Theorem cos9thpiminply 33768

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 2735	. . . 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 6878	. . . . 5 ⊢ (Poly1‘𝑄) = (Poly1‘(ℂfld ↾s ℚ))
5	2, 4	eqtri 2758	. . . 4 ⊢ 𝑃 = (Poly1‘(ℂfld ↾s ℚ))
6		cnfldbas 21317	. . . 4 ⊢ ℂ = (Base‘ℂfld)
7		cnfldfld 33304	. . . . 5 ⊢ ℂfld ∈ Field
8	7	a1i 11	. . . 4 ⊢ (⊤ → ℂfld ∈ Field)
9		cndrng 21359	. . . . . 6 ⊢ ℂfld ∈ DivRing
10		qsubdrg 21385	. . . . . . 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 20746	. . . . . 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 11186	. . . . . . . . . . . . 13 ⊢ i ∈ ℂ
20	19	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → i ∈ ℂ)
21		2cnd 12316	. . . . . . . . . . . . 13 ⊢ (⊤ → 2 ∈ ℂ)
22		picn 26417	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
23	22	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → π ∈ ℂ)
24	21, 23	mulcld 11253	. . . . . . . . . . . 12 ⊢ (⊤ → (2 · π) ∈ ℂ)
25	20, 24	mulcld 11253	. . . . . . . . . . 11 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
26		3cn 12319	. . . . . . . . . . . 12 ⊢ 3 ∈ ℂ
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℂ)
28		3ne0 12344	. . . . . . . . . . . 12 ⊢ 3 ≠ 0
29	28	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ≠ 0)
30	25, 27, 29	divcld 12015	. . . . . . . . . 10 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
31	30	efcld 16097	. . . . . . . . 9 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
32	18, 31	eqeltrid 2838	. . . . . . . 8 ⊢ (⊤ → 𝑂 ∈ ℂ)
33	27, 29	reccld 12008	. . . . . . . 8 ⊢ (⊤ → (1 / 3) ∈ ℂ)
34	32, 33	cxpcld 26667	. . . . . . 7 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ∈ ℂ)
35	17, 34	eqeltrid 2838	. . . . . 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 16111	. . . . . . . . . 10 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ≠ 0)
39	37, 38	eqnetrd 2999	. . . . . . . . 9 ⊢ (⊤ → 𝑂 ≠ 0)
40	32, 39, 33	cxpne0d 26672	. . . . . . . 8 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ≠ 0)
41	36, 40	eqnetrd 2999	. . . . . . 7 ⊢ (⊤ → 𝑍 ≠ 0)
42	35, 41	reccld 12008	. . . . . 6 ⊢ (⊤ → (1 / 𝑍) ∈ ℂ)
43	35, 42	addcld 11252	. . . . 5 ⊢ (⊤ → (𝑍 + (1 / 𝑍)) ∈ ℂ)
44	16, 43	eqeltrid 2838	. . . 4 ⊢ (⊤ → 𝐴 ∈ ℂ)
45		cnfld0 21353	. . . 4 ⊢ 0 = (0g‘ℂfld)
46		cos9thpiminply.m	. . . 4 ⊢ 𝑀 = (ℂfld minPoly ℚ)
47		eqid 2735	. . . 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 33767	. . . . 5 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝐴) = ((𝐴↑3) + ((-3 · 𝐴) + 1)))
56	18, 17, 16	cos9thpiminplylem5 33766	. . . . 5 ⊢ ((𝐴↑3) + ((-3 · 𝐴) + 1)) = 0
57	55, 56	eqtrdi 2786	. . . 4 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝐴) = 0)
58	3	qrng0 27582	. . . . 5 ⊢ 0 = (0g‘𝑄)
59		eqid 2735	. . . . 5 ⊢ (eval1‘𝑄) = (eval1‘𝑄)
60		eqid 2735	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
61	3	qfld 33237	. . . . . 6 ⊢ 𝑄 ∈ Field
62	61	a1i 11	. . . . 5 ⊢ (⊤ → 𝑄 ∈ Field)
63	3	qdrng 27581	. . . . . . . . . . 11 ⊢ 𝑄 ∈ DivRing
64	63	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 𝑄 ∈ DivRing)
65	64	drngringd 20695	. . . . . . . . 9 ⊢ (⊤ → 𝑄 ∈ Ring)
66	2	ply1ring 22181	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → 𝑃 ∈ Ring)
67	65, 66	syl 17	. . . . . . . 8 ⊢ (⊤ → 𝑃 ∈ Ring)
68	67	ringgrpd 20200	. . . . . . 7 ⊢ (⊤ → 𝑃 ∈ Grp)
