aks5lem8 - Mathbox for metakunt

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Theorem aks5lem8 42996

Description: Lemma for aks5. Clean up the conclusion. (Contributed by metakunt, 9-Aug-2025.)

**Hypotheses**
Ref	Expression
aks5lem7.1	⊢ (𝜑 → (♯‘(Base‘𝐾)) ∈ ℕ)
aks5lem7.2	⊢ 𝑃 = (chr‘𝐾)
aks5lem7.3	⊢ (𝜑 → 𝐾 ∈ Field)
aks5lem7.4	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks5lem7.5	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks5lem7.6	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
aks5lem7.7	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks5lem7.8	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks5lem7.9	⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
aks5lem7.10	⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))
aks5lem7.11	⊢ (𝜑 → 𝑅 ∥ ((♯‘(Base‘𝐾)) − 1))
aks5lem7.12	⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(._g‘(mulGrp‘𝑆))(𝑋(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~_QG 𝐿) = [((𝑁(._g‘(mulGrp‘𝑆))𝑋)(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~_QG 𝐿))
aks5lem7.13	⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
aks5lem7.14	⊢ 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁))
aks5lem7.15	⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(._g‘(mulGrp‘𝑆))𝑋)(-_g‘𝑆)(1_r‘𝑆))})
aks5lem7.16	⊢ 𝑋 = (var₁‘(ℤ/nℤ‘𝑁))

**Assertion**
Ref	Expression
aks5lem8	⊢ (𝜑 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝↑𝑛))

Distinct variable groups: 𝐴,𝑎 𝐴,𝑏 𝐾,𝑎 𝐾,𝑏 𝑁,𝑎 𝑁,𝑏 𝑃,𝑎 𝑃,𝑏 𝑅,𝑎 𝑅,𝑏 𝜑,𝑎 𝜑,𝑏 𝑛,𝑁,𝑝 𝑃,𝑛,𝑝 𝜑,𝑛,𝑝 Allowed substitution hints: 𝐴(𝑛, 𝑝) 𝑅(𝑛, 𝑝) 𝑆(𝑛, 𝑝, 𝑎, 𝑏) 𝐾(𝑛, 𝑝) 𝐿(𝑛, 𝑝, 𝑎, 𝑏) 𝑋(𝑛, 𝑝, 𝑎, 𝑏)

