval1‘𝑄)‘𝐹) “ {0}) = {𝑥 ∈ ℚ ∣ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0})
101	99, 100	syl 18	. . . . . 6 ⊢ (⊤ → (◡((eval1‘𝑄)‘𝐹) “ {0}) = {𝑥 ∈ ℚ ∣ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0})
102	59, 82	evl1fval1 22460	. . . . . . . . . . . . . . 15 ⊢ (eval1‘𝑄) = (𝑄 evalSub1 ℚ)
103	102	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → (eval1‘𝑄) = (𝑄 evalSub1 ℚ))
104	103	fveq1d 6884	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → ((eval1‘𝑄)‘𝐹) = ((𝑄 evalSub1 ℚ)‘𝐹))
105	104	fveq1d 6884	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) = (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥))
106		eqid 2769	. . . . . . . . . . . . . . 15 ⊢ (𝑄 evalSub1 ℚ) = (𝑄 evalSub1 ℚ)
107		cncrng 21512	. . . . . . . . . . . . . . . 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 1574	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑄 ∈ CRing
111	110	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℚ → 𝑄 ∈ CRing)
112	111	crngringd 20328	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℚ → 𝑄 ∈ Ring)
113	82	subrgid 20658	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 ∈ Ring → ℚ ∈ (SubRing‘𝑄))
114	112, 113	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℚ → ℚ ∈ (SubRing‘𝑄))
115	96	mptru 1574	. . . . . . . . . . . . . . . 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 33800	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → ((𝑄 evalSub1 ℚ)‘𝐹) = (((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ))
118	117	fveq1d 6884	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥) = ((((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ)‘𝑥))
119		fvres 6901	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → ((((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ)‘𝑥) = (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥))
120	118, 119	eqtr2d 2805	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥) = (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥))
121		qcn 12987	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℂ)
122	18, 17, 16, 3, 48, 49, 50, 2, 51, 52, 53, 54, 121	cos9thpiminplylem6 34122	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥) = ((𝑥↑3) + ((-3 · 𝑥) + 1)))
123	105, 120, 122	3eqtr2d 2810	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) = ((𝑥↑3) + ((-3 · 𝑥) + 1)))
124		id 23	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℚ)
125	124	cos9thpiminplylem2 34118	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℚ → ((𝑥↑3) + ((-3 · 𝑥) + 1)) ≠ 0)
126	123, 125	eqnetrd 3031	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) ≠ 0)
127	126	neneqd 2969	. . . . . . . . 9 ⊢ (𝑥 ∈ ℚ → ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
128	127	rgen 3087	. . . . . . . 8 ⊢ ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0
129	128	a1i 11	. . . . . . 7 ⊢ (⊤ → ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
130		rabeq0 4352	. . . . . . 7 ⊢ ({𝑥 ∈ ℚ ∣ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0} = ∅ ↔ ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
131	129, 130	sylibr 237	. . . . . 6 ⊢ (⊤ → {𝑥 ∈ ℚ ∣ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0} = ∅)
132	101, 131	eqtrd 2804	. . . . 5 ⊢ (⊤ → (◡((eval1‘𝑄)‘𝐹) “ {0}) = ∅)
133	54	a1i 11	. . . . . . 7 ⊢ (⊤ → 𝐹 = ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))))