69		eqid 2735	. . . . . . . . 9 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
70	69, 60	mgpbas 20103	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
71	69	ringmgp 20197	. . . . . . . . 9 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
72	67, 71	syl 17	. . . . . . . 8 ⊢ (⊤ → (mulGrp‘𝑃) ∈ Mnd)
73		3nn0 12517	. . . . . . . . 9 ⊢ 3 ∈ ℕ0
74	73	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℕ0)
75	52, 2, 60	vr1cl 22151	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
76	65, 75	syl 17	. . . . . . . 8 ⊢ (⊤ → 𝑋 ∈ (Base‘𝑃))
77	70, 50, 72, 74, 76	mulgnn0cld 19076	. . . . . . 7 ⊢ (⊤ → (3 ↑ 𝑋) ∈ (Base‘𝑃))
78	2	ply1sca 22186	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ DivRing → 𝑄 = (Scalar‘𝑃))
79	63, 78	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝑄 = (Scalar‘𝑃)
80	2	ply1lmod 22185	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ Ring → 𝑃 ∈ LMod)
81	65, 80	syl 17	. . . . . . . . . . 11 ⊢ (⊤ → 𝑃 ∈ LMod)
82	3	qrngbas 27580	. . . . . . . . . . 11 ⊢ ℚ = (Base‘𝑄)
83	51, 79, 67, 81, 82, 60	asclf 21840	. . . . . . . . . 10 ⊢ (⊤ → 𝐾:ℚ⟶(Base‘𝑃))
84	74	nn0zd 12612	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℤ)
85		zq 12968	. . . . . . . . . . 11 ⊢ (3 ∈ ℤ → 3 ∈ ℚ)
86		qnegcl 12980	. . . . . . . . . . 11 ⊢ (3 ∈ ℚ → -3 ∈ ℚ)
87	84, 85, 86	3syl 18	. . . . . . . . . 10 ⊢ (⊤ → -3 ∈ ℚ)
88	83, 87	ffvelcdmd 7074	. . . . . . . . 9 ⊢ (⊤ → (𝐾‘-3) ∈ (Base‘𝑃))
89	60, 49, 67, 88, 76	ringcld 20218	. . . . . . . 8 ⊢ (⊤ → ((𝐾‘-3) · 𝑋) ∈ (Base‘𝑃))
90		1zzd 12621	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℤ)
91		zq 12968	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
92	90, 91	syl 17	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℚ)
93	83, 92	ffvelcdmd 7074	. . . . . . . 8 ⊢ (⊤ → (𝐾‘1) ∈ (Base‘𝑃))
94	60, 48, 68, 89, 93	grpcld 18928	. . . . . . 7 ⊢ (⊤ → (((𝐾‘-3) · 𝑋) + (𝐾‘1)) ∈ (Base‘𝑃))
95	60, 48, 68, 77, 94	grpcld 18928	. . . . . 6 ⊢ (⊤ → ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))) ∈ (Base‘𝑃))
96	54, 95	eqeltrid 2838	. . . . 5 ⊢ (⊤ → 𝐹 ∈ (Base‘𝑃))
97	62	fldcrngd 20700	. . . . . . . . 9 ⊢ (⊤ → 𝑄 ∈ CRing)
98	59, 2, 60, 97, 82, 96	evl1fvf 33522	. . . . . . . 8 ⊢ (⊤ → ((eval1‘𝑄)‘𝐹):ℚ⟶ℚ)
99	98	ffnd 6706	. . . . . . 7 ⊢ (⊤ → ((eval1‘𝑄)‘𝐹) Fn ℚ)
100		fniniseg2 7051	. . . . . . 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 22267	. . . . . . . . . . . . . . 15 ⊢ (eval1‘𝑄) = (𝑄 evalSub1 ℚ)
103	102	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → (eval1‘𝑄) = (𝑄 evalSub1 ℚ))
104	103	fveq1d 6877	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → ((eval1‘𝑄)‘𝐹) = ((𝑄 evalSub1 ℚ)‘𝐹))
105	104	fveq1d 6877	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) = (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥))
106		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (𝑄 evalSub1 ℚ) = (𝑄 evalSub1 ℚ)
107		cncrng 21349	. . . . . . . . . . . . . . . 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 20204	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℚ → 𝑄 ∈ Ring)
113	82	subrgid 20531	. . . . . . . . . . . . . . . 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 33524	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → ((𝑄 evalSub1 ℚ)‘𝐹) = (((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ))
