Theorem 2sqr3minply 34323

Description: The polynomial ((𝑋↑3) − 2) is the minimal polynomial for (2↑_𝑐(1 / 3)) over ℚ, and its degree is 3. (Contributed by Thierry Arnoux, 14-Jun-2025.)

Hypotheses

Ref	Expression
2sqr3minply.q	⊢ 𝑄 = (ℂfld ↾s ℚ)
2sqr3minply.1	⊢ − = (-g‘𝑃)
2sqr3minply.2	⊢ ↑ = (.g‘(mulGrp‘𝑃))
2sqr3minply.p	⊢ 𝑃 = (Poly1‘𝑄)
2sqr3minply.k	⊢ 𝐾 = (algSc‘𝑃)
2sqr3minply.x	⊢ 𝑋 = (var1‘𝑄)
2sqr3minply.d	⊢ 𝐷 = (deg1‘𝑄)
2sqr3minply.f	⊢ 𝐹 = ((3 ↑ 𝑋) − (𝐾‘2))
2sqr3minply.a	⊢ 𝐴 = (2↑𝑐(1 / 3))
2sqr3minply.m	⊢ 𝑀 = (ℂfld minPoly ℚ)

Assertion

Ref	Expression
2sqr3minply	⊢ (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3)

Proof of Theorem 2sqr3minply

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

Step	Hyp	Ref	Expression
1		eqid 2760	. . . 4 ⊢ (ℂfld evalSub1 ℚ) = (ℂfld evalSub1 ℚ)
2		2sqr3minply.p	. . . . 5 ⊢ 𝑃 = (Poly1‘𝑄)
3		2sqr3minply.q	. . . . . 6 ⊢ 𝑄 = (ℂfld ↾s ℚ)
4	3	fveq2i 6884	. . . . 5 ⊢ (Poly1‘𝑄) = (Poly1‘(ℂfld ↾s ℚ))
5	2, 4	eqtri 2783	. . . 4 ⊢ 𝑃 = (Poly1‘(ℂfld ↾s ℚ))
6		cnfldbas 21621	. . . 4 ⊢ ℂ = (Base‘ℂfld)
7		cndrng 21646	. . . . . 6 ⊢ ℂfld ∈ DivRing
8		cncrng 21638	. . . . . 6 ⊢ ℂfld ∈ CRing
9		isfld 20932	. . . . . 6 ⊢ (ℂfld ∈ Field ↔ (ℂfld ∈ DivRing ∧ ℂfld ∈ CRing))
10	7, 8, 9	mpbir2an 724	. . . . 5 ⊢ ℂfld ∈ Field
11	10	a1i 11	. . . 4 ⊢ (⊤ → ℂfld ∈ Field)
12		qsubdrg 21664	. . . . . . 7 ⊢ (ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing)
13	12	simpli 489	. . . . . 6 ⊢ ℚ ∈ (SubRing‘ℂfld)
14	12	simpri 491	. . . . . 6 ⊢ (ℂfld ↾s ℚ) ∈ DivRing
15		issdrg 20984	. . . . . 6 ⊢ (ℚ ∈ (SubDRing‘ℂfld) ↔ (ℂfld ∈ DivRing ∧ ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing))
16	7, 13, 14, 15	mpbir3an 1360	. . . . 5 ⊢ ℚ ∈ (SubDRing‘ℂfld)
17	16	a1i 11	. . . 4 ⊢ (⊤ → ℚ ∈ (SubDRing‘ℂfld))
18		2sqr3minply.a	. . . . . 6 ⊢ 𝐴 = (2↑𝑐(1 / 3))
19		2cn 12365	. . . . . . 7 ⊢ 2 ∈ ℂ
20		3cn 12371	. . . . . . . 8 ⊢ 3 ∈ ℂ
21		3ne0 12399	. . . . . . . 8 ⊢ 3 ≠ 0
22	20, 21	reccli 11994	. . . . . . 7 ⊢ (1 / 3) ∈ ℂ
23		cxpcl 26943	. . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (1 / 3) ∈ ℂ) → (2↑𝑐(1 / 3)) ∈ ℂ)
24	19, 22, 23	mp2an 705	. . . . . 6 ⊢ (2↑𝑐(1 / 3)) ∈ ℂ
25	18, 24	eqeltri 2856	. . . . 5 ⊢ 𝐴 ∈ ℂ
26	25	a1i 11	. . . 4 ⊢ (⊤ → 𝐴 ∈ ℂ)
27		cnfld0 21641	. . . 4 ⊢ 0 = (0g‘ℂfld)
28		2sqr3minply.m	. . . 4 ⊢ 𝑀 = (ℂfld minPoly ℚ)
29		eqid 2760	. . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃)
30		2sqr3minply.f	. . . . . . . 8 ⊢ 𝐹 = ((3 ↑ 𝑋) − (𝐾‘2))
31	30	fveq2i 6884	. . . . . . 7 ⊢ ((ℂfld evalSub1 ℚ)‘𝐹) = ((ℂfld evalSub1 ℚ)‘((3 ↑ 𝑋) − (𝐾‘2)))
32	31	fveq1i 6882	. . . . . 6 ⊢ (((ℂfld evalSub1 ℚ)‘𝐹)‘𝐴) = (((ℂfld evalSub1 ℚ)‘((3 ↑ 𝑋) − (𝐾‘2)))‘𝐴)
33	32	a1i 11	. . . . 5 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝐴) = (((ℂfld evalSub1 ℚ)‘((3 ↑ 𝑋) − (𝐾‘2)))‘𝐴))
34		eqid 2760	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
35		2sqr3minply.1	. . . . . 6 ⊢ − = (-g‘𝑃)
36		cnfldsub 21645	. . . . . 6 ⊢ − = (-g‘ℂfld)
37	8	a1i 11	. . . . . 6 ⊢ (⊤ → ℂfld ∈ CRing)
38	13	a1i 11	. . . . . 6 ⊢ (⊤ → ℚ ∈ (SubRing‘ℂfld))
39		eqid 2760	. . . . . . . 8 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
40	39, 34	mgpbas 20304	. . . . . . 7 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
41		2sqr3minply.2	. . . . . . 7 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
42	3	qdrng 27888	. . . . . . . . . . 11 ⊢ 𝑄 ∈ DivRing
43	42	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 𝑄 ∈ DivRing)
44	43	drngringd 20927	. . . . . . . . 9 ⊢ (⊤ → 𝑄 ∈ Ring)
45	2	ply1ring 22504	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → 𝑃 ∈ Ring)
46	44, 45	syl 18	. . . . . . . 8 ⊢ (⊤ → 𝑃 ∈ Ring)
47	39	ringmgp 20404	. . . . . . . 8 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
48	46, 47	syl 18	. . . . . . 7 ⊢ (⊤ → (mulGrp‘𝑃) ∈ Mnd)
49		3nn0 12571	. . . . . . . 8 ⊢ 3 ∈ ℕ0
50	49	a1i 11	. . . . . . 7 ⊢ (⊤ → 3 ∈ ℕ0)
51		2sqr3minply.x	. . . . . . . . 9 ⊢ 𝑋 = (var1‘𝑄)
52	51, 2, 34	vr1cl 22474	. . . . . . . 8 ⊢ (𝑄 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
53	44, 52	syl 18	. . . . . . 7 ⊢ (⊤ → 𝑋 ∈ (Base‘𝑃))
54	40, 41, 48, 50, 53	mulgnn0cld 19244	. . . . . 6 ⊢ (⊤ → (3 ↑ 𝑋) ∈ (Base‘𝑃))
55		2sqr3minply.k	. . . . . . . 8 ⊢ 𝐾 = (algSc‘𝑃)
56	44	mptru 1577	. . . . . . . . 9 ⊢ 𝑄 ∈ Ring
57	2	ply1sca 22509	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → 𝑄 = (Scalar‘𝑃))
58	56, 57	ax-mp 5	. . . . . . . 8 ⊢ 𝑄 = (Scalar‘𝑃)
59	2	ply1lmod 22508	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → 𝑃 ∈ LMod)
60	44, 59	syl 18	. . . . . . . 8 ⊢ (⊤ → 𝑃 ∈ LMod)
61	3	qrngbas 27887	. . . . . . . 8 ⊢ ℚ = (Base‘𝑄)
62	55, 58, 46, 60, 61, 34	asclf 22128	. . . . . . 7 ⊢ (⊤ → 𝐾:ℚ⟶(Base‘𝑃))
63		2z 12675	. . . . . . . 8 ⊢ 2 ∈ ℤ
64		zq 13028	. . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∈ ℚ)
65	63, 64	mp1i 14	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℚ)
