P. 143 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 14201-14300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sqgt0d 14201	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 14202	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 14203	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 14204	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 14205	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 14206	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 14207	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 14208	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 14209	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	ltexp1d 14210	Elevating to a positive power does not affect inequalities. Similar to ltmul1d 13016 for exponentiation of positive reals. (Contributed by Steven Nguyen, 22-Aug-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴↑𝑁) < (𝐵↑𝑁)))
Theorem	ltexp1dd 14211	Raising both sides of 'less than' to the same positive integer preserves ordering. (Contributed by Steven Nguyen, 24-Aug-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → (𝐴↑𝑁) < (𝐵↑𝑁))
Theorem	exp11nnd 14212	The function elevating nonnegative reals to a positive integer is one-to-one. Similar to sq11d 14209 for positive real bases and positive integer exponents. The base cannot be generalized much further, since if 𝑁 is even then we have 𝐴↑𝑁 = -𝐴↑𝑁. (Contributed by SN, 14-Sep-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝐴↑𝑁) = (𝐵↑𝑁))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	mulsubdivbinom2 14213	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 14214	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 14215	The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
⊢ (;10↑2) = ;;100
Theorem	sq10e99m1 14216	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 14217	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)
5.6.8  Ordered pair theorem for nonnegative integers
Theorem	nn0le2msqi 14218	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 14219	A rather pretty lemma for nn0opthi 14221. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    ⇒   ⊢ (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶))
Theorem	nn0opthlem2 14220	Lemma for nn0opthi 14221. (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 14221	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 4575 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 14222	An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthi 14221. (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 14223	An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 14221. (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)))
5.6.9  Factorial function
Syntax	cfa 14224	Extend class notation to include the factorial of nonnegative integers.
class !
Definition	df-fac 14225	Define the factorial function on nonnegative integers. For example, (!‘5) = 120 because 1 · 2 · 3 · 4 · 5 = 120 (ex-fac 30541). 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 14226	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 14227	The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
⊢ (!‘0) = 1
Theorem	fac1 14228	The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
⊢ (!‘1) = 1
Theorem	facp1 14229	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 14230	The factorial of 2. (Contributed by NM, 17-Mar-2005.)
⊢ (!‘2) = 2
Theorem	fac3 14231	The factorial of 3. (Contributed by NM, 17-Mar-2005.)
⊢ (!‘3) = 6
Theorem	fac4 14232	The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (!‘4) = ;24
Theorem	facnn2 14233	Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)
⊢ (𝑁 ∈ ℕ → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁))
Theorem	faccl 14234	Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ)
Theorem	faccld 14235	Closure of the factorial function, deduction version of faccl 14234. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
Theorem	facmapnn 14236	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 14237	The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ≠ 0)
Theorem	facdiv 14238	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 14239	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 14240	Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (!‘𝑀) ≤ (!‘𝑁))
Theorem	faclbnd 14241	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 14242	A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) / 2) ≤ (!‘𝑁))
Theorem	faclbnd3 14243	A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀↑𝑁) ≤ ((𝑀↑𝑀) · (!‘𝑁)))
Theorem	faclbnd4lem1 14244	Lemma for faclbnd4 14248. 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 14245	Lemma for faclbnd4 14248. Use the weak deduction theorem to convert the hypotheses of faclbnd4lem1 14244 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 14246	Lemma for faclbnd4 14248. 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 14247	Lemma for faclbnd4 14248. 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 14248	Variant of faclbnd5 14249 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 14249	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 14250	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 14251	An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ≤ (𝑁↑𝑁))
Theorem	facavg 14252	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)))
5.6.10  The binomial coefficient operation
Syntax	cbc 14253	Extend class notation to include the binomial coefficient operation (combinatorial choose operation).
class C
Definition	df-bc 14254*	Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 30542). 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 14255	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 14256 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 14256	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 14257	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 14258	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 14259	Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 14274.) (Contributed by Mario Carneiro, 10-Mar-2014.)
⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+)
Theorem	bccmpl 14260	"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 14261	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 14262	The binomial coefficient " 0 choose 𝐾 " is 0 for a positive integer K. Note that (0C0) = 1 (see bcn0 14261). (Contributed by Alexander van der Vekens, 1-Jan-2018.)
⊢ (𝐾 ∈ ℕ → (0C𝐾) = 0)
Theorem	bcnn 14263	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 14264	Binomial coefficient: 𝑁 choose 1. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
⊢ (𝑁 ∈ ℕ0 → (𝑁C1) = 𝑁)
Theorem	bcnp1n 14265	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 14266	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 14267	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 14268	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 14269	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 14270	Binomial coefficient: 𝑁 choose 2. (Contributed by Mario Carneiro, 22-May-2014.)
⊢ (𝑁 ∈ ℕ0 → (𝑁C2) = ((𝑁 · (𝑁 − 1)) / 2))
Theorem	bcp1m1 14271	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 14272	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 14273	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 14274	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 14275	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 14276	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 14277	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 14278	The binomial coefficient of (𝑁 − 1) is 𝑁. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑁 ∈ ℕ0 → (𝑁C(𝑁 − 1)) = 𝑁)
Theorem	4bc3eq4 14279	The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
⊢ (4C3) = 4
Theorem	4bc2eq6 14280	The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
⊢ (4C2) = 6
5.6.11  The ` # ` (set size) function
Syntax	chash 14281	Extend the definition of a class to include the set size function.
class ♯
Definition	df-hash 14282	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 30543). (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ♯ = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞}))
Theorem	hashkf 14283	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 14284*	The value of the ♯ function in terms of the mapping 𝐺 from ω to ℕ0. The proof avoids the use of ax-ac 10370. (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 14285*	The converse of 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 14286	The value of the ♯ function on an infinite set. (Contributed by Mario Carneiro, 13-Jul-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞)
Theorem	hashbnd 14287	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 14288	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 14289	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 14288? (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 14290	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*)
Theorem	hashresfn 14291	Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 31-Jan-2017.)
⊢ (♯ ↾ 𝐴) Fn 𝐴
Theorem	dmhashres 14292	Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 12-Jan-2017.)
⊢ dom (♯ ↾ 𝐴) = 𝐴
Theorem	hashnn0pnf 14293	The value of the hash function for a set is either a nonnegative integer or positive infinity. TODO-AV: mark as OBSOLETE and replace it by hashxnn0 14290? (Contributed by Alexander van der Vekens, 6-Dec-2017.)
⊢ (𝑀 ∈ 𝑉 → ((♯‘𝑀) ∈ ℕ0 ∨ (♯‘𝑀) = +∞))
Theorem	hashnnn0genn0 14294	If the size of a set is not a nonnegative integer, it is greater than or equal to any nonnegative integer. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
⊢ ((𝑀 ∈ 𝑉 ∧ (♯‘𝑀) ∉ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (♯‘𝑀))
Theorem	hashnemnf 14295	The size of a set is never minus infinity. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ≠ -∞)
Theorem	hashv01gt1 14296	The size of a set is either 0 or 1 or greater than 1. (Contributed by Alexander van der Vekens, 29-Dec-2017.)
⊢ (𝑀 ∈ 𝑉 → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀)))
Theorem	hashfz1 14297	The set (1...𝑁) has 𝑁 elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
Theorem	hashen 14298	Two finite sets have the same number of elements iff they are equinumerous. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ 𝐴 ≈ 𝐵))
Theorem	hasheni 14299	Equinumerous sets have the same number of elements (even if they are not finite). (Contributed by Mario Carneiro, 15-Apr-2015.)
⊢ (𝐴 ≈ 𝐵 → (♯‘𝐴) = (♯‘𝐵))
Theorem	hasheqf1o 14300*	The size of two finite sets is equal if and only if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))