Theorem cos9thpiminply 33771

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 2729	. . . 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 6843	. . . . 5 ⊢ (Poly1‘𝑄) = (Poly1‘(ℂfld ↾s ℚ))
5	2, 4	eqtri 2752	. . . 4 ⊢ 𝑃 = (Poly1‘(ℂfld ↾s ℚ))
6		cnfldbas 21300	. . . 4 ⊢ ℂ = (Base‘ℂfld)
7		cnfldfld 33307	. . . . 5 ⊢ ℂfld ∈ Field
8	7	a1i 11	. . . 4 ⊢ (⊤ → ℂfld ∈ Field)
9		cndrng 21340	. . . . . 6 ⊢ ℂfld ∈ DivRing
10		qsubdrg 21361	. . . . . . 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 20708	. . . . . 6 ⊢ (ℚ ∈ (SubDRing‘ℂfld) ↔ (ℂfld ∈ DivRing ∧ ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing))
14	9, 11, 12, 13	mpbir3an 1342	. . . . 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 11103	. . . . . . . . . . . . 13 ⊢ i ∈ ℂ
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → i ∈ ℂ)
21		2cnd 12240	. . . . . . . . . . . . 13 ⊢ (⊤ → 2 ∈ ℂ)
22		picn 26400	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
23	22	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → π ∈ ℂ)
24	21, 23	mulcld 11170	. . . . . . . . . . . 12 ⊢ (⊤ → (2 · π) ∈ ℂ)
25	20, 24	mulcld 11170	. . . . . . . . . . 11 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
26		3cn 12243	. . . . . . . . . . . 12 ⊢ 3 ∈ ℂ
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℂ)
28		3ne0 12268	. . . . . . . . . . . 12 ⊢ 3 ≠ 0
29	28	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ≠ 0)
30	25, 27, 29	divcld 11934	. . . . . . . . . 10 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
31	30	efcld 16025	. . . . . . . . 9 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
32	18, 31	eqeltrid 2832	. . . . . . . 8 ⊢ (⊤ → 𝑂 ∈ ℂ)
33	27, 29	reccld 11927	. . . . . . . 8 ⊢ (⊤ → (1 / 3) ∈ ℂ)
34	32, 33	cxpcld 26650	. . . . . . 7 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ∈ ℂ)
35	17, 34	eqeltrid 2832	. . . . . 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 16039	. . . . . . . . . 10 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ≠ 0)
39	37, 38	eqnetrd 2992	. . . . . . . . 9 ⊢ (⊤ → 𝑂 ≠ 0)
40	32, 39, 33	cxpne0d 26655	. . . . . . . 8 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ≠ 0)
41	36, 40	eqnetrd 2992	. . . . . . 7 ⊢ (⊤ → 𝑍 ≠ 0)
42	35, 41	reccld 11927	. . . . . 6 ⊢ (⊤ → (1 / 𝑍) ∈ ℂ)
43	35, 42	addcld 11169	. . . . 5 ⊢ (⊤ → (𝑍 + (1 / 𝑍)) ∈ ℂ)
44	16, 43	eqeltrid 2832	. . . 4 ⊢ (⊤ → 𝐴 ∈ ℂ)
45		cnfld0 21334	. . . 4 ⊢ 0 = (0g‘ℂfld)
46		cos9thpiminply.m	. . . 4 ⊢ 𝑀 = (ℂfld minPoly ℚ)
47		eqid 2729	. . . 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 33770	. . . . 5 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝐴) = ((𝐴↑3) + ((-3 · 𝐴) + 1)))
56	18, 17, 16	cos9thpiminplylem5 33769	. . . . 5 ⊢ ((𝐴↑3) + ((-3 · 𝐴) + 1)) = 0
57	55, 56	eqtrdi 2780	. . . 4 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝐴) = 0)
58	3	qrng0 27565	. . . . 5 ⊢ 0 = (0g‘𝑄)
59		eqid 2729	. . . . 5 ⊢ (eval1‘𝑄) = (eval1‘𝑄)
60		eqid 2729	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
61	3	qfld 33263	. . . . . 6 ⊢ 𝑄 ∈ Field
