Theorem cos9thpiminply 34420

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 2761	. . . 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 6888	. . . . 5 ⊢ (Poly1‘𝑄) = (Poly1‘(ℂfld ↾s ℚ))
5	2, 4	eqtri 2784	. . . 4 ⊢ 𝑃 = (Poly1‘(ℂfld ↾s ℚ))
6		cnfldbas 21682	. . . 4 ⊢ ℂ = (Base‘ℂfld)
7		cnfldfld 33903	. . . . 5 ⊢ ℂfld ∈ Field
8	7	a1i 11	. . . 4 ⊢ (⊤ → ℂfld ∈ Field)
9		cndrng 21707	. . . . . 6 ⊢ ℂfld ∈ DivRing
10		qsubdrg 21725	. . . . . . 7 ⊢ (ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing)
11	10	simpli 489	. . . . . 6 ⊢ ℚ ∈ (SubRing‘ℂfld)
12	10	simpri 491	. . . . . 6 ⊢ (ℂfld ↾s ℚ) ∈ DivRing
13		issdrg 21045	. . . . . 6 ⊢ (ℚ ∈ (SubDRing‘ℂfld) ↔ (ℂfld ∈ DivRing ∧ ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing))
14	9, 11, 12, 13	mpbir3an 1360	. . . . 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 11259	. . . . . . . . . . . . 13 ⊢ i ∈ ℂ
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → i ∈ ℂ)
21		2cnd 12421	. . . . . . . . . . . . 13 ⊢ (⊤ → 2 ∈ ℂ)
22		picn 26785	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
23	22	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → π ∈ ℂ)
24	21, 23	mulcld 11329	. . . . . . . . . . . 12 ⊢ (⊤ → (2 · π) ∈ ℂ)
25	20, 24	mulcld 11329	. . . . . . . . . . 11 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
26		3cn 12424	. . . . . . . . . . . 12 ⊢ 3 ∈ ℂ
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℂ)
28		3ne0 12452	. . . . . . . . . . . 12 ⊢ 3 ≠ 0
29	28	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ≠ 0)
30	25, 27, 29	divcld 12093	. . . . . . . . . 10 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
31	30	efcld 16249	. . . . . . . . 9 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
32	18, 31	eqeltrid 2865	. . . . . . . 8 ⊢ (⊤ → 𝑂 ∈ ℂ)
33	27, 29	reccld 12086	. . . . . . . 8 ⊢ (⊤ → (1 / 3) ∈ ℂ)
34	32, 33	cxpcld 27036	. . . . . . 7 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ∈ ℂ)
35	17, 34	eqeltrid 2865	. . . . . 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 16263	. . . . . . . . . 10 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ≠ 0)
39	37, 38	eqnetrd 3023	. . . . . . . . 9 ⊢ (⊤ → 𝑂 ≠ 0)
40	32, 39, 33	cxpne0d 27041	. . . . . . . 8 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ≠ 0)
41	36, 40	eqnetrd 3023	. . . . . . 7 ⊢ (⊤ → 𝑍 ≠ 0)
42	35, 41	reccld 12086	. . . . . 6 ⊢ (⊤ → (1 / 𝑍) ∈ ℂ)
43	35, 42	addcld 11328	. . . . 5 ⊢ (⊤ → (𝑍 + (1 / 𝑍)) ∈ ℂ)
44	16, 43	eqeltrid 2865	. . . 4 ⊢ (⊤ → 𝐴 ∈ ℂ)
45		cnfld0 21702	. . . 4 ⊢ 0 = (0g‘ℂfld)
46		cos9thpiminply.m	. . . 4 ⊢ 𝑀 = (ℂfld minPoly ℚ)
47		eqid 2761	. . . 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 34419	. . . . 5 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝐴) = ((𝐴↑3) + (( -3 · 𝐴) + 1)))
56	18, 17, 16	cos9thpiminplylem5 34418	. . . . 5 ⊢ ((𝐴↑3) + (( -3 · 𝐴) + 1)) = 0
57	55, 56	eqtrdi 2812	. . . 4 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝐴) = 0)
58	3	qrng0 27948	. . . . 5 ⊢ 0 = (0g‘𝑄)
59		eqid 2761	. . . . 5 ⊢ (eval1‘𝑄) = (eval1‘𝑄)
60		eqid 2761	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
