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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lcm1un 39301 | Least common multiple of natural numbers up to 1 equals 1. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...1)) = 1 | ||
Theorem | lcm2un 39302 | Least common multiple of natural numbers up to 2 equals 2. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...2)) = 2 | ||
Theorem | lcm3un 39303 | Least common multiple of natural numbers up to 3 equals 6. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...3)) = 6 | ||
Theorem | lcm4un 39304 | Least common multiple of natural numbers up to 4 equals 12. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...4)) = ;12 | ||
Theorem | lcm5un 39305 | Least common multiple of natural numbers up to 5 equals 60. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...5)) = ;60 | ||
Theorem | lcm6un 39306 | Least common multiple of natural numbers up to 6 equals 60. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...6)) = ;60 | ||
Theorem | lcm7un 39307 | Least common multiple of natural numbers up to 7 equals 420. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...7)) = ;;420 | ||
Theorem | lcm8un 39308 | Least common multiple of natural numbers up to 8 equals 840. (Contributed by metakunt, 25-Apr-2024.) |
⊢ (lcm‘(1...8)) = ;;840 | ||
Theorem | 3factsumint1 39309* | 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 39310* | 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 39311* | 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 39312* | 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 39313* | 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 39314 | 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 39315 | 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 39316* | 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 39317* | 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 39318* | 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 39319* | 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 39320 | 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 39322. (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 39321 | 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 39322* | 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 39323 | 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 39324* | 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 39325* | (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 39326* | Induction step of lcmineqlem13 39329 (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 39327 | 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 39328* | 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 39329* | 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 39330 | 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 39331* | 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 39332 | 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 39333 | 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 39334 | 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 39335 | 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 39336 | 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 39337 | 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 39338 | 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 39339 | 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 39340 | 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 | 3lexlogpow5ineq1 39341 | First inequality in inequality chain, proposed by Mario Carneiro (Contributed by metakunt, 22-May-2024.) |