134	133	fveq2d 6886	. . . . . 6 ⊢ (⊤ → (𝐷‘𝐹) = (𝐷‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))))
135		1lt3 12416	. . . . . . . . 9 ⊢ 1 < 3
136	135	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1 < 3)
137		0lt1 11736	. . . . . . . . . . . 12 ⊢ 0 < 1
138	137	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 0 < 1)
139	138	gt0ne0d 11778	. . . . . . . . . . . 12 ⊢ (⊤ → 1 ≠ 0)
140	53, 2, 82, 51, 58	deg1scl 26239	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ Ring ∧ 1 ∈ ℚ ∧ 1 ≠ 0) → (𝐷‘(𝐾‘1)) = 0)
141	65, 92, 139, 140	syl3anc 1396	. . . . . . . . . . 11 ⊢ (⊤ → (𝐷‘(𝐾‘1)) = 0)
142		drngdomn 20833	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ Domn)
143	63, 142	mp1i 14	. . . . . . . . . . . . 13 ⊢ (⊤ → 𝑄 ∈ Domn)
144	27, 29	negne0d 11567	. . . . . . . . . . . . . 14 ⊢ (⊤ → -3 ≠ 0)
145	2, 51, 58, 47, 82	ply1scln0 22421	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ Ring ∧ -3 ∈ ℚ ∧ -3 ≠ 0) → (𝐾‘-3) ≠ (0g‘𝑃))
146	65, 87, 144, 145	syl3anc 1396	. . . . . . . . . . . . 13 ⊢ (⊤ → (𝐾‘-3) ≠ (0g‘𝑃))
147	107	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → ℂfld ∈ CRing)
148		drngnzr 20832	. . . . . . . . . . . . . . 15 ⊢ (ℂfld ∈ DivRing → ℂfld ∈ NzRing)
149	9, 148	mp1i 14	. . . . . . . . . . . . . 14 ⊢ (⊤ → ℂfld ∈ NzRing)
150	11	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → ℚ ∈ (SubRing‘ℂfld))
151	52, 47, 3, 2, 147, 149, 150	vr1nz 33828	. . . . . . . . . . . . 13 ⊢ (⊤ → 𝑋 ≠ (0g‘𝑃))
152	53, 2, 60, 49, 47, 143, 88, 146, 76, 151	deg1mul 26241	. . . . . . . . . . . 12 ⊢ (⊤ → (𝐷‘((𝐾‘-3) · 𝑋)) = ((𝐷‘(𝐾‘-3)) + (𝐷‘𝑋)))
153	53, 2, 82, 51, 58	deg1scl 26239	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ Ring ∧ -3 ∈ ℚ ∧ -3 ≠ 0) → (𝐷‘(𝐾‘-3)) = 0)
154	65, 87, 144, 153	syl3anc 1396	. . . . . . . . . . . . 13 ⊢ (⊤ → (𝐷‘(𝐾‘-3)) = 0)
155		drngnzr 20832	. . . . . . . . . . . . . . 15 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ NzRing)
156	63, 155	mp1i 14	. . . . . . . . . . . . . 14 ⊢ (⊤ → 𝑄 ∈ NzRing)
157	53, 2, 52, 156	deg1vr 33827	. . . . . . . . . . . . 13 ⊢ (⊤ → (𝐷‘𝑋) = 1)
158	154, 157	oveq12d 7429	. . . . . . . . . . . 12 ⊢ (⊤ → ((𝐷‘(𝐾‘-3)) + (𝐷‘𝑋)) = (0 + 1))
159		1cnd 11202	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℂ)
160	159	addlidd 11411	. . . . . . . . . . . 12 ⊢ (⊤ → (0 + 1) = 1)
161	152, 158, 160	3eqtrd 2808	. . . . . . . . . . 11 ⊢ (⊤ → (𝐷‘((𝐾‘-3) · 𝑋)) = 1)
162	138, 141, 161	3brtr4d 5147	. . . . . . . . . 10 ⊢ (⊤ → (𝐷‘(𝐾‘1)) < (𝐷‘((𝐾‘-3) · 𝑋)))
163	2, 53, 65, 60, 48, 89, 93, 162	deg1add 26229	. . . . . . . . 9 ⊢ (⊤ → (𝐷‘(((𝐾‘-3) · 𝑋) + (𝐾‘1))) = (𝐷‘((𝐾‘-3) · 𝑋)))
164	163, 161	eqtrd 2804	. . . . . . . 8 ⊢ (⊤ → (𝐷‘(((𝐾‘-3) · 𝑋) + (𝐾‘1))) = 1)
165	53, 2, 52, 69, 50	deg1pw 26247	. . . . . . . . 9 ⊢ ((𝑄 ∈ NzRing ∧ 3 ∈ ℕ0) → (𝐷‘(3 ↑ 𝑋)) = 3)
166	156, 74, 165	syl2anc 595	. . . . . . . 8 ⊢ (⊤ → (𝐷‘(3 ↑ 𝑋)) = 3)
167	136, 164, 166	3brtr4d 5147	. . . . . . 7 ⊢ (⊤ → (𝐷‘(((𝐾‘-3) · 𝑋) + (𝐾‘1))) < (𝐷‘(3 ↑ 𝑋)))
168	2, 53, 65, 60, 48, 77, 94, 167	deg1add 26229	. . . . . 6 ⊢ (⊤ → (𝐷‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))) = (𝐷‘(3 ↑ 𝑋)))
169	134, 168, 166	3eqtrd 2808	. . . . 5 ⊢ (⊤ → (𝐷‘𝐹) = 3)
170	58, 59, 53, 2, 60, 62, 96, 132, 169	ply1dg3rt0irred 33819	. . . 4 ⊢ (⊤ → 𝐹 ∈ (Irred‘𝑃))
171		eqid 2769	. . . . . . 7 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
