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	bernneq 14201	Bernoulli's inequality, due to Johan Bernoulli (1667-1748). (Contributed by NM, 21-Feb-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ -1 ≤ 𝐴) → (1 + (𝐴 · 𝑁)) ≤ ((1 + 𝐴)↑𝑁))
Theorem	bernneq2 14202	Variation of Bernoulli's inequality bernneq 14201. (Contributed by NM, 18-Oct-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴↑𝑁))
Theorem	bernneq3 14203	A corollary of bernneq 14201. (Contributed by Mario Carneiro, 11-Mar-2014.)
⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (𝑃↑𝑁))
Theorem	expnbnd 14204*	Exponentiation with a base greater than 1 has no upper bound. (Contributed by NM, 20-Oct-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵) → ∃𝑘 ∈ ℕ 𝐴 < (𝐵↑𝑘))
Theorem	expnlbnd 14205*	The reciprocal of exponentiation with a base greater than 1 has no positive lower bound. (Contributed by NM, 18-Jul-2008.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵) → ∃𝑘 ∈ ℕ (1 / (𝐵↑𝑘)) < 𝐴)
Theorem	expnlbnd2 14206*	The reciprocal of exponentiation with a base greater than 1 has no positive lower bound. (Contributed by NM, 18-Jul-2008.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(1 / (𝐵↑𝑘)) < 𝐴)
Theorem	expmulnbnd 14207*	Exponentiation with a base greater than 1 is not bounded by any linear function. (Contributed by Mario Carneiro, 31-Mar-2015.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵) → ∃𝑗 ∈ ℕ0 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐴 · 𝑘) < (𝐵↑𝑘))
Theorem	digit2 14208	Two ways to express the 𝐾 th digit in the decimal (when base 𝐵 = 10) expansion of a number 𝐴. 𝐾 = 1 corresponds to the first digit after the decimal point. (Contributed by NM, 25-Dec-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ ∧ 𝐾 ∈ ℕ) → ((⌊‘((𝐵↑𝐾) · 𝐴)) mod 𝐵) = ((⌊‘((𝐵↑𝐾) · 𝐴)) − (𝐵 · (⌊‘((𝐵↑(𝐾 − 1)) · 𝐴)))))
Theorem	digit1 14209	Two ways to express the 𝐾 th digit in the decimal expansion of a number 𝐴 (when base 𝐵 = 10). 𝐾 = 1 corresponds to the first digit after the decimal point. (Contributed by NM, 3-Jan-2009.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ ∧ 𝐾 ∈ ℕ) → ((⌊‘((𝐵↑𝐾) · 𝐴)) mod 𝐵) = (((⌊‘((𝐵↑𝐾) · 𝐴)) mod (𝐵↑𝐾)) − ((𝐵 · (⌊‘((𝐵↑(𝐾 − 1)) · 𝐴))) mod (𝐵↑𝐾))))
Theorem	modexp 14210	Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℝ+) ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑𝐶) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷))
Theorem	discr1 14211*	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 14212*	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 14213	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 14214	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 14215	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 14216	The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴↑2) ∈ ℕ)
Theorem	nnexpcld 14217	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 14218	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 14219	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 14220	The power of a positive number less 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 14221	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	sqgt0d 14222	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 14223	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 14224	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 14225	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 14226	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 14227	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 14228	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 14229	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 14230	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 14231	Elevating to a positive power does not affect inequalities. Similar to ltmul1d 13043 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 14232	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 14233	The function elevating nonnegative reals to a positive integer is one-to-one. Similar to sq11d 14230 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 14234	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 14235	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 14236	The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
⊢ (;10↑2) = ;;100
Theorem	sq10e99m1 14237	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 14238	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 14239	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 14240	A rather pretty lemma for nn0opthi 14242. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    ⇒   ⊢ (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶))
Theorem	nn0opthlem2 14241	Lemma for nn0opthi 14242. (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 14242	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 4599 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 14243	An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthi 14242. (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 14244	An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 14242. (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 14245	Extend class notation to include the factorial of nonnegative integers.
class !
Definition	df-fac 14246	Define the factorial function on nonnegative integers. For example, (!‘5) = 120 because 1 · 2 · 3 · 4 · 5 = 120 (ex-fac 30387). 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 14247	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 14248	The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
⊢ (!‘0) = 1
Theorem	fac1 14249	The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
⊢ (!‘1) = 1
Theorem	facp1 14250	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 14251	The factorial of 2. (Contributed by NM, 17-Mar-2005.)
⊢ (!‘2) = 2
Theorem	fac3 14252	The factorial of 3. (Contributed by NM, 17-Mar-2005.)
⊢ (!‘3) = 6
Theorem	fac4 14253	The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (!‘4) = ;24
Theorem	facnn2 14254	Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)
⊢ (𝑁 ∈ ℕ → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁))
Theorem	faccl 14255	Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ)
Theorem	faccld 14256	Closure of the factorial function, deduction version of faccl 14255. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
Theorem	facmapnn 14257	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 14258	The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ≠ 0)
Theorem	facdiv 14259	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 14260	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 14261	Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (!‘𝑀) ≤ (!‘𝑁))
Theorem	faclbnd 14262	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 14263	A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) / 2) ≤ (!‘𝑁))
Theorem	faclbnd3 14264	A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀↑𝑁) ≤ ((𝑀↑𝑀) · (!‘𝑁)))
Theorem	faclbnd4lem1 14265	Lemma for faclbnd4 14269. 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 14266	Lemma for faclbnd4 14269. Use the weak deduction theorem to convert the hypotheses of faclbnd4lem1 14265 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 14267	Lemma for faclbnd4 14269. 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 14268	Lemma for faclbnd4 14269. 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 14269	Variant of faclbnd5 14270 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 14270	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 14271	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 14272	An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ≤ (𝑁↑𝑁))
Theorem	facavg 14273	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 14274	Extend class notation to include the binomial coefficient operation (combinatorial choose operation).
class C
Definition	df-bc 14275*	Define the binomial coefficient operation. For example, (5C3) = 10 (ex-bc 30388). 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 14276	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 14277 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 14277	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 14278	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 14279	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 14280	Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 14295.) (Contributed by Mario Carneiro, 10-Mar-2014.)
⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+)
Theorem	bccmpl 14281	"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 14282	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 14283	The binomial coefficient " 0 choose 𝐾 " is 0 for a positive integer K. Note that (0C0) = 1 (see bcn0 14282). (Contributed by Alexander van der Vekens, 1-Jan-2018.)
⊢ (𝐾 ∈ ℕ → (0C𝐾) = 0)
Theorem	bcnn 14284	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 14285	Binomial coefficient: 𝑁 choose 1. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
⊢ (𝑁 ∈ ℕ0 → (𝑁C1) = 𝑁)
Theorem	bcnp1n 14286	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 14287	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 14288	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 14289	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 14290	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 14291	Binomial coefficient: 𝑁 choose 2. (Contributed by Mario Carneiro, 22-May-2014.)
⊢ (𝑁 ∈ ℕ0 → (𝑁C2) = ((𝑁 · (𝑁 − 1)) / 2))
Theorem	bcp1m1 14292	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 14293	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 14294	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 14295	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 14296	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 14297	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 14298	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 14299	The binomial coefficient of (𝑁 − 1) is 𝑁. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑁 ∈ ℕ0 → (𝑁C(𝑁 − 1)) = 𝑁)
Theorem	4bc3eq4 14300	The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
⊢ (4C3) = 4