66	62, 65	ffvelcdmd 7081	. . . . . 6 ⊢ (⊤ → (𝐾‘2) ∈ (Base‘𝑃))
67	1, 6, 2, 3, 34, 35, 36, 37, 38, 54, 66, 26	evls1subd 34015	. . . . 5 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘((3 ↑ 𝑋) − (𝐾‘2)))‘𝐴) = ((((ℂfld evalSub1 ℚ)‘(3 ↑ 𝑋))‘𝐴) − (((ℂfld evalSub1 ℚ)‘(𝐾‘2))‘𝐴)))
68		eqid 2760	. . . . . . . . . 10 ⊢ (.g‘(mulGrp‘ℂfld)) = (.g‘(mulGrp‘ℂfld))
69	1, 6, 2, 3, 34, 37, 38, 41, 68, 50, 53, 26	evls1expd 22624	. . . . . . . . 9 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘(3 ↑ 𝑋))‘𝐴) = (3(.g‘(mulGrp‘ℂfld))(((ℂfld evalSub1 ℚ)‘𝑋)‘𝐴)))
70	1, 51, 3, 6, 37, 38	evls1var 22595	. . . . . . . . . . . 12 ⊢ (⊤ → ((ℂfld evalSub1 ℚ)‘𝑋) = ( I ↾ ℂ))
71	70	fveq1d 6883	. . . . . . . . . . 11 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘𝑋)‘𝐴) = (( I ↾ ℂ)‘𝐴))
72		fvresi 7174	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (( I ↾ ℂ)‘𝐴) = 𝐴)
73	25, 72	mp1i 14	. . . . . . . . . . 11 ⊢ (⊤ → (( I ↾ ℂ)‘𝐴) = 𝐴)
74	71, 73	eqtrd 2795	. . . . . . . . . 10 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘𝑋)‘𝐴) = 𝐴)
75	74	oveq2d 7432	. . . . . . . . 9 ⊢ (⊤ → (3(.g‘(mulGrp‘ℂfld))(((ℂfld evalSub1 ℚ)‘𝑋)‘𝐴)) = (3(.g‘(mulGrp‘ℂfld))𝐴))
76		cnfldexp 21650	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 3 ∈ ℕ0) → (3(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑3))
77	26, 50, 76	syl2anc 596	. . . . . . . . 9 ⊢ (⊤ → (3(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑3))
78	69, 75, 77	3eqtrd 2799	. . . . . . . 8 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘(3 ↑ 𝑋))‘𝐴) = (𝐴↑3))
79	18	oveq1i 7426	. . . . . . . . 9 ⊢ (𝐴↑3) = ((2↑𝑐(1 / 3))↑3)
80		3nn 12369	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
81		cxproot 26959	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ 3 ∈ ℕ) → ((2↑𝑐(1 / 3))↑3) = 2)
82	19, 80, 81	mp2an 705	. . . . . . . . 9 ⊢ ((2↑𝑐(1 / 3))↑3) = 2
83	79, 82	eqtri 2783	. . . . . . . 8 ⊢ (𝐴↑3) = 2
84	78, 83	eqtrdi 2811	. . . . . . 7 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘(3 ↑ 𝑋))‘𝐴) = 2)
85	1, 2, 3, 6, 55, 37, 38, 65, 26	evls1scafv 22623	. . . . . . 7 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘(𝐾‘2))‘𝐴) = 2)
86	84, 85	oveq12d 7434	. . . . . 6 ⊢ (⊤ → ((((ℂfld evalSub1 ℚ)‘(3 ↑ 𝑋))‘𝐴) − (((ℂfld evalSub1 ℚ)‘(𝐾‘2))‘𝐴)) = (2 − 2))
87	19	subidi 11578	. . . . . 6 ⊢ (2 − 2) = 0
88	86, 87	eqtrdi 2811	. . . . 5 ⊢ (⊤ → ((((ℂfld evalSub1 ℚ)‘(3 ↑ 𝑋))‘𝐴) − (((ℂfld evalSub1 ℚ)‘(𝐾‘2))‘𝐴)) = 0)
89	33, 67, 88	3eqtrd 2799	. . . 4 ⊢ (⊤ → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝐴) = 0)
90	3	qrng0 27889	. . . . 5 ⊢ 0 = (0g‘𝑄)
91		eqid 2760	. . . . 5 ⊢ (eval1‘𝑄) = (eval1‘𝑄)
92		2sqr3minply.d	. . . . 5 ⊢ 𝐷 = (deg1‘𝑄)
93		fldsdrgfld 20994	. . . . . . . 8 ⊢ ((ℂfld ∈ Field ∧ ℚ ∈ (SubDRing‘ℂfld)) → (ℂfld ↾s ℚ) ∈ Field)
94	10, 16, 93	mp2an 705	. . . . . . 7 ⊢ (ℂfld ↾s ℚ) ∈ Field
95	3, 94	eqeltri 2856	. . . . . 6 ⊢ 𝑄 ∈ Field
96	95	a1i 11	. . . . 5 ⊢ (⊤ → 𝑄 ∈ Field)
97	46	ringgrpd 20408	. . . . . . 7 ⊢ (⊤ → 𝑃 ∈ Grp)
98	34, 35	grpsubcl 19169	. . . . . . 7 ⊢ ((𝑃 ∈ Grp ∧ (3 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝐾‘2) ∈ (Base‘𝑃)) → ((3 ↑ 𝑋) − (𝐾‘2)) ∈ (Base‘𝑃))
99	97, 54, 66, 98	syl3anc 1398	. . . . . 6 ⊢ (⊤ → ((3 ↑ 𝑋) − (𝐾‘2)) ∈ (Base‘𝑃))
100	30, 99	eqeltrid 2864	. . . . 5 ⊢ (⊤ → 𝐹 ∈ (Base‘𝑃))
101	96	fldcrngd 20934	. . . . . . . . 9 ⊢ (⊤ → 𝑄 ∈ CRing)
102	91, 2, 34, 101, 61, 100	evl1fvf 34006	. . . . . . . 8 ⊢ (⊤ → ((eval1‘𝑄)‘𝐹):ℚ⟶ℚ)
103	102	ffnd 6706	. . . . . . 7 ⊢ (⊤ → ((eval1‘𝑄)‘𝐹) Fn ℚ)
104		fniniseg2 7057	. . . . . . 7 ⊢ (((eval1‘𝑄)‘𝐹) Fn ℚ → (◡((eval1‘𝑄)‘𝐹) “ {0}) = {𝑥 ∈ ℚ ∣ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0})
105	103, 104	syl 18	. . . . . 6 ⊢ (⊤ → (◡((eval1‘𝑄)‘𝐹) “ {0}) = {𝑥 ∈ ℚ ∣ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0})
106		cnfldmul 21625	. . . . . . . . . . . . . . 15 ⊢ · = (.r‘ℂfld)
107	3, 106	ressmulr 17417	. . . . . . . . . . . . . 14 ⊢ (ℚ ∈ (SubRing‘ℂfld) → · = (.r‘𝑄))
108	13, 107	ax-mp 5	. . . . . . . . . . . . 13 ⊢ · = (.r‘𝑄)
109		cnfldadd 21623	. . . . . . . . . . . . . . 15 ⊢ + = (+g‘ℂfld)
110	3, 109	ressplusg 17401	. . . . . . . . . . . . . 14 ⊢ (ℚ ∈ (SubRing‘ℂfld) → + = (+g‘𝑄))
111	13, 110	ax-mp 5	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝑄)
112		eqid 2760	. . . . . . . . . . . . 13 ⊢ (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄))
113		eqid 2760	. . . . . . . . . . . . 13 ⊢ (coe1‘𝐹) = (coe1‘𝐹)
114	30	fveq2i 6884	. . . . . . . . . . . . . . . . . 18 ⊢ (coe1‘𝐹) = (coe1‘((3 ↑ 𝑋) − (𝐾‘2)))
115	114	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (coe1‘𝐹) = (coe1‘((3 ↑ 𝑋) − (𝐾‘2))))
116	30	fveq2i 6884	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷‘𝐹) = (𝐷‘((3 ↑ 𝑋) − (𝐾‘2)))
117	116	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (⊤ → (𝐷‘𝐹) = (𝐷‘((3 ↑ 𝑋) − (𝐾‘2))))