61	3	qfld 33859	. . . . . 6 ⊢ 𝑄 ∈ Field
62	61	a1i 11	. . . . 5 ⊢ (⊤ → 𝑄 ∈ Field)
63	3	qdrng 27947	. . . . . . . . . . 11 ⊢ 𝑄 ∈ DivRing
64	63	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 𝑄 ∈ DivRing)
65	64	drngringd 20988	. . . . . . . . 9 ⊢ (⊤ → 𝑄 ∈ Ring)
66	2	ply1ring 22565	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → 𝑃 ∈ Ring)
67	65, 66	syl 18	. . . . . . . 8 ⊢ (⊤ → 𝑃 ∈ Ring)
68	67	ringgrpd 20469	. . . . . . 7 ⊢ (⊤ → 𝑃 ∈ Grp)
69		eqid 2761	. . . . . . . . 9 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
70	69, 60	mgpbas 20365	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
71	69	ringmgp 20465	. . . . . . . . 9 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
72	67, 71	syl 18	. . . . . . . 8 ⊢ (⊤ → (mulGrp‘𝑃) ∈ Mnd)
73		3nn0 12624	. . . . . . . . 9 ⊢ 3 ∈ ℕ0
74	73	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℕ0)
75	52, 2, 60	vr1cl 22535	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
76	65, 75	syl 18	. . . . . . . 8 ⊢ (⊤ → 𝑋 ∈ (Base‘𝑃))
77	70, 50, 72, 74, 76	mulgnn0cld 19305	. . . . . . 7 ⊢ (⊤ → (3 ↑ 𝑋) ∈ (Base‘𝑃))
78	2	ply1sca 22570	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ DivRing → 𝑄 = (Scalar‘𝑃))
79	63, 78	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝑄 = (Scalar‘𝑃)
80	2	ply1lmod 22569	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ Ring → 𝑃 ∈ LMod)
81	65, 80	syl 18	. . . . . . . . . . 11 ⊢ (⊤ → 𝑃 ∈ LMod)
82	3	qrngbas 27946	. . . . . . . . . . 11 ⊢ ℚ = (Base‘𝑄)
83	51, 79, 67, 81, 82, 60	asclf 22189	. . . . . . . . . 10 ⊢ (⊤ → 𝐾:ℚ⟶(Base‘𝑃))
84	74	nn0zd 12718	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℤ)
85		zq 13081	. . . . . . . . . . 11 ⊢ (3 ∈ ℤ → 3 ∈ ℚ)
86		qnegcl 13094	. . . . . . . . . . 11 ⊢ (3 ∈ ℚ → -3 ∈ ℚ)
87	84, 85, 86	3syl 19	. . . . . . . . . 10 ⊢ (⊤ → -3 ∈ ℚ)
88	83, 87	ffvelcdmd 7085	. . . . . . . . 9 ⊢ (⊤ → (𝐾‘ -3) ∈ (Base‘𝑃))
89	60, 49, 67, 88, 76	ringcld 20484	. . . . . . . 8 ⊢ (⊤ → ((𝐾‘ -3) · 𝑋) ∈ (Base‘𝑃))
90		1zzd 12727	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℤ)
91		zq 13081	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
92	90, 91	syl 18	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℚ)
93	83, 92	ffvelcdmd 7085	. . . . . . . 8 ⊢ (⊤ → (𝐾‘1) ∈ (Base‘𝑃))
94	60, 48, 68, 89, 93	grpcld 19158	. . . . . . 7 ⊢ (⊤ → (((𝐾‘ -3) · 𝑋) + (𝐾‘1)) ∈ (Base‘𝑃))
95	60, 48, 68, 77, 94	grpcld 19158	. . . . . 6 ⊢ (⊤ → ((3 ↑ 𝑋) + (((𝐾‘ -3) · 𝑋) + (𝐾‘1))) ∈ (Base‘𝑃))
96	54, 95	eqeltrid 2865	. . . . 5 ⊢ (⊤ → 𝐹 ∈ (Base‘𝑃))
97	62	fldcrngd 20995	. . . . . . . . 9 ⊢ (⊤ → 𝑄 ∈ CRing)
98	59, 2, 60, 97, 82, 96	evl1fvf 34095	. . . . . . . 8 ⊢ (⊤ → ((eval1‘𝑄)‘𝐹):ℚ⟶ℚ)
99	98	ffnd 6710	. . . . . . 7 ⊢ (⊤ → ((eval1‘𝑄)‘𝐹) Fn ℚ)
100		fniniseg2 7061	. . . . . . 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 22649	. . . . . . . . . . . . . . 15 ⊢ (eval1‘𝑄) = (𝑄 evalSub1 ℚ)
103	102	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → (eval1‘𝑄) = (𝑄 evalSub1 ℚ))
104	103	fveq1d 6887	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → ((eval1‘𝑄)‘𝐹) = ((𝑄 evalSub1 ℚ)‘𝐹))
