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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | discr1 13601* | A nonnegative quadratic form has nonnegative leading coefficient. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ (((𝐴 · (𝑥↑2)) + (𝐵 · 𝑥)) + 𝐶)) & ⊢ 𝑋 = if(1 ≤ (((𝐵 + if(0 ≤ 𝐶, 𝐶, 0)) + 1) / -𝐴), (((𝐵 + if(0 ≤ 𝐶, 𝐶, 0)) + 1) / -𝐴), 1) ⇒ ⊢ (𝜑 → 0 ≤ 𝐴) | ||
Theorem | discr 13602* | If a quadratic polynomial with real coefficients is nonnegative for all values, then its discriminant is nonpositive. (Contributed by NM, 10-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ (((𝐴 · (𝑥↑2)) + (𝐵 · 𝑥)) + 𝐶)) ⇒ ⊢ (𝜑 → ((𝐵↑2) − (4 · (𝐴 · 𝐶))) ≤ 0) | ||
Theorem | expnngt1 13603 | If an integer power with a positive integer base is greater than 1, then the exponent is positive. (Contributed by AV, 28-Dec-2022.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ 1 < (𝐴↑𝐵)) → 𝐵 ∈ ℕ) | ||
Theorem | expnngt1b 13604 | An integer power with an integer base greater than 1 is greater than 1 iff the exponent is positive. (Contributed by AV, 28-Dec-2022.) |
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ ℤ) → (1 < (𝐴↑𝐵) ↔ 𝐵 ∈ ℕ)) | ||
Theorem | sqoddm1div8 13605 | A squared odd number minus 1 divided by 8 is the odd number multiplied with its successor divided by 2. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 = ((2 · 𝑁) + 1)) → (((𝑀↑2) − 1) / 8) = ((𝑁 · (𝑁 + 1)) / 2)) | ||
Theorem | nnsqcld 13606 | The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝐴↑2) ∈ ℕ) | ||
Theorem | nnexpcld 13607 | Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℕ) | ||
Theorem | nn0expcld 13608 | Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℕ0) | ||
Theorem | rpexpcld 13609 | Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ+) | ||
Theorem | ltexp2rd 13610 | The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 < 1) ⇒ ⊢ (𝜑 → (𝑀 < 𝑁 ↔ (𝐴↑𝑁) < (𝐴↑𝑀))) | ||
Theorem | reexpclzd 13611 | Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ) | ||
Theorem | resqcld 13612 | Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴↑2) ∈ ℝ) | ||
Theorem | sqge0d 13613 | A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴↑2)) | ||
Theorem | sqgt0d 13614 | The square of a nonzero real is positive. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → 0 < (𝐴↑2)) | ||
Theorem | ltexp2d 13615 | Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 1 < 𝐴) ⇒ ⊢ (𝜑 → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁))) | ||
Theorem | leexp2d 13616 | Ordering law for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 1 < 𝐴) ⇒ ⊢ (𝜑 → (𝑀 ≤ 𝑁 ↔ (𝐴↑𝑀) ≤ (𝐴↑𝑁))) | ||
Theorem | expcand 13617 | Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 1 < 𝐴) & ⊢ (𝜑 → (𝐴↑𝑀) = (𝐴↑𝑁)) ⇒ ⊢ (𝜑 → 𝑀 = 𝑁) | ||
Theorem | leexp2ad 13618 | Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 1 ≤ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) ⇒ ⊢ (𝜑 → (𝐴↑𝑀) ≤ (𝐴↑𝑁)) | ||
Theorem | leexp2rd 13619 | Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 ≤ 1) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ≤ (𝐴↑𝑀)) | ||
Theorem | lt2sqd 13620 | The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴↑2) < (𝐵↑2))) | ||
Theorem | le2sqd 13621 | The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴↑2) ≤ (𝐵↑2))) | ||
Theorem | sq11d 13622 | The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐵) & ⊢ (𝜑 → (𝐴↑2) = (𝐵↑2)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | mulsubdivbinom2 13623 | The square of a binomial with factor minus a number divided by a nonzero number. (Contributed by AV, 19-Jul-2021.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((((𝐶 · 𝐴) + 𝐵)↑2) − 𝐷) / 𝐶) = (((𝐶 · (𝐴↑2)) + (2 · (𝐴 · 𝐵))) + (((𝐵↑2) − 𝐷) / 𝐶))) | ||
Theorem | muldivbinom2 13624 | The square of a binomial with factor divided by a nonzero number. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((((𝐶 · 𝐴) + 𝐵)↑2) / 𝐶) = (((𝐶 · (𝐴↑2)) + (2 · (𝐴 · 𝐵))) + ((𝐵↑2) / 𝐶))) | ||