⊢ 7 < ((3 / 2)↑5) | ||
Theorem | 3lexlogpow5ineq2 39342 | Second inequality in inequality chain, proposed by Mario Carneiro. (Contributed by metakunt, 22-May-2024.) |
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 3 ≤ 𝑋) ⇒ ⊢ (𝜑 → ((3 / 2)↑5) ≤ ((2 logb 𝑋)↑5)) | ||
Theorem | 3lexlogpow5ineq3 39343 | 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 | intlewftc 39344* | 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 | aks4d1p1p1 39345* | 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 | 5bc2eq10 39346 | The value of 5 choose 2. (Contributed by metakunt, 8-Jun-2024.) |
⊢ (5C2) = ;10 | ||
Theorem | facp2 39347 | The factorial of a successor's successor. (Contributed by metakunt, 19-Apr-2024.) |
⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 2)) = ((!‘𝑁) · ((𝑁 + 1) · (𝑁 + 2)))) | ||
Theorem | 2np3bcnp1 39348 | Part of induction step for 2ap1caineq 39349. (Contributed by metakunt, 8-Jun-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (((2 · (𝑁 + 1)) + 1)C(𝑁 + 1)) = ((((2 · 𝑁) + 1)C𝑁) · (2 · (((2 · 𝑁) + 3) / (𝑁 + 2))))) | ||
Theorem | 2ap1caineq 39349 | Inequality for Theorem 6.6 for AKS. (Contributed by metakunt, 8-Jun-2024.) |
⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 2 ≤ 𝑁) ⇒ ⊢ (𝜑 → (2↑(𝑁 + 1)) < (((2 · 𝑁) + 1)C𝑁)) | ||
Theorem | metakunt1 39350* | A is an endomapping. (Contributed by metakunt, 23-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) ⇒ ⊢ (𝜑 → 𝐴:(1...𝑀)⟶(1...𝑀)) | ||
Theorem | metakunt2 39351* | A is an endomapping. (Contributed by metakunt, 23-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝐼, if(𝑥 < 𝐼, 𝑥, (𝑥 + 1)))) ⇒ ⊢ (𝜑 → 𝐴:(1...𝑀)⟶(1...𝑀)) | ||
Theorem | metakunt3 39352* | Value of A. (Contributed by metakunt, 23-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ (𝜑 → (𝐴‘𝑋) = if(𝑋 = 𝐼, 𝑀, if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)))) | ||
Theorem | metakunt4 39353* | Value of A. (Contributed by metakunt, 23-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝐼, if(𝑥 < 𝐼, 𝑥, (𝑥 + 1)))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ (𝜑 → (𝐴‘𝑋) = if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)))) | ||
Theorem | metakunt5 39354* | C is the left inverse for A. (Contributed by metakunt, 24-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐶‘(𝐴‘𝑋)) = 𝑋) | ||
Theorem | metakunt6 39355* | C is the left inverse for A. (Contributed by metakunt, 24-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐶‘(𝐴‘𝑋)) = 𝑋) | ||
Theorem | metakunt7 39356* | C is the left inverse for A. (Contributed by metakunt, 24-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝐼 < 𝑋) → ((𝐴‘𝑋) = (𝑋 − 1) ∧ ¬ (𝐴‘𝑋) = 𝑀 ∧ ¬ (𝐴‘𝑋) < 𝐼)) | ||
Theorem | metakunt8 39357* | C is the left inverse for A. (Contributed by metakunt, 24-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝐼 < 𝑋) → (𝐶‘(𝐴‘𝑋)) = 𝑋) | ||
Theorem | metakunt9 39358* | C is the left inverse for A. (Contributed by metakunt, 24-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)) = 𝑋) | ||
Theorem | metakunt10 39359* | C is the right inverse for A. (Contributed by metakunt, 24-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝑋 = 𝑀) → (𝐴‘(𝐶‘𝑋)) = 𝑋) | ||