62	61	a1i 11	. . . . 5 ⊢ (⊤ → 𝑄 ∈ Field)
63	3	qdrng 27564	. . . . . . . . . . 11 ⊢ 𝑄 ∈ DivRing
64	63	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 𝑄 ∈ DivRing)
65	64	drngringd 20657	. . . . . . . . 9 ⊢ (⊤ → 𝑄 ∈ Ring)
66	2	ply1ring 22165	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → 𝑃 ∈ Ring)
67	65, 66	syl 17	. . . . . . . 8 ⊢ (⊤ → 𝑃 ∈ Ring)
68	67	ringgrpd 20162	. . . . . . 7 ⊢ (⊤ → 𝑃 ∈ Grp)
69		eqid 2729	. . . . . . . . 9 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
70	69, 60	mgpbas 20065	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
71	69	ringmgp 20159	. . . . . . . . 9 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
72	67, 71	syl 17	. . . . . . . 8 ⊢ (⊤ → (mulGrp‘𝑃) ∈ Mnd)
73		3nn0 12436	. . . . . . . . 9 ⊢ 3 ∈ ℕ0
74	73	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℕ0)
75	52, 2, 60	vr1cl 22135	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
76	65, 75	syl 17	. . . . . . . 8 ⊢ (⊤ → 𝑋 ∈ (Base‘𝑃))
77	70, 50, 72, 74, 76	mulgnn0cld 19009	. . . . . . 7 ⊢ (⊤ → (3 ↑ 𝑋) ∈ (Base‘𝑃))
78	2	ply1sca 22170	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ DivRing → 𝑄 = (Scalar‘𝑃))
79	63, 78	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝑄 = (Scalar‘𝑃)
80	2	ply1lmod 22169	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ Ring → 𝑃 ∈ LMod)
81	65, 80	syl 17	. . . . . . . . . . 11 ⊢ (⊤ → 𝑃 ∈ LMod)
82	3	qrngbas 27563	. . . . . . . . . . 11 ⊢ ℚ = (Base‘𝑄)
83	51, 79, 67, 81, 82, 60	asclf 21824	. . . . . . . . . 10 ⊢ (⊤ → 𝐾:ℚ⟶(Base‘𝑃))
84	74	nn0zd 12531	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℤ)
85		zq 12889	. . . . . . . . . . 11 ⊢ (3 ∈ ℤ → 3 ∈ ℚ)
86		qnegcl 12901	. . . . . . . . . . 11 ⊢ (3 ∈ ℚ → -3 ∈ ℚ)
87	84, 85, 86	3syl 18	. . . . . . . . . 10 ⊢ (⊤ → -3 ∈ ℚ)
88	83, 87	ffvelcdmd 7039	. . . . . . . . 9 ⊢ (⊤ → (𝐾‘-3) ∈ (Base‘𝑃))
89	60, 49, 67, 88, 76	ringcld 20180	. . . . . . . 8 ⊢ (⊤ → ((𝐾‘-3) · 𝑋) ∈ (Base‘𝑃))
90		1zzd 12540	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℤ)
91		zq 12889	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
92	90, 91	syl 17	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℚ)
93	83, 92	ffvelcdmd 7039	. . . . . . . 8 ⊢ (⊤ → (𝐾‘1) ∈ (Base‘𝑃))
94	60, 48, 68, 89, 93	grpcld 18861	. . . . . . 7 ⊢ (⊤ → (((𝐾‘-3) · 𝑋) + (𝐾‘1)) ∈ (Base‘𝑃))
95	60, 48, 68, 77, 94	grpcld 18861	. . . . . 6 ⊢ (⊤ → ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))) ∈ (Base‘𝑃))
96	54, 95	eqeltrid 2832	. . . . 5 ⊢ (⊤ → 𝐹 ∈ (Base‘𝑃))
97	62	fldcrngd 20662	. . . . . . . . 9 ⊢ (⊤ → 𝑄 ∈ CRing)
98	59, 2, 60, 97, 82, 96	evl1fvf 33525	. . . . . . . 8 ⊢ (⊤ → ((eval1‘𝑄)‘𝐹):ℚ⟶ℚ)
99	98	ffnd 6671	. . . . . . 7 ⊢ (⊤ → ((eval1‘𝑄)‘𝐹) Fn ℚ)
100		fniniseg2 7016	. . . . . . 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 22251	. . . . . . . . . . . . . . 15 ⊢ (eval1‘𝑄) = (𝑄 evalSub1 ℚ)
103	102	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → (eval1‘𝑄) = (𝑄 evalSub1 ℚ))