118		3pos 12398	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 < 3
119	118	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⊤ → 0 < 3)
120		2ne0 12396	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ≠ 0
121	120	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⊤ → 2 ≠ 0)
122	92, 2, 61, 55, 90	deg1scl 26370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑄 ∈ Ring ∧ 2 ∈ ℚ ∧ 2 ≠ 0) → (𝐷‘(𝐾‘2)) = 0)
123	44, 65, 121, 122	syl3anc 1398	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⊤ → (𝐷‘(𝐾‘2)) = 0)
124		drngnzr 20941	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ NzRing)
125	42, 124	mp1i 14	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⊤ → 𝑄 ∈ NzRing)
126	92, 2, 51, 39, 41	deg1pw 26378	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑄 ∈ NzRing ∧ 3 ∈ ℕ0) → (𝐷‘(3 ↑ 𝑋)) = 3)
127	125, 50, 126	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⊤ → (𝐷‘(3 ↑ 𝑋)) = 3)
128	119, 123, 127	3brtr4d 5137	. . . . . . . . . . . . . . . . . . 19 ⊢ (⊤ → (𝐷‘(𝐾‘2)) < (𝐷‘(3 ↑ 𝑋)))
129	2, 92, 44, 34, 35, 54, 66, 128	deg1sub 26365	. . . . . . . . . . . . . . . . . 18 ⊢ (⊤ → (𝐷‘((3 ↑ 𝑋) − (𝐾‘2))) = (𝐷‘(3 ↑ 𝑋)))
130	117, 129, 127	3eqtrd 2799	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (𝐷‘𝐹) = 3)
131	115, 130	fveq12d 6888	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ((coe1‘𝐹)‘(𝐷‘𝐹)) = ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘3))
132		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (-g‘𝑄) = (-g‘𝑄)
133	2, 34, 35, 132	coe1subfv 22524	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑄 ∈ Ring ∧ (3 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝐾‘2) ∈ (Base‘𝑃)) ∧ 3 ∈ ℕ0) → ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘3) = (((coe1‘(3 ↑ 𝑋))‘3)(-g‘𝑄)((coe1‘(𝐾‘2))‘3)))
134	44, 54, 66, 50, 133	syl31anc 1400	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘3) = (((coe1‘(3 ↑ 𝑋))‘3)(-g‘𝑄)((coe1‘(𝐾‘2))‘3)))
135		subrgsubg 20768	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℚ ∈ (SubRing‘ℂfld) → ℚ ∈ (SubGrp‘ℂfld))
136	13, 135	mp1i 14	. . . . . . . . . . . . . . . . . 18 ⊢ (⊤ → ℚ ∈ (SubGrp‘ℂfld))
137		eqid 2760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (coe1‘(3 ↑ 𝑋)) = (coe1‘(3 ↑ 𝑋))
138	137, 34, 2, 61	coe1fvalcl 22469	. . . . . . . . . . . . . . . . . . 19 ⊢ (((3 ↑ 𝑋) ∈ (Base‘𝑃) ∧ 3 ∈ ℕ0) → ((coe1‘(3 ↑ 𝑋))‘3) ∈ ℚ)
139	54, 50, 138	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (⊤ → ((coe1‘(3 ↑ 𝑋))‘3) ∈ ℚ)
140		eqid 2760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (coe1‘(𝐾‘2)) = (coe1‘(𝐾‘2))
141	140, 34, 2, 61	coe1fvalcl 22469	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾‘2) ∈ (Base‘𝑃) ∧ 3 ∈ ℕ0) → ((coe1‘(𝐾‘2))‘3) ∈ ℚ)
142	66, 50, 141	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (⊤ → ((coe1‘(𝐾‘2))‘3) ∈ ℚ)
143	36, 3, 132	subgsub 19288	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℚ ∈ (SubGrp‘ℂfld) ∧ ((coe1‘(3 ↑ 𝑋))‘3) ∈ ℚ ∧ ((coe1‘(𝐾‘2))‘3) ∈ ℚ) → (((coe1‘(3 ↑ 𝑋))‘3) − ((coe1‘(𝐾‘2))‘3)) = (((coe1‘(3 ↑ 𝑋))‘3)(-g‘𝑄)((coe1‘(𝐾‘2))‘3)))
144	136, 139, 142, 143	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘3) − ((coe1‘(𝐾‘2))‘3)) = (((coe1‘(3 ↑ 𝑋))‘3)(-g‘𝑄)((coe1‘(𝐾‘2))‘3)))
145		iftrue 4488	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 3 → if(𝑖 = 3, 1, 0) = 1)
146	3	qrng1 27890	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 = (1r‘𝑄)
147	2, 51, 41, 44, 50, 90, 146	coe1mon 34030	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⊤ → (coe1‘(3 ↑ 𝑋)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 3, 1, 0)))
148		1cnd 11251	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⊤ → 1 ∈ ℂ)
149	145, 147, 50, 148	fvmptd4 7014	. . . . . . . . . . . . . . . . . . 19 ⊢ (⊤ → ((coe1‘(3 ↑ 𝑋))‘3) = 1)
150	21	neii 2957	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ¬ 3 = 0
151		eqeq1 2764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 3 → (𝑖 = 0 ↔ 3 = 0))
152	150, 151	mtbiri 330	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 3 → ¬ 𝑖 = 0)
153	152	iffalsed 4493	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 3 → if(𝑖 = 0, 2, 0) = 0)
154	2, 55, 61, 90	coe1scl 22545	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑄 ∈ Ring ∧ 2 ∈ ℚ) → (coe1‘(𝐾‘2)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, 2, 0)))
155	44, 65, 154	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⊤ → (coe1‘(𝐾‘2)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, 2, 0)))
156		0nn0 12568	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℕ0
157	156	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⊤ → 0 ∈ ℕ0)
158	153, 155, 50, 157	fvmptd4 7014	. . . . . . . . . . . . . . . . . . 19 ⊢ (⊤ → ((coe1‘(𝐾‘2))‘3) = 0)
159	149, 158	oveq12d 7434	. . . . . . . . . . . . . . . . . 18 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘3) − ((coe1‘(𝐾‘2))‘3)) = (1 − 0))