118	117	fveq1d 6877	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥) = ((((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ)‘𝑥))
119		fvres 6894	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → ((((ℂfld evalSub1 ℚ)‘𝐹) ↾ ℚ)‘𝑥) = (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥))
120	118, 119	eqtr2d 2771	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥) = (((𝑄 evalSub1 ℚ)‘𝐹)‘𝑥))
121		qcn 12977	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℂ)
122	18, 17, 16, 3, 48, 49, 50, 2, 51, 52, 53, 54, 121	cos9thpiminplylem6 33767	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑥) = ((𝑥↑3) + ((-3 · 𝑥) + 1)))
123	105, 120, 122	3eqtr2d 2776	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) = ((𝑥↑3) + ((-3 · 𝑥) + 1)))
124		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℚ)
125	124	cos9thpiminplylem2 33763	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℚ → ((𝑥↑3) + ((-3 · 𝑥) + 1)) ≠ 0)
126	123, 125	eqnetrd 2999	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) ≠ 0)
127	126	neneqd 2937	. . . . . . . . 9 ⊢ (𝑥 ∈ ℚ → ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
128	127	rgen 3053	. . . . . . . 8 ⊢ ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0
129	128	a1i 11	. . . . . . 7 ⊢ (⊤ → ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
130		rabeq0 4363	. . . . . . 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 2770	. . . . 5 ⊢ (⊤ → (◡((eval1‘𝑄)‘𝐹) “ {0}) = ∅)
133	54	a1i 11	. . . . . . 7 ⊢ (⊤ → 𝐹 = ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))))
134	133	fveq2d 6879	. . . . . 6 ⊢ (⊤ → (𝐷‘𝐹) = (𝐷‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))))
135		1lt3 12411	. . . . . . . . 9 ⊢ 1 < 3
136	135	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1 < 3)
137		0lt1 11757	. . . . . . . . . . . 12 ⊢ 0 < 1
138	137	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 0 < 1)
139	138	gt0ne0d 11799	. . . . . . . . . . . 12 ⊢ (⊤ → 1 ≠ 0)
140	53, 2, 82, 51, 58	deg1scl 26068	. . . . . . . . . . . 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 20707	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ Domn)
143	63, 142	mp1i 13	. . . . . . . . . . . . 13 ⊢ (⊤ → 𝑄 ∈ Domn)
144	27, 29	negne0d 11590	. . . . . . . . . . . . . 14 ⊢ (⊤ → -3 ≠ 0)
145	2, 51, 58, 47, 82	ply1scln0 22227	. . . . . . . . . . . . . 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 20706	. . . . . . . . . . . . . . 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 33549	. . . . . . . . . . . . 13 ⊢ (⊤ → 𝑋 ≠ (0g‘𝑃))
152	53, 2, 60, 49, 47, 143, 88, 146, 76, 151	deg1mul 26070	. . . . . . . . . . . 12 ⊢ (⊤ → (𝐷‘((𝐾‘-3) · 𝑋)) = ((𝐷‘(𝐾‘-3)) + (𝐷‘𝑋)))
153	53, 2, 82, 51, 58	deg1scl 26068	. . . . . . . . . . . . . 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 20706	. . . . . . . . . . . . . . 15 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ NzRing)
156	63, 155	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (⊤ → 𝑄 ∈ NzRing)
157	53, 2, 52, 156	deg1vr 33548	. . . . . . . . . . . . 13 ⊢ (⊤ → (𝐷‘𝑋) = 1)
158	154, 157	oveq12d 7421	. . . . . . . . . . . 12 ⊢ (⊤ → ((𝐷‘(𝐾‘-3)) + (𝐷‘𝑋)) = (0 + 1))
159		1cnd 11228	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℂ)
160	159	addlidd 11434	. . . . . . . . . . . 12 ⊢ (⊤ → (0 + 1) = 1)