**Proof of Theorem aks5lem8**
Step	Hyp	Ref	Expression
1		aks5lem7.4	. 2 ⊢ (𝜑 → 𝑃 ∈ ℙ)
2		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
3	2	oveq1d 7427	. . . 4 ⊢ ((𝜑 ∧ 𝑝 = 𝑃) → (𝑝↑𝑛) = (𝑃↑𝑛))
4	3	eqeq2d 2773	. . 3 ⊢ ((𝜑 ∧ 𝑝 = 𝑃) → (𝑁 = (𝑝↑𝑛) ↔ 𝑁 = (𝑃↑𝑛)))
5	4	rexbidv 3188	. 2 ⊢ ((𝜑 ∧ 𝑝 = 𝑃) → (∃𝑛 ∈ ℕ 𝑁 = (𝑝↑𝑛) ↔ ∃𝑛 ∈ ℕ 𝑁 = (𝑃↑𝑛)))
6		aks5lem7.1	. . . . 5 ⊢ (𝜑 → (♯‘(Base‘𝐾)) ∈ ℕ)
7		aks5lem7.2	. . . . 5 ⊢ 𝑃 = (chr‘𝐾)
8		aks5lem7.3	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Field)
9		aks5lem7.5	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℕ)
10		aks5lem7.6	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
11		aks5lem7.7	. . . . 5 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
12		aks5lem7.8	. . . . 5 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
13		aks5lem7.9	. . . . 5 ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
14		aks5lem7.10	. . . . 5 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))
15		aks5lem7.11	. . . . 5 ⊢ (𝜑 → 𝑅 ∥ ((♯‘(Base‘𝐾)) − 1))
16		aks5lem7.12	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(._g‘(mulGrp‘𝑆))(𝑋(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~_QG 𝐿) = [((𝑁(._g‘(mulGrp‘𝑆))𝑋)(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~_QG 𝐿))
17		aks5lem7.13	. . . . 5 ⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
18		aks5lem7.14	. . . . 5 ⊢ 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁))
19		aks5lem7.15	. . . . 5 ⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(._g‘(mulGrp‘𝑆))𝑋)(-_g‘𝑆)(1_r‘𝑆))})
20		aks5lem7.16	. . . . 5 ⊢ 𝑋 = (var₁‘(ℤ/nℤ‘𝑁))
21	6, 7, 8, 1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20	aks5lem7 42995	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁)))
22		eluzelz 12878	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
23	10, 22	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
24		0red 11217	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
25		3re 12327	. . . . . . . . 9 ⊢ 3 ∈ ℝ
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ∈ ℝ)
27	23	zred 12706	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ)
28		3pos 12355	. . . . . . . . 9 ⊢ 0 < 3
29	28	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 3)
30		eluzle 12881	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
31	10, 30	syl 18	. . . . . . . 8 ⊢ (𝜑 → 3 ≤ 𝑁)
32	24, 26, 27, 29, 31	ltletrd 11376	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑁)
33	23, 32	jca 520	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
34		elnnz 12607	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
35	33, 34	sylibr 237	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
36		pcprmpw 16949	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (∃𝑛 ∈ ℕ₀ 𝑁 = (𝑃↑𝑛) ↔ 𝑁 = (𝑃↑(𝑃 pCnt 𝑁))))
37	1, 35, 36	syl2anc 595	. . . 4 ⊢ (𝜑 → (∃𝑛 ∈ ℕ₀ 𝑁 = (𝑃↑𝑛) ↔ 𝑁 = (𝑃↑(𝑃 pCnt 𝑁))))
38	21, 37	mpbird 260	. . 3 ⊢ (𝜑 → ∃𝑛 ∈ ℕ₀ 𝑁 = (𝑃↑𝑛))
39		simprl 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑛 ∈ ℕ₀)
40	39	nn0zd 12622	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑛 ∈ ℤ)
41		simpr 489	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 0 < 𝑛) → 0 < 𝑛)
42	39	nn0red 12572	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑛 ∈ ℝ)
43		0red 11217	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 0 ∈ ℝ)
44	42, 43	lenltd 11362	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → (𝑛 ≤ 0 ↔ ¬ 0 < 𝑛))
45	44	bicomd 226	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → (¬ 0 < 𝑛 ↔ 𝑛 ≤ 0))
46	45	biimpd 232	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → (¬ 0 < 𝑛 → 𝑛 ≤ 0))
47	46	imp 411	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ ¬ 0 < 𝑛) → 𝑛 ≤ 0)
48		simpr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 ≤ 0) → 𝑛 ≤ 0)
49	39	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 ≤ 0) → 𝑛 ∈ ℕ₀)
50		nn0le0eq0 12538	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 ≤ 0 ↔ 𝑛 = 0))
51	50	bicomd 226	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 = 0 ↔ 𝑛 ≤ 0))
52	49, 51	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 ≤ 0) → (𝑛 = 0 ↔ 𝑛 ≤ 0))
53	48, 52	mpbird 260	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 ≤ 0) → 𝑛 = 0)
54		simplrr 789	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑁 = (𝑃↑𝑛))
55		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑛 = 0)
56	55	oveq2d 7428	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → (𝑃↑𝑛) = (𝑃↑0))
57	1	ad2antrr 738	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑃 ∈ ℙ)
58		prmnn 16738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
59	57, 58	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑃 ∈ ℕ)
60	59	nncnd 12255	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑃 ∈ ℂ)
61	60	exp0d 14183	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → (𝑃↑0) = 1)
62	56, 61	eqtrd 2797	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → (𝑃↑𝑛) = 1)
63	54, 62	eqtrd 2797	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑁 = 1)
64		1red 11215	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 1 ∈ ℝ)
65		1red 11215	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 1 ∈ ℝ)
66	35	nnred 12254	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℝ)
67		1lt3 12422	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 < 3
68	67	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 1 < 3)
69	65, 26, 66, 68, 31	ltletrd 11376	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 < 𝑁)
70	69	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 1 < 𝑁)
71	70	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 1 < 𝑁)
72	64, 71	ltned 11352	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 1 ≠ 𝑁)
73	72	necomd 3012	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑁 ≠ 1)
74	73	neneqd 2962	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → ¬ 𝑁 = 1)
75	63, 74	pm2.21dd 198	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 0 < 𝑛)
76	75	ex 417	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → (𝑛 = 0 → 0 < 𝑛))
77	76	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 ≤ 0) → (𝑛 = 0 → 0 < 𝑛))
78	53, 77	mpd 16	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 ≤ 0) → 0 < 𝑛)
79	78	ex 417	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → (𝑛 ≤ 0 → 0 < 𝑛))
80	79	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ ¬ 0 < 𝑛) → (𝑛 ≤ 0 → 0 < 𝑛))
81	47, 80	mpd 16	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ ¬ 0 < 𝑛) → 0 < 𝑛)
82	41, 81	pm2.61dan 824	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 0 < 𝑛)
83	40, 82	jca 520	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → (𝑛 ∈ ℤ ∧ 0 < 𝑛))
84		elnnz 12607	. . . 4 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℤ ∧ 0 < 𝑛))
85	83, 84	sylibr 237	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑛 ∈ ℕ)
86		simprr 784	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑁 = (𝑃↑𝑛))
87	38, 85, 86	reximssdv 3182	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝑁 = (𝑃↑𝑛))
88	1, 5, 87	rspcedvd 3582	1 ⊢ (𝜑 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝↑𝑛))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1569 ∈ wcel 2142 ∀wral 3078 ∃wrex 3088 {csn 4588 class class class wbr 5108 ‘cfv 6536 (class class class)co 7412 [cec 8690 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 < clt 11249 ≤ cle 11250 − cmin 11447 ℕcn 12239 2c2 12301 3c3 12302 ℕ₀cn0 12510 ℤcz 12597 ℤ_≥cuz 12868 ...cfz 13541 ⌊cfl 13830 ↑cexp 14104 ♯chash 14373 √csqrt 15291 ∥ cdvds 16316 gcd cgcd 16558 ℙcprime 16735 od_ℤcodz 16828 ϕcphi 16829 pCnt cpc 16902 Basecbs 17275 +_gcplusg 17316 -_gcsg 19008 ._gcmg 19139 ~_QG cqg 19194 mulGrpcmgp 20222 1_rcur 20269 Fieldcfield 20839 RSpancrsp 21342 ℤRHomczrh 21660 chrcchr 21662 ℤ/nℤczn 21663 var₁cv1 22347 Poly₁cpl1 22348 log_b clogb 26940