160		1m0e1 12409	. . . . . . . . . . . . . . . . . 18 ⊢ (1 − 0) = 1
161	159, 160	eqtrdi 2811	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘3) − ((coe1‘(𝐾‘2))‘3)) = 1)
162	144, 161	eqtr3d 2797	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘3)(-g‘𝑄)((coe1‘(𝐾‘2))‘3)) = 1)
163	131, 134, 162	3eqtrd 2799	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1)
164	130	fveq2d 6885	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ((coe1‘𝐹)‘(𝐷‘𝐹)) = ((coe1‘𝐹)‘3))
165	163, 164	eqtr3d 2797	. . . . . . . . . . . . . 14 ⊢ (⊤ → 1 = ((coe1‘𝐹)‘3))
166	165	mptru 1577	. . . . . . . . . . . . 13 ⊢ 1 = ((coe1‘𝐹)‘3)
167	115	fveq1d 6883	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ((coe1‘𝐹)‘2) = ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘2))
168		2nn0 12570	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℕ0
169	168	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → 2 ∈ ℕ0)
170	2, 34, 35, 132	coe1subfv 22524	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑄 ∈ Ring ∧ (3 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝐾‘2) ∈ (Base‘𝑃)) ∧ 2 ∈ ℕ0) → ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘2) = (((coe1‘(3 ↑ 𝑋))‘2)(-g‘𝑄)((coe1‘(𝐾‘2))‘2)))
171	44, 54, 66, 169, 170	syl31anc 1400	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘2) = (((coe1‘(3 ↑ 𝑋))‘2)(-g‘𝑄)((coe1‘(𝐾‘2))‘2)))
172		2re 12364	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 2 ∈ ℝ
173		2lt3 12463	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 2 < 3
174	172, 173	ltneii 11372	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ≠ 3
175		neeq1 3017	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 2 → (𝑖 ≠ 3 ↔ 2 ≠ 3))
176	174, 175	mpbiri 261	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 2 → 𝑖 ≠ 3)
177	176	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⊤ ∧ 𝑖 = 2) → 𝑖 ≠ 3)
178	177	neneqd 2960	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⊤ ∧ 𝑖 = 2) → ¬ 𝑖 = 3)
179	178	iffalsed 4493	. . . . . . . . . . . . . . . . . 18 ⊢ ((⊤ ∧ 𝑖 = 2) → if(𝑖 = 3, 1, 0) = 0)
180	147, 179, 169, 157	fvmptd 6997	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → ((coe1‘(3 ↑ 𝑋))‘2) = 0)
181		neeq1 3017	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 2 → (𝑖 ≠ 0 ↔ 2 ≠ 0))
182	120, 181	mpbiri 261	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 2 → 𝑖 ≠ 0)
183	182	neneqd 2960	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 2 → ¬ 𝑖 = 0)
184	183	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⊤ ∧ 𝑖 = 2) → ¬ 𝑖 = 0)
185	184	iffalsed 4493	. . . . . . . . . . . . . . . . . 18 ⊢ ((⊤ ∧ 𝑖 = 2) → if(𝑖 = 0, 2, 0) = 0)
186	155, 185, 169, 157	fvmptd 6997	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → ((coe1‘(𝐾‘2))‘2) = 0)
187	180, 186	oveq12d 7434	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘2)(-g‘𝑄)((coe1‘(𝐾‘2))‘2)) = (0(-g‘𝑄)0))
188	171, 187	eqtrd 2795	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘2) = (0(-g‘𝑄)0))
189	158, 142	eqeltrrd 2861	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → 0 ∈ ℚ)
190	36, 3, 132	subgsub 19288	. . . . . . . . . . . . . . . . 17 ⊢ ((ℚ ∈ (SubGrp‘ℂfld) ∧ 0 ∈ ℚ ∧ 0 ∈ ℚ) → (0 − 0) = (0(-g‘𝑄)0))
191	136, 189, 189, 190	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (0 − 0) = (0(-g‘𝑄)0))
192		0m0e0 12408	. . . . . . . . . . . . . . . 16 ⊢ (0 − 0) = 0
193	191, 192	eqtr3di 2810	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (0(-g‘𝑄)0) = 0)
194	167, 188, 193	3eqtrrd 2800	. . . . . . . . . . . . . 14 ⊢ (⊤ → 0 = ((coe1‘𝐹)‘2))
195	194	mptru 1577	. . . . . . . . . . . . 13 ⊢ 0 = ((coe1‘𝐹)‘2)
196	115	fveq1d 6883	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ((coe1‘𝐹)‘1) = ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘1))
197		1nn0 12569	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ0
198	197	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → 1 ∈ ℕ0)
199	2, 34, 35, 132	coe1subfv 22524	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑄 ∈ Ring ∧ (3 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝐾‘2) ∈ (Base‘𝑃)) ∧ 1 ∈ ℕ0) → ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘1) = (((coe1‘(3 ↑ 𝑋))‘1)(-g‘𝑄)((coe1‘(𝐾‘2))‘1)))
200	44, 54, 66, 198, 199	syl31anc 1400	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘1) = (((coe1‘(3 ↑ 𝑋))‘1)(-g‘𝑄)((coe1‘(𝐾‘2))‘1)))
201		1re 11257	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℝ
202		1lt3 12465	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 < 3
203	201, 202	ltneii 11372	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ≠ 3
204		neeq1 3017	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 1 → (𝑖 ≠ 3 ↔ 1 ≠ 3))
205	203, 204	mpbiri 261	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 1 → 𝑖 ≠ 3)
206	205	neneqd 2960	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 1 → ¬ 𝑖 = 3)
