aks5lem8 - Mathbox for metakunt

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Theorem aks5lem8 42183

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 7446	. . . 4 ⊢ ((𝜑 ∧ 𝑝 = 𝑃) → (𝑝↑𝑛) = (𝑃↑𝑛))
4	3	eqeq2d 2746	. . 3 ⊢ ((𝜑 ∧ 𝑝 = 𝑃) → (𝑁 = (𝑝↑𝑛) ↔ 𝑁 = (𝑃↑𝑛)))
5	4	rexbidv 3177	. 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 42182	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁)))
22		eluzelz 12886	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
23	10, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
24		0red 11262	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
25		3re 12344	. . . . . . . . 9 ⊢ 3 ∈ ℝ
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ∈ ℝ)
27	23	zred 12720	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ)
28		3pos 12369	. . . . . . . . 9 ⊢ 0 < 3
29	28	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 3)
30		eluzle 12889	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
31	10, 30	syl 17	. . . . . . . 8 ⊢ (𝜑 → 3 ≤ 𝑁)
32	24, 26, 27, 29, 31	ltletrd 11419	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑁)
33	23, 32	jca 511	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
34		elnnz 12621	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
35	33, 34	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
36		pcprmpw 16917	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (∃𝑛 ∈ ℕ₀ 𝑁 = (𝑃↑𝑛) ↔ 𝑁 = (𝑃↑(𝑃 pCnt 𝑁))))
37	1, 35, 36	syl2anc 584	. . . 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 12637	. . . . 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 12586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑛 ∈ ℝ)
43		0red 11262	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 0 ∈ ℝ)
44	42, 43	lenltd 11405	. . . . . . . . . 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 12552	. . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → (𝑃↑𝑛) = (𝑃↑0))
57	1	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑃 ∈ ℙ)
58		prmnn 16708	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
59	57, 58	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑃 ∈ ℕ)
60	59	nncnd 12280	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑃 ∈ ℂ)
61	60	exp0d 14177	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → (𝑃↑0) = 1)
62	56, 61	eqtrd 2775	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → (𝑃↑𝑛) = 1)
63	54, 62	eqtrd 2775	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑁 = 1)
64		1red 11260	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 1 ∈ ℝ)
65		1red 11260	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 1 ∈ ℝ)
66	35	nnred 12279	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℝ)
67		1lt3 12437	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 < 3
68	67	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 1 < 3)
69	65, 26, 66, 68, 31	ltletrd 11419	. . . . . . . . . . . . . . . . . 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 11395	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 1 ≠ 𝑁)
73	72	necomd 2994	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑁 ≠ 1)
74	73	neneqd 2943	. . . . . . . . . . . . 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 12621	. . . 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 3171	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝑁 = (𝑃↑𝑛))
88	1, 5, 87	rspcedvd 3624	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 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 {csn 4631 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 [cec 8742 ℝcr 11152 0cc0 11153 1c1 11154 · cmul 11158 < clt 11293 ≤ cle 11294 − cmin 11490 ℕcn 12264 2c2 12319 3c3 12320 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ...cfz 13544 ⌊cfl 13827 ↑cexp 14099 ♯chash 14366 √csqrt 15269 ∥ cdvds 16287 gcd cgcd 16528 ℙcprime 16705 od_ℤcodz 16797 ϕcphi 16798 pCnt cpc 16870 Basecbs 17245 +_gcplusg 17298 -_gcsg 18966 ._gcmg 19098 ~_QG cqg 19153 mulGrpcmgp 20152 1_rcur 20199 Fieldcfield 20747 RSpancrsp 21235 ℤRHomczrh 21528 chrcchr 21530 ℤ/nℤczn 21531 var₁cv1 22193 Poly₁cpl1 22194 log_b clogb 26822

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-prod 15937 df-fallfac 16040 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-dvds 16288 df-gcd 16529 df-prm 16706 df-odz 16799 df-phi 16800 df-pc 16871 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-pws 17496 df-xrs 17549 df-qtop 17554 df-imas 17555 df-qus 17556 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-nsg 19155 df-eqg 19156 df-ghm 19244 df-gim 19290 df-cntz 19348 df-od 19561 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-rim 20490 df-nzr 20530 df-subrng 20563 df-subrg 20587 df-rlreg 20711 df-domn 20712 df-idom 20713 df-drng 20748 df-field 20749 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-2idl 21278 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-chr 21534 df-zn 21535 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-evls 22116 df-evl 22117 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-coe1 22200 df-evls1 22335 df-evl1 22336 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-mdeg 26109 df-deg1 26110 df-mon1 26185 df-uc1p 26186 df-q1p 26187 df-r1p 26188 df-log 26613 df-cxp 26614 df-logb 26823 df-primroots 42074

This theorem is referenced by: aks5 42186

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