aks5lem8 - Mathbox for metakunt

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Theorem aks5lem8 42189

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 7402	. . . 4 ⊢ ((𝜑 ∧ 𝑝 = 𝑃) → (𝑝↑𝑛) = (𝑃↑𝑛))
4	3	eqeq2d 2740	. . 3 ⊢ ((𝜑 ∧ 𝑝 = 𝑃) → (𝑁 = (𝑝↑𝑛) ↔ 𝑁 = (𝑃↑𝑛)))
5	4	rexbidv 3157	. 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 42188	. . . 4 ⊢ (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁)))
22		eluzelz 12803	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
23	10, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
24		0red 11177	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
25		3re 12266	. . . . . . . . 9 ⊢ 3 ∈ ℝ
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ∈ ℝ)
27	23	zred 12638	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ)
28		3pos 12291	. . . . . . . . 9 ⊢ 0 < 3
29	28	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 3)
30		eluzle 12806	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
31	10, 30	syl 17	. . . . . . . 8 ⊢ (𝜑 → 3 ≤ 𝑁)
32	24, 26, 27, 29, 31	ltletrd 11334	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑁)
33	23, 32	jca 511	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
34		elnnz 12539	. . . . . 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 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 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑛 ∈ ℕ₀)
40	39	nn0zd 12555	. . . . 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 12504	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑛 ∈ ℝ)
43		0red 11177	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 0 ∈ ℝ)
44	42, 43	lenltd 11320	. . . . . . . . . 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 12470	. . . . . . . . . . . . 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 777	. . . . . . . . . . . . . 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 7403	. . . . . . . . . . . . . . 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 16644	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
59	57, 58	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑃 ∈ ℕ)
60	59	nncnd 12202	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑃 ∈ ℂ)
61	60	exp0d 14105	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → (𝑃↑0) = 1)
62	56, 61	eqtrd 2764	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → (𝑃↑𝑛) = 1)
63	54, 62	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑁 = 1)
64		1red 11175	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 1 ∈ ℝ)
65		1red 11175	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 1 ∈ ℝ)
66	35	nnred 12201	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℝ)
67		1lt3 12354	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 < 3
68	67	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 1 < 3)
69	65, 26, 66, 68, 31	ltletrd 11334	. . . . . . . . . . . . . . . . . 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 11310	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 1 ≠ 𝑁)
73	72	necomd 2980	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) ∧ 𝑛 = 0) → 𝑁 ≠ 1)
74	73	neneqd 2930	. . . . . . . . . . . . 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 812	. . . . 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 12539	. . . 4 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℤ ∧ 0 < 𝑛))
85	83, 84	sylibr 234	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑛 ∈ ℕ)
86		simprr 772	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ 𝑁 = (𝑃↑𝑛))) → 𝑁 = (𝑃↑𝑛))
87	38, 85, 86	reximssdv 3151	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝑁 = (𝑃↑𝑛))
88	1, 5, 87	rspcedvd 3590	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 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 {csn 4589 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 [cec 8669 ℝcr 11067 0cc0 11068 1c1 11069 · cmul 11073 < clt 11208 ≤ cle 11209 − cmin 11405 ℕcn 12186 2c2 12241 3c3 12242 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ...cfz 13468 ⌊cfl 13752 ↑cexp 14026 ♯chash 14295 √csqrt 15199 ∥ cdvds 16222 gcd cgcd 16464 ℙcprime 16641 od_ℤcodz 16733 ϕcphi 16734 pCnt cpc 16807 Basecbs 17179 +_gcplusg 17220 -_gcsg 18867 ._gcmg 18999 ~_QG cqg 19054 mulGrpcmgp 20049 1_rcur 20090 Fieldcfield 20639 RSpancrsp 21117 ℤRHomczrh 21409 chrcchr 21411 ℤ/nℤczn 21412 var₁cv1 22060 Poly₁cpl1 22061 log_b clogb 26674

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-disj 5075 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-ofr 7654 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-omul 8439 df-er 8671 df-ec 8673 df-qs 8677 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-prod 15870 df-fallfac 15973 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 df-dvds 16223 df-gcd 16465 df-prm 16642 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 17465 df-qtop 17470 df-imas 17471 df-qus 17472 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-nsg 19056 df-eqg 19057 df-ghm 19145 df-gim 19191 df-cntz 19249 df-od 19458 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-srg 20096 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-rhm 20381 df-rim 20382 df-nzr 20422 df-subrng 20455 df-subrg 20479 df-rlreg 20603 df-domn 20604 df-idom 20605 df-drng 20640 df-field 20641 df-lmod 20768 df-lss 20838 df-lsp 20878 df-sra 21080 df-rgmod 21081 df-lidl 21118 df-rsp 21119 df-2idl 21160 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-zring 21357 df-zrh 21413 df-chr 21415 df-zn 21416 df-assa 21762 df-asp 21763 df-ascl 21764 df-psr 21818 df-mvr 21819 df-mpl 21820 df-opsr 21822 df-evls 21981 df-evl 21982 df-psr1 22064 df-vr1 22065 df-ply1 22066 df-coe1 22067 df-evls1 22202 df-evl1 22203 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-mdeg 25960 df-deg1 25961 df-mon1 26036 df-uc1p 26037 df-q1p 26038 df-r1p 26039 df-log 26465 df-cxp 26466 df-logb 26675 df-primroots 42080

This theorem is referenced by: aks5 42192

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