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Mirrors > Home > MPE Home > Th. List > Mathboxes > aks5lem6 | Structured version Visualization version GIF version |
Description: Connect results of section 5 and Theorem 6.1 AKS. (Contributed by metakunt, 25-Jun-2025.) |
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 logb 𝑁))) |
aks5lem6.10 | ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘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)(1r‘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)) |
Ref | Expression |
---|---|
aks5lem6 | ⊢ (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁))) |
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 logb 𝑁))) | |
10 | aks5lem6.10 | . 2 ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘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 12886 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℤ) | |
15 | 6, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
16 | 0red 11262 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ) | |
17 | 3re 12344 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 3 ∈ ℝ) |
19 | 15 | zred 12720 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
20 | 3pos 12369 | . . . . . . . 8 ⊢ 0 < 3 | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 < 3) |
22 | eluzle 12889 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘3) → 3 ≤ 𝑁) | |
23 | 6, 22 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 3 ≤ 𝑁) |
24 | 16, 18, 19, 21, 23 | ltletrd 11419 | . . . . . 6 ⊢ (𝜑 → 0 < 𝑁) |
25 | 15, 24 | jca 511 | . . . . 5 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁)) |
26 | elnnz 12621 | . . . . 5 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁)) | |
27 | 25, 26 | sylibr 234 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
28 | 4, 27, 7 | 3jca 1127 | . . 3 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁)) |
29 | eqid 2735 | . . 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)(1r‘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 2744 | . . . . . . . . . . 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 7446 | . . . . . . . . 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 7447 | . . . . . . . 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 8784 | . . . . . . 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 2742 | . . . . . . . . . . . 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 7447 | . . . . . . . . 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 7446 | . . . . . . . 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 8784 | . . . . . . 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 2779 | . . . . . 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 3165 | . . . 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 42173 | . 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 42166 | 1 ⊢ (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {csn 4631 class class class wbr 5148 {copab 5210 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 [cec 8742 ℝcr 11152 0cc0 11153 1c1 11154 · cmul 11158 < clt 11293 ≤ cle 11294 ℕcn 12264 2c2 12319 3c3 12320 ℤcz 12611 ℤ≥cuz 12876 ...cfz 13544 ⌊cfl 13827 ↑cexp 14099 √csqrt 15269 ∥ cdvds 16287 gcd cgcd 16528 ℙcprime 16705 odℤcodz 16797 ϕcphi 16798 pCnt cpc 16870 Basecbs 17245 +gcplusg 17298 /s cqus 17552 -gcsg 18966 .gcmg 19098 ~QG cqg 19153 mulGrpcmgp 20152 1rcur 20199 RingIso crs 20487 Fieldcfield 20747 RSpancrsp 21235 ℤRHomczrh 21528 chrcchr 21530 ℤ/nℤczn 21531 var1cv1 22193 Poly1cpl1 22194 eval1ce1 22334 logb clogb 26822 PrimRoots cprimroots 42073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-prod 15937 df-fallfac 16040 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-dvds 16288 df-gcd 16529 df-prm 16706 df-odz 16799 df-phi 16800 df-pc 16871 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-pws 17496 df-xrs 17549 df-qtop 17554 df-imas 17555 df-qus 17556 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-nsg 19155 df-eqg 19156 df-ghm 19244 df-gim 19290 df-cntz 19348 df-od 19561 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-rim 20490 df-nzr 20530 df-subrng 20563 df-subrg 20587 df-rlreg 20711 df-domn 20712 df-idom 20713 df-drng 20748 df-field 20749 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-2idl 21278 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-chr 21534 df-zn 21535 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-evls 22116 df-evl 22117 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-coe1 22200 df-evls1 22335 df-evl1 22336 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-mdeg 26109 df-deg1 26110 df-mon1 26185 df-uc1p 26186 df-q1p 26187 df-r1p 26188 df-log 26613 df-cxp 26614 df-logb 26823 df-primroots 42074 |
This theorem is referenced by: aks5lem7 42182 |
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