![]() |
Mathbox for metakunt |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > aks5 | Structured version Visualization version GIF version |
Description: The AKS Primality test, given an integer 𝑁 greater than or equal to 3, find a coprime 𝑅 such that 𝑅 is big enough. Then, if a bunch of polynomial equalities in the residue ring hold then 𝑁 is a prime power. Currently depends on the axiom ax-exfinfld 42145, since we currently do not have the existence of finite fields in the database. (Contributed by metakunt, 16-Aug-2025.) |
Ref | Expression |
---|---|
aks5.1 | ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) |
aks5.2 | ⊢ 𝑋 = (var1‘(ℤ/nℤ‘𝑁)) |
aks5.3 | ⊢ 𝑆 = (Poly1‘(ℤ/nℤ‘𝑁)) |
aks5.4 | ⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))𝑋)(-g‘𝑆)(1r‘𝑆))}) |
aks5.5 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) |
aks5.6 | ⊢ (𝜑 → 𝑅 ∈ ℕ) |
aks5.7 | ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) |
aks5.8 | ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁)) |
aks5.9 | ⊢ (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)) |
aks5.10 | ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)(𝑎 gcd 𝑁) = 1) |
Ref | Expression |
---|---|
aks5 | ⊢ (𝜑 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝↑𝑛)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 770 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → (♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞))) | |
2 | simplr 768 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑞 ∈ ℙ) | |
3 | 2 | ad2antrr 725 | . . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑞 ∈ ℙ) |
4 | prmnn 16697 | . . . . . . 7 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑞 ∈ ℕ) |
6 | aks5.6 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℕ) | |
7 | 6 | ad2antrr 725 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑅 ∈ ℕ) |
8 | 2, 4 | syl 17 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑞 ∈ ℕ) |
9 | 8 | nnzd 12631 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑞 ∈ ℤ) |
10 | 7 | nnzd 12631 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑅 ∈ ℤ) |
11 | 9, 10 | gcdcomd 16537 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → (𝑞 gcd 𝑅) = (𝑅 gcd 𝑞)) |
12 | aks5.5 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) | |
13 | 12 | ad2antrr 725 | . . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑁 ∈ (ℤ≥‘3)) |
14 | eluzelz 12879 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℤ) | |
15 | 13, 14 | syl 17 | . . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑁 ∈ ℤ) |
16 | 10, 9, 15 | 3jca 1126 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → (𝑅 ∈ ℤ ∧ 𝑞 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
17 | 10, 15 | gcdcomd 16537 | . . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → (𝑅 gcd 𝑁) = (𝑁 gcd 𝑅)) |
18 | aks5.7 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) | |
19 | 18 | ad2antrr 725 | . . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → (𝑁 gcd 𝑅) = 1) |
20 | 17, 19 | eqtrd 2773 | . . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → (𝑅 gcd 𝑁) = 1) |
21 | simpr 484 | . . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑞 ∥ 𝑁) | |
22 | 20, 21 | jca 511 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → ((𝑅 gcd 𝑁) = 1 ∧ 𝑞 ∥ 𝑁)) |
23 | rpdvds 16683 | . . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℤ ∧ 𝑞 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑅 gcd 𝑁) = 1 ∧ 𝑞 ∥ 𝑁)) → (𝑅 gcd 𝑞) = 1) | |
24 | 16, 22, 23 | syl2anc 583 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → (𝑅 gcd 𝑞) = 1) |
25 | 11, 24 | eqtrd 2773 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → (𝑞 gcd 𝑅) = 1) |
26 | odzcl 16816 | . . . . . . . . 9 ⊢ ((𝑅 ∈ ℕ ∧ 𝑞 ∈ ℤ ∧ (𝑞 gcd 𝑅) = 1) → ((odℤ‘𝑅)‘𝑞) ∈ ℕ) | |
27 | 7, 9, 25, 26 | syl3anc 1369 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → ((odℤ‘𝑅)‘𝑞) ∈ ℕ) |