104	103	fveq1d 6842	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → ((eval1‘𝑄)‘𝐹) = ((𝑄 evalSub1 ℚ)‘𝐹))
105	104	fveq1d 6842	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) = (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥))
106		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝑄 evalSub1 ℚ) = (𝑄 evalSub1 ℚ)
107		cncrng 21330	. . . . . . . . . . . . . . . 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 1547	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑄 ∈ CRing
111	110	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℚ → 𝑄 ∈ CRing)
112	111	crngringd 20166	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℚ → 𝑄 ∈ Ring)
113	82	subrgid 20493	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 ∈ Ring → ℚ ∈ (SubRing‘𝑄))
114	112, 113	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℚ → ℚ ∈ (SubRing‘𝑄))
115	96	mptru 1547	. . . . . . . . . . . . . . . 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 33527	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → ((𝑄 evalSub1 ℚ)‘𝐹) = (((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ))
118	117	fveq1d 6842	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥) = ((((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ)‘𝑥))
119		fvres 6859	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → ((((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ)‘𝑥) = (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥))
120	118, 119	eqtr2d 2765	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥) = (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥))
121		qcn 12898	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℂ)
122	18, 17, 16, 3, 48, 49, 50, 2, 51, 52, 53, 54, 121	cos9thpiminplylem6 33770	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥) = ((𝑥↑3) + ((-3 · 𝑥) + 1)))
123	105, 120, 122	3eqtr2d 2770	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) = ((𝑥↑3) + ((-3 · 𝑥) + 1)))
124		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℚ)
125	124	cos9thpiminplylem2 33766	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℚ → ((𝑥↑3) + ((-3 · 𝑥) + 1)) ≠ 0)
126	123, 125	eqnetrd 2992	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) ≠ 0)
127	126	neneqd 2930	. . . . . . . . 9 ⊢ (𝑥 ∈ ℚ → ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
128	127	rgen 3046	. . . . . . . 8 ⊢ ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0
129	128	a1i 11	. . . . . . 7 ⊢ (⊤ → ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
130		rabeq0 4347	. . . . . . 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 2764	. . . . 5 ⊢ (⊤ → (◡((eval1‘𝑄)‘𝐹) “ {0}) = ∅)
133	54	a1i 11	. . . . . . 7 ⊢ (⊤ → 𝐹 = ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))))
134	133	fveq2d 6844	. . . . . 6 ⊢ (⊤ → (𝐷‘𝐹) = (𝐷‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))))
135		1lt3 12330	. . . . . . . . 9 ⊢ 1 < 3
136	135	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1 < 3)
137		0lt1 11676	. . . . . . . . . . . 12 ⊢ 0 < 1
138	137	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 0 < 1)