172	171, 47	irredn0 20505	. . . . . 6 ⊢ ((𝑃 ∈ Ring ∧ 𝐹 ∈ (Irred‘𝑃)) → 𝐹 ≠ (0g‘𝑃))
173	67, 170, 172	syl2anc 595	. . . . 5 ⊢ (⊤ → 𝐹 ≠ (0g‘𝑃))
174	169	fveq2d 6886	. . . . . 6 ⊢ (⊤ → ((coe1‘𝐹)‘(𝐷‘𝐹)) = ((coe1‘𝐹)‘3))
175	133	fveq2d 6886	. . . . . . . 8 ⊢ (⊤ → (coe1‘𝐹) = (coe1‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))))
176	175	fveq1d 6884	. . . . . . 7 ⊢ (⊤ → ((coe1‘𝐹)‘3) = ((coe1‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))))‘3))
177		cnfldadd 21497	. . . . . . . . . . 11 ⊢ + = (+g‘ℂfld)
178	3, 177	ressplusg 17344	. . . . . . . . . 10 ⊢ (ℚ ∈ (SubRing‘ℂfld) → + = (+g‘𝑄))
179	11, 178	ax-mp 5	. . . . . . . . 9 ⊢ + = (+g‘𝑄)
180	2, 60, 48, 179	coe1addfv 22395	. . . . . . . 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 1398	. . . . . . 7 ⊢ (⊤ → ((coe1‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))))‘3) = (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3)))
182		iftrue 4498	. . . . . . . . . 10 ⊢ (𝑖 = 3 → if(𝑖 = 3, 1, 0) = 1)
183	3	qrng1 27752	. . . . . . . . . . 11 ⊢ 1 = (1r‘𝑄)
184	2, 52, 50, 65, 74, 58, 183	coe1mon 33822	. . . . . . . . . 10 ⊢ (⊤ → (coe1‘(3 ↑ 𝑋)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 3, 1, 0)))
185	182, 184, 74, 159	fvmptd4 7015	. . . . . . . . 9 ⊢ (⊤ → ((coe1‘(3 ↑ 𝑋))‘3) = 1)
186	2, 60, 48, 179	coe1addfv 22395	. . . . . . . . . . 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 1398	. . . . . . . . . 10 ⊢ (⊤ → ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3) = (((coe1‘((𝐾‘-3) · 𝑋))‘3) + ((coe1‘(𝐾‘1))‘3)))
188	2	ply1assa 22328	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 ∈ CRing → 𝑃 ∈ AssAlg)
189	97, 188	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → 𝑃 ∈ AssAlg)
190		eqid 2769	. . . . . . . . . . . . . . . . 17 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
191	51, 79, 82, 60, 49, 190	asclmul1 22005	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ AssAlg ∧ -3 ∈ ℚ ∧ 𝑋 ∈ (Base‘𝑃)) → ((𝐾‘-3) · 𝑋) = (-3( ·𝑠 ‘𝑃)𝑋))
192	189, 87, 76, 191	syl3anc 1396	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ((𝐾‘-3) · 𝑋) = (-3( ·𝑠 ‘𝑃)𝑋))
193	70, 50	mulg1 19147	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ (Base‘𝑃) → (1 ↑ 𝑋) = 𝑋)
194	76, 193	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (1 ↑ 𝑋) = 𝑋)
195	194	oveq2d 7427	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)) = (-3( ·𝑠 ‘𝑃)𝑋))
196	192, 195	eqtr4d 2807	. . . . . . . . . . . . . 14 ⊢ (⊤ → ((𝐾‘-3) · 𝑋) = (-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))
197	196	fveq2d 6886	. . . . . . . . . . . . 13 ⊢ (⊤ → (coe1‘((𝐾‘-3) · 𝑋)) = (coe1‘(-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋))))
198	197	fveq1d 6884	. . . . . . . . . . . 12 ⊢ (⊤ → ((coe1‘((𝐾‘-3) · 𝑋))‘3) = ((coe1‘(-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))‘3))
199		1nn0 12520	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ0
200	199	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℕ0)
201		1red 11209	. . . . . . . . . . . . . 14 ⊢ (⊤ → 1 ∈ ℝ)
202	201, 136	ltned 11346	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ≠ 3)
203	58, 82, 2, 52, 190, 69, 50, 65, 87, 200, 74, 202	coe1tmfv2 22405	. . . . . . . . . . . 12 ⊢ (⊤ → ((coe1‘(-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))‘3) = 0)
204	198, 203	eqtrd 2804	. . . . . . . . . . 11 ⊢ (⊤ → ((coe1‘((𝐾‘-3) · 𝑋))‘3) = 0)