105	104	fveq1d 6887	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) = (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥))
106		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (𝑄 evalSub1 ℚ) = (𝑄 evalSub1 ℚ)
107		cncrng 21699	. . . . . . . . . . . . . . . 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 1577	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑄 ∈ CRing
111	110	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℚ → 𝑄 ∈ CRing)
112	111	crngringd 20473	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℚ → 𝑄 ∈ Ring)
113	82	subrgid 20825	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 ∈ Ring → ℚ ∈ (SubRing‘𝑄))
114	112, 113	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℚ → ℚ ∈ (SubRing‘𝑄))
115	96	mptru 1577	. . . . . . . . . . . . . . . 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 34097	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → ((𝑄 evalSub1 ℚ)‘𝐹) = (((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ))
118	117	fveq1d 6887	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥) = ((((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ)‘𝑥))
119		fvres 6904	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → ((((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ)‘𝑥) = (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥))
120	118, 119	eqtr2d 2797	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥) = (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥))
121		qcn 13090	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℂ)
122	18, 17, 16, 3, 48, 49, 50, 2, 51, 52, 53, 54, 121	cos9thpiminplylem6 34419	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥) = ((𝑥↑3) + (( -3 · 𝑥) + 1)))
123	105, 120, 122	3eqtr2d 2802	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) = ((𝑥↑3) + (( -3 · 𝑥) + 1)))
124		id 23	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℚ)
125	124	cos9thpiminplylem2 34415	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℚ → ((𝑥↑3) + (( -3 · 𝑥) + 1)) ≠ 0)
126	123, 125	eqnetrd 3023	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) ≠ 0)
127	126	neneqd 2961	. . . . . . . . 9 ⊢ (𝑥 ∈ ℚ → ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
128	127	rgen 3079	. . . . . . . 8 ⊢ ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0
129	128	a1i 11	. . . . . . 7 ⊢ (⊤ → ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
130		rabeq0 4338	. . . . . . 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 2796	. . . . 5 ⊢ (⊤ → (◡((eval1‘𝑄)‘𝐹) “ {0}) = ∅)
133	54	a1i 11	. . . . . . 7 ⊢ (⊤ → 𝐹 = ((3 ↑ 𝑋) + (((𝐾‘ -3) · 𝑋) + (𝐾‘1))))
134	133	fveq2d 6889	. . . . . 6 ⊢ (⊤ → (𝐷‘𝐹) = (𝐷‘((3 ↑ 𝑋) + (((𝐾‘ -3) · 𝑋) + (𝐾‘1)))))
135		1lt3 12518	. . . . . . . . 9 ⊢ 1 < 3
136	135	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1 < 3)
137		0lt1 11838	. . . . . . . . . . . 12 ⊢ 0 < 1
138	137	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 0 < 1)
139	138	gt0ne0d 11880	. . . . . . . . . . . 12 ⊢ (⊤ → 1 ≠ 0)
140	53, 2, 82, 51, 58	deg1scl 26431	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ Ring ∧ 1 ∈ ℚ ∧ 1 ≠ 0) → (𝐷‘(𝐾‘1)) = 0)