139	138	gt0ne0d 11718	. . . . . . . . . . . 12 ⊢ (⊤ → 1 ≠ 0)
140	53, 2, 82, 51, 58	deg1scl 26051	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ Ring ∧ 1 ∈ ℚ ∧ 1 ≠ 0) → (𝐷‘(𝐾‘1)) = 0)
141	65, 92, 139, 140	syl3anc 1373	. . . . . . . . . . 11 ⊢ (⊤ → (𝐷‘(𝐾‘1)) = 0)
142		drngdomn 20669	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ Domn)
143	63, 142	mp1i 13	. . . . . . . . . . . . 13 ⊢ (⊤ → 𝑄 ∈ Domn)
144	27, 29	negne0d 11507	. . . . . . . . . . . . . 14 ⊢ (⊤ → -3 ≠ 0)
145	2, 51, 58, 47, 82	ply1scln0 22211	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ Ring ∧ -3 ∈ ℚ ∧ -3 ≠ 0) → (𝐾‘-3) ≠ (0g‘𝑃))
146	65, 87, 144, 145	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (⊤ → (𝐾‘-3) ≠ (0g‘𝑃))
147	107	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → ℂfld ∈ CRing)
148		drngnzr 20668	. . . . . . . . . . . . . . 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 33552	. . . . . . . . . . . . 13 ⊢ (⊤ → 𝑋 ≠ (0g‘𝑃))
152	53, 2, 60, 49, 47, 143, 88, 146, 76, 151	deg1mul 26053	. . . . . . . . . . . 12 ⊢ (⊤ → (𝐷‘((𝐾‘-3) · 𝑋)) = ((𝐷‘(𝐾‘-3)) + (𝐷‘𝑋)))
153	53, 2, 82, 51, 58	deg1scl 26051	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ Ring ∧ -3 ∈ ℚ ∧ -3 ≠ 0) → (𝐷‘(𝐾‘-3)) = 0)
154	65, 87, 144, 153	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (⊤ → (𝐷‘(𝐾‘-3)) = 0)
155		drngnzr 20668	. . . . . . . . . . . . . . 15 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ NzRing)
156	63, 155	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (⊤ → 𝑄 ∈ NzRing)
157	53, 2, 52, 156	deg1vr 33551	. . . . . . . . . . . . 13 ⊢ (⊤ → (𝐷‘𝑋) = 1)
158	154, 157	oveq12d 7387	. . . . . . . . . . . 12 ⊢ (⊤ → ((𝐷‘(𝐾‘-3)) + (𝐷‘𝑋)) = (0 + 1))
159		1cnd 11145	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℂ)
160	159	addlidd 11351	. . . . . . . . . . . 12 ⊢ (⊤ → (0 + 1) = 1)
161	152, 158, 160	3eqtrd 2768	. . . . . . . . . . 11 ⊢ (⊤ → (𝐷‘((𝐾‘-3) · 𝑋)) = 1)
162	138, 141, 161	3brtr4d 5134	. . . . . . . . . 10 ⊢ (⊤ → (𝐷‘(𝐾‘1)) < (𝐷‘((𝐾‘-3) · 𝑋)))
163	2, 53, 65, 60, 48, 89, 93, 162	deg1add 26041	. . . . . . . . 9 ⊢ (⊤ → (𝐷‘(((𝐾‘-3) · 𝑋) + (𝐾‘1))) = (𝐷‘((𝐾‘-3) · 𝑋)))
164	163, 161	eqtrd 2764	. . . . . . . 8 ⊢ (⊤ → (𝐷‘(((𝐾‘-3) · 𝑋) + (𝐾‘1))) = 1)
165	53, 2, 52, 69, 50	deg1pw 26059	. . . . . . . . 9 ⊢ ((𝑄 ∈ NzRing ∧ 3 ∈ ℕ0) → (𝐷‘(3 ↑ 𝑋)) = 3)
166	156, 74, 165	syl2anc 584	. . . . . . . 8 ⊢ (⊤ → (𝐷‘(3 ↑ 𝑋)) = 3)
167	136, 164, 166	3brtr4d 5134	. . . . . . 7 ⊢ (⊤ → (𝐷‘(((𝐾‘-3) · 𝑋) + (𝐾‘1))) < (𝐷‘(3 ↑ 𝑋)))
168	2, 53, 65, 60, 48, 77, 94, 167	deg1add 26041	. . . . . 6 ⊢ (⊤ → (𝐷‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))) = (𝐷‘(3 ↑ 𝑋)))
169	134, 168, 166	3eqtrd 2768	. . . . 5 ⊢ (⊤ → (𝐷‘𝐹) = 3)
170	58, 59, 53, 2, 60, 62, 96, 132, 169	ply1dg3rt0irred 33544	. . . 4 ⊢ (⊤ → 𝐹 ∈ (Irred‘𝑃))
171		eqid 2729	. . . . . . 7 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
172	171, 47	irredn0 20343	. . . . . 6 ⊢ ((𝑃 ∈ Ring ∧ 𝐹 ∈ (Irred‘𝑃)) → 𝐹 ≠ (0g‘𝑃))
173	67, 170, 172	syl2anc 584	. . . . 5 ⊢ (⊤ → 𝐹 ≠ (0g‘𝑃))
