P. 109 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10801-10900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nn0ltexp2 10801	Special case of ltexp2 15177 which we use here because we haven't yet defined df-rpcxp 15095 which is used in the current proof of ltexp2 15177. (Contributed by Jim Kingdon, 7-Oct-2024.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁)))
Theorem	nn0leexp2 10802	Ordering law for exponentiation. (Contributed by Jim Kingdon, 9-Oct-2024.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ 1 < 𝐴) → (𝑀 ≤ 𝑁 ↔ (𝐴↑𝑀) ≤ (𝐴↑𝑁)))
Theorem	mulsubdivbinom2ap 10803	The square of a binomial with factor minus a number divided by a number apart from zero. (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	sq10 10804	The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
⊢ (;10↑2) = ;;100
Theorem	sq10e99m1 10805	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 10806	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	expcanlem 10807	Lemma for expcan 10808. Proving the order in one direction. (Contributed by Jim Kingdon, 29-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 1 < 𝐴)    ⇒   ⊢ (𝜑 → ((𝐴↑𝑀) ≤ (𝐴↑𝑁) → 𝑀 ≤ 𝑁))
Theorem	expcan 10808	Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑𝑀) = (𝐴↑𝑁) ↔ 𝑀 = 𝑁))
Theorem	expcand 10809	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	apexp1 10810	Exponentiation and apartness. (Contributed by Jim Kingdon, 9-Jul-2024.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) # (𝐵↑𝑁) → 𝐴 # 𝐵))
4.6.7  Ordered pair theorem for nonnegative integers
Theorem	nn0le2msqd 10811	The square function on nonnegative integers is monotonic. (Contributed by Jim Kingdon, 31-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵)))
Theorem	nn0opthlem1d 10812	A rather pretty lemma for nn0opth2 10816. (Contributed by Jim Kingdon, 31-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶)))
Theorem	nn0opthlem2d 10813	Lemma for nn0opth2 10816. (Contributed by Jim Kingdon, 31-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) < 𝐶 → ((𝐶 · 𝐶) + 𝐷) ≠ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵)))
Theorem	nn0opthd 10814	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 3631 that works for any set. (Contributed by Jim Kingdon, 31-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	nn0opth2d 10815	An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthd 10814. (Contributed by Jim Kingdon, 31-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	nn0opth2 10816	An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthd 10814. (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)))
4.6.8  Factorial function
Syntax	cfa 10817	Extend class notation to include the factorial of nonnegative integers.
class !
Definition	df-fac 10818	Define the factorial function on nonnegative integers. For example, (!‘5) = 120 because 1 · 2 · 3 · 4 · 5 = 120 (ex-fac 15374). 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 10819	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 10820	The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
⊢ (!‘0) = 1
Theorem	fac1 10821	The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
⊢ (!‘1) = 1
Theorem	facp1 10822	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 10823	The factorial of 2. (Contributed by NM, 17-Mar-2005.)
⊢ (!‘2) = 2
Theorem	fac3 10824	The factorial of 3. (Contributed by NM, 17-Mar-2005.)
⊢ (!‘3) = 6
Theorem	fac4 10825	The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (!‘4) = ;24
Theorem	facnn2 10826	Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)
⊢ (𝑁 ∈ ℕ → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁))
Theorem	faccl 10827	Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ)
Theorem	faccld 10828	Closure of the factorial function, deduction version of faccl 10827. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
Theorem	facne0 10829	The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ≠ 0)
Theorem	facdiv 10830	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 10831	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 10832	Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (!‘𝑀) ≤ (!‘𝑁))
Theorem	faclbnd 10833	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 10834	A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) / 2) ≤ (!‘𝑁))
Theorem	faclbnd3 10835	A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀↑𝑁) ≤ ((𝑀↑𝑀) · (!‘𝑁)))
Theorem	faclbnd6 10836	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 10837	An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ≤ (𝑁↑𝑁))
Theorem	facavg 10838	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)))
4.6.9  The binomial coefficient operation
Syntax	cbc 10839	Extend class notation to include the binomial coefficient operation (combinatorial choose operation).
class C
Definition	df-bc 10840*	Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 15375). 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". (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 10841	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 10842 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 10842	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 10843	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 10844	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 10845	Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 10860.) (Contributed by Mario Carneiro, 10-Mar-2014.)
⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+)
Theorem	bccmpl 10846	"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 10847	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 10848	The binomial coefficient " 0 choose 𝐾 " is 0 for a positive integer K. Note that (0C0) = 1 (see bcn0 10847). (Contributed by Alexander van der Vekens, 1-Jan-2018.)
⊢ (𝐾 ∈ ℕ → (0C𝐾) = 0)
Theorem	bcnn 10849	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 10850	Binomial coefficient: 𝑁 choose 1. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
⊢ (𝑁 ∈ ℕ0 → (𝑁C1) = 𝑁)
Theorem	bcnp1n 10851	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 10852	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 10853	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 10854	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 10855	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.) (Revised by Jim Kingdon, 23-Apr-2023.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℕ) → (𝑁C𝐾) = ((seq((𝑁 − 𝐾) + 1)( · , I )‘𝑁) / (!‘𝐾)))
Theorem	bcn2 10856	Binomial coefficient: 𝑁 choose 2. (Contributed by Mario Carneiro, 22-May-2014.)
