aks5lem8 - Mathbox for metakunt

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Theorem aks5lem8 42640

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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
3	2	oveq1d 7382	. . . 4 ⊢ ((𝜑 ∧ 𝑝 = 𝑃) → (𝑝↑𝑛) = (𝑃↑𝑛))
4	3	eqeq2d 2748	. . 3 ⊢ ((𝜑 ∧ 𝑝 = 𝑃) → (𝑁 = (𝑝↑𝑛) ↔ 𝑁 = (𝑃↑𝑛)))
5	4	rexbidv 3162	. 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 42639	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁)))
22		eluzelz 12798	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
23	10, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
24		0red 11147	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
25		3re 12261	. . . . . . . . 9 ⊢ 3 ∈ ℝ
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ∈ ℝ)
27	23	zred 12633	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ)
28		3pos 12286	. . . . . . . . 9 ⊢ 0 < 3
29	28	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 3)
30		eluzle 12801	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
31	10, 30	syl 17	. . . . . . . 8 ⊢ (𝜑 → 3 ≤ 𝑁)
32	24, 26, 27, 29, 31	ltletrd 11306	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑁)
33	23, 32	jca 511	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
34		elnnz 12534	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
35	33, 34	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
36		pcprmpw 16854	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (∃𝑛 ∈ ℕ₀ 𝑁 = (𝑃↑𝑛) ↔ 𝑁 = (𝑃↑(𝑃 pCnt 𝑁))))
37	1, 35, 36	syl2anc 585	. . . 4 ⊢ (𝜑 → (∃𝑛 ∈ ℕ₀ 𝑁 = (𝑃↑𝑛) ↔ 𝑁 = (𝑃↑(𝑃 pCnt 𝑁))))
38	21, 37	mpbird 257	. . 3 ⊢ (𝜑 → ∃𝑛 ∈ ℕ₀ 𝑁 = (𝑃↑𝑛))
39		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑛 ∈ ℕ₀)
40	39	nn0zd 12549	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑛 ∈ ℤ)
41		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 0 < 𝑛) → 0 < 𝑛)
42	39	nn0red 12499	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑛 ∈ ℝ)
43		0red 11147	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 0 ∈ ℝ)
44	42, 43	lenltd 11292	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → (𝑛 ≤ 0 ↔ ¬ 0 < 𝑛))
45	44	bicomd 223	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → (¬ 0 < 𝑛 ↔ 𝑛 ≤ 0))
46	45	biimpd 229	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → (¬ 0 < 𝑛 → 𝑛 ≤ 0))
47	46	imp 406	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ ¬ 0 < 𝑛) → 𝑛 ≤ 0)
48		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 ≤ 0) → 𝑛 ≤ 0)
49	39	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 ≤ 0) → 𝑛 ∈ ℕ₀)
50		nn0le0eq0 12465	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 ≤ 0 ↔ 𝑛 = 0))
51	50	bicomd 223	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 = 0 ↔ 𝑛 ≤ 0))
52	49, 51	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 ≤ 0) → (𝑛 = 0 ↔ 𝑛 ≤ 0))
53	48, 52	mpbird 257	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 ≤ 0) → 𝑛 = 0)
54		simplrr 778	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑁 = (𝑃↑𝑛))
55		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑛 = 0)
56	55	oveq2d 7383	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → (𝑃↑𝑛) = (𝑃↑0))
57	1	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑃 ∈ ℙ)
58		prmnn 16643	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
59	57, 58	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑃 ∈ ℕ)
60	59	nncnd 12190	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑃 ∈ ℂ)
61	60	exp0d 14102	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → (𝑃↑0) = 1)
62	56, 61	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → (𝑃↑𝑛) = 1)
63	54, 62	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑁 = 1)
64		1red 11145	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 1 ∈ ℝ)
65		1red 11145	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 1 ∈ ℝ)
66	35	nnred 12189	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℝ)
67		1lt3 12349	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 < 3
68	67	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 1 < 3)
69	65, 26, 66, 68, 31	ltletrd 11306	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 < 𝑁)
70	69	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 1 < 𝑁)
71	70	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 1 < 𝑁)
72	64, 71	ltned 11282	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 1 ≠ 𝑁)
73	72	necomd 2988	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑁 ≠ 1)
74	73	neneqd 2938	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → ¬ 𝑁 = 1)
75	63, 74	pm2.21dd 195	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 0 < 𝑛)
76	75	ex 412	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → (𝑛 = 0 → 0 < 𝑛))
77	76	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 ≤ 0) → (𝑛 = 0 → 0 < 𝑛))
78	53, 77	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 ≤ 0) → 0 < 𝑛)
79	78	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → (𝑛 ≤ 0 → 0 < 𝑛))
80	79	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ ¬ 0 < 𝑛) → (𝑛 ≤ 0 → 0 < 𝑛))
81	47, 80	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ ¬ 0 < 𝑛) → 0 < 𝑛)
82	41, 81	pm2.61dan 813	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 0 < 𝑛)
83	40, 82	jca 511	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → (𝑛 ∈ ℤ ∧ 0 < 𝑛))
84		elnnz 12534	. . . 4 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℤ ∧ 0 < 𝑛))
85	83, 84	sylibr 234	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑛 ∈ ℕ)
86		simprr 773	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑁 = (𝑃↑𝑛))
87	38, 85, 86	reximssdv 3156	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝑁 = (𝑃↑𝑛))
88	1, 5, 87	rspcedvd 3567	1 ⊢ (𝜑 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝↑𝑛))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 {csn 4568 class class class wbr 5086 ‘cfv 6499 (class class class)co 7367 [cec 8641 ℝcr 11037 0cc0 11038 1c1 11039 · cmul 11043 < clt 11179 ≤ cle 11180 − cmin 11377 ℕcn 12174 2c2 12236 3c3 12237 ℕ₀cn0 12437 ℤcz 12524 ℤ_≥cuz 12788 ...cfz 13461 ⌊cfl 13749 ↑cexp 14023 ♯chash 14292 √csqrt 15195 ∥ cdvds 16221 gcd cgcd 16463 ℙcprime 16640 od_ℤcodz 16733 ϕcphi 16734 pCnt cpc 16807 Basecbs 17179 +_gcplusg 17220 -_gcsg 18911 ._gcmg 19043 ~_QG cqg 19098 mulGrpcmgp 20121 1_rcur 20162 Fieldcfield 20707 RSpancrsp 21205 ℤRHomczrh 21479 chrcchr 21481 ℤ/nℤczn 21482 var₁cv1 22139 Poly₁cpl1 22140 log_b clogb 26728