Theorem | metakunt11 39360* | C is the right inverse for A. (Contributed by metakunt, 24-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ ((𝜑 ∧ 𝑋 < 𝐼) → (𝐴‘(𝐶‘𝑋)) = 𝑋) | ||
Theorem | metakunt12 39361* | C is the right inverse for A. (Contributed by metakunt, 25-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ ((𝜑 ∧ ¬ (𝑋 = 𝑀 ∨ 𝑋 < 𝐼)) → (𝐴‘(𝐶‘𝑋)) = 𝑋) | ||
Theorem | metakunt13 39362* | C is the right inverse for A. (Contributed by metakunt, 25-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ (𝜑 → (𝐴‘(𝐶‘𝑋)) = 𝑋) | ||
Theorem | metakunt14 39363* | A is a primitive permutation that moves the I-th element to the end and C is its inverse that moves the last element back to the I-th position. (Contributed by metakunt, 25-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) ⇒ ⊢ (𝜑 → (𝐴:(1...𝑀)–1-1-onto→(1...𝑀) ∧ ◡𝐴 = 𝐶)) | ||
Theorem | metakunt15 39364* | Construction of another permutation. (Contributed by metakunt, 25-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐹 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) ⇒ ⊢ (𝜑 → 𝐹:(1...(𝐼 − 1))–1-1-onto→(((𝑀 − 𝐼) + 1)...(𝑀 − 1))) | ||
Theorem | metakunt16 39365* | Construction of another permutation. (Contributed by metakunt, 25-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐹 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) ⇒ ⊢ (𝜑 → 𝐹:(𝐼...(𝑀 − 1))–1-1-onto→(1...(𝑀 − 𝐼))) | ||
Theorem | metakunt17 39366 | The union of three disjoint bijections is a bijection. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝐺:𝐴–1-1-onto→𝑋) & ⊢ (𝜑 → 𝐻:𝐵–1-1-onto→𝑌) & ⊢ (𝜑 → 𝐼:𝐶–1-1-onto→𝑍) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) & ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅) & ⊢ (𝜑 → (𝐵 ∩ 𝐶) = ∅) & ⊢ (𝜑 → (𝑋 ∩ 𝑌) = ∅) & ⊢ (𝜑 → (𝑋 ∩ 𝑍) = ∅) & ⊢ (𝜑 → (𝑌 ∩ 𝑍) = ∅) & ⊢ (𝜑 → 𝐹 = ((𝐺 ∪ 𝐻) ∪ 𝐼)) & ⊢ (𝜑 → 𝐷 = ((𝐴 ∪ 𝐵) ∪ 𝐶)) & ⊢ (𝜑 → 𝑊 = ((𝑋 ∪ 𝑌) ∪ 𝑍)) ⇒ ⊢ (𝜑 → 𝐹:𝐷–1-1-onto→𝑊) | ||
Theorem | metakunt18 39367 | Disjoint domains and codomains. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) ⇒ ⊢ (𝜑 → ((((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅) ∧ (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∩ (1...(𝑀 − 𝐼))) = ∅ ∧ ((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∩ {𝑀}) = ∅ ∧ ((1...(𝑀 − 𝐼)) ∩ {𝑀}) = ∅))) | ||
Theorem | metakunt19 39368* | Domains on restrictions of functions. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) & ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) & ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) ⇒ ⊢ (𝜑 → ((𝐶 Fn (1...(𝐼 − 1)) ∧ 𝐷 Fn (𝐼...(𝑀 − 1)) ∧ (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1)))) ∧ {〈𝑀, 𝑀〉} Fn {𝑀})) | ||
Theorem | metakunt20 39369* | Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) & ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) & ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) & ⊢ (𝜑 → 𝑋 = 𝑀) ⇒ ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {〈𝑀, 𝑀〉})‘𝑋)) | ||
Theorem | metakunt21 39370* | Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) & ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) & ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) & ⊢ (𝜑 → ¬ 𝑋 = 𝑀) & ⊢ (𝜑 → 𝑋 < 𝐼) ⇒ ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {〈𝑀, 𝑀〉})‘𝑋)) | ||
Theorem | metakunt22 39371* | Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) & ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) & ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) & ⊢ (𝜑 → ¬ 𝑋 = 𝑀) & ⊢ (𝜑 → ¬ 𝑋 < 𝐼) ⇒ ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {〈𝑀, 𝑀〉})‘𝑋)) | ||