28 | 27 | ad2antrr 725 | . . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → ((odℤ‘𝑅)‘𝑞) ∈ ℕ) |
29 | 28 | nnnn0d 12578 | . . . . . 6 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → ((odℤ‘𝑅)‘𝑞) ∈ ℕ0) |
30 | 5, 29 | nnexpcld 14270 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → (𝑞↑((odℤ‘𝑅)‘𝑞)) ∈ ℕ) |
31 | 1, 30 | eqeltrd 2837 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → (♯‘(Base‘𝑘)) ∈ ℕ) |
32 | eqid 2733 | . . . 4 ⊢ (chr‘𝑘) = (chr‘𝑘) | |
33 | simplr 768 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑘 ∈ Field) | |
34 | simprr 772 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → (chr‘𝑘) = 𝑞) | |
35 | 34, 3 | eqeltrd 2837 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → (chr‘𝑘) ∈ ℙ) |
36 | 6 | ad4antr 731 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑅 ∈ ℕ) |
37 | 12 | ad4antr 731 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑁 ∈ (ℤ≥‘3)) |
38 | simpllr 775 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑞 ∥ 𝑁) | |
39 | 34, 38 | eqbrtrd 5171 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → (chr‘𝑘) ∥ 𝑁) |
40 | 18 | ad4antr 731 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → (𝑁 gcd 𝑅) = 1) |
41 | aks5.1 | . . . 4 ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁))) | |
42 | aks5.8 | . . . . 5 ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁)) | |
43 | 42 | ad4antr 731 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁)) |
44 | 5 | nnzd 12631 | . . . . . 6 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑞 ∈ ℤ) |
45 | 25 | ad2antrr 725 | . . . . . 6 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → (𝑞 gcd 𝑅) = 1) |
46 | odzid 16817 | . . . . . 6 ⊢ ((𝑅 ∈ ℕ ∧ 𝑞 ∈ ℤ ∧ (𝑞 gcd 𝑅) = 1) → 𝑅 ∥ ((𝑞↑((odℤ‘𝑅)‘𝑞)) − 1)) | |
47 | 36, 44, 45, 46 | syl3anc 1369 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑅 ∥ ((𝑞↑((odℤ‘𝑅)‘𝑞)) − 1)) |
48 | 1 | eqcomd 2739 | . . . . . 6 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → (𝑞↑((odℤ‘𝑅)‘𝑞)) = (♯‘(Base‘𝑘))) |
49 | 48 | oveq1d 7440 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → ((𝑞↑((odℤ‘𝑅)‘𝑞)) − 1) = ((♯‘(Base‘𝑘)) − 1)) |
50 | 47, 49 | breqtrd 5175 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑅 ∥ ((♯‘(Base‘𝑘)) − 1)) |
51 | aks5.9 | . . . . 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)) | |
52 | 51 | ad4antr 731 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → ∀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)) |
53 | aks5.10 | . . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)(𝑎 gcd 𝑁) = 1) | |
54 | 53 | ad4antr 731 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → ∀𝑎 ∈ (1...𝐴)(𝑎 gcd 𝑁) = 1) |
55 | aks5.3 | . . . 4 ⊢ 𝑆 = (Poly1‘(ℤ/nℤ‘𝑁)) | |
56 | aks5.4 | . . . 4 ⊢ 𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))𝑋)(-g‘𝑆)(1r‘𝑆))}) | |
57 | aks5.2 | . . . 4 ⊢ 𝑋 = (var1‘(ℤ/nℤ‘𝑁)) | |
58 | 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 50, 52, 54, 55, 56, 57 | aks5lem8 42144 | . . 3 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝↑𝑛)) |
59 | 2, 27 | exfinfldd 42146 | . . 3 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) |
60 | 58, 59 | r19.29a 3158 | . 2 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝↑𝑛)) |
61 | uzuzle23 12922 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ (ℤ≥‘2)) | |
62 | 12, 61 | syl 17 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2)) |
63 | exprmfct 16727 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → ∃𝑞 ∈ ℙ 𝑞 ∥ 𝑁) | |
64 | 62, 63 | syl 17 | . 2 ⊢ (𝜑 → ∃𝑞 ∈ ℙ 𝑞 ∥ 𝑁) |
