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	mulexpd 14201	Nonnegative 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.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁)))
Theorem	znsqcld 14202	The square of a nonzero integer is a positive integer. (Contributed by Thierry Arnoux, 2-Feb-2020.)
⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ≠ 0)    ⇒   ⊢ (𝜑 → (𝑁↑2) ∈ ℕ)
Theorem	reexpcld 14203	Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ)
Theorem	expge0d 14204	A nonnegative real raised to a nonnegative integer is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴↑𝑁))
Theorem	expge1d 14205	A real greater than or equal to 1 raised to a nonnegative integer is greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 1 ≤ 𝐴)    ⇒   ⊢ (𝜑 → 1 ≤ (𝐴↑𝑁))
Theorem	ltexp2a 14206	Exponent ordering relationship for exponentiation of a fixed real base greater than 1 to integer exponents. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 < 𝐴 ∧ 𝑀 < 𝑁)) → (𝐴↑𝑀) < (𝐴↑𝑁))
Theorem	expmordi 14207	Base ordering relationship for exponentiation of nonnegative reals to a fixed positive integer power. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 < 𝐵) ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) < (𝐵↑𝑁))
Theorem	rpexpmord 14208	Base ordering relationship for exponentiation of positive reals to a fixed positive integer exponent. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐴↑𝑁) < (𝐵↑𝑁)))
Theorem	expcan 14209	Cancellation law for integer exponentiation of reals. (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	ltexp2 14210	Strict ordering law for exponentiation of a fixed real base greater than 1 to integer exponents. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁)))
Theorem	leexp2 14211	Ordering law for exponentiation of a fixed real base greater than 1 to integer exponents. (Contributed by Mario Carneiro, 26-Apr-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 ≤ 𝑁 ↔ (𝐴↑𝑀) ≤ (𝐴↑𝑁)))
Theorem	leexp2a 14212	Weak ordering relationship for exponentiation of a fixed real base greater than or equal to 1 to integer exponents. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑀) ≤ (𝐴↑𝑁))
Theorem	ltexp2r 14213	The integer powers of a fixed positive real smaller than 1 decrease as the exponent increases. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
⊢ (((𝐴 ∈ ℝ+ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝑀 < 𝑁 ↔ (𝐴↑𝑁) < (𝐴↑𝑀)))
Theorem	leexp2r 14214	Weak ordering relationship for exponentiation of a fixed real base in [0, 1] to integer exponents. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑁) ≤ (𝐴↑𝑀))
Theorem	leexp1a 14215	Weak base ordering relationship for exponentiation of real bases to a fixed nonnegative integer exponent. (Contributed by NM, 18-Dec-2005.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑁 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → (𝐴↑𝑁) ≤ (𝐵↑𝑁))
Theorem	exple1 14216	A real between 0 and 1 inclusive raised to a nonnegative integer power is less than or equal to 1. (Contributed by Paul Chapman, 29-Dec-2007.) (Revised by Mario Carneiro, 5-Jun-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1) ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ≤ 1)
Theorem	expubnd 14217	An upper bound on 𝐴↑𝑁 when 2 ≤ 𝐴. (Contributed by NM, 19-Dec-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 2 ≤ 𝐴) → (𝐴↑𝑁) ≤ ((2↑𝑁) · ((𝐴 − 1)↑𝑁)))
Theorem	sumsqeq0 14218	The sum of two squres of reals is zero if and only if both reals are zero. (Contributed by NM, 29-Apr-2005.) (Revised by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 28-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 = 0 ∧ 𝐵 = 0) ↔ ((𝐴↑2) + (𝐵↑2)) = 0))
Theorem	sqvali 14219	Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴↑2) = (𝐴 · 𝐴)
Theorem	sqcli 14220	Closure of square. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴↑2) ∈ ℂ
Theorem	sqeq0i 14221	A complex number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ ((𝐴↑2) = 0 ↔ 𝐴 = 0)
Theorem	sqrecii 14222	The square of a reciprocal is the reciprocal of the square. (Contributed by NM, 17-Sep-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    ⇒   ⊢ ((1 / 𝐴)↑2) = (1 / (𝐴↑2))
Theorem	sqmuli 14223	Distribution of squaring over multiplication. (Contributed by NM, 3-Sep-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))
Theorem	sqdivi 14224	Distribution of squaring over division. (Contributed by NM, 20-Aug-2001.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))
Theorem	resqcli 14225	Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (𝐴↑2) ∈ ℝ
Theorem	sqgt0i 14226	The square of a nonzero real is positive. (Contributed by NM, 17-Sep-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (𝐴 ≠ 0 → 0 < (𝐴↑2))
