aks5lem6 - Mathbox for metakunt

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Theorem aks5lem6 43508

Description: Connect results of section 5 and Theorem 6.1 AKS. (Contributed by metakunt, 25-Jun-2025.)

**Hypotheses**
Ref	Expression
aks5lem6.1	⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly₁‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘𝑓)‘𝑦)) = (((eval₁‘𝐾)‘𝑓)‘(𝑒(._g‘(mulGrp‘𝐾))𝑦)))}
aks5lem6.2	⊢ 𝑃 = (chr‘𝐾)
aks5lem6.3	⊢ (𝜑 → 𝐾 ∈ Field)
aks5lem6.4	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks5lem6.5	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks5lem6.6	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
aks5lem6.7	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks5lem6.8	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks5lem6.9	⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
aks5lem6.10	⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))
aks5lem6.11	⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(._g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
aks5lem6.12	⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
aks5lem6.13	⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
aks5lem6.14	⊢ 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁))
aks5lem6.15	⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘𝑆)(1_r‘𝑆))})
aks5lem6.16	⊢ 𝑋 = (var₁‘(ℤ/nℤ‘𝑁))
aks5lem6.17	⊢ (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))

**Assertion**
Ref	Expression
aks5lem6	⊢ (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁)))

Distinct variable groups: ∼ ,𝑎 𝐴,𝑎,𝑒,𝑓,𝑦 𝐴,𝑏 𝑥,𝐴,𝑦 𝐾,𝑎,𝑒,𝑓,𝑦 𝐾,𝑏 𝑥,𝐾 𝑒,𝐿,𝑦 𝑀,𝑎,𝑦 𝑀,𝑏 𝑥,𝑀 𝑁,𝑎,𝑒,𝑓,𝑦 𝑁,𝑏 𝑥,𝑁 𝑃,𝑎,𝑒,𝑓,𝑦 𝑃,𝑏 𝑥,𝑃 𝑅,𝑎,𝑒,𝑓,𝑦 𝑥,𝑅 𝑆,𝑒,𝑦 𝜑,𝑎,𝑦 𝜑,𝑏 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑒, 𝑓) ∼ (𝑥, 𝑦, 𝑒, 𝑓, 𝑏) 𝑅(𝑏) 𝑆(𝑥, 𝑓, 𝑎, 𝑏) 𝐿(𝑥, 𝑓, 𝑎, 𝑏) 𝑀(𝑒, 𝑓) 𝑋(𝑥, 𝑦, 𝑒, 𝑓, 𝑎, 𝑏)

