P. 90 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8901-9000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	expcl2lemap 8901*	Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)
⊢ 𝐹 ⊆ ℂ    &   ⊢ ((x ∈ 𝐹 ∧ y ∈ 𝐹) → (x · y) ∈ 𝐹)    &   ⊢ 1 ∈ 𝐹    &   ⊢ ((x ∈ 𝐹 ∧ x # 0) → (1 / x) ∈ 𝐹)    ⇒   ⊢ ((A ∈ 𝐹 ∧ A # 0 ∧ B ∈ ℤ) → (A↑B) ∈ 𝐹)
Theorem	nnexpcl 8902	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
⊢ ((A ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈ ℕ)
Theorem	nn0expcl 8903	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
⊢ ((A ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈ ℕ0)
Theorem	zexpcl 8904	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
⊢ ((A ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈ ℤ)
Theorem	qexpcl 8905	Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
⊢ ((A ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈ ℚ)
Theorem	reexpcl 8906	Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
⊢ ((A ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈ ℝ)
Theorem	expcl 8907	Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈ ℂ)
Theorem	rpexpcl 8908	Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
⊢ ((A ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (A↑𝑁) ∈ ℝ+)
Theorem	reexpclzap 8909	Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((A ∈ ℝ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑𝑁) ∈ ℝ)
Theorem	qexpclz 8910	Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
⊢ ((A ∈ ℚ ∧ A ≠ 0 ∧ 𝑁 ∈ ℤ) → (A↑𝑁) ∈ ℚ)
Theorem	m1expcl2 8911	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1})
Theorem	m1expcl 8912	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℤ)
Theorem	expclzaplem 8913*	Closure law for integer exponentiation. Lemma for expclzap 8914 and expap0i 8921. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑𝑁) ∈ {z ∈ ℂ ∣ z # 0})
Theorem	expclzap 8914	Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑𝑁) ∈ ℂ)
Theorem	nn0expcli 8915	Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ A ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (A↑𝑁) ∈ ℕ0
Theorem	nn0sqcl 8916	The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ (A ∈ ℕ0 → (A↑2) ∈ ℕ0)
Theorem	expm1t 8917	Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ) → (A↑𝑁) = ((A↑(𝑁 − 1)) · A))
Theorem	1exp 8918	Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1)
Theorem	expap0 8919	Positive integer exponentiation is apart from zero iff its mantissa is apart from zero. That it is easier to prove this first, and then prove expeq0 8920 in terms of it, rather than the other way around, is perhaps an illustration of the maxim "In constructive analysis, the apartness is more basic [ than ] equality." ([Geuvers], p. 1). (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((A↑𝑁) # 0 ↔ A # 0))
Theorem	expeq0 8920	Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.)
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((A↑𝑁) = 0 ↔ A = 0))
Theorem	expap0i 8921	Integer exponentiation is apart from zero if its mantissa is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑𝑁) # 0)
Theorem	expgt0 8922	Nonnegative integer exponentiation with a positive mantissa is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((A ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 0 < A) → 0 < (A↑𝑁))
Theorem	expnegzap 8923	Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑-𝑁) = (1 / (A↑𝑁)))
Theorem	0exp 8924	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
Theorem	expge0 8925	Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((A ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ A) → 0 ≤ (A↑𝑁))
Theorem	expge1 8926	Nonnegative integer exponentiation with a mantissa greater than or equal to 1 is greater than or equal to 1. (Contributed by NM, 21-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((A ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 1 ≤ A) → 1 ≤ (A↑𝑁))
Theorem	expgt1 8927	Positive integer exponentiation with a mantissa greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((A ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < A) → 1 < (A↑𝑁))
Theorem	mulexp 8928	Positive integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 13-Feb-2005.)
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((A · B)↑𝑁) = ((A↑𝑁) · (B↑𝑁)))
Theorem	mulexpzap 8929	Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (B ∈ ℂ ∧ B # 0) ∧ 𝑁 ∈ ℤ) → ((A · B)↑𝑁) = ((A↑𝑁) · (B↑𝑁)))
Theorem	exprecap 8930	Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → ((1 / A)↑𝑁) = (1 / (A↑𝑁)))
Theorem	expadd 8931	Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.)