161	152, 158, 160	3eqtrd 2774	. . . . . . . . . . 11 ⊢ (⊤ → (𝐷‘((𝐾‘-3) · 𝑋)) = 1)
162	138, 141, 161	3brtr4d 5151	. . . . . . . . . 10 ⊢ (⊤ → (𝐷‘(𝐾‘1)) < (𝐷‘((𝐾‘-3) · 𝑋)))
163	2, 53, 65, 60, 48, 89, 93, 162	deg1add 26058	. . . . . . . . 9 ⊢ (⊤ → (𝐷‘(((𝐾‘-3) · 𝑋) + (𝐾‘1))) = (𝐷‘((𝐾‘-3) · 𝑋)))
164	163, 161	eqtrd 2770	. . . . . . . 8 ⊢ (⊤ → (𝐷‘(((𝐾‘-3) · 𝑋) + (𝐾‘1))) = 1)
165	53, 2, 52, 69, 50	deg1pw 26076	. . . . . . . . 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 5151	. . . . . . 7 ⊢ (⊤ → (𝐷‘(((𝐾‘-3) · 𝑋) + (𝐾‘1))) < (𝐷‘(3 ↑ 𝑋)))
168	2, 53, 65, 60, 48, 77, 94, 167	deg1add 26058	. . . . . 6 ⊢ (⊤ → (𝐷‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))) = (𝐷‘(3 ↑ 𝑋)))
169	134, 168, 166	3eqtrd 2774	. . . . 5 ⊢ (⊤ → (𝐷‘𝐹) = 3)
170	58, 59, 53, 2, 60, 62, 96, 132, 169	ply1dg3rt0irred 33541	. . . 4 ⊢ (⊤ → 𝐹 ∈ (Irred‘𝑃))
171		eqid 2735	. . . . . . 7 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
172	171, 47	irredn0 20381	. . . . . 6 ⊢ ((𝑃 ∈ Ring ∧ 𝐹 ∈ (Irred‘𝑃)) → 𝐹 ≠ (0g‘𝑃))
173	67, 170, 172	syl2anc 584	. . . . 5 ⊢ (⊤ → 𝐹 ≠ (0g‘𝑃))
174	169	fveq2d 6879	. . . . . 6 ⊢ (⊤ → ((coe1‘𝐹)‘(𝐷‘𝐹)) = ((coe1‘𝐹)‘3))
175	133	fveq2d 6879	. . . . . . . 8 ⊢ (⊤ → (coe1‘𝐹) = (coe1‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))))
176	175	fveq1d 6877	. . . . . . 7 ⊢ (⊤ → ((coe1‘𝐹)‘3) = ((coe1‘((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1))))‘3))
177		cnfldadd 21319	. . . . . . . . . . 11 ⊢ + = (+g‘ℂfld)
178	3, 177	ressplusg 17303	. . . . . . . . . 10 ⊢ (ℚ ∈ (SubRing‘ℂfld) → + = (+g‘𝑄))
179	11, 178	ax-mp 5	. . . . . . . . 9 ⊢ + = (+g‘𝑄)
180	2, 60, 48, 179	coe1addfv 22200	. . . . . . . 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 4506	. . . . . . . . . 10 ⊢ (𝑖 = 3 → if(𝑖 = 3, 1, 0) = 1)
183	3	qrng1 27583	. . . . . . . . . . 11 ⊢ 1 = (1r‘𝑄)
184	2, 52, 50, 65, 74, 58, 183	coe1mon 33544	. . . . . . . . . 10 ⊢ (⊤ → (coe1‘(3 ↑ 𝑋)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 3, 1, 0)))
185	182, 184, 74, 159	fvmptd4 7009	. . . . . . . . 9 ⊢ (⊤ → ((coe1‘(3 ↑ 𝑋))‘3) = 1)
186	2, 60, 48, 179	coe1addfv 22200	. . . . . . . . . . 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 22133	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 ∈ CRing → 𝑃 ∈ AssAlg)
189	97, 188	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → 𝑃 ∈ AssAlg)
190		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
191	51, 79, 82, 60, 49, 190	asclmul1 21844	. . . . . . . . . . . . . . . 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 19062	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ (Base‘𝑃) → (1 ↑ 𝑋) = 𝑋)
194	76, 193	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (1 ↑ 𝑋) = 𝑋)
195	194	oveq2d 7419	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)) = (-3( ·𝑠 ‘𝑃)𝑋))
196	192, 195	eqtr4d 2773	. . . . . . . . . . . . . 14 ⊢ (⊤ → ((𝐾‘-3) · 𝑋) = (-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))
197	196	fveq2d 6879	. . . . . . . . . . . . 13 ⊢ (⊤ → (coe1‘((𝐾‘-3) · 𝑋)) = (coe1‘(-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋))))
198	197	fveq1d 6877	. . . . . . . . . . . 12 ⊢ (⊤ → ((coe1‘((𝐾‘-3) · 𝑋))‘3) = ((coe1‘(-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))‘3))
199		1nn0 12515	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ0
