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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 12889 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℤ) | |
| 15 | 6, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) | 
| 16 | 0red 11265 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ) | |
| 17 | 3re 12347 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
| 18 | 17 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 3 ∈ ℝ) | 
| 19 | 15 | zred 12724 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ) | 
| 20 | 3pos 12372 | . . . . . . . 8 ⊢ 0 < 3 | |
| 21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 < 3) | 
| 22 | eluzle 12892 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘3) → 3 ≤ 𝑁) | |
| 23 | 6, 22 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 3 ≤ 𝑁) | 
| 24 | 16, 18, 19, 21, 23 | ltletrd 11422 | . . . . . 6 ⊢ (𝜑 → 0 < 𝑁) | 
| 25 | 15, 24 | jca 511 | . . . . 5 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁)) | 
| 26 | elnnz 12625 | . . . . 5 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁)) | |
| 27 | 25, 26 | sylibr 234 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| 28 | 4, 27, 7 | 3jca 1128 | . . 3 ⊢ (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁)) | 
| 29 | eqid 2736 | . . 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 2745 | . . . . . . . . . . 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 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)) → ((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)) = (𝑋(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))) | 
| 37 | 36 | oveq2d 7448 | . . . . . . . 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 8786 | . . . . . . 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 2743 | . . . . . . . . . . . 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 7448 | . . . . . . . . 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 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))𝑋)(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)) = ((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))) | 
| 45 | 44 | eceq1d 8786 | . . . . . . 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 2780 | . . . . . 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 3166 | . . . 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 42193 | . 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 42186 | 1 ⊢ (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 {csn 4625 class class class wbr 5142 {copab 5204 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 [cec 8744 ℝcr 11155 0cc0 11156 1c1 11157 · cmul 11161 < clt 11296 ≤ cle 11297 ℕcn 12267 2c2 12322 3c3 12323 ℤcz 12615 ℤ≥cuz 12879 ...cfz 13548 ⌊cfl 13831 ↑cexp 14103 √csqrt 15273 ∥ cdvds 16291 gcd cgcd 16532 ℙcprime 16709 odℤcodz 16801 ϕcphi 16802 pCnt cpc 16875 Basecbs 17248 +gcplusg 17298 /s cqus 17551 -gcsg 18954 .gcmg 19086 ~QG cqg 19141 mulGrpcmgp 20138 1rcur 20179 RingIso crs 20471 Fieldcfield 20731 RSpancrsp 21218 ℤRHomczrh 21511 chrcchr 21513 ℤ/nℤczn 21514 var1cv1 22178 Poly1cpl1 22179 eval1ce1 22319 logb clogb 26808 PrimRoots cprimroots 42093 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-ofr 7699 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-prod 15941 df-fallfac 16044 df-ef 16104 df-sin 16106 df-cos 16107 df-pi 16109 df-dvds 16292 df-gcd 16533 df-prm 16710 df-odz 16803 df-phi 16804 df-pc 16876 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 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 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-pws 17495 df-xrs 17548 df-qtop 17553 df-imas 17554 df-qus 17555 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-nsg 19143 df-eqg 19144 df-ghm 19232 df-gim 19278 df-cntz 19336 df-od 19547 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-srg 20185 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-dvr 20402 df-rhm 20473 df-rim 20474 df-nzr 20514 df-subrng 20547 df-subrg 20571 df-rlreg 20695 df-domn 20696 df-idom 20697 df-drng 20732 df-field 20733 df-lmod 20861 df-lss 20931 df-lsp 20971 df-sra 21173 df-rgmod 21174 df-lidl 21219 df-rsp 21220 df-2idl 21261 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-zring 21459 df-zrh 21515 df-chr 21517 df-zn 21518 df-assa 21874 df-asp 21875 df-ascl 21876 df-psr 21930 df-mvr 21931 df-mpl 21932 df-opsr 21934 df-evls 22099 df-evl 22100 df-psr1 22182 df-vr1 22183 df-ply1 22184 df-coe1 22185 df-evls1 22320 df-evl1 22321 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903 df-mdeg 26095 df-deg1 26096 df-mon1 26171 df-uc1p 26172 df-q1p 26173 df-r1p 26174 df-log 26599 df-cxp 26600 df-logb 26809 df-primroots 42094 | 
| This theorem is referenced by: aks5lem7 42202 | 
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