**Proof of Theorem aks5lem6**
Step	Hyp	Ref	Expression
1		aks5lem6.1	. 2 ⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly₁‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘𝑓)‘𝑦)) = (((eval₁‘𝐾)‘𝑓)‘(𝑒(._g‘(mulGrp‘𝐾))𝑦)))}
2		aks5lem6.2	. 2 ⊢ 𝑃 = (chr‘𝐾)
3		aks5lem6.3	. 2 ⊢ (𝜑 → 𝐾 ∈ Field)
4		aks5lem6.4	. 2 ⊢ (𝜑 → 𝑃 ∈ ℙ)
5		aks5lem6.5	. 2 ⊢ (𝜑 → 𝑅 ∈ ℕ)
6		aks5lem6.6	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
7		aks5lem6.7	. 2 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
8		aks5lem6.8	. 2 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
9		aks5lem6.9	. 2 ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
10		aks5lem6.10	. 2 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))
11		aks5lem6.11	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(._g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
12		aks5lem6.12	. 2 ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
13		aks5lem6.13	. 2 ⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
14		eluzelz 13015	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
15	6, 14	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
16		0red 11350	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
17		3re 12463	. . . . . . . 8 ⊢ 3 ∈ ℝ
18	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → 3 ∈ ℝ)
19	15	zred 12843	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
20		3pos 12491	. . . . . . . 8 ⊢ 0 < 3
21	20	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 3)
22		eluzle 13018	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
23	6, 22	syl 18	. . . . . . 7 ⊢ (𝜑 → 3 ≤ 𝑁)
24	16, 18, 19, 21, 23	ltletrd 11509	. . . . . 6 ⊢ (𝜑 → 0 < 𝑁)
25	15, 24	jca 521	. . . . 5 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
26		elnnz 12743	. . . . 5 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
27	25, 26	sylibr 237	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
28	4, 27, 7	3jca 1146	. . 3 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))
29		eqid 2763	. . 3 ⊢ (𝑆 /_s (𝑆 ~_QG 𝐿)) = (𝑆 /_s (𝑆 ~_QG 𝐿))
30		aks5lem6.15	. . 3 ⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(-_g‘𝑆)(1_r‘𝑆))})
31		aks5lem6.14	. . 3 ⊢ 𝑆 = (Poly₁‘(ℤ/nℤ‘𝑁))
32		aks5lem6.17	. . . 4 ⊢ (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))
33		aks5lem6.16	. . . . . . . . . . . 12 ⊢ 𝑋 = (var₁‘(ℤ/nℤ‘𝑁))
34	33	eqcomi 2772	. . . . . . . . . . 11 ⊢ (var₁‘(ℤ/nℤ‘𝑁)) = 𝑋
35	34	a1i 11	. . . . . . . . . 10 ⊢ (((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)) → (var₁‘(ℤ/nℤ‘𝑁)) = 𝑋)
36	35	oveq1d 7466	. . . . . . . . 9 ⊢ (((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)) → ((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎)) = (𝑋(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))
37	36	oveq2d 7467	. . . . . . . 8 ⊢ (((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)) → (𝑁(._g‘(mulGrp‘𝑆))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎))) = (𝑁(._g‘(mulGrp‘𝑆))(𝑋(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎))))
38	37	eceq1d 8796	. . . . . . 7 ⊢ (((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)) → [(𝑁(._g‘(mulGrp‘𝑆))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~_QG 𝐿) = [(𝑁(._g‘(mulGrp‘𝑆))(𝑋(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~_QG 𝐿))
39		simpr 490	. . . . . . 7 ⊢ (((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)) → [(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))
40		eqcom 2770	. . . . . . . . . . . 12 ⊢ ((var₁‘(ℤ/nℤ‘𝑁)) = 𝑋 ↔ 𝑋 = (var₁‘(ℤ/nℤ‘𝑁)))
41	40	imbi2i 339	. . . . . . . . . . 11 ⊢ ((((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)) → (var₁‘(ℤ/nℤ‘𝑁)) = 𝑋) ↔ (((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)) → 𝑋 = (var₁‘(ℤ/nℤ‘𝑁))))
42	35, 41	mpbi 233	. . . . . . . . . 10 ⊢ (((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)) → 𝑋 = (var₁‘(ℤ/nℤ‘𝑁)))
43	42	oveq2d 7467	. . . . . . . . 9 ⊢ (((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)) → (𝑁(._g‘(mulGrp‘𝑆))𝑋) = (𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁))))
44	43	oveq1d 7466	. . . . . . . 8 ⊢ (((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)) → ((𝑁(._g‘(mulGrp‘𝑆))𝑋)(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎)) = ((𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))
45	44	eceq1d 8796	. . . . . . 7 ⊢ (((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)) → [((𝑁(._g‘(mulGrp‘𝑆))𝑋)(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~_QG 𝐿) = [((𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~_QG 𝐿))
46	38, 39, 45	3eqtrd 2802	. . . . . 6 ⊢ (((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)) → [(𝑁(._g‘(mulGrp‘𝑆))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~_QG 𝐿) = [((𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~_QG 𝐿))
47	46	ex 418	. . . . 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) → [(𝑁(._g‘(mulGrp‘𝑆))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~_QG 𝐿) = [((𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~_QG 𝐿)))
48	47	ralimdva 3178	. . . 4 ⊢ (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) → ∀𝑎 ∈ (1...𝐴)[(𝑁(._g‘(mulGrp‘𝑆))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~_QG 𝐿) = [((𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~_QG 𝐿)))
49	32, 48	mpd 16	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(._g‘(mulGrp‘𝑆))((var₁‘(ℤ/nℤ‘𝑁))(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~_QG 𝐿) = [((𝑁(._g‘(mulGrp‘𝑆))(var₁‘(ℤ/nℤ‘𝑁)))(+_g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~_QG 𝐿))
50	3, 2, 28, 29, 30, 5, 1, 31, 49	aks5lem5a 43507	. 2 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
51	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 50	aks6d1c7 43500	1 ⊢ (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 {csn 4588 class class class wbr 5108 {copab 5173 ↦ cmpt 5192 ‘cfv 6566 (class class class)co 7451 [cec 8753 ℝcr 11238 0cc0 11239 1c1 11240 · cmul 11244 < clt 11382 ≤ cle 11383 ℕcn 12375 2c2 12437 3c3 12438 ℤcz 12733 ℤ_≥cuz 13005 ...cfz 13679 ⌊cfl 13970 ↑cexp 14244 √csqrt 15443 ∥ cdvds 16466 gcd cgcd 16708 ℙcprime 16890 od_ℤcodz 16984 ϕcphi 16985 pCnt cpc 17058 Basecbs 17431 +_gcplusg 17472 /_s cqus 17727 -_gcsg 19196 ._gcmg 19327 ~_QG cqg 19382 mulGrpcmgp 20411 1_rcur 20458 RingIso crs 20751 Fieldcfield 21032 RSpancrsp 21536 ℤRHomczrh 21860 chrcchr 21862 ℤ/nℤczn 21863 var₁cv1 22549 Poly₁cpl1 22550 eval₁ce1 22687 log_b clogb 27149 PrimRoots cprimroots 43407