141	65, 92, 139, 140	syl3anc 1398	. . . . . . . . . . 11 ⊢ (⊤ → (𝐷‘(𝐾‘1)) = 0)
142		drngdomn 21003	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ Domn)
143	63, 142	mp1i 14	. . . . . . . . . . . . 13 ⊢ (⊤ → 𝑄 ∈ Domn)
144	27, 29	negne0d 11667	. . . . . . . . . . . . . 14 ⊢ (⊤ → -3 ≠ 0)
145	2, 51, 58, 47, 82	ply1scln0 22610	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ Ring ∧ -3 ∈ ℚ ∧ -3 ≠ 0) → (𝐾‘ -3) ≠ (0g‘𝑃))
146	65, 87, 144, 145	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (⊤ → (𝐾‘ -3) ≠ (0g‘𝑃))
147	107	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → ℂfld ∈ CRing)
148		drngnzr 21002	. . . . . . . . . . . . . . 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 34125	. . . . . . . . . . . . 13 ⊢ (⊤ → 𝑋 ≠ (0g‘𝑃))
152	53, 2, 60, 49, 47, 143, 88, 146, 76, 151	deg1mul 26433	. . . . . . . . . . . 12 ⊢ (⊤ → (𝐷‘((𝐾‘ -3) · 𝑋)) = ((𝐷‘(𝐾‘ -3)) + (𝐷‘𝑋)))
153	53, 2, 82, 51, 58	deg1scl 26431	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ Ring ∧ -3 ∈ ℚ ∧ -3 ≠ 0) → (𝐷‘(𝐾‘ -3)) = 0)
154	65, 87, 144, 153	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (⊤ → (𝐷‘(𝐾‘ -3)) = 0)
155		drngnzr 21002	. . . . . . . . . . . . . . 15 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ NzRing)
156	63, 155	mp1i 14	. . . . . . . . . . . . . 14 ⊢ (⊤ → 𝑄 ∈ NzRing)
157	53, 2, 52, 156	deg1vr 34124	. . . . . . . . . . . . 13 ⊢ (⊤ → (𝐷‘𝑋) = 1)
158	154, 157	oveq12d 7438	. . . . . . . . . . . 12 ⊢ (⊤ → ((𝐷‘(𝐾‘ -3)) + (𝐷‘𝑋)) = (0 + 1))
159		1cnd 11302	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℂ)
160	159	addlidd 11511	. . . . . . . . . . . 12 ⊢ (⊤ → (0 + 1) = 1)
161	152, 158, 160	3eqtrd 2800	. . . . . . . . . . 11 ⊢ (⊤ → (𝐷‘((𝐾‘ -3) · 𝑋)) = 1)
162	138, 141, 161	3brtr4d 5137	. . . . . . . . . 10 ⊢ (⊤ → (𝐷‘(𝐾‘1)) < (𝐷‘((𝐾‘ -3) · 𝑋)))
163	2, 53, 65, 60, 48, 89, 93, 162	deg1add 26421	. . . . . . . . 9 ⊢ (⊤ → (𝐷‘(((𝐾‘ -3) · 𝑋) + (𝐾‘1))) = (𝐷‘((𝐾‘ -3) · 𝑋)))
164	163, 161	eqtrd 2796	. . . . . . . 8 ⊢ (⊤ → (𝐷‘(((𝐾‘ -3) · 𝑋) + (𝐾‘1))) = 1)
165	53, 2, 52, 69, 50	deg1pw 26439	. . . . . . . . 9 ⊢ ((𝑄 ∈ NzRing ∧ 3 ∈ ℕ0) → (𝐷‘(3 ↑ 𝑋)) = 3)
166	156, 74, 165	syl2anc 596	. . . . . . . 8 ⊢ (⊤ → (𝐷‘(3 ↑ 𝑋)) = 3)
167	136, 164, 166	3brtr4d 5137	. . . . . . 7 ⊢ (⊤ → (𝐷‘(((𝐾‘ -3) · 𝑋) + (𝐾‘1))) < (𝐷‘(3 ↑ 𝑋)))
168	2, 53, 65, 60, 48, 77, 94, 167	deg1add 26421	. . . . . 6 ⊢ (⊤ → (𝐷‘((3 ↑ 𝑋) + (((𝐾‘ -3) · 𝑋) + (𝐾‘1)))) = (𝐷‘(3 ↑ 𝑋)))
169	134, 168, 166	3eqtrd 2800	. . . . 5 ⊢ (⊤ → (𝐷‘𝐹) = 3)
170	58, 59, 53, 2, 60, 62, 96, 132, 169	ply1dg3rt0irred 34116	. . . 4 ⊢ (⊤ → 𝐹 ∈ (Irred‘𝑃))
171		eqid 2761	. . . . . . 7 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
172	171, 47	irredn0 20653	. . . . . 6 ⊢ ((𝑃 ∈ Ring ∧ 𝐹 ∈ (Irred‘𝑃)) → 𝐹 ≠ (0g‘𝑃))
173	67, 170, 172	syl2anc 596	. . . . 5 ⊢ (⊤ → 𝐹 ≠ (0g‘𝑃))
174	169	fveq2d 6889	. . . . . 6 ⊢ (⊤ → ((coe1‘𝐹)‘(𝐷‘𝐹)) = ((coe1‘𝐹)‘3))
175	133	fveq2d 6889	. . . . . . . 8 ⊢ (⊤ → (coe1‘𝐹) = (coe1‘((3 ↑ 𝑋) + (((𝐾‘ -3) · 𝑋) + (𝐾‘1)))))
176	175	fveq1d 6887	. . . . . . 7 ⊢ (⊤ → ((coe1‘𝐹)‘3) = ((coe1‘((3 ↑ 𝑋) + (((𝐾‘ -3) · 𝑋) + (𝐾‘1))))‘3))