207	206	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⊤ ∧ 𝑖 = 1) → ¬ 𝑖 = 3)
208	207	iffalsed 4493	. . . . . . . . . . . . . . . . . 18 ⊢ ((⊤ ∧ 𝑖 = 1) → if(𝑖 = 3, 1, 0) = 0)
209	147, 208, 198, 157	fvmptd 6997	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → ((coe1‘(3 ↑ 𝑋))‘1) = 0)
210		ax-1ne0 11218	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ≠ 0
211		neeq1 3017	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 1 → (𝑖 ≠ 0 ↔ 1 ≠ 0))
212	210, 211	mpbiri 261	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 1 → 𝑖 ≠ 0)
213	212	neneqd 2960	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 1 → ¬ 𝑖 = 0)
214	213	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⊤ ∧ 𝑖 = 1) → ¬ 𝑖 = 0)
215	214	iffalsed 4493	. . . . . . . . . . . . . . . . . 18 ⊢ ((⊤ ∧ 𝑖 = 1) → if(𝑖 = 0, 2, 0) = 0)
216	155, 215, 198, 157	fvmptd 6997	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → ((coe1‘(𝐾‘2))‘1) = 0)
217	209, 216	oveq12d 7434	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘1)(-g‘𝑄)((coe1‘(𝐾‘2))‘1)) = (0(-g‘𝑄)0))
218	200, 217	eqtrd 2795	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘1) = (0(-g‘𝑄)0))
219	196, 218, 193	3eqtrrd 2800	. . . . . . . . . . . . . 14 ⊢ (⊤ → 0 = ((coe1‘𝐹)‘1))
220	219	mptru 1577	. . . . . . . . . . . . 13 ⊢ 0 = ((coe1‘𝐹)‘1)
221	115	fveq1d 6883	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ((coe1‘𝐹)‘0) = ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘0))
222	2, 34, 35, 132	coe1subfv 22524	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑄 ∈ Ring ∧ (3 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝐾‘2) ∈ (Base‘𝑃)) ∧ 0 ∈ ℕ0) → ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘0) = (((coe1‘(3 ↑ 𝑋))‘0)(-g‘𝑄)((coe1‘(𝐾‘2))‘0)))
223	44, 54, 66, 157, 222	syl31anc 1400	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘0) = (((coe1‘(3 ↑ 𝑋))‘0)(-g‘𝑄)((coe1‘(𝐾‘2))‘0)))
224	21	necomi 3009	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ≠ 3
225		neeq1 3017	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 0 → (𝑖 ≠ 3 ↔ 0 ≠ 3))
226	224, 225	mpbiri 261	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 0 → 𝑖 ≠ 3)
227	226	neneqd 2960	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 0 → ¬ 𝑖 = 3)
228	227	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⊤ ∧ 𝑖 = 0) → ¬ 𝑖 = 3)
229	228	iffalsed 4493	. . . . . . . . . . . . . . . . . 18 ⊢ ((⊤ ∧ 𝑖 = 0) → if(𝑖 = 3, 1, 0) = 0)
230	147, 229, 157, 157	fvmptd 6997	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → ((coe1‘(3 ↑ 𝑋))‘0) = 0)
231		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⊤ ∧ 𝑖 = 0) → 𝑖 = 0)
232	231	iftrued 4490	. . . . . . . . . . . . . . . . . 18 ⊢ ((⊤ ∧ 𝑖 = 0) → if(𝑖 = 0, 2, 0) = 2)
233	155, 232, 157, 169	fvmptd 6997	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → ((coe1‘(𝐾‘2))‘0) = 2)
234	230, 233	oveq12d 7434	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (((coe1‘(3 ↑ 𝑋))‘0)(-g‘𝑄)((coe1‘(𝐾‘2))‘0)) = (0(-g‘𝑄)2))
235	223, 234	eqtrd 2795	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ((coe1‘((3 ↑ 𝑋) − (𝐾‘2)))‘0) = (0(-g‘𝑄)2))
236		df-neg 11493	. . . . . . . . . . . . . . . 16 ⊢ -2 = (0 − 2)
237	36, 3, 132	subgsub 19288	. . . . . . . . . . . . . . . . 17 ⊢ ((ℚ ∈ (SubGrp‘ℂfld) ∧ 0 ∈ ℚ ∧ 2 ∈ ℚ) → (0 − 2) = (0(-g‘𝑄)2))
238	136, 189, 65, 237	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → (0 − 2) = (0(-g‘𝑄)2))
239	236, 238	eqtr2id 2808	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (0(-g‘𝑄)2) = -2)
240	221, 235, 239	3eqtrrd 2800	. . . . . . . . . . . . . 14 ⊢ (⊤ → -2 = ((coe1‘𝐹)‘0))
241	240	mptru 1577	. . . . . . . . . . . . 13 ⊢ -2 = ((coe1‘𝐹)‘0)
242	95	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → 𝑄 ∈ Field)
243	242	fldcrngd 20934	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑄 ∈ CRing)
244	100	mptru 1577	. . . . . . . . . . . . . 14 ⊢ 𝐹 ∈ (Base‘𝑃)
245	244	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝐹 ∈ (Base‘𝑃))
246	130	mptru 1577	. . . . . . . . . . . . . 14 ⊢ (𝐷‘𝐹) = 3
247	246	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → (𝐷‘𝐹) = 3)
248		id 23	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℚ)
249	2, 91, 61, 34, 108, 111, 112, 113, 92, 166, 195, 220, 241, 243, 245, 247, 248	evl1deg3 34021	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) = (((1 · (3(.g‘(mulGrp‘𝑄))𝑥)) + (0 · (2(.g‘(mulGrp‘𝑄))𝑥))) + ((0 · 𝑥) + -2)))
250		qsscn 13034	. . . . . . . . . . . . . . . . . 18 ⊢ ℚ ⊆ ℂ
251		eqid 2760	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((mulGrp‘ℂfld) ↾s ℚ) = ((mulGrp‘ℂfld) ↾s ℚ)
252		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
253	252, 6	mgpbas 20304	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℂ = (Base‘(mulGrp‘ℂfld))
