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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 42181, 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 771 | . . . . 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 769 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑞 ∈ ℙ) | |
3 | 2 | ad2antrr 726 | . . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑞 ∈ ℙ) |
4 | prmnn 16707 | . . . . . . 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 726 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑅 ∈ ℕ) |
8 | 2, 4 | syl 17 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑞 ∈ ℕ) |
9 | 8 | nnzd 12636 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑞 ∈ ℤ) |
10 | 7 | nnzd 12636 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑅 ∈ ℤ) |
11 | 9, 10 | gcdcomd 16547 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → (𝑞 gcd 𝑅) = (𝑅 gcd 𝑞)) |
12 | aks5.5 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) | |
13 | 12 | ad2antrr 726 | . . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑁 ∈ (ℤ≥‘3)) |
14 | eluzelz 12884 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℤ) | |
15 | 13, 14 | syl 17 | . . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → 𝑁 ∈ ℤ) |
16 | 10, 9, 15 | 3jca 1129 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → (𝑅 ∈ ℤ ∧ 𝑞 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
17 | 10, 15 | gcdcomd 16547 | . . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → (𝑅 gcd 𝑁) = (𝑁 gcd 𝑅)) |
18 | aks5.7 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) | |
19 | 18 | ad2antrr 726 | . . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → (𝑁 gcd 𝑅) = 1) |
20 | 17, 19 | eqtrd 2776 | . . . . . . . . . . . 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 16693 | . . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℤ ∧ 𝑞 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑅 gcd 𝑁) = 1 ∧ 𝑞 ∥ 𝑁)) → (𝑅 gcd 𝑞) = 1) | |
24 | 16, 22, 23 | syl2anc 584 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → (𝑅 gcd 𝑞) = 1) |
25 | 11, 24 | eqtrd 2776 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → (𝑞 gcd 𝑅) = 1) |
26 | odzcl 16827 | . . . . . . . . 9 ⊢ ((𝑅 ∈ ℕ ∧ 𝑞 ∈ ℤ ∧ (𝑞 gcd 𝑅) = 1) → ((odℤ‘𝑅)‘𝑞) ∈ ℕ) | |
27 | 7, 9, 25, 26 | syl3anc 1373 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → ((odℤ‘𝑅)‘𝑞) ∈ ℕ) |
28 | 27 | ad2antrr 726 | . . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → ((odℤ‘𝑅)‘𝑞) ∈ ℕ) |
29 | 28 | nnnn0d 12583 | . . . . . 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 14280 | . . . . 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 2840 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → (♯‘(Base‘𝑘)) ∈ ℕ) |
32 | eqid 2736 | . . . 4 ⊢ (chr‘𝑘) = (chr‘𝑘) | |
33 | simplr 769 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑘 ∈ Field) | |
34 | simprr 773 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → (chr‘𝑘) = 𝑞) | |
35 | 34, 3 | eqeltrd 2840 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → (chr‘𝑘) ∈ ℙ) |
36 | 6 | ad4antr 732 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑅 ∈ ℕ) |
37 | 12 | ad4antr 732 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑁 ∈ (ℤ≥‘3)) |
38 | simpllr 776 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑞 ∥ 𝑁) | |
39 | 34, 38 | eqbrtrd 5163 | . . . 4 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → (chr‘𝑘) ∥ 𝑁) |
40 | 18 | ad4antr 732 | . . . 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 732 | . . . 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 12636 | . . . . . 6 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → 𝑞 ∈ ℤ) |
45 | 25 | ad2antrr 726 | . . . . . 6 ⊢ (((((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) ∧ 𝑘 ∈ Field) ∧ ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) → (𝑞 gcd 𝑅) = 1) |
46 | odzid 16828 | . . . . . 6 ⊢ ((𝑅 ∈ ℕ ∧ 𝑞 ∈ ℤ ∧ (𝑞 gcd 𝑅) = 1) → 𝑅 ∥ ((𝑞↑((odℤ‘𝑅)‘𝑞)) − 1)) | |
47 | 36, 44, 45, 46 | syl3anc 1373 | . . . . 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 2742 | . . . . . 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 7444 | . . . . 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 5167 | . . . 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 732 | . . . 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 732 | . . . 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 42180 | . . 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 42182 | . . 3 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑞↑((odℤ‘𝑅)‘𝑞)) ∧ (chr‘𝑘) = 𝑞)) |
60 | 58, 59 | r19.29a 3161 | . 2 ⊢ (((𝜑 ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ 𝑁) → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝↑𝑛)) |
61 | uzuzle23 12927 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ (ℤ≥‘2)) | |
62 | 12, 61 | syl 17 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2)) |
63 | exprmfct 16737 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → ∃𝑞 ∈ ℙ 𝑞 ∥ 𝑁) | |
64 | 62, 63 | syl 17 | . 2 ⊢ (𝜑 → ∃𝑞 ∈ ℙ 𝑞 ∥ 𝑁) |
65 | 60, 64 | r19.29a 3161 | 1 ⊢ (𝜑 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝↑𝑛)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3060 ∃wrex 3069 {csn 4624 class class class wbr 5141 ‘cfv 6559 (class class class)co 7429 [cec 8739 1c1 11152 · cmul 11156 < clt 11291 − cmin 11488 ℕcn 12262 2c2 12317 3c3 12318 ℤcz 12609 ℤ≥cuz 12874 ...cfz 13543 ⌊cfl 13826 ↑cexp 14098 ♯chash 14365 √csqrt 15268 ∥ cdvds 16286 gcd cgcd 16527 ℙcprime 16704 odℤcodz 16796 ϕcphi 16797 Basecbs 17243 +gcplusg 17293 -gcsg 18949 .gcmg 19081 ~QG cqg 19136 mulGrpcmgp 20133 1rcur 20174 Fieldcfield 20722 RSpancrsp 21209 ℤRHomczrh 21502 chrcchr 21504 ℤ/nℤczn 21505 var1cv1 22167 Poly1cpl1 22168 logb clogb 26797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-inf2 9677 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-pre-sup 11229 ax-addf 11230 ax-mulf 11231 ax-exfinfld 42181 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 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 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-disj 5109 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 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 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-of 7694 df-ofr 7695 df-om 7884 df-1st 8010 df-2nd 8011 df-supp 8182 df-tpos 8247 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-2o 8503 df-oadd 8506 df-omul 8507 df-er 8741 df-ec 8743 df-qs 8747 df-map 8864 df-pm 8865 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9398 df-fi 9447 df-sup 9478 df-inf 9479 df-oi 9546 df-dju 9937 df-card 9975 df-acn 9978 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-div 11917 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-xnn0 12596 df-z 12610 df-dec 12730 df-uz 12875 df-q 12987 df-rp 13031 df-xneg 13150 df-xadd 13151 df-xmul 13152 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-prod 15936 df-fallfac 16039 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-dvds 16287 df-gcd 16528 df-prm 16705 df-odz 16798 df-phi 16799 df-pc 16871 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-starv 17308 df-sca 17309 df-vsca 17310 df-ip 17311 df-tset 17312 df-ple 17313 df-ds 17315 df-unif 17316 df-hom 17317 df-cco 17318 df-rest 17463 df-topn 17464 df-0g 17482 df-gsum 17483 df-topgen 17484 df-pt 17485 df-prds 17488 df-pws 17490 df-xrs 17543 df-qtop 17548 df-imas 17549 df-qus 17550 df-xps 17551 df-mre 17625 df-mrc 17626 df-acs 17628 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-mhm 18792 df-submnd 18793 df-grp 18950 df-minusg 18951 df-sbg 18952 df-mulg 19082 df-subg 19137 df-nsg 19138 df-eqg 19139 df-ghm 19227 df-gim 19273 df-cntz 19331 df-od 19542 df-cmn 19796 df-abl 19797 df-mgp 20134 df-rng 20146 df-ur 20175 df-srg 20180 df-ring 20228 df-cring 20229 df-oppr 20326 df-dvdsr 20349 df-unit 20350 df-invr 20380 df-dvr 20393 df-rhm 20464 df-rim 20465 df-nzr 20505 df-subrng 20538 df-subrg 20562 df-rlreg 20686 df-domn 20687 df-idom 20688 df-drng 20723 df-field 20724 df-lmod 20852 df-lss 20922 df-lsp 20962 df-sra 21164 df-rgmod 21165 df-lidl 21210 df-rsp 21211 df-2idl 21252 df-psmet 21348 df-xmet 21349 df-met 21350 df-bl 21351 df-mopn 21352 df-fbas 21353 df-fg 21354 df-cnfld 21357 df-zring 21450 df-zrh 21506 df-chr 21508 df-zn 21509 df-assa 21865 df-asp 21866 df-ascl 21867 df-psr 21921 df-mvr 21922 df-mpl 21923 df-opsr 21925 df-evls 22090 df-evl 22091 df-psr1 22171 df-vr1 22172 df-ply1 22173 df-coe1 22174 df-evls1 22309 df-evl1 22310 df-top 22890 df-topon 22907 df-topsp 22929 df-bases 22943 df-cld 23017 df-ntr 23018 df-cls 23019 df-nei 23096 df-lp 23134 df-perf 23135 df-cn 23225 df-cnp 23226 df-haus 23313 df-tx 23560 df-hmeo 23753 df-fil 23844 df-fm 23936 df-flim 23937 df-flf 23938 df-xms 24320 df-ms 24321 df-tms 24322 df-cncf 24894 df-limc 25891 df-dv 25892 df-mdeg 26084 df-deg1 26085 df-mon1 26160 df-uc1p 26161 df-q1p 26162 df-r1p 26163 df-log 26588 df-cxp 26589 df-logb 26798 df-primroots 42071 |
This theorem is referenced by: (None) |
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