177		cnfldadd 21684	. . . . . . . . . . 11 ⊢ + = (+g‘ℂfld)
178	3, 177	ressplusg 17462	. . . . . . . . . 10 ⊢ (ℚ ∈ (SubRing‘ℂfld) → + = (+g‘𝑄))
179	11, 178	ax-mp 5	. . . . . . . . 9 ⊢ + = (+g‘𝑄)
180	2, 60, 48, 179	coe1addfv 22584	. . . . . . . 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 1400	. . . . . . 7 ⊢ (⊤ → ((coe1‘((3 ↑ 𝑋) + (((𝐾‘ -3) · 𝑋) + (𝐾‘1))))‘3) = (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘ -3) · 𝑋) + (𝐾‘1)))‘3)))
182		iftrue 4488	. . . . . . . . . 10 ⊢ (𝑖 = 3 → if(𝑖 = 3, 1, 0) = 1)
183	3	qrng1 27949	. . . . . . . . . . 11 ⊢ 1 = (1r‘𝑄)
184	2, 52, 50, 65, 74, 58, 183	coe1mon 34119	. . . . . . . . . 10 ⊢ (⊤ → (coe1‘(3 ↑ 𝑋)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 3, 1, 0)))
185	182, 184, 74, 159	fvmptd4 7018	. . . . . . . . 9 ⊢ (⊤ → ((coe1‘(3 ↑ 𝑋))‘3) = 1)
186	2, 60, 48, 179	coe1addfv 22584	. . . . . . . . . . 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 1400	. . . . . . . . . 10 ⊢ (⊤ → ((coe1‘(((𝐾‘ -3) · 𝑋) + (𝐾‘1)))‘3) = (((coe1‘((𝐾‘ -3) · 𝑋))‘3) + ((coe1‘(𝐾‘1))‘3)))
188	2	ply1assa 22517	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 ∈ CRing → 𝑃 ∈ AssAlg)
189	97, 188	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → 𝑃 ∈ AssAlg)
190		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
191	51, 79, 82, 60, 49, 190	asclmul1 22194	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ AssAlg ∧ -3 ∈ ℚ ∧ 𝑋 ∈ (Base‘𝑃)) → ((𝐾‘ -3) · 𝑋) = ( -3( ·𝑠 ‘𝑃)𝑋))
192	189, 87, 76, 191	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ((𝐾‘ -3) · 𝑋) = ( -3( ·𝑠 ‘𝑃)𝑋))
193	70, 50	mulg1 19291	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ (Base‘𝑃) → (1 ↑ 𝑋) = 𝑋)
194	76, 193	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (1 ↑ 𝑋) = 𝑋)
195	194	oveq2d 7436	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ( -3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)) = ( -3( ·𝑠 ‘𝑃)𝑋))
196	192, 195	eqtr4d 2799	. . . . . . . . . . . . . 14 ⊢ (⊤ → ((𝐾‘ -3) · 𝑋) = ( -3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))
197	196	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ (⊤ → (coe1‘((𝐾‘ -3) · 𝑋)) = (coe1‘( -3( ·𝑠 ‘𝑃)(1 ↑ 𝑋))))
198	197	fveq1d 6887	. . . . . . . . . . . 12 ⊢ (⊤ → ((coe1‘((𝐾‘ -3) · 𝑋))‘3) = ((coe1‘( -3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))‘3))
199		1nn0 12622	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ0
200	199	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℕ0)
201		1red 11309	. . . . . . . . . . . . . 14 ⊢ (⊤ → 1 ∈ ℝ)
202	201, 136	ltned 11446	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ≠ 3)
203	58, 82, 2, 52, 190, 69, 50, 65, 87, 200, 74, 202	coe1tmfv2 22594	. . . . . . . . . . . 12 ⊢ (⊤ → ((coe1‘( -3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))‘3) = 0)
204	198, 203	eqtrd 2796	. . . . . . . . . . 11 ⊢ (⊤ → ((coe1‘((𝐾‘ -3) · 𝑋))‘3) = 0)
205	2, 51, 82, 58	coe1scl 22606	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ Ring ∧ 1 ∈ ℚ) → (coe1‘(𝐾‘1)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, 1, 0)))