Theorem | metakunt23 39372* | B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) & ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) & ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) ⇒ ⊢ (𝜑 → (𝐵‘𝑋) = (((𝐶 ∪ 𝐷) ∪ {〈𝑀, 𝑀〉})‘𝑋)) | ||
Theorem | metakunt24 39373 | Technical condition such that metakunt17 39366 holds (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) ⇒ ⊢ (𝜑 → ((((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅ ∧ (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}) ∧ (1...𝑀) = (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀 − 𝐼))) ∪ {𝑀}))) | ||
Theorem | metakunt25 39374* | B is a permutation. (Contributed by metakunt, 28-May-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) ⇒ ⊢ (𝜑 → 𝐵:(1...𝑀)–1-1-onto→(1...𝑀)) | ||
Theorem | metakunt26 39375* | Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) & ⊢ (𝜑 → 𝑋 = 𝐼) ⇒ ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑋) | ||
Theorem | metakunt27 39376* | Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) & ⊢ (𝜑 → ¬ 𝑋 = 𝐼) & ⊢ (𝜑 → 𝑋 < 𝐼) ⇒ ⊢ (𝜑 → (𝐵‘(𝐴‘𝑋)) = (𝑋 + (𝑀 − 𝐼))) | ||
Theorem | metakunt28 39377* | Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) & ⊢ (𝜑 → ¬ 𝑋 = 𝐼) & ⊢ (𝜑 → ¬ 𝑋 < 𝐼) ⇒ ⊢ (𝜑 → (𝐵‘(𝐴‘𝑋)) = (𝑋 − 𝐼)) | ||
Theorem | metakunt29 39378* | Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) & ⊢ (𝜑 → ¬ 𝑋 = 𝐼) & ⊢ (𝜑 → 𝑋 < 𝐼) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ 𝐻 = if(𝐼 ≤ (𝑋 + (𝑀 − 𝐼)), 1, 0) ⇒ ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = ((𝑋 + (𝑀 − 𝐼)) + 𝐻)) | ||
Theorem | metakunt30 39379* | Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) & ⊢ (𝜑 → ¬ 𝑋 = 𝐼) & ⊢ (𝜑 → ¬ 𝑋 < 𝐼) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ 𝐻 = if(𝐼 ≤ (𝑋 − 𝐼), 1, 0) ⇒ ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = ((𝑋 − 𝐼) + 𝐻)) | ||
Theorem | metakunt31 39380* | Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ 𝐺 = if(𝐼 ≤ (𝑋 + (𝑀 − 𝐼)), 1, 0) & ⊢ 𝐻 = if(𝐼 ≤ (𝑋 − 𝐼), 1, 0) & ⊢ 𝑅 = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))) ⇒ ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅) | ||
Theorem | metakunt32 39381* | Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) & ⊢ 𝐷 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑥, if(𝑥 < 𝐼, ((𝑥 + (𝑀 − 𝐼)) + if(𝐼 ≤ (𝑥 + (𝑀 − 𝐼)), 1, 0)), ((𝑥 − 𝐼) + if(𝐼 ≤ (𝑥 − 𝐼), 1, 0))))) & ⊢ 𝐺 = if(𝐼 ≤ (𝑋 + (𝑀 − 𝐼)), 1, 0) & ⊢ 𝐻 = if(𝐼 ≤ (𝑋 − 𝐼), 1, 0) & ⊢ 𝑅 = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻))) ⇒ ⊢ (𝜑 → (𝐷‘𝑋) = 𝑅) | ||
Theorem | metakunt33 39382* | Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) & ⊢ 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) & ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) & ⊢ 𝐷 = (𝑤 ∈ (1...𝑀) ↦ if(𝑤 = 𝐼, 𝑤, if(𝑤 < 𝐼, ((𝑤 + (𝑀 − 𝐼)) + if(𝐼 ≤ (𝑤 + (𝑀 − 𝐼)), 1, 0)), ((𝑤 − 𝐼) + if(𝐼 ≤ (𝑤 − 𝐼), 1, 0))))) ⇒ ⊢ (𝜑 → (𝐶 ∘ (𝐵 ∘ 𝐴)) = 𝐷) | ||
Theorem | metakunt34 39383* | 𝐷 is a permutation. (Contributed by metakunt, 18-Jul-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ≤ 𝑀) & ⊢ 𝐷 = (𝑤 ∈ (1...𝑀) ↦ if(𝑤 = 𝐼, 𝑤, if(𝑤 < 𝐼, ((𝑤 + (𝑀 − 𝐼)) + if(𝐼 ≤ (𝑤 + (𝑀 − 𝐼)), 1, 0)), ((𝑤 − 𝐼) + if(𝐼 ≤ (𝑤 − 𝐼), 1, 0))))) ⇒ ⊢ (𝜑 → 𝐷:(1...𝑀)–1-1-onto→(1...𝑀)) | ||
Theorem | andiff 39384 | Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.) |
⊢ (𝜑 → (𝜒 → 𝜃)) & ⊢ (𝜓 → (𝜃 → 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) | ||
Theorem | fac2xp3 39385 | Factorial of 2x+3, sublemma for sublemma for AKS. (Contributed by metakunt, 19-Apr-2024.) |
⊢ (𝑥 ∈ ℕ0 → (!‘((2 · 𝑥) + 3)) = ((!‘((2 · 𝑥) + 1)) · (((2 · 𝑥) + 2) · ((2 · 𝑥) + 3)))) | ||
Theorem | prodsplit 39386* | Product split into two factors, original by Steven Nguyen. (Contributed by metakunt, 21-Apr-2024.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 𝐾))) → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...(𝑁 + 𝐾))𝐴 = (∏𝑘 ∈ (𝑀...𝑁)𝐴 · ∏𝑘 ∈ ((𝑁 + 1)...(𝑁 + 𝐾))𝐴)) | ||
Theorem | 2xp3dxp2ge1d 39387 | 2x+3 is greater than or equal to x+2 for x >= -1, a deduction version (Contributed by metakunt, 21-Apr-2024.) |
⊢ (𝜑 → 𝑋 ∈ (-1[,)+∞)) ⇒ ⊢ (𝜑 → 1 ≤ (((2 · 𝑋) + 3) / (𝑋 + 2))) | ||
Theorem | factwoffsmonot 39388 | A factorial with offset is monotonely increasing. (Contributed by metakunt, 20-Apr-2024.) |
⊢ (((𝑋 ∈ ℕ0 ∧ 𝑌 ∈ ℕ0 ∧ 𝑋 ≤ 𝑌) ∧ 𝑁 ∈ ℕ0) → (!‘(𝑋 + 𝑁)) ≤ (!‘(𝑌 + 𝑁))) | ||
Theorem | ioin9i8 39389 | Miscellaneous inference creating a biconditional from an implied converse implication. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ (𝜑 → (𝜓 ∨ 𝜒)) & ⊢ (𝜒 → ¬ 𝜃) & ⊢ (𝜓 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 ↔ 𝜃)) | ||
Theorem | jaodd 39390 | Double deduction form of jaoi 854. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜏) → 𝜃))) | ||
Theorem | sylibda 39391 | A syllogism deduction. (Contributed by SN, 16-Jul-2024.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜃) | ||
Theorem | nsb 39392 | Generalization rule for negated wff. (Contributed by Steven Nguyen, 3-May-2023.) |
⊢ ¬ 𝜑 ⇒ ⊢ ¬ [𝑥 / 𝑦]𝜑 | ||
Theorem | sbtd 39393* | A true statement is true upon substitution (deduction). A similar proof is possible for icht 43969. (Contributed by SN, 4-May-2024.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → [𝑡 / 𝑥]𝜓) | ||
Theorem | sbn1 39394 | One direction of sbn 2283, using fewer axioms. Compare 19.2 1981. (Contributed by Steven Nguyen, 18-Aug-2023.) |
⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑) | ||
Theorem | sbor2 39395 | One direction of sbor 2312, using fewer axioms. Compare 19.33 1885. (Contributed by Steven Nguyen, 18-Aug-2023.) |
⊢ (([𝑡 / 𝑥]𝜑 ∨ [𝑡 / 𝑥]𝜓) → [𝑡 / 𝑥](𝜑 ∨ 𝜓)) | ||
Theorem | 19.9dev 39396* | 19.9d 2201 in the case of an existential quantifier, avoiding the ax-10 2142 from nfex 2332 that would be used for the hypothesis of 19.9d 2201, at the cost of an additional DV condition on 𝑦, 𝜑. (Contributed by SN, 26-May-2024.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑦𝜓)) | ||
Theorem | 3rspcedvd 39397* | Triple application of rspcedvd 3574. (Contributed by Steven Nguyen, 27-Feb-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝜒 ↔ 𝜃)) & ⊢ ((𝜑 ∧ 𝑧 = 𝐶) → (𝜃 ↔ 𝜏)) & ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜓) | ||
Theorem | rabeqcda 39398* | When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3626. (Contributed by Steven Nguyen, 7-Jun-2023.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴) | ||
Theorem | rabdif 39399* | Move difference in and out of a restricted class abstraction. (Contributed by Steven Nguyen, 6-Jun-2023.) |
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑} | ||
Theorem | sn-axrep5v 39400* | A condensed form of axrep5 5160. (Contributed by SN, 21-Sep-2023.) |
⊢ (∀𝑤 ∈ 𝑥 ∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑)) |
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