174	169	fveq2d 6844	. . . . . 6 ⊢ (⊤ → ((coe1‘𝐹)‘(𝐷‘𝐹)) = ((coe1‘𝐹)‘3))
175	133	fveq2d 6844	. . . . . . . 8 ⊢ (⊤ → (coe1‘𝐹) = (coe1‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))))
176	175	fveq1d 6842	. . . . . . 7 ⊢ (⊤ → ((coe1‘𝐹)‘3) = ((coe1‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))))‘3))
177		cnfldadd 21302	. . . . . . . . . . 11 ⊢ + = (+g‘ℂfld)
178	3, 177	ressplusg 17230	. . . . . . . . . 10 ⊢ (ℚ ∈ (SubRing‘ℂfld) → + = (+g‘𝑄))
179	11, 178	ax-mp 5	. . . . . . . . 9 ⊢ + = (+g‘𝑄)
180	2, 60, 48, 179	coe1addfv 22184	. . . . . . . 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 1375	. . . . . . 7 ⊢ (⊤ → ((coe1‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))))‘3) = (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3)))
182		iftrue 4490	. . . . . . . . . 10 ⊢ (𝑖 = 3 → if(𝑖 = 3, 1, 0) = 1)
183	3	qrng1 27566	. . . . . . . . . . 11 ⊢ 1 = (1r‘𝑄)
184	2, 52, 50, 65, 74, 58, 183	coe1mon 33547	. . . . . . . . . 10 ⊢ (⊤ → (coe1‘(3 ↑ 𝑋)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 3, 1, 0)))
185	182, 184, 74, 159	fvmptd4 6974	. . . . . . . . 9 ⊢ (⊤ → ((coe1‘(3 ↑ 𝑋))‘3) = 1)
186	2, 60, 48, 179	coe1addfv 22184	. . . . . . . . . . 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 1375	. . . . . . . . . 10 ⊢ (⊤ → ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3) = (((coe1‘((𝐾‘-3) · 𝑋))‘3) + ((coe1‘(𝐾‘1))‘3)))
188	2	ply1assa 22117	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 ∈ CRing → 𝑃 ∈ AssAlg)
189	97, 188	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → 𝑃 ∈ AssAlg)
190		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
191	51, 79, 82, 60, 49, 190	asclmul1 21828	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ AssAlg ∧ -3 ∈ ℚ ∧ 𝑋 ∈ (Base‘𝑃)) → ((𝐾‘-3) · 𝑋) = (-3( ·𝑠 ‘𝑃)𝑋))
192	189, 87, 76, 191	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ((𝐾‘-3) · 𝑋) = (-3( ·𝑠 ‘𝑃)𝑋))
193	70, 50	mulg1 18995	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ (Base‘𝑃) → (1 ↑ 𝑋) = 𝑋)
194	76, 193	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (1 ↑ 𝑋) = 𝑋)
195	194	oveq2d 7385	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)) = (-3( ·𝑠 ‘𝑃)𝑋))
196	192, 195	eqtr4d 2767	. . . . . . . . . . . . . 14 ⊢ (⊤ → ((𝐾‘-3) · 𝑋) = (-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))
197	196	fveq2d 6844	. . . . . . . . . . . . 13 ⊢ (⊤ → (coe1‘((𝐾‘-3) · 𝑋)) = (coe1‘(-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋))))
198	197	fveq1d 6842	. . . . . . . . . . . 12 ⊢ (⊤ → ((coe1‘((𝐾‘-3) · 𝑋))‘3) = ((coe1‘(-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))‘3))
199		1nn0 12434	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ0
200	199	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℕ0)
201		1red 11151	. . . . . . . . . . . . . 14 ⊢ (⊤ → 1 ∈ ℝ)
202	201, 136	ltned 11286	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ≠ 3)
203	58, 82, 2, 52, 190, 69, 50, 65, 87, 200, 74, 202	coe1tmfv2 22194	. . . . . . . . . . . 12 ⊢ (⊤ → ((coe1‘(-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))‘3) = 0)
