aks5lem6 - Mathbox for metakunt

	Mathbox for metakunt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > aks5lem6		Structured version Visualization version GIF version

Theorem aks5lem6 43379

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 12982	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
15	6, 14	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
16		0red 11318	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
17		3re 12430	. . . . . . . 8 ⊢ 3 ∈ ℝ
18	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → 3 ∈ ℝ)
19	15	zred 12810	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
20		3pos 12458	. . . . . . . 8 ⊢ 0 < 3
21	20	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 3)
22		eluzle 12985	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
23	6, 22	syl 18	. . . . . . 7 ⊢ (𝜑 → 3 ≤ 𝑁)
24	16, 18, 19, 21, 23	ltletrd 11477	. . . . . 6 ⊢ (𝜑 → 0 < 𝑁)
25	15, 24	jca 521	. . . . 5 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
26		elnnz 12710	. . . . 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 2760	. . 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 2769	. . . . . . . . . . 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 7439	. . . . . . . . 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 7440	. . . . . . . 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 8766	. . . . . . 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 2767	. . . . . . . . . . . 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 7440	. . . . . . . . 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 7439	. . . . . . . 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 8766	. . . . . . 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 2799	. . . . . 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 3174	. . . 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 43378	. 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 43371	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 1569 ∈ wcel 2144 ∀wral 3076 {csn 4584 class class class wbr 5102 {copab 5166 ↦ cmpt 5185 ‘cfv 6543 (class class class)co 7424 [cec 8723 ℝcr 11206 0cc0 11207 1c1 11208 · cmul 11212 < clt 11350 ≤ cle 11351 ℕcn 12342 2c2 12404 3c3 12405 ℤcz 12700 ℤ_≥cuz 12972 ...cfz 13646 ⌊cfl 13937 ↑cexp 14211 √csqrt 15409 ∥ cdvds 16431 gcd cgcd 16673 ℙcprime 16855 od_ℤcodz 16949 ϕcphi 16950 pCnt cpc 17023 Basecbs 17396 +_gcplusg 17437 /_s cqus 17686 -_gcsg 19155 ._gcmg 19286 ~_QG cqg 19341 mulGrpcmgp 20369 1_rcur 20416 RingIso crs 20709 Fieldcfield 20990 RSpancrsp 21494 ℤRHomczrh 21814 chrcchr 21816 ℤ/nℤczn 21817 var₁cv1 22503 Poly₁cpl1 22504 eval₁ce1 22641 log_b clogb 27101 PrimRoots cprimroots 43278

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-inf2 9649 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286 ax-mulf 11287

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-ofr 7698 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8185 df-tpos 8251 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-oadd 8488 df-er 8725 df-ec 8727 df-qs 8731 df-map 8857 df-pm 8858 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9362 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9511 df-dju 9989 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-xnn0 12687 df-z 12701 df-dec 12822 df-uz 12973 df-q 13083 df-rp 13128 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-ioo 13487 df-ioc 13488 df-ico 13489 df-icc 13490 df-fz 13647 df-fzo 13796 df-fl 13939 df-mod 14017 df-seq 14152 df-exp 14212 df-fac 14425 df-bc 14454 df-hash 14482 df-shft 15229 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-limsup 15647 df-clim 15664 df-rlim 15665 df-sum 15863 df-prod 16082 df-fallfac 16183 df-ef 16242 df-sin 16244 df-cos 16245 df-pi 16247 df-dvds 16432 df-gcd 16674 df-prm 16856 df-odz 16951 df-phi 16952 df-pc 17024 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-rest 17602 df-topn 17603 df-0g 17621 df-gsum 17622 df-topgen 17623 df-pt 17624 df-prds 17627 df-pws 17629 df-xrs 17683 df-qtop 17688 df-imas 17689 df-qus 17690 df-xps 17691 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-mhm 18987 df-submnd 18988 df-grp 19156 df-minusg 19157 df-sbg 19158 df-mulg 19287 df-subg 19342 df-nsg 19343 df-eqg 19344 df-ghm 19437 df-gim 19482 df-cntz 19540 df-od 19751 df-cmn 20005 df-abl 20006 df-mgp 20370 df-rng 20384 df-ur 20417 df-srg 20422 df-ring 20470 df-cring 20471 df-oppr 20576 df-dvdsr 20596 df-unit 20597 df-invr 20627 df-dvr 20640 df-rhm 20711 df-rim 20712 df-nzr 20772 df-subrng 20807 df-subrg 20831 df-rlreg 20955 df-domn 20956 df-idom 20957 df-drng 20991 df-field 20992 df-lmod 21146 df-lss 21216 df-lsp 21256 df-sra 21457 df-rgmod 21458 df-lidl 21495 df-rsp 21496 df-2idl 21552 df-psmet 21679 df-xmet 21680 df-met 21681 df-bl 21682 df-mopn 21683 df-fbas 21684 df-fg 21685 df-cnfld 21688 df-zring 21762 df-zrh 21818 df-chr 21820 df-zn 21821 df-assa 22170 df-asp 22171 df-ascl 22172 df-psr 22226 df-mvr 22227 df-mpl 22228 df-opsr 22230 df-evls 22392 df-evl 22393 df-psr1 22507 df-vr1 22508 df-ply1 22509 df-coe1 22510 df-evls1 22642 df-evl1 22643 df-top 23221 df-topon 23238 df-topsp 23260 df-bases 23273 df-cld 23346 df-ntr 23347 df-cls 23348 df-nei 23425 df-lp 23463 df-perf 23464 df-cn 23554 df-cnp 23555 df-haus 23642 df-tx 23890 df-hmeo 24083 df-fil 24174 df-fm 24266 df-flim 24267 df-flf 24268 df-xms 24648 df-ms 24649 df-tms 24650 df-cncf 25208 df-limc 26195 df-dv 26196 df-mdeg 26382 df-deg1 26383 df-mon1 26458 df-uc1p 26459 df-q1p 26460 df-r1p 26461 df-log 26893 df-cxp 26894 df-logb 27102 df-primroots 43279

This theorem is used by: aks5lem7 43387

W3C validator