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Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexpp1zapd 10401 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴𝑁) · 𝐴))
 
Theoremexpm1apd 10402 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴↑(𝑁 − 1)) = ((𝐴𝑁) / 𝐴))
 
Theoremexpsubapd 10403 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐴↑(𝑀𝑁)) = ((𝐴𝑀) / (𝐴𝑁)))
 
Theoremsqmuld 10404 Distribution of square over multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2)))
 
Theoremsqdivapd 10405 Distribution of square over division. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))
 
Theoremexpdivapd 10406 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝐴 / 𝐵)↑𝑁) = ((𝐴𝑁) / (𝐵𝑁)))
 
Theoremmulexpd 10407 Positive integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝐴 · 𝐵)↑𝑁) = ((𝐴𝑁) · (𝐵𝑁)))
 
Theorem0expd 10408 Value of zero raised to a positive integer power. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → (0↑𝑁) = 0)
 
Theoremreexpcld 10409 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑁) ∈ ℝ)
 
Theoremexpge0d 10410 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → 0 ≤ (𝐴𝑁))
 
Theoremexpge1d 10411 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → 1 ≤ 𝐴)       (𝜑 → 1 ≤ (𝐴𝑁))
 
Theoremsqoddm1div8 10412 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))
 
Theoremnnsqcld 10413 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → (𝐴↑2) ∈ ℕ)
 
Theoremnnexpcld 10414 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑁) ∈ ℕ)
 
Theoremnn0expcld 10415 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑁) ∈ ℕ0)
 
Theoremrpexpcld 10416 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴𝑁) ∈ ℝ+)
 
Theoremreexpclzapd 10417 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴𝑁) ∈ ℝ)
 
Theoremresqcld 10418 Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴↑2) ∈ ℝ)
 
Theoremsqge0d 10419 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → 0 ≤ (𝐴↑2))
 
Theoremsqgt0apd 10420 The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 # 0)       (𝜑 → 0 < (𝐴↑2))
 
Theoremleexp2ad 10421 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 ≤ 𝐴)    &   (𝜑𝑁 ∈ (ℤ𝑀))       (𝜑 → (𝐴𝑀) ≤ (𝐴𝑁))
 
Theoremleexp2rd 10422 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 ≤ 1)       (𝜑 → (𝐴𝑁) ≤ (𝐴𝑀))
 
Theoremlt2sqd 10423 The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐴↑2) < (𝐵↑2)))
 
Theoremle2sqd 10424 The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → (𝐴𝐵 ↔ (𝐴↑2) ≤ (𝐵↑2)))
 
Theoremsq11d 10425 The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 ≤ 𝐵)    &   (𝜑 → (𝐴↑2) = (𝐵↑2))       (𝜑𝐴 = 𝐵)
 
Theoremsq11ap 10426 Analogue to sq11 10333 but for apartness. (Contributed by Jim Kingdon, 12-Aug-2021.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) # (𝐵↑2) ↔ 𝐴 # 𝐵))
 
Theoremsq10 10427 The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
(10↑2) = 100
 
Theoremsq10e99m1 10428 The square of 10 is 99 plus 1. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
(10↑2) = (99 + 1)
 
Theorem3dec 10429 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)
 
Theoremexpcanlem 10430 Lemma for expcan 10431. Proving the order in one direction. (Contributed by Jim Kingdon, 29-Jan-2022.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑 → 1 < 𝐴)       (𝜑 → ((𝐴𝑀) ≤ (𝐴𝑁) → 𝑀𝑁))
 
Theoremexpcan 10431 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) → ((𝐴𝑀) = (𝐴𝑁) ↔ 𝑀 = 𝑁))
 
Theoremexpcand 10432 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑 → 1 < 𝐴)    &   (𝜑 → (𝐴𝑀) = (𝐴𝑁))       (𝜑𝑀 = 𝑁)
 
4.6.7  Ordered pair theorem for nonnegative integers
 
Theoremnn0le2msqd 10433 The square function on nonnegative integers is monotonic. (Contributed by Jim Kingdon, 31-Oct-2021.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)       (𝜑 → (𝐴𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵)))
 
Theoremnn0opthlem1d 10434 A rather pretty lemma for nn0opth2 10438. (Contributed by Jim Kingdon, 31-Oct-2021.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐶 ∈ ℕ0)       (𝜑 → (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶)))
 
Theoremnn0opthlem2d 10435 Lemma for nn0opth2 10438. (Contributed by Jim Kingdon, 31-Oct-2021.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)    &   (𝜑𝐶 ∈ ℕ0)    &   (𝜑𝐷 ∈ ℕ0)       (𝜑 → ((𝐴 + 𝐵) < 𝐶 → ((𝐶 · 𝐶) + 𝐷) ≠ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵)))
 