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-ofr 7632 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-prod 15869 df-fallfac 15972 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-dvds 16222 df-gcd 16464 df-prm 16641 df-odz 16735 df-phi 16736 df-pc 16808 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-pws 17412 df-xrs 17466 df-qtop 17471 df-imas 17472 df-qus 17473 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-nsg 19100 df-eqg 19101 df-ghm 19188 df-gim 19234 df-cntz 19292 df-od 19503 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-srg 20168 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-rhm 20452 df-rim 20453 df-nzr 20490 df-subrng 20523 df-subrg 20547 df-rlreg 20671 df-domn 20672 df-idom 20673 df-drng 20708 df-field 20709 df-lmod 20857 df-lss 20927 df-lsp 20967 df-sra 21168 df-rgmod 21169 df-lidl 21206 df-rsp 21207 df-2idl 21248 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-zring 21427 df-zrh 21483 df-chr 21485 df-zn 21486 df-assa 21833 df-asp 21834 df-ascl 21835 df-psr 21889 df-mvr 21890 df-mpl 21891 df-opsr 21893 df-evls 22052 df-evl 22053 df-psr1 22143 df-vr1 22144 df-ply1 22145 df-coe1 22146 df-evls1 22280 df-evl1 22281 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-mdeg 26020 df-deg1 26021 df-mon1 26096 df-uc1p 26097 df-q1p 26098 df-r1p 26099 df-log 26520 df-cxp 26521 df-logb 26729 df-primroots 42531

This theorem is referenced by: aks5 42643

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