204	198, 203	eqtrd 2764	. . . . . . . . . . 11 ⊢ (⊤ → ((coe1‘((𝐾‘-3) · 𝑋))‘3) = 0)
205	2, 51, 82, 58	coe1scl 22206	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ Ring ∧ 1 ∈ ℚ) → (coe1‘(𝐾‘1)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, 1, 0)))
206	65, 92, 205	syl2anc 584	. . . . . . . . . . . 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 2992	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑖 = 3) → 𝑖 ≠ 0)
210	209	neneqd 2930	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑖 = 3) → ¬ 𝑖 = 0)
211	210	iffalsed 4495	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑖 = 3) → if(𝑖 = 0, 1, 0) = 0)
212		0zd 12517	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ∈ ℤ)
213	206, 211, 74, 212	fvmptd 6957	. . . . . . . . . . 11 ⊢ (⊤ → ((coe1‘(𝐾‘1))‘3) = 0)
214	204, 213	oveq12d 7387	. . . . . . . . . 10 ⊢ (⊤ → (((coe1‘((𝐾‘-3) · 𝑋))‘3) + ((coe1‘(𝐾‘1))‘3)) = (0 + 0))
215		00id 11325	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
216	215	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (0 + 0) = 0)
217	187, 214, 216	3eqtrd 2768	. . . . . . . . 9 ⊢ (⊤ → ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3) = 0)
218	185, 217	oveq12d 7387	. . . . . . . 8 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3)) = (1 + 0))
219	159	addridd 11350	. . . . . . . 8 ⊢ (⊤ → (1 + 0) = 1)
220	218, 219	eqtrd 2764	. . . . . . 7 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3)) = 1)
221	176, 181, 220	3eqtrd 2768	. . . . . 6 ⊢ (⊤ → ((coe1‘𝐹)‘3) = 1)
222	174, 221	eqtrd 2764	. . . . 5 ⊢ (⊤ → ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1)
223	3	fveq2i 6843	. . . . . . 7 ⊢ (Monic1p‘𝑄) = (Monic1p‘(ℂfld ↾s ℚ))
224	223	eqcomi 2738	. . . . . 6 ⊢ (Monic1p‘(ℂfld ↾s ℚ)) = (Monic1p‘𝑄)
225	2, 60, 47, 53, 224, 183	ismon1p 26081	. . . . 5 ⊢ (𝐹 ∈ (Monic1p‘(ℂfld ↾s ℚ)) ↔ (𝐹 ∈ (Base‘𝑃) ∧ 𝐹 ≠ (0g‘𝑃) ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1))
226	96, 173, 222, 225	syl3anbrc 1344	. . . 4 ⊢ (⊤ → 𝐹 ∈ (Monic1p‘(ℂfld ↾s ℚ)))
227	1, 5, 6, 8, 15, 44, 45, 46, 47, 57, 170, 226	irredminply 33699	. . 3 ⊢ (⊤ → 𝐹 = (𝑀‘𝐴))
228	227, 169	jca 511	. 2 ⊢ (⊤ → (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3))
229	228	mptru 1547	1 ⊢ (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3402  ∅c0 4292  ifcif 4484  {csn 4585   class class class wbr 5102   ↦ cmpt 5183  ^◡ccnv 5630   ↾ cres 5633   “ cima 5634   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369  ℂcc 11042  0cc0 11044  1c1 11045  ici 11046   + caddc 11047   · cmul 11049   < clt 11184  -cneg 11382   / cdiv 11811  2c2 12217  3c3 12218  ℕ₀cn0 12418  ℤcz 12505  ℚcq 12883  ↑cexp 14002  expce 16003  πcpi 16008  Basecbs 17155   ↾_s cress 17176  +_gcplusg 17196  ._rcmulr 17197  Scalarcsca 17199   ·_𝑠 cvsca 17200  0_gc0g 17378  Mndcmnd 18643  ._gcmg 18981  mulGrpcmgp 20060  Ringcrg 20153  CRingccrg 20154  Irredcir 20276  NzRingcnzr 20432  SubRingcsubrg 20489  Domncdomn 20612  DivRingcdr 20649  Fieldcfield 20650  SubDRingcsdrg 20706  LModclmod 20798  ℂ_fldccnfld 21296  AssAlgcasa 21792  algSccascl 21794  var₁cv1 22093  Poly₁cpl1 22094  coe₁cco1 22095   evalSub₁ ces1 22233  eval₁ce1 22234  deg₁cdg1 25992  Monic_1pcmn1 26064  ↑_𝑐ccxp 26497   minPoly cminply 33682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122  ax-addf 11123  ax-mulf 11124