Theoremnn0opthd 10436 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 3506 that works for any set. (Contributed by Jim Kingdon, 31-Oct-2021.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)    &   (𝜑𝐶 ∈ ℕ0)    &   (𝜑𝐷 ∈ ℕ0)       (𝜑 → ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 
Theoremnn0opth2d 10437 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthd 10436. (Contributed by Jim Kingdon, 31-Oct-2021.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)    &   (𝜑𝐶 ∈ ℕ0)    &   (𝜑𝐷 ∈ ℕ0)       (𝜑 → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 
Theoremnn0opth2 10438 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthd 10436. (Contributed by NM, 22-Jul-2004.)
(((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) ∧ (𝐶 ∈ ℕ0𝐷 ∈ ℕ0)) → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 
4.6.8  Factorial function
 
Syntaxcfa 10439 Extend class notation to include the factorial of nonnegative integers.
class !
 
Definitiondf-fac 10440 Define the factorial function on nonnegative integers. For example, (!‘5) = 120 because 1 · 2 · 3 · 4 · 5 = 120 (ex-fac 12867). In the literature, the factorial function is written as a postscript exclamation point. (Contributed by NM, 2-Dec-2004.)
! = ({⟨0, 1⟩} ∪ seq1( · , I ))
 
Theoremfacnn 10441 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))
 
Theoremfac0 10442 The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
(!‘0) = 1
 
Theoremfac1 10443 The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
(!‘1) = 1
 
Theoremfacp1 10444 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)))
 
Theoremfac2 10445 The factorial of 2. (Contributed by NM, 17-Mar-2005.)
(!‘2) = 2
 
Theoremfac3 10446 The factorial of 3. (Contributed by NM, 17-Mar-2005.)
(!‘3) = 6
 
Theoremfac4 10447 The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
(!‘4) = 24
 
Theoremfacnn2 10448 Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)
(𝑁 ∈ ℕ → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁))
 
Theoremfaccl 10449 Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)
(𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ)
 
Theoremfaccld 10450 Closure of the factorial function, deduction version of faccl 10449. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑁 ∈ ℕ0)       (𝜑 → (!‘𝑁) ∈ ℕ)
 
Theoremfacne0 10451 The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)
(𝑁 ∈ ℕ0 → (!‘𝑁) ≠ 0)
 
Theoremfacdiv 10452 A positive integer divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑁𝑀) → ((!‘𝑀) / 𝑁) ∈ ℕ)
 
Theoremfacndiv 10453 No positive integer (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.)
(((𝑀 ∈ ℕ0𝑁 ∈ ℕ) ∧ (1 < 𝑁𝑁𝑀)) → ¬ (((!‘𝑀) + 1) / 𝑁) ∈ ℤ)
 
Theoremfacwordi 10454 Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (!‘𝑀) ≤ (!‘𝑁))
 
Theoremfaclbnd 10455 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀𝑀) · (!‘𝑁)))
 
Theoremfaclbnd2 10456 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
(𝑁 ∈ ℕ0 → ((2↑𝑁) / 2) ≤ (!‘𝑁))
 
Theoremfaclbnd3 10457 A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁) ≤ ((𝑀𝑀) · (!‘𝑁)))
 
Theoremfaclbnd6 10458 Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → ((!‘𝑁) · ((𝑁 + 1)↑𝑀)) ≤ (!‘(𝑁 + 𝑀)))
 
Theoremfacubnd 10459 An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)
(𝑁 ∈ ℕ0 → (!‘𝑁) ≤ (𝑁𝑁))
 
Theoremfacavg 10460 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 ∈ ℕ0𝑁 ∈ ℕ0) → (!‘(⌊‘((𝑀 + 𝑁) / 2))) ≤ ((!‘𝑀) · (!‘𝑁)))
 
4.6.9  The binomial coefficient operation
 
Syntaxcbc 10461 Extend class notation to include the binomial coefficient operation (combinatorial choose operation).
class C
 
Definitiondf-bc 10462* Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 12868).

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 ≤ 𝑘𝑛 does not hold. (Contributed by NM, 10-Jul-2005.)