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318 ax-mulf 11319

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-ofr 7726 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-tpos 8281 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-er 8755 df-ec 8757 df-qs 8761 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-xnn0 12720 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-shft 15263 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-limsup 15682 df-clim 15699 df-rlim 15700 df-sum 15898 df-prod 16117 df-fallfac 16218 df-ef 16277 df-sin 16279 df-cos 16280 df-pi 16282 df-dvds 16467 df-gcd 16709 df-prm 16891 df-odz 16986 df-phi 16987 df-pc 17059 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-pws 17670 df-xrs 17724 df-qtop 17729 df-imas 17730 df-qus 17731 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-mhm 19028 df-submnd 19029 df-grp 19197 df-minusg 19198 df-sbg 19199 df-mulg 19328 df-subg 19383 df-nsg 19384 df-eqg 19385 df-ghm 19478 df-gim 19523 df-cntz 19582 df-od 19793 df-cmn 20047 df-abl 20048 df-mgp 20412 df-rng 20426 df-ur 20459 df-srg 20464 df-ring 20512 df-cring 20513 df-oppr 20618 df-dvdsr 20638 df-unit 20639 df-invr 20669 df-dvr 20682 df-rhm 20753 df-rim 20754 df-nzr 20814 df-subrng 20849 df-subrg 20873 df-rlreg 20997 df-domn 20998 df-idom 20999 df-drng 21033 df-field 21034 df-lmod 21188 df-lss 21258 df-lsp 21298 df-sra 21499 df-rgmod 21500 df-lidl 21537 df-rsp 21538 df-2idl 21594 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-zring 21807 df-zrh 21864 df-chr 21866 df-zn 21867 df-assa 22216 df-asp 22217 df-ascl 22218 df-psr 22272 df-mvr 22273 df-mpl 22274 df-opsr 22276 df-evls 22438 df-evl 22439 df-psr1 22553 df-vr1 22554 df-ply1 22555 df-coe1 22556 df-evls1 22688 df-evl1 22689 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-limc 26241 df-dv 26242 df-mdeg 26428 df-deg1 26429 df-mon1 26504 df-uc1p 26505 df-q1p 26506 df-r1p 26507 df-log 26941 df-cxp 26942 df-logb 27150 df-primroots 43408

This theorem is used by: aks5lem7 43516

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