205	2, 51, 82, 58	coe1scl 22417	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ Ring ∧ 1 ∈ ℚ) → (coe1‘(𝐾‘1)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, 1, 0)))
206	65, 92, 205	syl2anc 595	. . . . . . . . . . . 12 ⊢ (⊤ → (coe1‘(𝐾‘1)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, 1, 0)))
207		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑖 = 3) → 𝑖 = 3)
208	28	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑖 = 3) → 3 ≠ 0)
209	207, 208	eqnetrd 3031	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑖 = 3) → 𝑖 ≠ 0)
210	209	neneqd 2969	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑖 = 3) → ¬ 𝑖 = 0)
211	210	iffalsed 4503	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑖 = 3) → if(𝑖 = 0, 1, 0) = 0)
212		0zd 12603	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ∈ ℤ)
213	206, 211, 74, 212	fvmptd 6998	. . . . . . . . . . 11 ⊢ (⊤ → ((coe1‘(𝐾‘1))‘3) = 0)
214	204, 213	oveq12d 7429	. . . . . . . . . 10 ⊢ (⊤ → (((coe1‘((𝐾‘-3) · 𝑋))‘3) + ((coe1‘(𝐾‘1))‘3)) = (0 + 0))
215		00id 11385	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
216	215	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (0 + 0) = 0)
217	187, 214, 216	3eqtrd 2808	. . . . . . . . 9 ⊢ (⊤ → ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3) = 0)
218	185, 217	oveq12d 7429	. . . . . . . 8 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3)) = (1 + 0))
219	159	addridd 11410	. . . . . . . 8 ⊢ (⊤ → (1 + 0) = 1)
220	218, 219	eqtrd 2804	. . . . . . 7 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3)) = 1)
221	176, 181, 220	3eqtrd 2808	. . . . . 6 ⊢ (⊤ → ((coe1‘𝐹)‘3) = 1)
222	174, 221	eqtrd 2804	. . . . 5 ⊢ (⊤ → ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1)
223	3	fveq2i 6885	. . . . . . 7 ⊢ (Monic1p‘𝑄) = (Monic1p‘(ℂfld ↾s ℚ))
224	223	eqcomi 2778	. . . . . 6 ⊢ (Monic1p‘(ℂfld ↾s ℚ)) = (Monic1p‘𝑄)
225	2, 60, 47, 53, 224, 183	ismon1p 26269	. . . . 5 ⊢ (𝐹 ∈ (Monic1p‘(ℂfld ↾s ℚ)) ↔ (𝐹 ∈ (Base‘𝑃) ∧ 𝐹 ≠ (0g‘𝑃) ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1))
226	96, 173, 222, 225	syl3anbrc 1360	. . . 4 ⊢ (⊤ → 𝐹 ∈ (Monic1p‘(ℂfld ↾s ℚ)))
227	1, 5, 6, 8, 15, 44, 45, 46, 47, 57, 170, 226	irredminply 34051	. . 3 ⊢ (⊤ → 𝐹 = (𝑀‘𝐴))
228	227, 169	jca 520	. 2 ⊢ (⊤ → (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3))
229	228	mptru 1574	1 ⊢ (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3)

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1567  ⊤wtru 1568   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  {crab 3423  ∅c0 4294  ifcif 4492  {csn 4594   class class class wbr 5113   ↦ cmpt 5196  ^◡ccnv 5661   ↾ cres 5664   “ cima 5665   Fn wfn 6532  ‘cfv 6537  (class class class)co 7411  ℂcc 11098  0cc0 11100  1c1 11101  ici 11102   + caddc 11103   · cmul 11105   < clt 11243  -cneg 11442   / cdiv 11871  2c2 12295  3c3 12296  ℕ₀cn0 12504  ℤcz 12591  ℚcq 12972  ↑cexp 14097  expce 16115  πcpi 16120  Basecbs 17269   ↾_s cress 17290  +_gcplusg 17310  ._rcmulr 17311  Scalarcsca 17313   ·_𝑠 cvsca 17314  0_gc0g 17492  Mndcmnd 18792  ._gcmg 19133  mulGrpcmgp 20216  Ringcrg 20315  CRingccrg 20316  Irredcir 20438  NzRingcnzr 20595  SubRingcsubrg 20654  Domncdomn 20777  DivRingcdr 20813  Fieldcfield 20814  SubDRingcsdrg 20867  LModclmod 20959  ℂ_fldccnfld 21491  AssAlgcasa 21969  algSccascl 21971  var₁cv1 22305  Poly₁cpl1 22306  coe₁cco1 22307   evalSub₁ ces1 22442  eval₁ce1 22443  deg₁cdg1 26180  Monic_1pcmn1 26252  ↑_𝑐ccxp 26686   minPoly cminply 34034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9610  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177  ax-pre-sup 11178  ax-addf 11179  ax-mulf 11180