254	251, 253	ressbas2 17355	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℚ ⊆ ℂ → ℚ = (Base‘((mulGrp‘ℂfld) ↾s ℚ)))
255	250, 254	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℚ = (Base‘((mulGrp‘ℂfld) ↾s ℚ))
256	3, 252	mgpress 20309	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℂfld ∈ DivRing ∧ ℚ ∈ (SubRing‘ℂfld)) → ((mulGrp‘ℂfld) ↾s ℚ) = (mulGrp‘𝑄))
257	7, 13, 256	mp2an 705	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((mulGrp‘ℂfld) ↾s ℚ) = (mulGrp‘𝑄)
258	257	fveq2i 6884	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘((mulGrp‘ℂfld) ↾s ℚ)) = (Base‘(mulGrp‘𝑄))
259	255, 258	eqtri 2783	. . . . . . . . . . . . . . . . . . 19 ⊢ ℚ = (Base‘(mulGrp‘𝑄))
260		eqid 2760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
261	260	ringmgp 20404	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑄 ∈ Ring → (mulGrp‘𝑄) ∈ Mnd)
262	56, 261	mp1i 14	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℚ → (mulGrp‘𝑄) ∈ Mnd)
263	49	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℚ → 3 ∈ ℕ0)
264	259, 112, 262, 263, 248	mulgnn0cld 19244	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℚ → (3(.g‘(mulGrp‘𝑄))𝑥) ∈ ℚ)
265	250, 264	sselid 3929	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℚ → (3(.g‘(mulGrp‘𝑄))𝑥) ∈ ℂ)
266	265	mullidd 11276	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℚ → (1 · (3(.g‘(mulGrp‘𝑄))𝑥)) = (3(.g‘(mulGrp‘𝑄))𝑥))
267	257	eqcomi 2769	. . . . . . . . . . . . . . . . 17 ⊢ (mulGrp‘𝑄) = ((mulGrp‘ℂfld) ↾s ℚ)
268	250, 253	sseqtri 3979	. . . . . . . . . . . . . . . . . 18 ⊢ ℚ ⊆ (Base‘(mulGrp‘ℂfld))
269	268	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℚ → ℚ ⊆ (Base‘(mulGrp‘ℂfld)))
270	80	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℚ → 3 ∈ ℕ)
271	267, 269, 248, 270	ressmulgnnd 19227	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℚ → (3(.g‘(mulGrp‘𝑄))𝑥) = (3(.g‘(mulGrp‘ℂfld))𝑥))
272		qcn 13037	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℂ)
273		cnfldexp 21650	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 3 ∈ ℕ0) → (3(.g‘(mulGrp‘ℂfld))𝑥) = (𝑥↑3))
274	272, 263, 273	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℚ → (3(.g‘(mulGrp‘ℂfld))𝑥) = (𝑥↑3))
275	266, 271, 274	3eqtrd 2799	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℚ → (1 · (3(.g‘(mulGrp‘𝑄))𝑥)) = (𝑥↑3))
276	168	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℚ → 2 ∈ ℕ0)
277	259, 112, 262, 276, 248	mulgnn0cld 19244	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℚ → (2(.g‘(mulGrp‘𝑄))𝑥) ∈ ℚ)
278	250, 277	sselid 3929	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℚ → (2(.g‘(mulGrp‘𝑄))𝑥) ∈ ℂ)
279	278	mul02d 11457	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℚ → (0 · (2(.g‘(mulGrp‘𝑄))𝑥)) = 0)
280	275, 279	oveq12d 7434	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → ((1 · (3(.g‘(mulGrp‘𝑄))𝑥)) + (0 · (2(.g‘(mulGrp‘𝑄))𝑥))) = ((𝑥↑3) + 0))
281	272, 263	expcld 14235	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℚ → (𝑥↑3) ∈ ℂ)
282	281	addridd 11459	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → ((𝑥↑3) + 0) = (𝑥↑3))
283	280, 282	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → ((1 · (3(.g‘(mulGrp‘𝑄))𝑥)) + (0 · (2(.g‘(mulGrp‘𝑄))𝑥))) = (𝑥↑3))
284	272	mul02d 11457	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℚ → (0 · 𝑥) = 0)
285	284	oveq1d 7431	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → ((0 · 𝑥) + -2) = (0 + -2))
286	19	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℚ → 2 ∈ ℂ)
287	286	negcld 11605	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℚ → -2 ∈ ℂ)
288	287	addlidd 11460	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → (0 + -2) = -2)
289	285, 288	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → ((0 · 𝑥) + -2) = -2)
290	283, 289	oveq12d 7434	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → (((1 · (3(.g‘(mulGrp‘𝑄))𝑥)) + (0 · (2(.g‘(mulGrp‘𝑄))𝑥))) + ((0 · 𝑥) + -2)) = ((𝑥↑3) + -2))
291	281, 286	negsubd 11624	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → ((𝑥↑3) + -2) = ((𝑥↑3) − 2))
292	249, 290, 291	3eqtrd 2799	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) = ((𝑥↑3) − 2))
293		2prm 16807	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℙ
294		3z 12676	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℤ
295		3re 12370	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
296	172, 295, 173	ltleii 11382	. . . . . . . . . . . . . . . 16 ⊢ 2 ≤ 3
297	63	eluz1i 12920	. . . . . . . . . . . . . . . 16 ⊢ (3 ∈ (ℤ≥‘2) ↔ (3 ∈ ℤ ∧ 2 ≤ 3))
298	294, 296, 297	mpbir2an 724	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ (ℤ≥‘2)
299		rtprmirr 27029	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℙ ∧ 3 ∈ (ℤ≥‘2)) → (2↑𝑐(1 / 3)) ∈ (ℝ ∖ ℚ))
300	293, 298, 299	mp2an 705	. . . . . . . . . . . . . 14 ⊢ (2↑𝑐(1 / 3)) ∈ (ℝ ∖ ℚ)