Theorem | sq10 13625 | The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
⊢ (;10↑2) = ;;100 | ||
Theorem | sq10e99m1 13626 | The square of 10 is 99 plus 1. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
⊢ (;10↑2) = (;99 + 1) | ||
Theorem | 3dec 13627 | A "decimal constructor" which is used to build up "decimal integers" or "numeric terms" in base 10 with 3 "digits". (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 ⇒ ⊢ ;;𝐴𝐵𝐶 = ((((;10↑2) · 𝐴) + (;10 · 𝐵)) + 𝐶) | ||
Theorem | nn0le2msqi 13628 | The square function on nonnegative integers is monotonic. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 ⇒ ⊢ (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵)) | ||
Theorem | nn0opthlem1 13629 | A rather pretty lemma for nn0opthi 13631. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 ⇒ ⊢ (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶)) | ||
Theorem | nn0opthlem2 13630 | Lemma for nn0opthi 13631. (Contributed by Raph Levien, 10-Dec-2002.) (Revised by Scott Fenton, 8-Sep-2010.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 ⇒ ⊢ ((𝐴 + 𝐵) < 𝐶 → ((𝐶 · 𝐶) + 𝐷) ≠ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵)) | ||
Theorem | nn0opthi 13631 | An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. We can represent an ordered pair of nonnegative integers 𝐴 and 𝐵 by (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵). If two such ordered pairs are equal, their first elements are equal and their second elements are equal. Contrast this ordered pair representation with the standard one df-op 4574 that works for any set. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Scott Fenton, 8-Sep-2010.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 ⇒ ⊢ ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | nn0opth2i 13632 | An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthi 13631. (Contributed by NM, 22-Jul-2004.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 ⇒ ⊢ ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | nn0opth2 13633 | An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 13631. (Contributed by NM, 22-Jul-2004.) |
⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0)) → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Syntax | cfa 13634 | Extend class notation to include the factorial of nonnegative integers. |
class ! | ||
Definition | df-fac 13635 | Define the factorial function on nonnegative integers. For example, (!‘5) = 120 because 1 · 2 · 3 · 4 · 5 = 120 (ex-fac 28230). In the literature, the factorial function is written as a postscript exclamation point. (Contributed by NM, 2-Dec-2004.) |
⊢ ! = ({〈0, 1〉} ∪ seq1( · , I )) | ||
Theorem | facnn 13636 | Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) | ||
Theorem | fac0 13637 | The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
⊢ (!‘0) = 1 | ||
Theorem | fac1 13638 | The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
⊢ (!‘1) = 1 | ||
Theorem | facp1 13639 | The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) | ||
Theorem | fac2 13640 | The factorial of 2. (Contributed by NM, 17-Mar-2005.) |
⊢ (!‘2) = 2 | ||
Theorem | fac3 13641 | The factorial of 3. (Contributed by NM, 17-Mar-2005.) |
⊢ (!‘3) = 6 | ||
Theorem | fac4 13642 | The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (!‘4) = ;24 | ||
Theorem | facnn2 13643 | Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.) |
⊢ (𝑁 ∈ ℕ → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁)) | ||
Theorem | faccl 13644 | Closure of the factorial function. (Contributed by NM, 2-Dec-2004.) |
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | ||
Theorem | faccld 13645 | Closure of the factorial function, deduction version of faccl 13644. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (!‘𝑁) ∈ ℕ) | ||