200	199	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℕ0)
201		1red 11234	. . . . . . . . . . . . . 14 ⊢ (⊤ → 1 ∈ ℝ)
202	201, 136	ltned 11369	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ≠ 3)
203	58, 82, 2, 52, 190, 69, 50, 65, 87, 200, 74, 202	coe1tmfv2 22210	. . . . . . . . . . . 12 ⊢ (⊤ → ((coe1‘(-3( ·𝑠 ‘𝑃)(1 ↑ 𝑋)))‘3) = 0)
204	198, 203	eqtrd 2770	. . . . . . . . . . 11 ⊢ (⊤ → ((coe1‘((𝐾‘-3) · 𝑋))‘3) = 0)
205	2, 51, 82, 58	coe1scl 22222	. . . . . . . . . . . . 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 2999	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑖 = 3) → 𝑖 ≠ 0)
210	209	neneqd 2937	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑖 = 3) → ¬ 𝑖 = 0)
211	210	iffalsed 4511	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑖 = 3) → if(𝑖 = 0, 1, 0) = 0)
212		0zd 12598	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ∈ ℤ)
213	206, 211, 74, 212	fvmptd 6992	. . . . . . . . . . 11 ⊢ (⊤ → ((coe1‘(𝐾‘1))‘3) = 0)
214	204, 213	oveq12d 7421	. . . . . . . . . 10 ⊢ (⊤ → (((coe1‘((𝐾‘-3) · 𝑋))‘3) + ((coe1‘(𝐾‘1))‘3)) = (0 + 0))
215		00id 11408	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
216	215	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (0 + 0) = 0)
217	187, 214, 216	3eqtrd 2774	. . . . . . . . 9 ⊢ (⊤ → ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3) = 0)
218	185, 217	oveq12d 7421	. . . . . . . 8 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3)) = (1 + 0))
219	159	addridd 11433	. . . . . . . 8 ⊢ (⊤ → (1 + 0) = 1)
220	218, 219	eqtrd 2770	. . . . . . 7 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘3) + ((coe1‘(((𝐾‘-3) · 𝑋) + (𝐾‘1)))‘3)) = 1)
221	176, 181, 220	3eqtrd 2774	. . . . . 6 ⊢ (⊤ → ((coe1‘𝐹)‘3) = 1)
222	174, 221	eqtrd 2770	. . . . 5 ⊢ (⊤ → ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1)
223	3	fveq2i 6878	. . . . . . 7 ⊢ (Monic1p‘𝑄) = (Monic1p‘(ℂfld ↾s ℚ))
224	223	eqcomi 2744	. . . . . 6 ⊢ (Monic1p‘(ℂfld ↾s ℚ)) = (Monic1p‘𝑄)
225	2, 60, 47, 53, 224, 183	ismon1p 26098	. . . . 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 33696	. . 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 2108   ≠ wne 2932  ∀wral 3051  {crab 3415  ∅c0 4308  ifcif 4500  {csn 4601   class class class wbr 5119   ↦ cmpt 5201  ^◡ccnv 5653   ↾ cres 5656   “ cima 5657   Fn wfn 6525  ‘cfv 6530  (class class class)co 7403  ℂcc 11125  0cc0 11127  1c1 11128  ici 11129   + caddc 11130   · cmul 11132   < clt 11267  -cneg 11465   / cdiv 11892  2c2 12293  3c3 12294  ℕ₀cn0 12499  ℤcz 12586  ℚcq 12962  ↑cexp 14077  expce 16075  πcpi 16080  Basecbs 17226   ↾_s cress 17249  +_gcplusg 17269  ._rcmulr 17270  Scalarcsca 17272   ·_𝑠 cvsca 17273  0_gc0g 17451  Mndcmnd 18710  ._gcmg 19048  mulGrpcmgp 20098  Ringcrg 20191  CRingccrg 20192  Irredcir 20314  NzRingcnzr 20470  SubRingcsubrg 20527  Domncdomn 20650  DivRingcdr 20687  Fieldcfield 20688  SubDRingcsdrg 20744  LModclmod 20815  ℂ_fldccnfld 21313  AssAlgcasa 21808  algSccascl 21810  var₁cv1 22109  Poly₁cpl1 22110  coe₁cco1 22111   evalSub₁ ces1 22249  eval₁ce1 22250  deg₁cdg1 26009  Monic_1pcmn1 26081  ↑_𝑐ccxp 26514   minPoly cminply 33679

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-inf2 9653  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205  ax-addf 11206  ax-mulf 11207