301		eldifn 4079	. . . . . . . . . . . . . 14 ⊢ ((2↑𝑐(1 / 3)) ∈ (ℝ ∖ ℚ) → ¬ (2↑𝑐(1 / 3)) ∈ ℚ)
302	300, 301	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ¬ (2↑𝑐(1 / 3)) ∈ ℚ
303		nelne2 3053	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℚ ∧ ¬ (2↑𝑐(1 / 3)) ∈ ℚ) → 𝑥 ≠ (2↑𝑐(1 / 3)))
304	302, 303	mpan2 704	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → 𝑥 ≠ (2↑𝑐(1 / 3)))
305		qre 13027	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ)
306	305	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → 𝑥 ∈ ℝ)
307		2pos 12394	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 2
308	281, 286	subeq0ad 11626	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℚ → (((𝑥↑3) − 2) = 0 ↔ (𝑥↑3) = 2))
309	308	biimpa 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → (𝑥↑3) = 2)
310	307, 309	breqtrrid 5143	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → 0 < (𝑥↑3))
311	80	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → 3 ∈ ℕ)
312		n2dvds3 16486	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 2 ∥ 3
313	312	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → ¬ 2 ∥ 3)
314	306, 311, 313	expgt0b 33319	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → (0 < 𝑥 ↔ 0 < (𝑥↑3)))
315	310, 314	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → 0 < 𝑥)
316	306, 315	elrpd 13108	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → 𝑥 ∈ ℝ+)
317	295	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → 3 ∈ ℝ)
318	22	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → (1 / 3) ∈ ℂ)
319	316, 317, 318	cxpmuld 27006	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → (𝑥↑𝑐(3 · (1 / 3))) = ((𝑥↑𝑐3)↑𝑐(1 / 3)))
320	20	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℚ → 3 ∈ ℂ)
321	21	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℚ → 3 ≠ 0)
322	320, 321	recidd 12035	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℚ → (3 · (1 / 3)) = 1)
323	322	oveq2d 7432	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℚ → (𝑥↑𝑐(3 · (1 / 3))) = (𝑥↑𝑐1))
324	272	cxp1d 26975	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℚ → (𝑥↑𝑐1) = 𝑥)
325	323, 324	eqtrd 2795	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℚ → (𝑥↑𝑐(3 · (1 / 3))) = 𝑥)
326	325	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → (𝑥↑𝑐(3 · (1 / 3))) = 𝑥)
327		cxpexp 26937	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 3 ∈ ℕ0) → (𝑥↑𝑐3) = (𝑥↑3))
328	272, 263, 327	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℚ → (𝑥↑𝑐3) = (𝑥↑3))
329	328	oveq1d 7431	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℚ → ((𝑥↑𝑐3)↑𝑐(1 / 3)) = ((𝑥↑3)↑𝑐(1 / 3)))
330	329	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → ((𝑥↑𝑐3)↑𝑐(1 / 3)) = ((𝑥↑3)↑𝑐(1 / 3)))
331	319, 326, 330	3eqtr3rd 2804	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → ((𝑥↑3)↑𝑐(1 / 3)) = 𝑥)
332	309	oveq1d 7431	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → ((𝑥↑3)↑𝑐(1 / 3)) = (2↑𝑐(1 / 3)))
333	331, 332	eqtr3d 2797	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℚ ∧ ((𝑥↑3) − 2) = 0) → 𝑥 = (2↑𝑐(1 / 3)))
334	304, 333	mteqand 3046	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℚ → ((𝑥↑3) − 2) ≠ 0)
335	292, 334	eqnetrd 3022	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℚ → (((eval1‘𝑄)‘𝐹)‘𝑥) ≠ 0)
336	335	neneqd 2960	. . . . . . . . 9 ⊢ (𝑥 ∈ ℚ → ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
337	336	rgen 3078	. . . . . . . 8 ⊢ ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0
338	337	a1i 11	. . . . . . 7 ⊢ (⊤ → ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
339		rabeq0 4338	. . . . . . 7 ⊢ ({𝑥 ∈ ℚ ∣ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0} = ∅ ↔ ∀𝑥 ∈ ℚ ¬ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0)
340	338, 339	sylibr 237	. . . . . 6 ⊢ (⊤ → {𝑥 ∈ ℚ ∣ (((eval1‘𝑄)‘𝐹)‘𝑥) = 0} = ∅)
341	105, 340	eqtrd 2795	. . . . 5 ⊢ (⊤ → (◡((eval1‘𝑄)‘𝐹) “ {0}) = ∅)
342	90, 91, 92, 2, 34, 96, 100, 341, 130	ply1dg3rt0irred 34027	. . . 4 ⊢ (⊤ → 𝐹 ∈ (Irred‘𝑃))
343		eqid 2760	. . . . . . 7 ⊢ (Irred‘𝑃) = (Irred‘𝑃)
344	343, 29	irredn0 20592	. . . . . 6 ⊢ ((𝑃 ∈ Ring ∧ 𝐹 ∈ (Irred‘𝑃)) → 𝐹 ≠ (0g‘𝑃))
345	46, 342, 344	syl2anc 596	. . . . 5 ⊢ (⊤ → 𝐹 ≠ (0g‘𝑃))
346	3	fveq2i 6884	. . . . . . 7 ⊢ (deg1‘𝑄) = (deg1‘(ℂfld ↾s ℚ))
347	92, 346	eqtri 2783	. . . . . 6 ⊢ 𝐷 = (deg1‘(ℂfld ↾s ℚ))
348		eqid 2760	. . . . . 6 ⊢ (Monic1p‘(ℂfld ↾s ℚ)) = (Monic1p‘(ℂfld ↾s ℚ))
349		eqid 2760	. . . . . . 7 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ)
350	349	qrng1 27890	. . . . . 6 ⊢ 1 = (1r‘(ℂfld ↾s ℚ))
351	5, 34, 29, 347, 348, 350	ismon1p 26400	. . . . 5 ⊢ (𝐹 ∈ (Monic1p‘(ℂfld ↾s ℚ)) ↔ (𝐹 ∈ (Base‘𝑃) ∧ 𝐹 ≠ (0g‘𝑃) ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1))
352	100, 345, 163, 351	syl3anbrc 1362	. . . 4 ⊢ (⊤ → 𝐹 ∈ (Monic1p‘(ℂfld ↾s ℚ)))
353	1, 5, 6, 11, 17, 26, 27, 28, 29, 89, 342, 352	irredminply 34259	. . 3 ⊢ (⊤ → 𝐹 = (𝑀‘𝐴))