Theorem	sqge0i 14227	The square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 0 ≤ (𝐴↑2)
Theorem	lt2sqi 14228	The square function on nonnegative reals is increasing. (Contributed by NM, 12-Sep-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 < 𝐵 ↔ (𝐴↑2) < (𝐵↑2)))
Theorem	le2sqi 14229	The square function on nonnegative reals is nondecreasing. (Contributed by NM, 12-Sep-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 ≤ 𝐵 ↔ (𝐴↑2) ≤ (𝐵↑2)))
Theorem	sq11i 14230	The square function is one-to-one for nonnegative reals. (Contributed by NM, 27-Oct-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
Theorem	sq0 14231	The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
⊢ (0↑2) = 0
Theorem	sq0i 14232	If a number is zero, then its square is zero. (Contributed by FL, 10-Dec-2006.)
⊢ (𝐴 = 0 → (𝐴↑2) = 0)
Theorem	sq0id 14233	If a number is zero, then its square is zero. Deduction form of sq0i 14232. Converse of sqeq0d 14185. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 = 0)    ⇒   ⊢ (𝜑 → (𝐴↑2) = 0)
Theorem	sq1 14234	The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
⊢ (1↑2) = 1
Theorem	neg1sqe1 14235	The square of -1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (-1↑2) = 1
Theorem	sq2 14236	The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
⊢ (2↑2) = 4
Theorem	sq3 14237	The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
⊢ (3↑2) = 9
Theorem	sq4e2t8 14238	The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)
⊢ (4↑2) = (2 · 8)
Theorem	cu2 14239	The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
⊢ (2↑3) = 8
Theorem	irec 14240	The reciprocal of i. (Contributed by NM, 11-Oct-1999.)
⊢ (1 / i) = -i
Theorem	i2 14241	i squared. (Contributed by NM, 6-May-1999.)
⊢ (i↑2) = -1
Theorem	i3 14242	i cubed. (Contributed by NM, 31-Jan-2007.)
⊢ (i↑3) = -i
Theorem	i4 14243	i to the fourth power. (Contributed by NM, 31-Jan-2007.)
⊢ (i↑4) = 1
Theorem	nnlesq 14244	A positive integer is less than or equal to its square. For general integers, see zzlesq 14245. (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.)
⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁↑2))
Theorem	zzlesq 14245	An integer is less than or equal to its square. (Contributed by BJ, 6-Feb-2025.)
⊢ (𝑁 ∈ ℤ → 𝑁 ≤ (𝑁↑2))
Theorem	iexpcyc 14246	Taking i to the 𝐾-th power is the same as using the 𝐾 mod 4 -th power instead, by i4 14243. (Contributed by Mario Carneiro, 7-Jul-2014.)
⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾))
Theorem	expnass 14247	A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)
⊢ ((3↑3)↑3) < (3↑(3↑3))
Theorem	sqlecan 14248	Cancel one factor of a square in a ≤ comparison. Unlike lemul1 12119, the common factor 𝐴 may be zero. (Contributed by NM, 17-Jan-2008.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) ≤ (𝐵 · 𝐴) ↔ 𝐴 ≤ 𝐵))
Theorem	subsq 14249	Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴 − 𝐵)))
Theorem	subsq2 14250	Express the difference of the squares of two numbers as a polynomial in the difference of the numbers. (Contributed by NM, 21-Feb-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) − (𝐵↑2)) = (((𝐴 − 𝐵)↑2) + ((2 · 𝐵) · (𝐴 − 𝐵))))
Theorem	binom2i 14251	The square of a binomial. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵)↑2) = (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2))
Theorem	subsqi 14252	Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴 − 𝐵))
Theorem	sqeqori 14253	The squares of two complex numbers are equal iff one number equals the other or its negative. Lemma 15-4.7 of [Gleason] p. 311 and its converse. (Contributed by NM, 15-Jan-2006.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))
Theorem	subsq0i 14254	The two solutions to the difference of squares set equal to zero. (Contributed by NM, 25-Apr-2006.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (((𝐴↑2) − (𝐵↑2)) = 0 ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))
Theorem	sqeqor 14255	The squares of two complex numbers are equal iff one number equals the other or its negative. Lemma 15-4.7 of [Gleason] p. 311 and its converse. (Contributed by Paul Chapman, 15-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
Theorem	binom2 14256	The square of a binomial. (Contributed by FL, 10-Dec-2006.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑2) = (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)))
Theorem	binom2d 14257	Deduction form of binom2 14256. (Contributed by Igor Ieskov, 14-Dec-2023.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵)↑2) = (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)))
Theorem	binom21 14258	Special case of binom2 14256 where 𝐵 = 1. (Contributed by Scott Fenton, 11-May-2014.)
⊢ (𝐴 ∈ ℂ → ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · 𝐴)) + 1))
Theorem	binom2sub 14259	Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵)↑2) = (((𝐴↑2) − (2 · (𝐴 · 𝐵))) + (𝐵↑2)))
Theorem	binom2sub1 14260	Special case of binom2sub 14259 where 𝐵 = 1. (Contributed by AV, 2-Aug-2021.)
⊢ (𝐴 ∈ ℂ → ((𝐴 − 1)↑2) = (((𝐴↑2) − (2 · 𝐴)) + 1))
Theorem	binom2subi 14261	Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((𝐴 − 𝐵)↑2) = (((𝐴↑2) − (2 · (𝐴 · 𝐵))) + (𝐵↑2))
Theorem	mulbinom2 14262	The square of a binomial with factor. (Contributed by AV, 19-Jul-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐶 · 𝐴) + 𝐵)↑2) = ((((𝐶 · 𝐴)↑2) + ((2 · 𝐶) · (𝐴 · 𝐵))) + (𝐵↑2)))
Theorem	binom3 14263	The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑3) = (((𝐴↑3) + (3 · ((𝐴↑2) · 𝐵))) + ((3 · (𝐴 · (𝐵↑2))) + (𝐵↑3))))
Theorem	sq01 14264	If a complex number equals its square, it must be 0 or 1. (Contributed by NM, 6-Jun-2006.)
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) = 𝐴 ↔ (𝐴 = 0 ∨ 𝐴 = 1)))
Theorem	zesq 14265	An integer is even iff its square is even. (Contributed by Mario Carneiro, 12-Sep-2015.)
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ↔ ((𝑁↑2) / 2) ∈ ℤ))
Theorem	nnesq 14266	A positive integer is even iff its square is even. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ↔ ((𝑁↑2) / 2) ∈ ℕ))
Theorem	crreczi 14267	Reciprocal of a complex number in terms of real and imaginary components. Remark in [Apostol] p. 361. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Jeff Hankins, 16-Dec-2013.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) → (1 / (𝐴 + (i · 𝐵))) = ((𝐴 − (i · 𝐵)) / ((𝐴↑2) + (𝐵↑2))))
Theorem	bernneq 14268	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 14269	Variation of Bernoulli's inequality bernneq 14268. (Contributed by NM, 18-Oct-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴↑𝑁))
Theorem	bernneq3 14270	A corollary of bernneq 14268. (Contributed by Mario Carneiro, 11-Mar-2014.)
⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (𝑃↑𝑁))
Theorem	expnbnd 14271*	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 14272*	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 14273*	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 14274*	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 14275	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 14276	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 14277	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 14278*	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 14279*	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 14280	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 14281	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 14282	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 14283	The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴↑2) ∈ ℕ)
Theorem	nnexpcld 14284	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 14285	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 14286	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 14287	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 14288	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 14289	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 14290	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 14291	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 14292	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 14293	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 14294	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 14295	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 14296	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 14297	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 14298	Elevating to a positive power does not affect inequalities. Similar to ltmul1d 13118 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 14299	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 14300	The function elevating nonnegative reals to a positive integer is one-to-one. Similar to sq11d 14297 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)