aks5lem6 - Mathbox for metakunt

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Theorem aks5lem6 43017

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 12888	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
15	6, 14	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
16		0red 11226	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
17		3re 12336	. . . . . . . 8 ⊢ 3 ∈ ℝ
18	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → 3 ∈ ℝ)
19	15	zred 12716	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
20		3pos 12364	. . . . . . . 8 ⊢ 0 < 3
21	20	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 3)
22		eluzle 12891	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
23	6, 22	syl 18	. . . . . . 7 ⊢ (𝜑 → 3 ≤ 𝑁)
24	16, 18, 19, 21, 23	ltletrd 11385	. . . . . 6 ⊢ (𝜑 → 0 < 𝑁)
25	15, 24	jca 521	. . . . 5 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
26		elnnz 12616	. . . . 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 2765	. . 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 2774	. . . . . . . . . . 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 7434	. . . . . . . . 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 7435	. . . . . . . 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 8741	. . . . . . 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 2772	. . . . . . . . . . . 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 7435	. . . . . . . . 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 7434	. . . . . . . 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 8741	. . . . . . 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 2804	. . . . . 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 3179	. . . 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 43016	. 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 43009	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 2146 ∀wral 3081 {csn 4591 class class class wbr 5111 {copab 5175 ↦ cmpt 5194 ‘cfv 6540 (class class class)co 7419 [cec 8698 ℝcr 11114 0cc0 11115 1c1 11116 · cmul 11120 < clt 11258 ≤ cle 11259 ℕcn 12248 2c2 12310 3c3 12311 ℤcz 12606 ℤ_≥cuz 12878 ...cfz 13551 ⌊cfl 13841 ↑cexp 14115 √csqrt 15308 ∥ cdvds 16332 gcd cgcd 16574 ℙcprime 16751 od_ℤcodz 16844 ϕcphi 16845 pCnt cpc 16918 Basecbs 17291 +_gcplusg 17332 /_s cqus 17581 -_gcsg 19046 ._gcmg 19177 ~_QG cqg 19232 mulGrpcmgp 20260 1_rcur 20307 RingIso crs 20598 Fieldcfield 20878 RSpancrsp 21381 ℤRHomczrh 21699 chrcchr 21701 ℤ/nℤczn 21702 var₁cv1 22386 Poly₁cpl1 22387 eval₁ce1 22524 log_b clogb 26980 PrimRoots cprimroots 42916

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194 ax-mulf 11195

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 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-ofr 7685 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-er 8700 df-ec 8702 df-qs 8706 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-dju 9903 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-xnn0 12593 df-z 12607 df-dec 12728 df-uz 12879 df-q 12989 df-rp 13033 df-xneg 13153 df-xadd 13154 df-xmul 13155 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13552 df-fzo 13700 df-fl 13843 df-mod 13921 df-seq 14056 df-exp 14116 df-fac 14328 df-bc 14357 df-hash 14385 df-shft 15128 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-limsup 15546 df-clim 15563 df-rlim 15564 df-sum 15762 df-prod 15981 df-fallfac 16084 df-ef 16143 df-sin 16145 df-cos 16146 df-pi 16148 df-dvds 16333 df-gcd 16575 df-prm 16752 df-odz 16846 df-phi 16847 df-pc 16919 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-starv 17347 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-hom 17356 df-cco 17357 df-rest 17497 df-topn 17498 df-0g 17516 df-gsum 17517 df-topgen 17518 df-pt 17519 df-prds 17522 df-pws 17524 df-xrs 17578 df-qtop 17583 df-imas 17584 df-qus 17585 df-xps 17586 df-mre 17660 df-mrc 17661 df-acs 17663 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-mhm 18878 df-submnd 18879 df-grp 19047 df-minusg 19048 df-sbg 19049 df-mulg 19178 df-subg 19233 df-nsg 19234 df-eqg 19235 df-ghm 19328 df-gim 19373 df-cntz 19431 df-od 19642 df-cmn 19896 df-abl 19897 df-mgp 20261 df-rng 20275 df-ur 20308 df-srg 20313 df-ring 20361 df-cring 20362 df-oppr 20465 df-dvdsr 20485 df-unit 20486 df-invr 20516 df-dvr 20529 df-rhm 20600 df-rim 20601 df-nzr 20660 df-subrng 20695 df-subrg 20719 df-rlreg 20843 df-domn 20844 df-idom 20845 df-drng 20879 df-field 20880 df-lmod 21033 df-lss 21103 df-lsp 21143 df-sra 21344 df-rgmod 21345 df-lidl 21382 df-rsp 21383 df-2idl 21439 df-psmet 21564 df-xmet 21565 df-met 21566 df-bl 21567 df-mopn 21568 df-fbas 21569 df-fg 21570 df-cnfld 21573 df-zring 21647 df-zrh 21703 df-chr 21705 df-zn 21706 df-assa 22053 df-asp 22054 df-ascl 22055 df-psr 22109 df-mvr 22110 df-mpl 22111 df-opsr 22113 df-evls 22275 df-evl 22276 df-psr1 22390 df-vr1 22391 df-ply1 22392 df-coe1 22393 df-evls1 22525 df-evl1 22526 df-top 23101 df-topon 23118 df-topsp 23140 df-bases 23153 df-cld 23226 df-ntr 23227 df-cls 23228 df-nei 23305 df-lp 23343 df-perf 23344 df-cn 23434 df-cnp 23435 df-haus 23522 df-tx 23770 df-hmeo 23963 df-fil 24054 df-fm 24146 df-flim 24147 df-flf 24148 df-xms 24528 df-ms 24529 df-tms 24530 df-cncf 25088 df-limc 26076 df-dv 26077 df-mdeg 26263 df-deg1 26264 df-mon1 26339 df-uc1p 26340 df-q1p 26341 df-r1p 26342 df-log 26772 df-cxp 26773 df-logb 26981 df-primroots 42917

This theorem is used by: aks5lem7 43025

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