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-ofr 7676  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8157  df-tpos 8222  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8694  df-map 8826  df-pm 8827  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9322  df-fi 9371  df-sup 9402  df-inf 9403  df-oi 9472  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-div 11872  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-uz 12863  df-q 12973  df-rp 13017  df-xneg 13137  df-xadd 13138  df-xmul 13139  df-ioo 13376  df-ioc 13377  df-ico 13378  df-icc 13379  df-fz 13536  df-fzo 13683  df-fl 13825  df-mod 13903  df-seq 14038  df-exp 14098  df-fac 14310  df-bc 14339  df-hash 14367  df-shft 15104  df-sgn 15124  df-cj 15150  df-re 15151  df-im 15152  df-sqrt 15286  df-abs 15287  df-limsup 15522  df-clim 15539  df-rlim 15540  df-sum 15738  df-ef 16121  df-sin 16123  df-cos 16124  df-pi 16126  df-dvds 16311  df-gcd 16553  df-prm 16730  df-struct 17207  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-mulr 17324  df-starv 17325  df-sca 17326  df-vsca 17327  df-ip 17328  df-tset 17329  df-ple 17330  df-ds 17332  df-unif 17333  df-hom 17334  df-cco 17335  df-rest 17475  df-topn 17476  df-0g 17494  df-gsum 17495  df-topgen 17496  df-pt 17497  df-prds 17500  df-pws 17502  df-xrs 17556  df-qtop 17561  df-imas 17562  df-xps 17564  df-mre 17638  df-mrc 17639  df-acs 17641  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-mhm 18841  df-submnd 18842  df-grp 19003  df-minusg 19004  df-sbg 19005  df-mulg 19134  df-subg 19189  df-ghm 19284  df-cntz 19387  df-cmn 19852  df-abl 19853  df-mgp 20217  df-rng 20231  df-ur 20264  df-srg 20269  df-ring 20317  df-cring 20318  df-oppr 20419  df-dvdsr 20439  df-unit 20440  df-irred 20441  df-invr 20470  df-dvr 20483  df-rhm 20554  df-nzr 20596  df-subrng 20631  df-subrg 20655  df-rlreg 20779  df-domn 20780  df-idom 20781  df-drng 20815  df-field 20816  df-sdrg 20868  df-lmod 20961  df-lss 21031  df-lsp 21071  df-sra 21272  df-rgmod 21273  df-lidl 21310  df-rsp 21311  df-psmet 21483  df-xmet 21484  df-met 21485  df-bl 21486  df-mopn 21487  df-fbas 21488  df-fg 21489  df-cnfld 21492  df-assa 21972  df-asp 21973  df-ascl 21974  df-psr 22028  df-mvr 22029  df-mpl 22030  df-opsr 22032  df-evls 22194  df-evl 22195  df-psr1 22309  df-vr1 22310  df-ply1 22311  df-coe1 22312  df-evls1 22444  df-evl1 22445  df-top 23020  df-topon 23037  df-topsp 23059  df-bases 23072  df-cld 23145  df-ntr 23146  df-cls 23147  df-nei 23224  df-lp 23262  df-perf 23263  df-cn 23353  df-cnp 23354  df-haus 23441  df-tx 23688  df-hmeo 23881  df-fil 23972  df-fm 24064  df-flim 24065  df-flf 24066  df-xms 24446  df-ms 24447  df-tms 24448  df-cncf 25006  df-limc 25994  df-dv 25995  df-mdeg 26181  df-deg1 26182  df-mon1 26257  df-uc1p 26258  df-q1p 26259  df-r1p 26260  df-ig1p 26261  df-log 26687  df-cxp 26688  df-irng 34019  df-minply 34035

This theorem is referenced by:  cos9thpinconstrlem2  34125