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-disj 5076 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-ofr 7677 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-tpos 8220 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-oadd 8455 df-omul 8456 df-er 8692 df-ec 8694 df-qs 8698 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9894 df-card 9932 df-acn 9935 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-xnn0 12584 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-fac 14317 df-bc 14346 df-hash 14374 df-shft 15111 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-limsup 15529 df-clim 15546 df-rlim 15547 df-sum 15745 df-prod 15965 df-fallfac 16068 df-ef 16127 df-sin 16129 df-cos 16130 df-pi 16132 df-dvds 16317 df-gcd 16559 df-prm 16736 df-odz 16830 df-phi 16831 df-pc 16903 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-pws 17508 df-xrs 17562 df-qtop 17567 df-imas 17568 df-qus 17569 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-mhm 18847 df-submnd 18848 df-grp 19009 df-minusg 19010 df-sbg 19011 df-mulg 19140 df-subg 19195 df-nsg 19196 df-eqg 19197 df-ghm 19290 df-gim 19335 df-cntz 19393 df-od 19604 df-cmn 19858 df-abl 19859 df-mgp 20223 df-rng 20237 df-ur 20270 df-srg 20275 df-ring 20323 df-cring 20324 df-oppr 20426 df-dvdsr 20446 df-unit 20447 df-invr 20477 df-dvr 20490 df-rhm 20561 df-rim 20562 df-nzr 20621 df-subrng 20656 df-subrg 20680 df-rlreg 20804 df-domn 20805 df-idom 20806 df-drng 20840 df-field 20841 df-lmod 20994 df-lss 21064 df-lsp 21104 df-sra 21305 df-rgmod 21306 df-lidl 21343 df-rsp 21344 df-2idl 21400 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-zring 21608 df-zrh 21664 df-chr 21666 df-zn 21667 df-assa 22014 df-asp 22015 df-ascl 22016 df-psr 22070 df-mvr 22071 df-mpl 22072 df-opsr 22074 df-evls 22236 df-evl 22237 df-psr1 22351 df-vr1 22352 df-ply1 22353 df-coe1 22354 df-evls1 22486 df-evl1 22487 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-lp 23304 df-perf 23305 df-cn 23395 df-cnp 23396 df-haus 23483 df-tx 23730 df-hmeo 23923 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-xms 24488 df-ms 24489 df-tms 24490 df-cncf 25048 df-limc 26036 df-dv 26037 df-mdeg 26223 df-deg1 26224 df-mon1 26299 df-uc1p 26300 df-q1p 26301 df-r1p 26302 df-log 26732 df-cxp 26733 df-logb 26941 df-primroots 42887

This theorem is used by: aks5 42999

W3C validator