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-ofr 7634  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-tpos 8182  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-fi 9338  df-sup 9369  df-inf 9370  df-oi 9439  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-q 12884  df-rp 12928  df-xneg 13048  df-xadd 13049  df-xmul 13050  df-ioo 13286  df-ioc 13287  df-ico 13288  df-icc 13289  df-fz 13445  df-fzo 13592  df-fl 13730  df-mod 13808  df-seq 13943  df-exp 14003  df-fac 14215  df-bc 14244  df-hash 14272  df-shft 15009  df-sgn 15029  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-limsup 15413  df-clim 15430  df-rlim 15431  df-sum 15629  df-ef 16009  df-sin 16011  df-cos 16012  df-pi 16014  df-dvds 16199  df-gcd 16441  df-prm 16618  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-rest 17361  df-topn 17362  df-0g 17380  df-gsum 17381  df-topgen 17382  df-pt 17383  df-prds 17386  df-pws 17388  df-xrs 17441  df-qtop 17446  df-imas 17447  df-xps 17449  df-mre 17523  df-mrc 17524  df-acs 17526  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-mhm 18692  df-submnd 18693  df-grp 18850  df-minusg 18851  df-sbg 18852  df-mulg 18982  df-subg 19037  df-ghm 19127  df-cntz 19231  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-srg 20107  df-ring 20155  df-cring 20156  df-oppr 20257  df-dvdsr 20277  df-unit 20278  df-irred 20279  df-invr 20308  df-dvr 20321  df-rhm 20392  df-nzr 20433  df-subrng 20466  df-subrg 20490  df-rlreg 20614  df-domn 20615  df-idom 20616  df-drng 20651  df-field 20652  df-sdrg 20707  df-lmod 20800  df-lss 20870  df-lsp 20910  df-sra 21112  df-rgmod 21113  df-lidl 21150  df-rsp 21151  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-fbas 21293  df-fg 21294  df-cnfld 21297  df-assa 21795  df-asp 21796  df-ascl 21797  df-psr 21851  df-mvr 21852  df-mpl 21853  df-opsr 21855  df-evls 22014  df-evl 22015  df-psr1 22097  df-vr1 22098  df-ply1 22099  df-coe1 22100  df-evls1 22235  df-evl1 22236  df-top 22814  df-topon 22831  df-topsp 22853  df-bases 22866  df-cld 22939  df-ntr 22940  df-cls 22941  df-nei 23018  df-lp 23056  df-perf 23057  df-cn 23147  df-cnp 23148  df-haus 23235  df-tx 23482  df-hmeo 23675  df-fil 23766  df-fm 23858  df-flim 23859  df-flf 23860  df-xms 24241  df-ms 24242  df-tms 24243  df-cncf 24804  df-limc 25800  df-dv 25801  df-mdeg 25993  df-deg1 25994  df-mon1 26069  df-uc1p 26070  df-q1p 26071  df-r1p 26072  df-ig1p 26073  df-log 26498  df-cxp 26499  df-irng 33672  df-minply 33683

This theorem is referenced by:  cos9thpinconstrlem2  33773