206	65, 92, 205	syl2anc 596	. . . . . . . . . . . 12 ⊢ (⊤ → (coe1‘(𝐾‘1)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, 1, 0)))
207		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑖 = 3) → 𝑖 = 3)
208	28	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑖 = 3) → 3 ≠ 0)
209	207, 208	eqnetrd 3023	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑖 = 3) → 𝑖 ≠ 0)
210	209	neneqd 2961	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑖 = 3) → ¬ 𝑖 = 0)
211	210	iffalsed 4493	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑖 = 3) → if(𝑖 = 0, 1, 0) = 0)
212		0zd 12705	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ∈ ℤ)
213	206, 211, 74, 212	fvmptd 7001	. . . . . . . . . . 11 ⊢ (⊤ → ((coe1‘(𝐾‘1))‘3) = 0)
214	204, 213	oveq12d 7438	. . . . . . . . . 10 ⊢ (⊤ → (((coe1‘((𝐾‘ -3) · 𝑋))‘3) + ((coe1‘(𝐾‘1))‘3)) = (0 + 0))
215		00id 11485	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
216	215	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (0 + 0) = 0)
217	187, 214, 216	3eqtrd 2800	. . . . . . . . 9 ⊢ (⊤ → ((coe1‘(((𝐾‘ -3) · 𝑋) + (𝐾‘1)))‘3) = 0)
218	185, 217	oveq12d 7438	. . . . . . . 8 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘ -3) · 𝑋) + (𝐾‘1)))‘3)) = (1 + 0))
219	159	addridd 11510	. . . . . . . 8 ⊢ (⊤ → (1 + 0) = 1)
220	218, 219	eqtrd 2796	. . . . . . 7 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘ -3) · 𝑋) + (𝐾‘1)))‘3)) = 1)
221	176, 181, 220	3eqtrd 2800	. . . . . 6 ⊢ (⊤ → ((coe1‘𝐹)‘3) = 1)
222	174, 221	eqtrd 2796	. . . . 5 ⊢ (⊤ → ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1)
223	3	fveq2i 6888	. . . . . . 7 ⊢ (Monic1p‘𝑄) = (Monic1p‘(ℂfld ↾s ℚ))
224	223	eqcomi 2770	. . . . . 6 ⊢ (Monic1p‘(ℂfld ↾s ℚ)) = (Monic1p‘𝑄)
225	2, 60, 47, 53, 224, 183	ismon1p 26461	. . . . 5 ⊢ (𝐹 ∈ (Monic1p‘(ℂfld ↾s ℚ)) ↔ (𝐹 ∈ (Base‘𝑃) ∧ 𝐹 ≠ (0g‘𝑃) ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1))
226	96, 173, 222, 225	syl3anbrc 1362	. . . 4 ⊢ (⊤ → 𝐹 ∈ (Monic1p‘(ℂfld ↾s ℚ)))
227	1, 5, 6, 8, 15, 44, 45, 46, 47, 57, 170, 226	irredminply 34348	. . 3 ⊢ (⊤ → 𝐹 = (𝑀‘𝐴))
228	227, 169	jca 521	. 2 ⊢ (⊤ → (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3))
229	228	mptru 1577	1 ⊢ (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  {crab 3413  ∅c0 4279  ifcif 4482  {csn 4584   class class class wbr 5103   ↦ cmpt 5186  ^◡ccnv 5650   ↾ cres 5653   “ cima 5654   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  ℂcc 11198  0cc0 11200  1c1 11201  ici 11202   + caddc 11203   · cmul 11205   < clt 11343   -cneg 11542   / cdiv 11973  2c2 12397  3c3 12398  ℕ₀cn0 12606  ℤcz 12693  ℚcq 13075  ↑cexp 14204  expce 16227  πcpi 16232  Basecbs 17387   ↾_s cress 17408  +_gcplusg 17428  ._rcmulr 17429  Scalarcsca 17431   ·_𝑠 cvsca 17432  0_gc0g 17610  Mndcmnd 18923  ._gcmg 19277  mulGrpcmgp 20360  Ringcrg 20459  CRingccrg 20460  Irredcir 20586  NzRingcnzr 20762  SubRingcsubrg 20821  Domncdomn 20944  DivRingcdr 20980  Fieldcfield 20981  SubDRingcsdrg 21043  LModclmod 21135  ℂ_fldccnfld 21678  AssAlgcasa 22158  algSccascl 22160  var₁cv1 22494  Poly₁cpl1 22495  coe₁cco1 22496   evalSub₁ ces1 22631  eval₁ce1 22632  deg₁cdg1 26372  Monic_1pcmn1 26444  ↑_𝑐ccxp 26883   minPoly cminply 34331