C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0))
 
Theorembcval 10463 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 ≤ 𝐾𝑁 does not hold. See bcval2 10464 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → (𝑁C𝐾) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁𝐾)) · (!‘𝐾))), 0))
 
Theorembcval2 10464 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) / ((!‘(𝑁𝐾)) · (!‘𝐾))))
 
Theorembcval3 10465 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 ∈ ℕ0𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0)
 
Theorembcval4 10466 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 ∈ ℕ0𝐾 ∈ ℤ ∧ (𝐾 < 0 ∨ 𝑁 < 𝐾)) → (𝑁C𝐾) = 0)
 
Theorembcrpcl 10467 Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 10482.) (Contributed by Mario Carneiro, 10-Mar-2014.)
(𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+)
 
Theorembccmpl 10468 "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.)
((𝑁 ∈ ℕ0𝐾 ∈ ℤ) → (𝑁C𝐾) = (𝑁C(𝑁𝐾)))
 
Theorembcn0 10469 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)
 
Theorembc0k 10470 The binomial coefficient " 0 choose 𝐾 " is 0 for a positive integer K. Note that (0C0) = 1 (see bcn0 10469). (Contributed by Alexander van der Vekens, 1-Jan-2018.)
(𝐾 ∈ ℕ → (0C𝐾) = 0)
 
Theorembcnn 10471 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)
 
Theorembcn1 10472 Binomial coefficient: 𝑁 choose 1. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
(𝑁 ∈ ℕ0 → (𝑁C1) = 𝑁)
 
Theorembcnp1n 10473 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))
 
Theorembcm1k 10474 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)))
 
Theorembcp1n 10475 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))))
 
Theorembcp1nk 10476 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))))
 
Theorembcval5 10477 Write out the top and bottom parts of the binomial coefficient (𝑁C𝐾) = (𝑁 · (𝑁 − 1) · ... · ((𝑁𝐾) + 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 ∈ ℕ0𝐾 ∈ ℕ) → (𝑁C𝐾) = ((seq((𝑁𝐾) + 1)( · , I )‘𝑁) / (!‘𝐾)))
 
Theorembcn2 10478 Binomial coefficient: 𝑁 choose 2. (Contributed by Mario Carneiro, 22-May-2014.)
(𝑁 ∈ ℕ0 → (𝑁C2) = ((𝑁 · (𝑁 − 1)) / 2))
 
Theorembcp1m1 10479 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))
 
Theorembcpasc 10480 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 ∈ ℕ0𝐾 ∈ ℤ) → ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾))
 
Theorembccl 10481 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 ∈ ℕ0𝐾 ∈ ℤ) → (𝑁C𝐾) ∈ ℕ0)
 
Theorembccl2 10482 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)
 
Theorembcn2m1 10483 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))
 
Theorembcn2p1 10484 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))
 
Theorempermnn 10485 The number of permutations of 𝑁𝑅 objects from a collection of 𝑁 objects is a positive integer. (Contributed by Jason Orendorff, 24-Jan-2007.)
(𝑅 ∈ (0...𝑁) → ((!‘𝑁) / (!‘𝑅)) ∈ ℕ)
 
Theorembcnm1 10486 The binomial coefficent of (𝑁 − 1) is 𝑁. (Contributed by Scott Fenton, 16-May-2014.)
(𝑁 ∈ ℕ0 → (𝑁C(𝑁 − 1)) = 𝑁)
 
Theorem4bc3eq4 10487 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
(4C3) = 4
 
Theorem4bc2eq6 10488 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
(4C2) = 6
 
4.6.10  The ` # ` (set size) function
 
Syntaxchash 10489 Extend the definition of a class to include the set size function.
class
 
Definitiondf-ihash 10490* Define the set size function , which gives the cardinality of a finite set as a member of 0, 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 8312). 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}) ∣ 𝑦𝑥}))
 
Theoremhashinfuni 10491* The ordinal size of an infinite set is ω. (Contributed by Jim Kingdon, 20-Feb-2022.)
(ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = ω)
 
Theoremhashinfom 10492 The value of the function on an infinite set. (Contributed by Jim Kingdon, 20-Feb-2022.)
(ω ≼ 𝐴 → (♯‘𝐴) = +∞)
 
Theoremhashennnuni 10493* 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 ∧ 𝑁𝐴) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = 𝑁)
 
Theoremhashennn 10494* The size of a set equinumerous to an element of ω. (Contributed by Jim Kingdon, 21-Feb-2022.)
((𝑁 ∈ ω ∧ 𝑁𝐴) → (♯‘𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))
 
Theoremhashcl 10495 Closure of the function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)
(𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
 
Theoremhashfiv01gt1 10496 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)))
 
Theoremhashfz1 10497 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)
 
Theoremhashen 10498 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) ↔ 𝐴𝐵))
 
Theoremhasheqf1o 10499* 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:𝐴1-1-onto𝐵))
 
Theoremfiinfnf1o 10500* There is no bijection between a finite set and an infinite set. By infnfi 6757 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:𝐴1-1-onto𝐵)
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