⊢ ((A ∈ ℂ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (A↑(𝑀 + 𝑁)) = ((A↑𝑀) · (A↑𝑁)))
Theorem	expaddzaplem 8932	Lemma for expaddzap 8933. (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (𝑀 ∈ ℝ ∧ -𝑀 ∈ ℕ) ∧ 𝑁 ∈ ℕ0) → (A↑(𝑀 + 𝑁)) = ((A↑𝑀) · (A↑𝑁)))
Theorem	expaddzap 8933	Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (A↑(𝑀 + 𝑁)) = ((A↑𝑀) · (A↑𝑁)))
Theorem	expmul 8934	Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.)
⊢ ((A ∈ ℂ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (A↑(𝑀 · 𝑁)) = ((A↑𝑀)↑𝑁))
Theorem	expmulzap 8935	Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (A↑(𝑀 · 𝑁)) = ((A↑𝑀)↑𝑁))
Theorem	expsubap 8936	Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (A↑(𝑀 − 𝑁)) = ((A↑𝑀) / (A↑𝑁)))
Theorem	expp1zap 8937	Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑(𝑁 + 1)) = ((A↑𝑁) · A))
Theorem	expm1ap 8938	Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑(𝑁 − 1)) = ((A↑𝑁) / A))
Theorem	expdivap 8939	Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ ((A ∈ ℂ ∧ (B ∈ ℂ ∧ B # 0) ∧ 𝑁 ∈ ℕ0) → ((A / B)↑𝑁) = ((A↑𝑁) / (B↑𝑁)))
Theorem	ltexp2a 8940	Ordering relationship for exponentiation. (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	leexp2a 8941	Weak ordering relationship for exponentiation. (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	leexp2r 8942	Weak ordering relationship for exponentiation. (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 8943	Weak mantissa ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
⊢ (((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝑁 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ B)) → (A↑𝑁) ≤ (B↑𝑁))
Theorem	exple1 8944	Nonnegative integer exponentiation with a mantissa between 0 and 1 inclusive 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 8945	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	sqval 8946	Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
⊢ (A ∈ ℂ → (A↑2) = (A · A))
Theorem	sqneg 8947	The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
⊢ (A ∈ ℂ → (-A↑2) = (A↑2))
Theorem	sqsubswap 8948	Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → ((A − B)↑2) = ((B − A)↑2))
Theorem	sqcl 8949	Closure of square. (Contributed by NM, 10-Aug-1999.)
⊢ (A ∈ ℂ → (A↑2) ∈ ℂ)
Theorem	sqmul 8950	Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → ((A · B)↑2) = ((A↑2) · (B↑2)))
Theorem	sqeq0 8951	A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
⊢ (A ∈ ℂ → ((A↑2) = 0 ↔ A = 0))
Theorem	sqdivap 8952	Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → ((A / B)↑2) = ((A↑2) / (B↑2)))
Theorem	sqne0 8953	A number is nonzero iff its square is nonzero. (Contributed by NM, 11-Mar-2006.)
⊢ (A ∈ ℂ → ((A↑2) ≠ 0 ↔ A ≠ 0))
Theorem	resqcl 8954	Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
⊢ (A ∈ ℝ → (A↑2) ∈ ℝ)
Theorem	sqgt0ap 8955	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ ((A ∈ ℝ ∧ A # 0) → 0 < (A↑2))
Theorem	nnsqcl 8956	The naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (A ∈ ℕ → (A↑2) ∈ ℕ)
Theorem	zsqcl 8957	Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (A ∈ ℤ → (A↑2) ∈ ℤ)
Theorem	qsqcl 8958	The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
⊢ (A ∈ ℚ → (A↑2) ∈ ℚ)
Theorem	sq11 8959	The square function is one-to-one for nonnegative reals. (Contributed by NM, 8-Apr-2001.) (Proof shortened by Mario Carneiro, 28-May-2016.)
⊢ (((A ∈ ℝ ∧ 0 ≤ A) ∧ (B ∈ ℝ ∧ 0 ≤ B)) → ((A↑2) = (B↑2) ↔ A = B))
Theorem	lt2sq 8960	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006.)
⊢ (((A ∈ ℝ ∧ 0 ≤ A) ∧ (B ∈ ℝ ∧ 0 ≤ B)) → (A < B ↔ (A↑2) < (B↑2)))
Theorem	le2sq 8961	The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.)
⊢ (((A ∈ ℝ ∧ 0 ≤ A) ∧ (B ∈ ℝ ∧ 0 ≤ B)) → (A ≤ B ↔ (A↑2) ≤ (B↑2)))
Theorem	le2sq2 8962	The square of a 'less than or equal to' ordering. (Contributed by NM, 21-Mar-2008.)
⊢ (((A ∈ ℝ ∧ 0 ≤ A) ∧ (B ∈ ℝ ∧ A ≤ B)) → (A↑2) ≤ (B↑2))
Theorem	sqge0 8963	A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
⊢ (A ∈ ℝ → 0 ≤ (A↑2))
Theorem	zsqcl2 8964	The square of an integer is a nonnegative integer. (Contributed by Mario Carneiro, 18-Apr-2014.) (Revised by Mario Carneiro, 14-Jul-2014.)
⊢ (A ∈ ℤ → (A↑2) ∈ ℕ0)
Theorem	sumsqeq0 8965	Two real numbers are equal to 0 iff their Euclidean norm is. (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 8966	Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
⊢ A ∈ ℂ    ⇒   ⊢ (A↑2) = (A · A)
Theorem	sqcli 8967	Closure of square. (Contributed by NM, 2-Aug-1999.)
⊢ A ∈ ℂ    ⇒   ⊢ (A↑2) ∈ ℂ
Theorem	sqeq0i 8968	A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
⊢ A ∈ ℂ    ⇒   ⊢ ((A↑2) = 0 ↔ A = 0)
Theorem	sqmuli 8969	Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)
⊢ A ∈ ℂ    &   ⊢ B ∈ ℂ    ⇒   ⊢ ((A · B)↑2) = ((A↑2) · (B↑2))
Theorem	sqdivapi 8970	Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ A ∈ ℂ    &   ⊢ B ∈ ℂ    &   ⊢ B # 0    ⇒   ⊢ ((A / B)↑2) = ((A↑2) / (B↑2))
Theorem	resqcli 8971	Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
⊢ A ∈ ℝ    ⇒   ⊢ (A↑2) ∈ ℝ
Theorem	sqgt0api 8972	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ A ∈ ℝ    ⇒   ⊢ (A # 0 → 0 < (A↑2))
Theorem	sqge0i 8973	A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
⊢ A ∈ ℝ    ⇒   ⊢ 0 ≤ (A↑2)
Theorem	lt2sqi 8974	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 12-Sep-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ∧ 0 ≤ B) → (A < B ↔ (A↑2) < (B↑2)))
Theorem	le2sqi 8975	The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ∧ 0 ≤ B) → (A ≤ B ↔ (A↑2) ≤ (B↑2)))
Theorem	sq11i 8976	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 8977	The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
⊢ (0↑2) = 0
Theorem	sq0i 8978	If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
⊢ (A = 0 → (A↑2) = 0)
Theorem	sq0id 8979	If a number is zero, its square is zero. Deduction form of sq0i 8978. Converse of sqeq0d 9013. (Contributed by David Moews, 28-Feb-2017.)
⊢ (φ → A = 0)    ⇒   ⊢ (φ → (A↑2) = 0)
Theorem	sq1 8980	The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
⊢ (1↑2) = 1
Theorem	neg1sqe1 8981	-1 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (-1↑2) = 1
Theorem	sq2 8982	The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
⊢ (2↑2) = 4
Theorem	sq3 8983	The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
⊢ (3↑2) = 9
Theorem	cu2 8984	The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
⊢ (2↑3) = 8
Theorem	irec 8985	The reciprocal of i. (Contributed by NM, 11-Oct-1999.)
⊢ (1 / i) = -i
Theorem	i2 8986	i squared. (Contributed by NM, 6-May-1999.)
⊢ (i↑2) = -1
Theorem	i3 8987	i cubed. (Contributed by NM, 31-Jan-2007.)
⊢ (i↑3) = -i
Theorem	i4 8988	i to the fourth power. (Contributed by NM, 31-Jan-2007.)
⊢ (i↑4) = 1
Theorem	nnlesq 8989	A positive integer is less than or equal to its square. (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.)
⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁↑2))
Theorem	expnass 8990	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	subsq 8991	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 8992	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 8993	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 8994	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	binom2 8995	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	binom21 8996	Special case of binom2 8995 where B = 1. (Contributed by Scott Fenton, 11-May-2014.)
⊢ (A ∈ ℂ → ((A + 1)↑2) = (((A↑2) + (2 · A)) + 1))
Theorem	binom2sub 8997	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	binom2subi 8998	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	binom3 8999	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	zesq 9000	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))