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-inf2 9642  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278  ax-addf 11279  ax-mulf 11280

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-ofr 7694  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8178  df-tpos 8243  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-map 8849  df-pm 8850  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9354  df-fi 9403  df-sup 9434  df-inf 9435  df-oi 9504  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-q 13076  df-rp 13121  df-xneg 13241  df-xadd 13242  df-xmul 13243  df-ioo 13480  df-ioc 13481  df-ico 13482  df-icc 13483  df-fz 13640  df-fzo 13789  df-fl 13932  df-mod 14010  df-seq 14145  df-exp 14205  df-fac 14418  df-bc 14447  df-hash 14475  df-shft 15220  df-sgn 15240  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-limsup 15638  df-clim 15655  df-rlim 15656  df-sum 15854  df-ef 16233  df-sin 16235  df-cos 16236  df-pi 16238  df-dvds 16423  df-gcd 16665  df-prm 16847  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-starv 17443  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-unif 17451  df-hom 17452  df-cco 17453  df-rest 17593  df-topn 17594  df-0g 17612  df-gsum 17613  df-topgen 17614  df-pt 17615  df-prds 17618  df-pws 17620  df-xrs 17674  df-qtop 17679  df-imas 17680  df-xps 17682  df-mre 17756  df-mrc 17757  df-acs 17759  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-submnd 18979  df-grp 19147  df-minusg 19148  df-sbg 19149  df-mulg 19278  df-subg 19333  df-ghm 19428  df-cntz 19531  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-srg 20413  df-ring 20461  df-cring 20462  df-oppr 20567  df-dvdsr 20587  df-unit 20588  df-irred 20589  df-invr 20618  df-dvr 20631  df-rhm 20702  df-nzr 20763  df-subrng 20798  df-subrg 20822  df-rlreg 20946  df-domn 20947  df-idom 20948  df-drng 20982  df-field 20983  df-sdrg 21044  df-lmod 21137  df-lss 21207  df-lsp 21247  df-sra 21448  df-rgmod 21449  df-lidl 21486  df-rsp 21487  df-psmet 21670  df-xmet 21671  df-met 21672  df-bl 21673  df-mopn 21674  df-fbas 21675  df-fg 21676  df-cnfld 21679  df-assa 22161  df-asp 22162  df-ascl 22163  df-psr 22217  df-mvr 22218  df-mpl 22219  df-opsr 22221  df-evls 22383  df-evl 22384  df-psr1 22498  df-vr1 22499  df-ply1 22500  df-coe1 22501  df-evls1 22633  df-evl1 22634  df-top 23212  df-topon 23229  df-topsp 23251  df-bases 23264  df-cld 23337  df-ntr 23338  df-cls 23339  df-nei 23416  df-lp 23454  df-perf 23455  df-cn 23545  df-cnp 23546  df-haus 23633  df-tx 23881  df-hmeo 24074  df-fil 24165  df-fm 24257  df-flim 24258  df-flf 24259  df-xms 24639  df-ms 24640  df-tms 24641  df-cncf 25199  df-limc 26186  df-dv 26187  df-mdeg 26373  df-deg1 26374  df-mon1 26449  df-uc1p 26450  df-q1p 26451  df-r1p 26452  df-ig1p 26453  df-log 26884  df-cxp 26885  df-irng 34316  df-minply 34332

This theorem is used by:  cos9thpinconstrlem2  34422