354	353, 130	jca 521	. 2 ⊢ (⊤ → (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3))
355	354	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 2955  ∀wral 3076  {crab 3412   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  ifcif 4482  {csn 4584   class class class wbr 5103   ↦ cmpt 5186   I cid 5549  ^◡ccnv 5654   ↾ cres 5657   “ cima 5658   Fn wfn 6530  ‘cfv 6535  (class class class)co 7416  ℂcc 11147  ℝcr 11148  0cc0 11149  1c1 11150   + caddc 11152   · cmul 11154   < clt 11292   ≤ cle 11293   − cmin 11490  -cneg 11491   / cdiv 11920  ℕcn 12282  2c2 12344  3c3 12345  ℕ₀cn0 12553  ℤcz 12640  ℤ_≥cuz 12912  ℚcq 13022  ↑cexp 14150   ∥ cdvds 16367  ℙcprime 16786  Basecbs 17326   ↾_s cress 17347  +_gcplusg 17367  ._rcmulr 17368  Scalarcsca 17370  0_gc0g 17549  Mndcmnd 18862  Grpcgrp 19083  -_gcsg 19085  ._gcmg 19216  SubGrpcsubg 19269  mulGrpcmgp 20299  Ringcrg 20398  CRingccrg 20399  Irredcir 20525  NzRingcnzr 20701  SubRingcsubrg 20760  DivRingcdr 20919  Fieldcfield 20920  SubDRingcsdrg 20982  LModclmod 21074  ℂ_fldccnfld 21617  algSccascl 22099  var₁cv1 22433  Poly₁cpl1 22434  coe₁cco1 22435   evalSub₁ ces1 22570  eval₁ce1 22571  deg₁cdg1 26311  Monic_1pcmn1 26383  ↑_𝑐ccxp 26824   minPoly cminply 34242

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 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-inf2 9627  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226  ax-pre-sup 11227  ax-addf 11228  ax-mulf 11229

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  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 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7684  df-ofr 7685  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8164  df-tpos 8229  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-2o 8463  df-er 8703  df-map 8835  df-pm 8836  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9339  df-fi 9388  df-sup 9419  df-inf 9420  df-oi 9489  df-card 9969  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11492  df-neg 11493  df-div 11921  df-nn 12283  df-2 12352  df-3 12353  df-4 12354  df-5 12355  df-6 12356  df-7 12357  df-8 12358  df-9 12359  df-n0 12554  df-z 12641  df-dec 12762  df-uz 12913  df-q 13023  df-rp 13068  df-xneg 13188  df-xadd 13189  df-xmul 13190  df-ioo 13427  df-ioc 13428  df-ico 13429  df-icc 13430  df-fz 13587  df-fzo 13735  df-fl 13878  df-mod 13956  df-seq 14091  df-exp 14151  df-fac 14363  df-bc 14392  df-hash 14420  df-shft 15165  df-cj 15211  df-re 15212  df-im 15213  df-sqrt 15347  df-abs 15348  df-limsup 15583  df-clim 15600  df-rlim 15601  df-sum 15799  df-ef 16178  df-sin 16180  df-cos 16181  df-pi 16183  df-dvds 16368  df-gcd 16610  df-prm 16787  df-numer 16851  df-denom 16852  df-struct 17264  df-sets 17281  df-slot 17299  df-ndx 17311  df-base 17327  df-ress 17348  df-plusg 17380  df-mulr 17381  df-starv 17382  df-sca 17383  df-vsca 17384  df-ip 17385  df-tset 17386  df-ple 17387  df-ds 17389  df-unif 17390  df-hom 17391  df-cco 17392  df-rest 17532  df-topn 17533  df-0g 17551  df-gsum 17552  df-topgen 17553  df-pt 17554  df-prds 17557  df-pws 17559  df-xrs 17613  df-qtop 17618  df-imas 17619  df-xps 17621  df-mre 17695  df-mrc 17696  df-acs 17698  df-mgm 18755  df-sgrp 18847  df-mnd 18863  df-mhm 18917  df-submnd 18918  df-grp 19086  df-minusg 19087  df-sbg 19088  df-mulg 19217  df-subg 19272  df-ghm 19367  df-cntz 19470  df-cmn 19935  df-abl 19936  df-mgp 20300  df-rng 20314  df-ur 20347  df-srg 20352  df-ring 20400  df-cring 20401  df-oppr 20506  df-dvdsr 20526  df-unit 20527  df-irred 20528  df-invr 20557  df-dvr 20570  df-rhm 20641  df-nzr 20702  df-subrng 20737  df-subrg 20761  df-rlreg 20885  df-domn 20886  df-idom 20887  df-drng 20921  df-field 20922  df-sdrg 20983  df-lmod 21076  df-lss 21146  df-lsp 21186  df-sra 21387  df-rgmod 21388  df-lidl 21425  df-rsp 21426  df-psmet 21609  df-xmet 21610  df-met 21611  df-bl 21612  df-mopn 21613  df-fbas 21614  df-fg 21615  df-cnfld 21618  df-assa 22100  df-asp 22101  df-ascl 22102  df-psr 22156  df-mvr 22157  df-mpl 22158  df-opsr 22160  df-evls 22322  df-evl 22323  df-psr1 22437  df-vr1 22438  df-ply1 22439  df-coe1 22440  df-evls1 22572  df-evl1 22573  df-top 23151  df-topon 23168  df-topsp 23190  df-bases 23203  df-cld 23276  df-ntr 23277  df-cls 23278  df-nei 23355  df-lp 23393  df-perf 23394  df-cn 23484  df-cnp 23485  df-haus 23572  df-tx 23820  df-hmeo 24013  df-fil 24104  df-fm 24196  df-flim 24197  df-flf 24198  df-xms 24578  df-ms 24579  df-tms 24580  df-cncf 25138  df-limc 26125  df-dv 26126  df-mdeg 26312  df-deg1 26313  df-mon1 26388  df-uc1p 26389  df-q1p 26390  df-r1p 26391  df-ig1p 26392  df-log 26825  df-cxp 26826  df-irng 34227  df-minply 34243

This theorem is used by:  2sqr3nconstr  34324