Theorem | facmapnn 13646 | The factorial function restricted to positive integers is a mapping from the positive integers to the positive integers. (Contributed by AV, 8-Aug-2020.) |
⊢ (𝑛 ∈ ℕ ↦ (!‘𝑛)) ∈ (ℕ ↑m ℕ) | ||
Theorem | facne0 13647 | The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.) |
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ≠ 0) | ||
Theorem | facdiv 13648 | A positive integer divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀) → ((!‘𝑀) / 𝑁) ∈ ℕ) | ||
Theorem | facndiv 13649 | No positive integer (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.) |
⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) ∧ (1 < 𝑁 ∧ 𝑁 ≤ 𝑀)) → ¬ (((!‘𝑀) + 1) / 𝑁) ∈ ℤ) | ||
Theorem | facwordi 13650 | Ordering property of factorial. (Contributed by NM, 9-Dec-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (!‘𝑀) ≤ (!‘𝑁)) | ||
Theorem | faclbnd 13651 | A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁))) | ||
Theorem | faclbnd2 13652 | A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.) |
⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) / 2) ≤ (!‘𝑁)) | ||
Theorem | faclbnd3 13653 | A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀↑𝑁) ≤ ((𝑀↑𝑀) · (!‘𝑁))) | ||
Theorem | faclbnd4lem1 13654 | Lemma for faclbnd4 13658. Prepare the induction step. (Contributed by NM, 20-Dec-2005.) |
⊢ 𝑁 ∈ ℕ & ⊢ 𝐾 ∈ ℕ0 & ⊢ 𝑀 ∈ ℕ0 ⇒ ⊢ ((((𝑁 − 1)↑𝐾) · (𝑀↑(𝑁 − 1))) ≤ (((2↑(𝐾↑2)) · (𝑀↑(𝑀 + 𝐾))) · (!‘(𝑁 − 1))) → ((𝑁↑(𝐾 + 1)) · (𝑀↑𝑁)) ≤ (((2↑((𝐾 + 1)↑2)) · (𝑀↑(𝑀 + (𝐾 + 1)))) · (!‘𝑁))) | ||
Theorem | faclbnd4lem2 13655 | Lemma for faclbnd4 13658. Use the weak deduction theorem to convert the hypotheses of faclbnd4lem1 13654 to antecedents. (Contributed by NM, 23-Dec-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → ((((𝑁 − 1)↑𝐾) · (𝑀↑(𝑁 − 1))) ≤ (((2↑(𝐾↑2)) · (𝑀↑(𝑀 + 𝐾))) · (!‘(𝑁 − 1))) → ((𝑁↑(𝐾 + 1)) · (𝑀↑𝑁)) ≤ (((2↑((𝐾 + 1)↑2)) · (𝑀↑(𝑀 + (𝐾 + 1)))) · (!‘𝑁)))) | ||
Theorem | faclbnd4lem3 13656 | Lemma for faclbnd4 13658. The 𝑁 = 0 case. (Contributed by NM, 23-Dec-2005.) |
⊢ (((𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0) ∧ 𝑁 = 0) → ((𝑁↑𝐾) · (𝑀↑𝑁)) ≤ (((2↑(𝐾↑2)) · (𝑀↑(𝑀 + 𝐾))) · (!‘𝑁))) | ||
Theorem | faclbnd4lem4 13657 | Lemma for faclbnd4 13658. Prove the 0 < 𝑁 case by induction on 𝐾. (Contributed by NM, 19-Dec-2005.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → ((𝑁↑𝐾) · (𝑀↑𝑁)) ≤ (((2↑(𝐾↑2)) · (𝑀↑(𝑀 + 𝐾))) · (!‘𝑁))) | ||
Theorem | faclbnd4 13658 | Variant of faclbnd5 13659 providing a non-strict lower bound. (Contributed by NM, 23-Dec-2005.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → ((𝑁↑𝐾) · (𝑀↑𝑁)) ≤ (((2↑(𝐾↑2)) · (𝑀↑(𝑀 + 𝐾))) · (!‘𝑁))) | ||
Theorem | faclbnd5 13659 | The factorial function grows faster than powers and exponentiations. If we consider 𝐾 and 𝑀 to be constants, the right-hand side of the inequality is a constant times 𝑁-factorial. (Contributed by NM, 24-Dec-2005.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ∧ 𝑀 ∈ ℕ) → ((𝑁↑𝐾) · (𝑀↑𝑁)) < ((2 · ((2↑(𝐾↑2)) · (𝑀↑(𝑀 + 𝐾)))) · (!‘𝑁))) | ||
Theorem | faclbnd6 13660 | Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → ((!‘𝑁) · ((𝑁 + 1)↑𝑀)) ≤ (!‘(𝑁 + 𝑀))) | ||
Theorem | facubnd 13661 | An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.) |
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ≤ (𝑁↑𝑁)) | ||
Theorem | facavg 13662 | The product of two factorials is greater than or equal to the factorial of (the floor of) their average. (Contributed by NM, 9-Dec-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (!‘(⌊‘((𝑀 + 𝑁) / 2))) ≤ ((!‘𝑀) · (!‘𝑁))) | ||
Syntax | cbc 13663 | Extend class notation to include the binomial coefficient operation (combinatorial choose operation). |
class C | ||
Definition | df-bc 13664* |
Define the binomial coefficient operation. For example,
(5C3) = 10 (ex-bc 28231).
In the literature, this function is often written as a column vector of the two arguments, or with the arguments as subscripts before and after the letter "C". The expression (𝑁C𝐾) is read "𝑁 choose 𝐾". Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 ≤ 𝑘 ≤ 𝑛 does not hold. (Contributed by NM, 10-Jul-2005.) |
⊢ C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)) | ||
Theorem | bcval 13665 | Value of the binomial coefficient, 𝑁 choose 𝐾. Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 ≤ 𝐾 ≤ 𝑁 does not hold. See bcval2 13666 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))), 0)) | ||
Theorem | bcval2 13666 | Value of the binomial coefficient, 𝑁 choose 𝐾, in its standard domain. (Contributed by NM, 9-Jun-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) | ||
Theorem | bcval3 13667 | Value of the binomial coefficient, 𝑁 choose 𝐾, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0) | ||
Theorem | bcval4 13668 | Value of the binomial coefficient, 𝑁 choose 𝐾, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ (𝐾 < 0 ∨ 𝑁 < 𝐾)) → (𝑁C𝐾) = 0) | ||
Theorem | bcrpcl 13669 | Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 13684.) (Contributed by Mario Carneiro, 10-Mar-2014.) |
⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+) | ||
Theorem | bccmpl 13670 | "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) = (𝑁C(𝑁 − 𝐾))) | ||
Theorem | bcn0 13671 | 𝑁 choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁C0) = 1) | ||
Theorem | bc0k 13672 | The binomial coefficient " 0 choose 𝐾 " is 0 for a positive integer K. Note that (0C0) = 1 (see bcn0 13671). (Contributed by Alexander van der Vekens, 1-Jan-2018.) |
⊢ (𝐾 ∈ ℕ → (0C𝐾) = 0) | ||
Theorem | bcnn 13673 | 𝑁 choose 𝑁 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁C𝑁) = 1) | ||
Theorem | bcn1 13674 | Binomial coefficient: 𝑁 choose 1. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁C1) = 𝑁) | ||
Theorem | bcnp1n 13675 | Binomial coefficient: 𝑁 + 1 choose 𝑁. (Contributed by NM, 20-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1)C𝑁) = (𝑁 + 1)) | ||
Theorem | bcm1k 13676 | The proportion of one binomial coefficient to another with 𝐾 decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.) |
⊢ (𝐾 ∈ (1...𝑁) → (𝑁C𝐾) = ((𝑁C(𝐾 − 1)) · ((𝑁 − (𝐾 − 1)) / 𝐾))) | ||
Theorem | bcp1n 13677 | The proportion of one binomial coefficient to another with 𝑁 increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.) |
⊢ (𝐾 ∈ (0...𝑁) → ((𝑁 + 1)C𝐾) = ((𝑁C𝐾) · ((𝑁 + 1) / ((𝑁 + 1) − 𝐾)))) | ||
Theorem | bcp1nk 13678 | The proportion of one binomial coefficient to another with 𝑁 and 𝐾 increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.) |
⊢ (𝐾 ∈ (0...𝑁) → ((𝑁 + 1)C(𝐾 + 1)) = ((𝑁C𝐾) · ((𝑁 + 1) / (𝐾 + 1)))) | ||
Theorem | bcval5 13679 | Write out the top and bottom parts of the binomial coefficient (𝑁C𝐾) = (𝑁 · (𝑁 − 1) · ... · ((𝑁 − 𝐾) + 1)) / 𝐾! explicitly. In this form, it is valid even for 𝑁 < 𝐾, although it is no longer valid for nonpositive 𝐾. (Contributed by Mario Carneiro, 22-May-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℕ) → (𝑁C𝐾) = ((seq((𝑁 − 𝐾) + 1)( · , I )‘𝑁) / (!‘𝐾))) | ||
Theorem | bcn2 13680 | Binomial coefficient: 𝑁 choose 2. (Contributed by Mario Carneiro, 22-May-2014.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁C2) = ((𝑁 · (𝑁 − 1)) / 2)) | ||
Theorem | bcp1m1 13681 | Compute the binomial coefficient of (𝑁 + 1) over (𝑁 − 1) (Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro, 22-May-2014.) |
⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1)C(𝑁 − 1)) = (((𝑁 + 1) · 𝑁) / 2)) | ||
Theorem | bcpasc 13682 | Pascal's rule for the binomial coefficient, generalized to all integers 𝐾. Equation 2 of [Gleason] p. 295. (Contributed by NM, 13-Jul-2005.) (Revised by Mario Carneiro, 10-Mar-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾)) | ||
Theorem | bccl 13683 | A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 9-Nov-2013.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) ∈ ℕ0) | ||
Theorem | bccl2 13684 | A binomial coefficient, in its standard domain, is a positive integer. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 10-Mar-2014.) |
⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℕ) | ||
Theorem | bcn2m1 13685 | Compute the binomial coefficient "𝑁 choose 2 " from "(𝑁 − 1) choose 2 ": (N-1) + ( (N-1) 2 ) = ( N 2 ). (Contributed by Alexander van der Vekens, 7-Jan-2018.) |
⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + ((𝑁 − 1)C2)) = (𝑁C2)) | ||
Theorem | bcn2p1 13686 | Compute the binomial coefficient "(𝑁 + 1) choose 2 " from "𝑁 choose 2 ": N + ( N 2 ) = ( (N+1) 2 ). (Contributed by Alexander van der Vekens, 8-Jan-2018.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁 + (𝑁C2)) = ((𝑁 + 1)C2)) | ||
Theorem | permnn 13687 | The number of permutations of 𝑁 − 𝑅 objects from a collection of 𝑁 objects is a positive integer. (Contributed by Jason Orendorff, 24-Jan-2007.) |
⊢ (𝑅 ∈ (0...𝑁) → ((!‘𝑁) / (!‘𝑅)) ∈ ℕ) | ||
Theorem | bcnm1 13688 | The binomial coefficent of (𝑁 − 1) is 𝑁. (Contributed by Scott Fenton, 16-May-2014.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁C(𝑁 − 1)) = 𝑁) | ||
Theorem | 4bc3eq4 13689 | The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.) |
⊢ (4C3) = 4 | ||
Theorem | 4bc2eq6 13690 | The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.) |
⊢ (4C2) = 6 | ||
Syntax | chash 13691 | Extend the definition of a class to include the set size function. |
class ♯ | ||
Definition | df-hash 13692 | Define the set size function ♯, which gives the cardinality of a finite set as a member of ℕ0, and assigns all infinite sets the value +∞. For example, (♯‘{0, 1, 2}) = 3 (ex-hash 28232). (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ ♯ = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞})) | ||
Theorem | hashkf 13693 | The finite part of the size function maps all finite sets to their cardinality, as members of ℕ0. (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 26-Dec-2014.) |
⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) & ⊢ 𝐾 = (𝐺 ∘ card) ⇒ ⊢ 𝐾:Fin⟶ℕ0 | ||
Theorem | hashgval 13694* | The value of the ♯ function in terms of the mapping 𝐺 from ω to ℕ0. The proof avoids the use of ax-ac 9881. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 26-Dec-2014.) |
⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ⇒ ⊢ (𝐴 ∈ Fin → (𝐺‘(card‘𝐴)) = (♯‘𝐴)) | ||
Theorem | hashginv 13695* | ◡𝐺 maps the size function's value to card. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.) |
⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ⇒ ⊢ (𝐴 ∈ Fin → (◡𝐺‘(♯‘𝐴)) = (card‘𝐴)) | ||
Theorem | hashinf 13696 | The value of the ♯ function on an infinite set. (Contributed by Mario Carneiro, 13-Jul-2014.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | ||
Theorem | hashbnd 13697 | If 𝐴 has size bounded by an integer 𝐵, then 𝐴 is finite. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ℕ0 ∧ (♯‘𝐴) ≤ 𝐵) → 𝐴 ∈ Fin) | ||
Theorem | hashfxnn0 13698 | The size function is a function into the extended nonnegative integers. (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by AV, 10-Dec-2020.) |
⊢ ♯:V⟶ℕ0* | ||
Theorem | hashf 13699 | The size function maps all finite sets to their cardinality, as members of ℕ0, and infinite sets to +∞. TODO-AV: mark as OBSOLETE and replace it by hashfxnn0 13698? (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 13-Jul-2014.) (Proof shortened by AV, 24-Oct-2021.) |
⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | ||
Theorem | hashxnn0 13700 | The value of the hash function for a set is an extended nonnegative integer. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 10-Dec-2020.) |
⊢ (𝑀 ∈ 𝑉 → (♯‘𝑀) ∈ ℕ0*) |
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