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-isom 6539  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-of 7669  df-ofr 7670  df-om 7860  df-1st 7986  df-2nd 7987  df-supp 8158  df-tpos 8223  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-2o 8479  df-er 8717  df-map 8840  df-pm 8841  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9372  df-fi 9421  df-sup 9452  df-inf 9453  df-oi 9522  df-card 9951  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-div 11893  df-nn 12239  df-2 12301  df-3 12302  df-4 12303  df-5 12304  df-6 12305  df-7 12306  df-8 12307  df-9 12308  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12851  df-q 12963  df-rp 13007  df-xneg 13126  df-xadd 13127  df-xmul 13128  df-ioo 13364  df-ioc 13365  df-ico 13366  df-icc 13367  df-fz 13523  df-fzo 13670  df-fl 13807  df-mod 13885  df-seq 14018  df-exp 14078  df-fac 14290  df-bc 14319  df-hash 14347  df-shft 15084  df-sgn 15104  df-cj 15116  df-re 15117  df-im 15118  df-sqrt 15252  df-abs 15253  df-limsup 15485  df-clim 15502  df-rlim 15503  df-sum 15701  df-ef 16081  df-sin 16083  df-cos 16084  df-pi 16086  df-dvds 16271  df-gcd 16512  df-prm 16689  df-struct 17164  df-sets 17181  df-slot 17199  df-ndx 17211  df-base 17227  df-ress 17250  df-plusg 17282  df-mulr 17283  df-starv 17284  df-sca 17285  df-vsca 17286  df-ip 17287  df-tset 17288  df-ple 17289  df-ds 17291  df-unif 17292  df-hom 17293  df-cco 17294  df-rest 17434  df-topn 17435  df-0g 17453  df-gsum 17454  df-topgen 17455  df-pt 17456  df-prds 17459  df-pws 17461  df-xrs 17514  df-qtop 17519  df-imas 17520  df-xps 17522  df-mre 17596  df-mrc 17597  df-acs 17599  df-mgm 18616  df-sgrp 18695  df-mnd 18711  df-mhm 18759  df-submnd 18760  df-grp 18917  df-minusg 18918  df-sbg 18919  df-mulg 19049  df-subg 19104  df-ghm 19194  df-cntz 19298  df-cmn 19761  df-abl 19762  df-mgp 20099  df-rng 20111  df-ur 20140  df-srg 20145  df-ring 20193  df-cring 20194  df-oppr 20295  df-dvdsr 20315  df-unit 20316  df-irred 20317  df-invr 20346  df-dvr 20359  df-rhm 20430  df-nzr 20471  df-subrng 20504  df-subrg 20528  df-rlreg 20652  df-domn 20653  df-idom 20654  df-drng 20689  df-field 20690  df-sdrg 20745  df-lmod 20817  df-lss 20887  df-lsp 20927  df-sra 21129  df-rgmod 21130  df-lidl 21167  df-rsp 21168  df-psmet 21305  df-xmet 21306  df-met 21307  df-bl 21308  df-mopn 21309  df-fbas 21310  df-fg 21311  df-cnfld 21314  df-assa 21811  df-asp 21812  df-ascl 21813  df-psr 21867  df-mvr 21868  df-mpl 21869  df-opsr 21871  df-evls 22030  df-evl 22031  df-psr1 22113  df-vr1 22114  df-ply1 22115  df-coe1 22116  df-evls1 22251  df-evl1 22252  df-top 22830  df-topon 22847  df-topsp 22869  df-bases 22882  df-cld 22955  df-ntr 22956  df-cls 22957  df-nei 23034  df-lp 23072  df-perf 23073  df-cn 23163  df-cnp 23164  df-haus 23251  df-tx 23498  df-hmeo 23691  df-fil 23782  df-fm 23874  df-flim 23875  df-flf 23876  df-xms 24257  df-ms 24258  df-tms 24259  df-cncf 24820  df-limc 25817  df-dv 25818  df-mdeg 26010  df-deg1 26011  df-mon1 26086  df-uc1p 26087  df-q1p 26088  df-r1p 26089  df-ig1p 26090  df-log 26515  df-cxp 26516  df-irng 33671  df-minply 33680

This theorem is referenced by: (None)