65 | 60, 64 | r19.29a 3158 | 1 ⊢ (𝜑 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝↑𝑛)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1535 ∈ wcel 2104 ∀wral 3057 ∃wrex 3066 {csn 4630 class class class wbr 5149 ‘cfv 6558 (class class class)co 7425 [cec 8736 1c1 11147 · cmul 11151 < clt 11286 − cmin 11483 ℕcn 12257 2c2 12312 3c3 12313 ℤcz 12604 ℤ≥cuz 12869 ...cfz 13537 ⌊cfl 13816 ↑cexp 14088 ♯chash 14355 √csqrt 15258 ∥ cdvds 16276 gcd cgcd 16517 ℙcprime 16694 odℤcodz 16786 ϕcphi 16787 Basecbs 17234 +gcplusg 17287 -gcsg 18951 .gcmg 19083 ~QG cqg 19138 mulGrpcmgp 20137 1rcur 20184 Fieldcfield 20728 RSpancrsp 21216 ℤRHomczrh 21509 chrcchr 21511 ℤ/nℤczn 21512 var1cv1 22174 Poly1cpl1 22175 logb clogb 26803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226 ax-exfinfld 42145 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 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 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4915 df-int 4954 df-iun 5000 df-iin 5001 df-disj 5117 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-of 7691 df-ofr 7692 df-om 7881 df-1st 8007 df-2nd 8008 df-supp 8179 df-tpos 8244 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-1o 8499 df-2o 8500 df-oadd 8503 df-omul 8504 df-er 8738 df-ec 8740 df-qs 8744 df-map 8861 df-pm 8862 df-ixp 8931 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-acn 9973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11485 df-neg 11486 df-div 11912 df-nn 12258 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12518 df-xnn0 12591 df-z 12605 df-dec 12725 df-uz 12870 df-q 12982 df-rp 13026 df-xneg 13145 df-xadd 13146 df-xmul 13147 df-ioo 13381 df-ioc 13382 df-ico 13383 df-icc 13384 df-fz 13538 df-fzo 13682 df-fl 13818 df-mod 13896 df-seq 14029 df-exp 14089 df-fac 14299 df-bc 14328 df-hash 14356 df-shft 15092 df-cj 15124 df-re 15125 df-im 15126 df-sqrt 15260 df-abs 15261 df-limsup 15493 df-clim 15510 df-rlim 15511 df-sum 15709 df-prod 15926 df-fallfac 16029 df-ef 16089 df-sin 16091 df-cos 16092 df-pi 16094 df-dvds 16277 df-gcd 16518 df-prm 16695 df-odz 16788 df-phi 16789 df-pc 16860 df-struct 17170 df-sets 17187 df-slot 17205 df-ndx 17217 df-base 17235 df-ress 17264 df-plusg 17300 df-mulr 17301 df-starv 17302 df-sca 17303 df-vsca 17304 df-ip 17305 df-tset 17306 df-ple 17307 df-ds 17309 df-unif 17310 df-hom 17311 df-cco 17312 df-rest 17458 df-topn 17459 df-0g 17477 df-gsum 17478 df-topgen 17479 df-pt 17480 df-prds 17483 df-pws 17485 df-xrs 17538 df-qtop 17543 df-imas 17544 df-qus 17545 df-xps 17546 df-mre 17620 df-mrc 17621 df-acs 17623 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18794 df-submnd 18795 df-grp 18952 df-minusg 18953 df-sbg 18954 df-mulg 19084 df-subg 19139 df-nsg 19140 df-eqg 19141 df-ghm 19229 df-gim 19275 df-cntz 19333 df-od 19546 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20156 df-ur 20185 df-srg 20190 df-ring 20238 df-cring 20239 df-oppr 20336 df-dvdsr 20359 df-unit 20360 df-invr 20390 df-dvr 20403 df-rhm 20474 df-rim 20475 df-nzr 20515 df-subrng 20548 df-subrg 20573 df-rlreg 20692 df-domn 20693 df-idom 20694 df-drng 20729 df-field 20730 df-lmod 20858 df-lss 20929 df-lsp 20969 df-sra 21171 df-rgmod 21172 df-lidl 21217 df-rsp 21218 df-2idl 21259 df-psmet 21355 df-xmet 21356 df-met 21357 df-bl 21358 df-mopn 21359 df-fbas 21360 df-fg 21361 df-cnfld 21364 df-zring 21457 df-zrh 21513 df-chr 21515 df-zn 21516 df-assa 21872 df-asp 21873 df-ascl 21874 df-psr 21928 df-mvr 21929 df-mpl 21930 df-opsr 21932 df-evls 22097 df-evl 22098 df-psr1 22178 df-vr1 22179 df-ply1 22180 df-coe1 22181 df-evls1 22316 df-evl1 22317 df-top 22897 df-topon 22914 df-topsp 22936 df-bases 22950 df-cld 23024 df-ntr 23025 df-cls 23026 df-nei 23103 df-lp 23141 df-perf 23142 df-cn 23232 df-cnp 23233 df-haus 23320 df-tx 23567 df-hmeo 23760 df-fil 23851 df-fm 23943 df-flim 23944 df-flf 23945 df-xms 24327 df-ms 24328 df-tms 24329 df-cncf 24899 df-limc 25897 df-dv 25898 df-mdeg 26090 df-deg1 26091 df-mon1 26166 df-uc1p 26167 df-q1p 26168 df-r1p 26169 df-log 26594 df-cxp 26595 df-logb 26804 df-primroots 42035 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |