![]() |
Metamath
Proof Explorer Theorem List (p. 421 of 491) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30946) |
![]() (30947-32469) |
![]() (32470-49035) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lcm8un 42001 | Least common multiple of natural numbers up to 8 equals 840. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...8)) = ;;840 | ||
Theorem | 3factsumint1 42002* | Move constants out of integrals or sums and/or commute sum and integral. (Contributed by metakunt, 26-Apr-2024.) |
⊢ 𝐴 = (𝐿[,]𝑈) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ)) ⇒ ⊢ (𝜑 → ∫𝐴Σ𝑘 ∈ 𝐵 (𝐹 · (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴(𝐹 · (𝐺 · 𝐻)) d𝑥) | ||
Theorem | 3factsumint2 42003* | Move constants out of integrals or sums and/or commute sum and integral. (Contributed by metakunt, 26-Apr-2024.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐵 ∫𝐴(𝐹 · (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴(𝐺 · (𝐹 · 𝐻)) d𝑥) | ||
Theorem | 3factsumint3 42004* | Move constants out of integrals or sums and/or commute sum and integral. (Contributed by metakunt, 26-Apr-2024.) |
⊢ 𝐴 = (𝐿[,]𝑈) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ)) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐵 ∫𝐴(𝐺 · (𝐹 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 (𝐺 · ∫𝐴(𝐹 · 𝐻) d𝑥)) | ||
Theorem | 3factsumint4 42005* | Move constants out of integrals or sums and/or commute sum and integral. (Contributed by metakunt, 26-Apr-2024.) |
⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ) ⇒ ⊢ (𝜑 → ∫𝐴Σ𝑘 ∈ 𝐵 (𝐹 · (𝐺 · 𝐻)) d𝑥 = ∫𝐴(𝐹 · Σ𝑘 ∈ 𝐵 (𝐺 · 𝐻)) d𝑥) | ||
Theorem | 3factsumint 42006* | Helpful equation for lcm inequality proof. (Contributed by metakunt, 26-Apr-2024.) |
⊢ 𝐴 = (𝐿[,]𝑈) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐿 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ)) ⇒ ⊢ (𝜑 → ∫𝐴(𝐹 · Σ𝑘 ∈ 𝐵 (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 (𝐺 · ∫𝐴(𝐹 · 𝐻) d𝑥)) | ||
Theorem | resopunitintvd 42007 | Restrict continuous function on open unit interval. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐴) ∈ (ℂ–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ (0(,)1) ↦ 𝐴) ∈ ((0(,)1)–cn→ℂ)) | ||
Theorem | resclunitintvd 42008 | Restrict continuous function on closed unit interval. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐴) ∈ (ℂ–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ (0[,]1) ↦ 𝐴) ∈ ((0[,]1)–cn→ℂ)) | ||
Theorem | resdvopclptsd 42009* | Restrict derivative on unit interval. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℝ D (𝑥 ∈ (0[,]1) ↦ 𝐴)) = (𝑥 ∈ (0(,)1) ↦ 𝐵)) | ||
Theorem | lcmineqlem1 42010* | Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 29-Apr-2024.) |
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) d𝑥) | ||
Theorem | lcmineqlem2 42011* | Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 29-Apr-2024.) |
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥)) | ||
Theorem | lcmineqlem3 42012* | Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 30-Apr-2024.) |
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (1 / (𝑀 + 𝑘)))) | ||
Theorem | lcmineqlem4 42013 | Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. F is found in lcmineqlem6 42015. (Contributed by metakunt, 10-May-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) & ⊢ (𝜑 → 𝐾 ∈ (0...(𝑁 − 𝑀))) ⇒ ⊢ (𝜑 → ((lcm‘(1...𝑁)) / (𝑀 + 𝐾)) ∈ ℤ) | ||
Theorem | lcmineqlem5 42014 | Technical lemma for reciprocal multiplication in deduction form. (Contributed by metakunt, 10-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 · (𝐵 · (1 / 𝐶))) = (𝐵 · (𝐴 / 𝐶))) | ||
Theorem | lcmineqlem6 42015* | Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 10-May-2024.) |
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → ((lcm‘(1...𝑁)) · 𝐹) ∈ ℤ) | ||
Theorem | lcmineqlem7 42016 | Derivative of 1-x for chain rule application. (Contributed by metakunt, 12-May-2024.) |
⊢ (ℂ D (𝑥 ∈ ℂ ↦ (1 − 𝑥))) = (𝑥 ∈ ℂ ↦ -1) | ||
Theorem | lcmineqlem8 42017* | Derivative of (1-x)^(N-M). (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 < 𝑁) ⇒ ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ ((1 − 𝑥)↑(𝑁 − 𝑀)))) = (𝑥 ∈ ℂ ↦ (-(𝑁 − 𝑀) · ((1 − 𝑥)↑((𝑁 − 𝑀) − 1))))) | ||
Theorem | lcmineqlem9 42018* | (1-x)^(N-M) is continuous. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((1 − 𝑥)↑(𝑁 − 𝑀))) ∈ (ℂ–cn→ℂ)) | ||
Theorem | lcmineqlem10 42019* | Induction step of lcmineqlem13 42022 (deduction form). (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 < 𝑁) ⇒ ⊢ (𝜑 → ∫(0[,]1)((𝑥↑((𝑀 + 1) − 1)) · ((1 − 𝑥)↑(𝑁 − (𝑀 + 1)))) d𝑥 = ((𝑀 / (𝑁 − 𝑀)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥)) | ||
Theorem | lcmineqlem11 42020 | Induction step, continuation for binomial coefficients. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 < 𝑁) ⇒ ⊢ (𝜑 → (1 / ((𝑀 + 1) · (𝑁C(𝑀 + 1)))) = ((𝑀 / (𝑁 − 𝑀)) · (1 / (𝑀 · (𝑁C𝑀))))) | ||
Theorem | lcmineqlem12 42021* | Base case for induction. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ∫(0[,]1)((𝑡↑(1 − 1)) · ((1 − 𝑡)↑(𝑁 − 1))) d𝑡 = (1 / (1 · (𝑁C1)))) | ||
Theorem | lcmineqlem13 42022* | Induction proof for lcm integral. (Contributed by metakunt, 12-May-2024.) |
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝐹 = (1 / (𝑀 · (𝑁C𝑀)))) | ||
Theorem | lcmineqlem14 42023 | Technical lemma for inequality estimate. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℕ) & ⊢ (𝜑 → 𝐷 ∈ ℕ) & ⊢ (𝜑 → 𝐸 ∈ ℕ) & ⊢ (𝜑 → (𝐴 · 𝐶) ∥ 𝐷) & ⊢ (𝜑 → (𝐵 · 𝐶) ∥ 𝐸) & ⊢ (𝜑 → 𝐷 ∥ 𝐸) & ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) ∥ 𝐸) | ||
Theorem | lcmineqlem15 42024* | F times the least common multiple of 1 to n is a natural number. (Contributed by metakunt, 10-May-2024.) |
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → ((lcm‘(1...𝑁)) · 𝐹) ∈ ℕ) | ||
Theorem | lcmineqlem16 42025 | Technical divisibility lemma. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → (𝑀 · (𝑁C𝑀)) ∥ (lcm‘(1...𝑁))) | ||
Theorem | lcmineqlem17 42026 | Inequality of 2^{2n}. (Contributed by metakunt, 29-Apr-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (2↑(2 · 𝑁)) ≤ (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))) | ||
Theorem | lcmineqlem18 42027 | Technical lemma to shift factors in binomial coefficient. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ((𝑁 + 1) · (((2 · 𝑁) + 1)C(𝑁 + 1))) = (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁))) | ||
Theorem | lcmineqlem19 42028 | Dividing implies inequality for lcm inequality lemma. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ((𝑁 · ((2 · 𝑁) + 1)) · ((2 · 𝑁)C𝑁)) ∥ (lcm‘(1...((2 · 𝑁) + 1)))) | ||
Theorem | lcmineqlem20 42029 | Inequality for lcm lemma. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝑁 · (2↑(2 · 𝑁))) ≤ (lcm‘(1...((2 · 𝑁) + 1)))) | ||
Theorem | lcmineqlem21 42030 | The lcm inequality lemma without base cases 7 and 8. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 4 ≤ 𝑁) ⇒ ⊢ (𝜑 → (2↑((2 · 𝑁) + 2)) ≤ (lcm‘(1...((2 · 𝑁) + 1)))) | ||
Theorem | lcmineqlem22 42031 | The lcm inequality lemma without base cases 7 and 8. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 4 ≤ 𝑁) ⇒ ⊢ (𝜑 → ((2↑((2 · 𝑁) + 1)) ≤ (lcm‘(1...((2 · 𝑁) + 1))) ∧ (2↑((2 · 𝑁) + 2)) ≤ (lcm‘(1...((2 · 𝑁) + 2))))) | ||
Theorem | lcmineqlem23 42032 | Penultimate step to the lcm inequality lemma. (Contributed by metakunt, 12-May-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 9 ≤ 𝑁) ⇒ ⊢ (𝜑 → (2↑𝑁) ≤ (lcm‘(1...𝑁))) | ||
Theorem | lcmineqlem 42033 | The least common multiple inequality lemma, a central result for future use. Theorem 3.1 from https://www3.nd.edu/%7eandyp/notes/AKS.pdf (Contributed by metakunt, 16-May-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 7 ≤ 𝑁) ⇒ ⊢ (𝜑 → (2↑𝑁) ≤ (lcm‘(1...𝑁))) | ||
Theorem | 3exp7 42034 | 3 to the power of 7 equals 2187. (Contributed by metakunt, 21-Aug-2024.) |
⊢ (3↑7) = ;;;2187 | ||
Theorem | 3lexlogpow5ineq1 42035 | First inequality in inequality chain, proposed by Mario Carneiro (Contributed by metakunt, 22-May-2024.) |
⊢ 9 < ((;11 / 7)↑5) | ||
Theorem | 3lexlogpow5ineq2 42036 | Second inequality in inequality chain, proposed by Mario Carneiro. (Contributed by metakunt, 22-May-2024.) |
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 3 ≤ 𝑋) ⇒ ⊢ (𝜑 → ((;11 / 7)↑5) ≤ ((2 logb 𝑋)↑5)) | ||
Theorem | 3lexlogpow5ineq4 42037 | Sharper logarithm inequality chain. (Contributed by metakunt, 21-Aug-2024.) |
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 3 ≤ 𝑋) ⇒ ⊢ (𝜑 → 9 < ((2 logb 𝑋)↑5)) | ||
Theorem | 3lexlogpow5ineq3 42038 | Combined inequality chain for a specific power of the binary logarithm, proposed by Mario Carneiro. (Contributed by metakunt, 22-May-2024.) |
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 3 ≤ 𝑋) ⇒ ⊢ (𝜑 → 7 < ((2 logb 𝑋)↑5)) | ||
Theorem | 3lexlogpow2ineq1 42039 | Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.) |
⊢ ((3 / 2) < (2 logb 3) ∧ (2 logb 3) < (5 / 3)) | ||
Theorem | 3lexlogpow2ineq2 42040 | Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.) |
⊢ (2 < ((2 logb 3)↑2) ∧ ((2 logb 3)↑2) < 3) | ||
Theorem | 3lexlogpow5ineq5 42041 | Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.) |
⊢ ((2 logb 3)↑5) ≤ ;15 | ||
Theorem | intlewftc 42042* | Inequality inference by invoking fundamental theorem of calculus. (Contributed by metakunt, 22-Jul-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → 𝐷 = (ℝ D 𝐹)) & ⊢ (𝜑 → 𝐸 = (ℝ D 𝐺)) & ⊢ (𝜑 → 𝐷 ∈ ((𝐴(,)𝐵)–cn→ℝ)) & ⊢ (𝜑 → 𝐸 ∈ ((𝐴(,)𝐵)–cn→ℝ)) & ⊢ (𝜑 → 𝐷 ∈ 𝐿1) & ⊢ (𝜑 → 𝐸 ∈ 𝐿1) & ⊢ (𝜑 → 𝐷 = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃)) & ⊢ (𝜑 → 𝐸 = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑃 ≤ 𝑄) & ⊢ (𝜑 → (𝐹‘𝐴) ≤ (𝐺‘𝐴)) ⇒ ⊢ (𝜑 → (𝐹‘𝐵) ≤ (𝐺‘𝐵)) | ||
Theorem | aks4d1lem1 42043 | Technical lemma to reduce proof size. (Contributed by metakunt, 14-Nov-2024.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) ⇒ ⊢ (𝜑 → (𝐵 ∈ ℕ ∧ 9 < 𝐵)) | ||
Theorem | aks4d1p1p1 42044* | Exponential law for finite products, special case. (Contributed by metakunt, 22-Jul-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ (1...𝑁)(𝐴↑𝑐𝑘) = (𝐴↑𝑐Σ𝑘 ∈ (1...𝑁)𝑘)) | ||
Theorem | dvrelog2 42045* | The derivative of the logarithm, ftc2 26099 version. (Contributed by metakunt, 11-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (log‘𝑥)) & ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) ⇒ ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺) | ||
Theorem | dvrelog3 42046* | The derivative of the logarithm on an open interval. (Contributed by metakunt, 11-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)) & ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) ⇒ ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺) | ||
Theorem | dvrelog2b 42047* | Derivative of the binary logarithm. (Contributed by metakunt, 11-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 logb 𝑥)) & ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) ⇒ ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺) | ||
Theorem | 0nonelalab 42048 | Technical lemma for open interval. (Contributed by metakunt, 12-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) ⇒ ⊢ (𝜑 → 0 ≠ 𝐶) | ||
Theorem | dvrelogpow2b 42049* | Derivative of the power of the binary logarithm. (Contributed by metakunt, 12-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 logb 𝑥)↑𝑁)) & ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥))) & ⊢ 𝐶 = (𝑁 / ((log‘2)↑𝑁)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺) | ||
Theorem | aks4d1p1p3 42050 | Bound of a ceiling of the binary logarithm to the fifth power. (Contributed by metakunt, 19-Aug-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) & ⊢ (𝜑 → 3 ≤ 𝑁) ⇒ ⊢ (𝜑 → (𝑁↑𝑐(⌊‘(2 logb 𝐵))) < (𝑁↑𝑐(2 logb (((2 logb 𝑁)↑5) + 1)))) | ||
Theorem | aks4d1p1p2 42051* | Rewrite 𝐴 in more suitable form. (Contributed by metakunt, 19-Aug-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) & ⊢ (𝜑 → 3 ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝐴 < (𝑁↑𝑐(((2 logb (((2 logb 𝑁)↑5) + 1)) + (((2 logb 𝑁)↑2) / 2)) + (((2 logb 𝑁)↑4) / 2)))) | ||
Theorem | aks4d1p1p4 42052* | Technical step for inequality. The hard work is in to prove the final hypothesis. (Contributed by metakunt, 19-Aug-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) & ⊢ (𝜑 → 3 ≤ 𝑁) & ⊢ 𝐶 = (2 logb (((2 logb 𝑁)↑5) + 1)) & ⊢ 𝐷 = ((2 logb 𝑁)↑2) & ⊢ 𝐸 = ((2 logb 𝑁)↑4) & ⊢ (𝜑 → ((2 · 𝐶) + 𝐷) ≤ 𝐸) ⇒ ⊢ (𝜑 → 𝐴 < (2↑𝐵)) | ||
Theorem | dvle2 42053* | Collapsed dvle 26060. (Contributed by metakunt, 19-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐸)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐹)) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐺)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐻)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐹 ≤ 𝐻) & ⊢ (𝑥 = 𝐴 → 𝐸 = 𝑃) & ⊢ (𝑥 = 𝐴 → 𝐺 = 𝑄) & ⊢ (𝑥 = 𝐵 → 𝐸 = 𝑅) & ⊢ (𝑥 = 𝐵 → 𝐺 = 𝑆) & ⊢ (𝜑 → 𝑃 ≤ 𝑄) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝑅 ≤ 𝑆) | ||
Theorem | aks4d1p1p6 42054* | Inequality lift to differentiable functions for a term in AKS inequality lemma. (Contributed by metakunt, 19-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 3 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 · (2 logb (((2 logb 𝑥)↑5) + 1))) + ((2 logb 𝑥)↑2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 · ((1 / ((((2 logb 𝑥)↑5) + 1) · (log‘2))) · (((5 · ((2 logb 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0))) + ((2 / ((log‘2)↑2)) · (((log‘𝑥)↑(2 − 1)) / 𝑥))))) | ||
Theorem | aks4d1p1p7 42055 | Bound of intermediary of inequality step. (Contributed by metakunt, 19-Aug-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 4 ≤ 𝐴) ⇒ ⊢ (𝜑 → ((2 · ((1 / ((((2 logb 𝐴)↑5) + 1) · (log‘2))) · (((5 · ((2 logb 𝐴)↑4)) · (1 / (𝐴 · (log‘2)))) + 0))) + ((2 / ((log‘2)↑2)) · (((log‘𝐴)↑(2 − 1)) / 𝐴))) ≤ ((4 / ((log‘2)↑4)) · (((log‘𝐴)↑3) / 𝐴))) | ||
Theorem | aks4d1p1p5 42056* | Show inequality for existence of a non-divisor. (Contributed by metakunt, 19-Aug-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) & ⊢ (𝜑 → 4 ≤ 𝑁) & ⊢ 𝐶 = (2 logb (((2 logb 𝑁)↑5) + 1)) & ⊢ 𝐷 = ((2 logb 𝑁)↑2) & ⊢ 𝐸 = ((2 logb 𝑁)↑4) ⇒ ⊢ (𝜑 → 𝐴 < (2↑𝐵)) | ||
Theorem | aks4d1p1 42057* | Show inequality for existence of a non-divisor. (Contributed by metakunt, 21-Aug-2024.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) ⇒ ⊢ (𝜑 → 𝐴 < (2↑𝐵)) | ||
Theorem | aks4d1p2 42058 | Technical lemma for existence of non-divisor. (Contributed by metakunt, 27-Oct-2024.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) ⇒ ⊢ (𝜑 → (2↑𝐵) ≤ (lcm‘(1...𝐵))) | ||
Theorem | aks4d1p3 42059* | There exists a small enough number such that it does not divide 𝐴. (Contributed by metakunt, 27-Oct-2024.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴) | ||
Theorem | aks4d1p4 42060* | There exists a small enough number such that it does not divide 𝐴. (Contributed by metakunt, 28-Oct-2024.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) & ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) ⇒ ⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴)) | ||
Theorem | aks4d1p5 42061* | Show that 𝑁 and 𝑅 are coprime for AKS existence theorem. Precondition will be eliminated in further theorem. (Contributed by metakunt, 30-Oct-2024.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) & ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) & ⊢ (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴) → ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴) ⇒ ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) | ||
Theorem | aks4d1p6 42062* | The maximal prime power exponent is smaller than the binary logarithm floor of 𝐵. (Contributed by metakunt, 30-Oct-2024.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) & ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑃 ∥ 𝑅) & ⊢ 𝐾 = (𝑃 pCnt 𝑅) ⇒ ⊢ (𝜑 → 𝐾 ≤ (⌊‘(2 logb 𝐵))) | ||
Theorem | aks4d1p7d1 42063* | Technical step in AKS lemma 4.1 (Contributed by metakunt, 31-Oct-2024.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) & ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) & ⊢ (𝜑 → ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁)) ⇒ ⊢ (𝜑 → 𝑅 ∥ (𝑁↑(⌊‘(2 logb 𝐵)))) | ||
Theorem | aks4d1p7 42064* | Technical step in AKS lemma 4.1 (Contributed by metakunt, 31-Oct-2024.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) & ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ ℙ (𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁)) | ||
Theorem | aks4d1p8d1 42065 | If a prime divides one number 𝑀, but not another number 𝑁, then it divides the quotient of 𝑀 and the gcd of 𝑀 and 𝑁. (Contributed by Thierry Arnoux, 10-Nov-2024.) |
⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∥ 𝑀) & ⊢ (𝜑 → ¬ 𝑃 ∥ 𝑁) ⇒ ⊢ (𝜑 → 𝑃 ∥ (𝑀 / (𝑀 gcd 𝑁))) | ||
Theorem | aks4d1p8d2 42066 | Any prime power dividing a positive integer is less than that integer if that integer has another prime factor. (Contributed by metakunt, 13-Nov-2024.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑄 ∈ ℙ) & ⊢ (𝜑 → 𝑃 ∥ 𝑅) & ⊢ (𝜑 → 𝑄 ∥ 𝑅) & ⊢ (𝜑 → ¬ 𝑃 ∥ 𝑁) & ⊢ (𝜑 → 𝑄 ∥ 𝑁) ⇒ ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < 𝑅) | ||
Theorem | aks4d1p8d3 42067 | The remainder of a division with its maximal prime power is coprime with that prime power. (Contributed by metakunt, 13-Nov-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑃 ∥ 𝑁) ⇒ ⊢ (𝜑 → ((𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) gcd (𝑃↑(𝑃 pCnt 𝑁))) = 1) | ||
Theorem | aks4d1p8 42068* | Show that 𝑁 and 𝑅 are coprime for AKS existence theorem, with eliminated hypothesis. (Contributed by metakunt, 10-Nov-2024.) (Proof sketch by Thierry Arnoux.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) & ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) ⇒ ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) | ||
Theorem | aks4d1p9 42069* | Show that the order is bound by the squared binary logarithm. (Contributed by metakunt, 14-Nov-2024.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) & ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) ⇒ ⊢ (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁)) | ||
Theorem | aks4d1 42070* | Lemma 4.1 from https://www3.nd.edu/%7eandyp/notes/AKS.pdf, existence of a polynomially bounded number by the digit size of 𝑁 that asserts the polynomial subspace that we need to search to guarantee that 𝑁 is prime. Eventually we want to show that the polynomial searching space is bounded by degree 𝐵. (Contributed by metakunt, 14-Nov-2024.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) & ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ (1...𝐵)((𝑁 gcd 𝑟) = 1 ∧ ((2 logb 𝑁)↑2) < ((odℤ‘𝑟)‘𝑁))) | ||
Theorem | fldhmf1 42071 | A field homomorphism is injective. This follows immediately from the definition of the ring homomorphism that sends the multiplicative identity to the multiplicative identity. (Contributed by metakunt, 7-Jan-2025.) |
⊢ (𝜑 → 𝐾 ∈ Field) & ⊢ (𝜑 → 𝐿 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (𝐾 RingHom 𝐿)) & ⊢ 𝐴 = (Base‘𝐾) & ⊢ 𝐵 = (Base‘𝐿) ⇒ ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵) | ||
Syntax | cprimroots 42072 | Define the class of primitive roots. (Contributed by metakunt, 25-Apr-2025.) |
class PrimRoots | ||
Definition | df-primroots 42073* | A 𝑟-th primitive root is a root of unity such that the exponent divides 𝑟. (Contributed by metakunt, 25-Apr-2025.) |
⊢ PrimRoots = (𝑟 ∈ CMnd, 𝑘 ∈ ℕ0 ↦ ⦋(Base‘𝑟) / 𝑏⦌{𝑎 ∈ 𝑏 ∣ ((𝑘(.g‘𝑟)𝑎) = (0g‘𝑟) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑟)𝑎) = (0g‘𝑟) → 𝑘 ∥ 𝑙))}) | ||
Theorem | isprimroot 42074* | The value of a primitive root. (Contributed by metakunt, 25-Apr-2025.) |
⊢ (𝜑 → 𝑅 ∈ CMnd) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ ↑ = (.g‘𝑅) ⇒ ⊢ (𝜑 → (𝑀 ∈ (𝑅 PrimRoots 𝐾) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾 ↑ 𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 ↑ 𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙)))) | ||
Theorem | isprimroot2 42075 | Alternative way of creating primitive roots. (Contributed by metakunt, 14-Jul-2025.) |
⊢ (𝜑 → 𝑅 ∈ CMnd) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (Base‘𝑅)) & ⊢ (𝜑 → ((od‘𝑅)‘𝑀) = 𝐾) ⇒ ⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾)) | ||
Theorem | mndmolinv 42076* | An element of a monoid that has a right inverse has at most one left inverse. (Contributed by metakunt, 25-Apr-2025.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (𝐴(+g‘𝑀)𝑥) = (0g‘𝑀)) ⇒ ⊢ (𝜑 → ∃*𝑥 ∈ 𝐵 (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀)) | ||
Theorem | linvh 42077* | If an element has a unique left inverse, then the value satisfies the left inverse value equation. (Contributed by metakunt, 25-Apr-2025.) |
⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅)) & ⊢ (𝜑 → ∃!𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑋) = (0g‘𝑅)) ⇒ ⊢ (𝜑 → (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋) = (0g‘𝑅)) | ||
Theorem | primrootsunit1 42078* | Primitive roots have left inverses. (Contributed by metakunt, 25-Apr-2025.) |
⊢ (𝜑 → 𝑅 ∈ CMnd) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)} ⇒ ⊢ (𝜑 → ((𝑅 PrimRoots 𝐾) = ((𝑅 ↾s 𝑈) PrimRoots 𝐾) ∧ (𝑅 ↾s 𝑈) ∈ Abel)) | ||
Theorem | primrootsunit 42079* | Primitive roots have left inverses. (Contributed by metakunt, 25-Apr-2025.) |
⊢ (𝜑 → 𝑅 ∈ CMnd) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)} ⇒ ⊢ (𝜑 → ((𝑅 PrimRoots 𝐾) = ((𝑅 ↾s 𝑈) PrimRoots 𝐾) ∧ (𝑅 ↾s 𝑈) ∈ Abel)) | ||
Theorem | primrootscoprmpow 42080* | Coprime powers of primitive roots are primitive roots. (Contributed by metakunt, 25-Apr-2025.) |
⊢ (𝜑 → 𝑅 ∈ CMnd) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝐸 ∈ ℕ) & ⊢ (𝜑 → (𝐸 gcd 𝐾) = 1) & ⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾)) & ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)} ⇒ ⊢ (𝜑 → (𝐸(.g‘𝑅)𝑀) ∈ (𝑅 PrimRoots 𝐾)) | ||
Theorem | posbezout 42081* | Bezout's identity restricted on positive integers in all but one variable. (Contributed by metakunt, 26-Apr-2025.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℤ (𝐴 gcd 𝐵) = ((𝐴 · 𝑥) + (𝐵 · 𝑦))) | ||
Theorem | primrootscoprf 42082* | Coprime powers of primitive roots are primitive roots, as a function. (Contributed by metakunt, 26-Apr-2025.) |
⊢ 𝐹 = (𝑚 ∈ (𝑅 PrimRoots 𝐾) ↦ (𝐸(.g‘𝑅)𝑚)) & ⊢ (𝜑 → 𝑅 ∈ CMnd) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝐸 ∈ ℕ) & ⊢ (𝜑 → (𝐸 gcd 𝐾) = 1) ⇒ ⊢ (𝜑 → 𝐹:(𝑅 PrimRoots 𝐾)⟶(𝑅 PrimRoots 𝐾)) | ||
Theorem | primrootscoprbij 42083* | A bijection between coprime powers of primitive roots and primitive roots. (Contributed by metakunt, 26-Apr-2025.) |
⊢ 𝐹 = (𝑚 ∈ (𝑅 PrimRoots 𝐾) ↦ (𝐼(.g‘𝑅)𝑚)) & ⊢ (𝜑 → 𝑅 ∈ CMnd) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐽 ∈ ℕ) & ⊢ (𝜑 → 𝑍 ∈ ℤ) & ⊢ (𝜑 → 1 = ((𝐼 · 𝐽) + (𝐾 · 𝑍))) & ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)} ⇒ ⊢ (𝜑 → 𝐹:(𝑅 PrimRoots 𝐾)–1-1-onto→(𝑅 PrimRoots 𝐾)) | ||
Theorem | primrootscoprbij2 42084* | A bijection between coprime powers of primitive roots and primitive roots. (Contributed by metakunt, 26-Apr-2025.) |
⊢ 𝐹 = (𝑚 ∈ (𝑅 PrimRoots 𝐾) ↦ (𝐼(.g‘𝑅)𝑚)) & ⊢ (𝜑 → 𝑅 ∈ CMnd) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → (𝐼 gcd 𝐾) = 1) ⇒ ⊢ (𝜑 → 𝐹:(𝑅 PrimRoots 𝐾)–1-1-onto→(𝑅 PrimRoots 𝐾)) | ||
Theorem | remexz 42085* | Division with rest. (Contributed by metakunt, 15-May-2025.) |
⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (0...(𝐴 − 1))𝑁 = ((𝑥 · 𝐴) + 𝑦)) | ||
