aks5lem6 - Mathbox for metakunt

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Theorem aks5lem6 42648

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 12792	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
15	6, 14	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
16		0red 11141	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
17		3re 12255	. . . . . . . 8 ⊢ 3 ∈ ℝ
18	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → 3 ∈ ℝ)
19	15	zred 12627	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
20		3pos 12280	. . . . . . . 8 ⊢ 0 < 3
21	20	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 3)
22		eluzle 12795	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
23	6, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → 3 ≤ 𝑁)
24	16, 18, 19, 21, 23	ltletrd 11300	. . . . . 6 ⊢ (𝜑 → 0 < 𝑁)
25	15, 24	jca 511	. . . . 5 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
26		elnnz 12528	. . . . 5 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
27	25, 26	sylibr 234	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
28	4, 27, 7	3jca 1129	. . 3 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))
29		eqid 2737	. . 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 2746	. . . . . . . . . . 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 7376	. . . . . . . . 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 7377	. . . . . . . 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 8678	. . . . . . 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 484	. . . . . . 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 2744	. . . . . . . . . . . 12 ⊢ ((var₁‘(ℤ/nℤ‘𝑁)) = 𝑋 ↔ 𝑋 = (var₁‘(ℤ/nℤ‘𝑁)))
41	40	imbi2i 336	. . . . . . . . . . 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 230	. . . . . . . . . 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 7377	. . . . . . . . 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 7376	. . . . . . . 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 8678	. . . . . . 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 2776	. . . . . 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 412	. . . . 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 3150	. . . 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 15	. . 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 42647	. 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 42640	1 ⊢ (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 {csn 4568 class class class wbr 5086 {copab 5148 ↦ cmpt 5167 ‘cfv 6493 (class class class)co 7361 [cec 8635 ℝcr 11031 0cc0 11032 1c1 11033 · cmul 11037 < clt 11173 ≤ cle 11174 ℕcn 12168 2c2 12230 3c3 12231 ℤcz 12518 ℤ_≥cuz 12782 ...cfz 13455 ⌊cfl 13743 ↑cexp 14017 √csqrt 15189 ∥ cdvds 16215 gcd cgcd 16457 ℙcprime 16634 od_ℤcodz 16727 ϕcphi 16728 pCnt cpc 16801 Basecbs 17173 +_gcplusg 17214 /_s cqus 17463 -_gcsg 18905 ._gcmg 19037 ~_QG cqg 19092 mulGrpcmgp 20115 1_rcur 20156 RingIso crs 20444 Fieldcfield 20701 RSpancrsp 21200 ℤRHomczrh 21492 chrcchr 21494 ℤ/nℤczn 21495 var₁cv1 22152 Poly₁cpl1 22153 eval₁ce1 22292 log_b clogb 26744 PrimRoots cprimroots 42547

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 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111 ax-mulf 11112

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-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-ofr 7626 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-ec 8639 df-qs 8643 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9819 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-xnn0 12505 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13296 df-ioc 13297 df-ico 13298 df-icc 13299 df-fz 13456 df-fzo 13603 df-fl 13745 df-mod 13823 df-seq 13958 df-exp 14018 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15023 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-limsup 15427 df-clim 15444 df-rlim 15445 df-sum 15643 df-prod 15863 df-fallfac 15966 df-ef 16026 df-sin 16028 df-cos 16029 df-pi 16031 df-dvds 16216 df-gcd 16458 df-prm 16635 df-odz 16729 df-phi 16730 df-pc 16802 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-hom 17238 df-cco 17239 df-rest 17379 df-topn 17380 df-0g 17398 df-gsum 17399 df-topgen 17400 df-pt 17401 df-prds 17404 df-pws 17406 df-xrs 17460 df-qtop 17465 df-imas 17466 df-qus 17467 df-xps 17468 df-mre 17542 df-mrc 17543 df-acs 17545 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-mhm 18745 df-submnd 18746 df-grp 18906 df-minusg 18907 df-sbg 18908 df-mulg 19038 df-subg 19093 df-nsg 19094 df-eqg 19095 df-ghm 19182 df-gim 19228 df-cntz 19286 df-od 19497 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-srg 20162 df-ring 20210 df-cring 20211 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-invr 20362 df-dvr 20375 df-rhm 20446 df-rim 20447 df-nzr 20484 df-subrng 20517 df-subrg 20541 df-rlreg 20665 df-domn 20666 df-idom 20667 df-drng 20702 df-field 20703 df-lmod 20851 df-lss 20921 df-lsp 20961 df-sra 21163 df-rgmod 21164 df-lidl 21201 df-rsp 21202 df-2idl 21243 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-zring 21440 df-zrh 21496 df-chr 21498 df-zn 21499 df-assa 21846 df-asp 21847 df-ascl 21848 df-psr 21902 df-mvr 21903 df-mpl 21904 df-opsr 21906 df-evls 22065 df-evl 22066 df-psr1 22156 df-vr1 22157 df-ply1 22158 df-coe1 22159 df-evls1 22293 df-evl1 22294 df-top 22872 df-topon 22889 df-topsp 22911 df-bases 22924 df-cld 22997 df-ntr 22998 df-cls 22999 df-nei 23076 df-lp 23114 df-perf 23115 df-cn 23205 df-cnp 23206 df-haus 23293 df-tx 23540 df-hmeo 23733 df-fil 23824 df-fm 23916 df-flim 23917 df-flf 23918 df-xms 24298 df-ms 24299 df-tms 24300 df-cncf 24858 df-limc 25846 df-dv 25847 df-mdeg 26033 df-deg1 26034 df-mon1 26109 df-uc1p 26110 df-q1p 26111 df-r1p 26112 df-log 26536 df-cxp 26537 df-logb 26745 df-primroots 42548

This theorem is referenced by: aks5lem7 42656

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