⊢ (𝑁 ∈ ℕ0 → (𝑁C2) = ((𝑁 · (𝑁 − 1)) / 2))
Theorem	bcp1m1 10857	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 10858	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 10859	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 10860	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 10861	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 10862	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 10863	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 10864	The binomial coefficent of (𝑁 − 1) is 𝑁. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑁 ∈ ℕ0 → (𝑁C(𝑁 − 1)) = 𝑁)
Theorem	4bc3eq4 10865	The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
⊢ (4C3) = 4
Theorem	4bc2eq6 10866	The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
⊢ (4C2) = 6
4.6.10  The ` # ` (set size) function
Syntax	chash 10867	Extend the definition of a class to include the set size function.
class ♯
Definition	df-ihash 10868*	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. Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8609). We adopt the former notation from Corollary 8.2.4 of [AczelRathjen], p. 80 (although that work only defines it for finite sets). This definition (in terms of ∪ and ≼) is not taken directly from the literature, but for finite sets should be equivalent to the conventional definition that the size of a finite set is the unique natural number which is equinumerous to the given set. (Contributed by Jim Kingdon, 19-Feb-2022.)
⊢ ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
Theorem	hashinfuni 10869*	The ordinal size of an infinite set is ω. (Contributed by Jim Kingdon, 20-Feb-2022.)
⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω)
Theorem	hashinfom 10870	The value of the ♯ function on an infinite set. (Contributed by Jim Kingdon, 20-Feb-2022.)
⊢ (ω ≼ 𝐴 → (♯‘𝐴) = +∞)
Theorem	hashennnuni 10871*	The ordinal size of a set equinumerous to an element of ω is that element of ω. (Contributed by Jim Kingdon, 20-Feb-2022.)
⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = 𝑁)
Theorem	hashennn 10872*	The size of a set equinumerous to an element of ω. (Contributed by Jim Kingdon, 21-Feb-2022.)
⊢ ((𝑁 ∈ ω ∧ 𝑁 ≈ 𝐴) → (♯‘𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))
Theorem	hashcl 10873	Closure of the ♯ function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)
⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
Theorem	hashfiv01gt1 10874	The size of a finite set is either 0 or 1 or greater than 1. (Contributed by Jim Kingdon, 21-Feb-2022.)
⊢ (𝑀 ∈ Fin → ((♯‘𝑀) = 0 ∨ (♯‘𝑀) = 1 ∨ 1 < (♯‘𝑀)))
Theorem	hashfz1 10875	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 10876	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	hasheqf1o 10877*	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))
Theorem	fiinfnf1o 10878*	There is no bijection between a finite set and an infinite set. By infnfi 6956 the theorem would also hold if "infinite" were expressed as ω ≼ 𝐵. (Contributed by Alexander van der Vekens, 25-Dec-2017.)
⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
Theorem	fihasheqf1oi 10879	The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → (♯‘𝐴) = (♯‘𝐵))
Theorem	fihashf1rn 10880	The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Jim Kingdon, 21-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐹) = (♯‘ran 𝐹))
Theorem	fihasheqf1od 10881	The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)    ⇒   ⊢ (𝜑 → (♯‘𝐴) = (♯‘𝐵))
Theorem	fz1eqb 10882	Two possibly-empty 1-based finite sets of sequential integers are equal iff their endpoints are equal. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 29-Mar-2014.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((1...𝑀) = (1...𝑁) ↔ 𝑀 = 𝑁))
Theorem	filtinf 10883	The size of an infinite set is greater than the size of a finite set. (Contributed by Jim Kingdon, 21-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ ω ≼ 𝐵) → (♯‘𝐴) < (♯‘𝐵))
Theorem	isfinite4im 10884	A finite set is equinumerous to the range of integers from one up to the hash value of the set. (Contributed by Jim Kingdon, 22-Feb-2022.)
⊢ (𝐴 ∈ Fin → (1...(♯‘𝐴)) ≈ 𝐴)
Theorem	fihasheq0 10885	Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
Theorem	fihashneq0 10886	Two ways of saying a finite set is not empty. Also, "A is inhabited" would be equivalent by fin0 6946. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
⊢ (𝐴 ∈ Fin → (0 < (♯‘𝐴) ↔ 𝐴 ≠ ∅))
Theorem	hashnncl 10887	Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ 𝐴 ≠ ∅))
Theorem	hash0 10888	The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)
⊢ (♯‘∅) = 0
Theorem	fihashelne0d 10889	A finite set with an element has nonzero size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → ¬ (♯‘𝐴) = 0)
Theorem	hashsng 10890	The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = 1)
Theorem	fihashen1 10891	A finite set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1o))
Theorem	fihashfn 10892	A function on a finite set is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.) (Intuitionized by Jim Kingdon, 24-Feb-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (♯‘𝐹) = (♯‘𝐴))
Theorem	fseq1hash 10893	The value of the size function on a finite 1-based sequence. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)
Theorem	omgadd 10894	Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.)
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))
Theorem	fihashdom 10895	Dominance relation for the size function. (Contributed by Jim Kingdon, 24-Feb-2022.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))
Theorem	hashunlem 10896	Lemma for hashun 10897. Ordinal size of the union. (Contributed by Jim Kingdon, 25-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → 𝑀 ∈ ω)    &   ⊢ (𝜑 → 𝐴 ≈ 𝑁)    &   ⊢ (𝜑 → 𝐵 ≈ 𝑀)    ⇒   ⊢ (𝜑 → (𝐴 ∪ 𝐵) ≈ (𝑁 +o 𝑀))
Theorem	hashun 10897	The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
Theorem	1elfz0hash 10898	1 is an element of the finite set of sequential nonnegative integers bounded by the size of a nonempty finite set. (Contributed by AV, 9-May-2020.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → 1 ∈ (0...(♯‘𝐴)))
Theorem	hashunsng 10899	The size of the union of a finite set with a disjoint singleton is one more than the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.)
⊢ (𝐵 ∈ 𝑉 → ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ 𝐴) → (♯‘(𝐴 ∪ {𝐵})) = ((♯‘𝐴) + 1)))
Theorem	hashprg 10900	The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV, 18-Sep-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ 𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))