Theorem | primrootlekpowne0 42086 | There is no smaller power of a primitive root that sends it to the neutral element. (Contributed by metakunt, 15-May-2025.) |
⊢ (𝜑 → 𝑅 ∈ CMnd) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾)) & ⊢ (𝜑 → 𝑁 ∈ (1...(𝐾 − 1))) ⇒ ⊢ (𝜑 → (𝑁(.g‘𝑅)𝑀) ≠ (0g‘𝑅)) | ||
Theorem | primrootspoweq0 42087* | The power of a 𝑅-th primitive root is zero if and only if it divides 𝑅. (Contributed by metakunt, 15-May-2025.) |
⊢ (𝜑 → 𝑅 ∈ CMnd) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾)) & ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)} & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → ((𝑁(.g‘(𝑅 ↾s 𝑈))𝑀) = (0g‘(𝑅 ↾s 𝑈)) ↔ 𝐾 ∥ 𝑁)) | ||
Theorem | aks6d1c1p1 42088* | Definition of the introspective relation. (Contributed by metakunt, 25-Apr-2025.) |
⊢ ∼ = {〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))} & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐸 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝐸 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦)))) | ||
Theorem | aks6d1c1p1rcl 42089* | Reverse closure of the introspective relation. (Contributed by metakunt, 25-Apr-2025.) |
⊢ ∼ = {〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))} & ⊢ (𝜑 → 𝐸 ∼ 𝐹) ⇒ ⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵)) | ||
Theorem | aks6d1c1p2 42090* | 𝑃 and linear factors are introspective. (Contributed by metakunt, 25-Apr-2025.) |
⊢ ∼ = {〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))} & ⊢ 𝑆 = (Poly1‘𝐾) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑋 = (var1‘𝐾) & ⊢ 𝑊 = (mulGrp‘𝑆) & ⊢ 𝑉 = (mulGrp‘𝐾) & ⊢ ↑ = (.g‘𝑉) & ⊢ 𝐶 = (algSc‘𝑆) & ⊢ 𝐷 = (.g‘𝑊) & ⊢ 𝑃 = (chr‘𝐾) & ⊢ 𝑂 = (eval1‘𝐾) & ⊢ + = (+g‘𝑆) & ⊢ (𝜑 → 𝐾 ∈ Field) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) & ⊢ (𝜑 → 𝑃 ∥ 𝑁) & ⊢ 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) & ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝑃 ∼ 𝐹) | ||
Theorem | aks6d1c1p3 42091* | In a field with a Frobenius isomorphism (read: algebraic closure or finite field), 𝑁 and linear factors are introspective. (Contributed by metakunt, 25-Apr-2025.) |
⊢ ∼ = {〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))} & ⊢ 𝑆 = (Poly1‘𝐾) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑋 = (var1‘𝐾) & ⊢ 𝑊 = (mulGrp‘𝑆) & ⊢ 𝑉 = (mulGrp‘𝐾) & ⊢ ↑ = (.g‘𝑉) & ⊢ 𝐶 = (algSc‘𝑆) & ⊢ 𝐷 = (.g‘𝑊) & ⊢ 𝑃 = (chr‘𝐾) & ⊢ 𝑂 = (eval1‘𝐾) & ⊢ + = (+g‘𝑆) & ⊢ (𝜑 → 𝐾 ∈ Field) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) & ⊢ (𝜑 → 𝑃 ∥ 𝑁) & ⊢ 𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴))) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∼ 𝐹) & ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingIso 𝐾)) ⇒ ⊢ (𝜑 → (𝑁 / 𝑃) ∼ 𝐹) | ||
Theorem | aks6d1c1p4 42092* | The product of polynomials is introspective. (Contributed by metakunt, 25-Apr-2025.) |
⊢ ∼ = {〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))} & ⊢ 𝑆 = (Poly1‘𝐾) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑋 = (var1‘𝐾) & ⊢ 𝑊 = (mulGrp‘𝑆) & ⊢ 𝑉 = (mulGrp‘𝐾) & ⊢ ↑ = (.g‘𝑉) & ⊢ 𝐶 = (algSc‘𝑆) & ⊢ 𝐷 = (.g‘𝑊) & ⊢ 𝑃 = (chr‘𝐾) & ⊢ 𝑂 = (eval1‘𝐾) & ⊢ + = (+g‘𝑆) & ⊢ (𝜑 → 𝐾 ∈ Field) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) & ⊢ (𝜑 → 𝑃 ∥ 𝑁) & ⊢ (𝜑 → 𝐸 ∼ 𝐹) & ⊢ (𝜑 → 𝐸 ∼ 𝐺) ⇒ ⊢ (𝜑 → 𝐸 ∼ (𝐹(+g‘𝑊)𝐺)) | ||
Theorem | aks6d1c1p5 42093* | The product of exponents is introspective. (Contributed by metakunt, 26-Apr-2025.) |
⊢ ∼ = {〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))} & ⊢ 𝑆 = (Poly1‘𝐾) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑋 = (var1‘𝐾) & ⊢ 𝑊 = (mulGrp‘𝑆) & ⊢ 𝑉 = (mulGrp‘𝐾) & ⊢ ↑ = (.g‘𝑉) & ⊢ 𝐶 = (algSc‘𝑆) & ⊢ 𝑃 = (chr‘𝐾) & ⊢ 𝑂 = (eval1‘𝐾) & ⊢ + = (+g‘𝑆) & ⊢ (𝜑 → 𝐾 ∈ Field) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → (𝐸 gcd 𝑅) = 1) & ⊢ (𝜑 → 𝑃 ∥ 𝑁) & ⊢ (𝜑 → 𝐷 ∼ 𝐹) & ⊢ (𝜑 → 𝐸 ∼ 𝐹) ⇒ ⊢ (𝜑 → (𝐷 · 𝐸) ∼ 𝐹) | ||
Theorem | aks6d1c1p7 42094* | 𝑋 is introspective to all positive integers. (Contributed by metakunt, 30-Apr-2025.) |
⊢ ∼ = {〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))} & ⊢ 𝑆 = (Poly1‘𝐾) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑋 = (var1‘𝐾) & ⊢ 𝑉 = (mulGrp‘𝐾) & ⊢ ↑ = (.g‘𝑉) & ⊢ 𝑃 = (chr‘𝐾) & ⊢ 𝑂 = (eval1‘𝐾) & ⊢ (𝜑 → 𝐾 ∈ Field) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∥ 𝑁) & ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) & ⊢ (𝜑 → 𝐿 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝐿 ∼ 𝑋) | ||
Theorem | aks6d1c1p6 42095* | If a polynomials 𝐹 is introspective to 𝐸, then so are its powers. (Contributed by metakunt, 30-Apr-2025.) |
⊢ ∼ = {〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))} & ⊢ 𝑆 = (Poly1‘𝐾) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑋 = (var1‘𝐾) & ⊢ 𝑊 = (mulGrp‘𝑆) & ⊢ 𝑉 = (mulGrp‘𝐾) & ⊢ ↑ = (.g‘𝑉) & ⊢ 𝐶 = (algSc‘𝑆) & ⊢ 𝐷 = (.g‘𝑊) & ⊢ 𝑃 = (chr‘𝐾) & ⊢ 𝑂 = (eval1‘𝐾) & ⊢ + = (+g‘𝑆) & ⊢ (𝜑 → 𝐾 ∈ Field) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∥ 𝑁) & ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) & ⊢ (𝜑 → 𝐸 ∼ 𝐹) & ⊢ (𝜑 → 𝐿 ∈ ℕ0) ⇒ ⊢ (𝜑 → 𝐸 ∼ (𝐿𝐷𝐹)) | ||
Theorem | aks6d1c1p8 42096* | If a number 𝐸 is introspective to 𝐹, then so are its powers. (Contributed by metakunt, 30-Apr-2025.) |
⊢ ∼ = {〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))} & ⊢ 𝑆 = (Poly1‘𝐾) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑋 = (var1‘𝐾) & ⊢ 𝑊 = (mulGrp‘𝑆) & ⊢ 𝑉 = (mulGrp‘𝐾) & ⊢ ↑ = (.g‘𝑉) & ⊢ 𝐶 = (algSc‘𝑆) & ⊢ 𝐷 = (.g‘𝑊) & ⊢ 𝑃 = (chr‘𝐾) & ⊢ 𝑂 = (eval1‘𝐾) & ⊢ + = (+g‘𝑆) & ⊢ (𝜑 → 𝐾 ∈ Field) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∥ 𝑁) & ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) & ⊢ (𝜑 → 𝐸 ∼ 𝐹) & ⊢ (𝜑 → 𝐿 ∈ ℕ0) & ⊢ (𝜑 → (𝐸 gcd 𝑅) = 1) ⇒ ⊢ (𝜑 → (𝐸↑𝐿) ∼ 𝐹) | ||
Theorem | aks6d1c1 42097* | Claim 1 of Theorem 6.1 https://www3.nd.edu/%7eandyp/notes/AKS.pdf. (Contributed by metakunt, 30-Apr-2025.) |
⊢ ∼ = {〈𝑒, 𝑓〉 ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))} & ⊢ 𝑆 = (Poly1‘𝐾) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑋 = (var1‘𝐾) & ⊢ 𝑊 = (mulGrp‘𝑆) & ⊢ 𝑉 = (mulGrp‘𝐾) & ⊢ ↑ = (.g‘𝑉) & ⊢ 𝐶 = (algSc‘𝑆) & ⊢ 𝐷 = (.g‘𝑊) & ⊢ 𝑃 = (chr‘𝐾) & ⊢ 𝑂 = (eval1‘𝐾) & ⊢ + = (+g‘𝑆) & ⊢ (𝜑 → 𝐾 ∈ Field) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∥ 𝑁) & ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1) & ⊢ (𝜑 → 𝐹:(0...𝐴)⟶ℕ0) & ⊢ 𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) & ⊢ (𝜑 → 𝐴 ∈ ℕ0) & ⊢ (𝜑 → 𝑈 ∈ ℕ0) & ⊢ (𝜑 → 𝐿 ∈ ℕ0) & ⊢ 𝐸 = ((𝑃↑𝑈) · ((𝑁 / 𝑃)↑𝐿)) & ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎)))) & ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingIso 𝐾)) ⇒ ⊢ (𝜑 → 𝐸 ∼ (𝐺‘𝐹)) | ||
Theorem | evl1gprodd 42098* | Polynomial evaluation builder for a finite group product of polynomials. (Contributed by metakunt, 29-Apr-2025.) |
⊢ 𝑂 = (eval1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑄 = (mulGrp‘𝑃) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ 𝑆 = (mulGrp‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝑈) & ⊢ (𝜑 → 𝑁 ∈ Fin) ⇒ ⊢ (𝜑 → ((𝑂‘(𝑄 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝑌) = (𝑆 Σg (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌)))) | ||
Theorem | aks6d1c2p1 42099* | In the AKS-theorem the subset defined by 𝐸 takes values in the positive integers. (Contributed by metakunt, 7-Jan-2025.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑃 ∥ 𝑁) & ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) ⇒ ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ) | ||
Theorem | aks6d1c2p2 42100* | Injective condition for countability argument assuming that 𝑁 is not a prime power. (Contributed by metakunt, 7-Jan-2025.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑃 ∥ 𝑁) & ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) & ⊢ (𝜑 → 𝑄 ∈ ℙ) & ⊢ (𝜑 → 𝑄 ∥ 𝑁) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) ⇒ ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)–1-1→ℕ) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |