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	iseqfn 8901*	The sequence builder function is a function. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (φ → 𝑀 ∈ ℤ)    &   ⊢ (φ → 𝑆 ∈ 𝑉)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)    ⇒   ⊢ (φ → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ≥‘𝑀))
Theorem	iseq1 8902*	Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 31-May-2020.)
⊢ (φ → 𝑀 ∈ ℤ)    &   ⊢ (φ → 𝑆 ∈ 𝑉)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)    ⇒   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹‘𝑀))
Theorem	iseqcl 8903*	Closure properties of the recursive sequence builder. (Contributed by Jim Kingdon, 1-Jun-2020.)
⊢ (φ → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (φ → 𝑆 ∈ 𝑉)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)    ⇒   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) ∈ 𝑆)
Theorem	iseqp1 8904*	Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 31-May-2020.)
⊢ (φ → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (φ → 𝑆 ∈ 𝑉)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)    ⇒   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1))))
Theorem	iseqfveq2 8905*	Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
⊢ (φ → 𝐾 ∈ (ℤ≥‘𝑀))    &   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺‘𝐾))    &   ⊢ (φ → 𝑆 ∈ 𝑉)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝐾)) → (𝐺‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)    &   ⊢ (φ → 𝑁 ∈ (ℤ≥‘𝐾))    &   ⊢ ((φ ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))
Theorem	iseqfeq2 8906*	Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
⊢ (φ → 𝐾 ∈ (ℤ≥‘𝑀))    &   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺‘𝐾))    &   ⊢ (φ → 𝑆 ∈ 𝑉)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝐾)) → (𝐺‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)    &   ⊢ ((φ ∧ 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆) ↾ (ℤ≥‘𝐾)) = seq𝐾( + , 𝐺, 𝑆))
Theorem	iseqfveq 8907*	Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
⊢ (φ → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((φ ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))    &   ⊢ (φ → 𝑆 ∈ 𝑉)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐹‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ x ∈ (ℤ≥‘𝑀)) → (𝐺‘x) ∈ 𝑆)    &   ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x + y) ∈ 𝑆)    ⇒   ⊢ (φ → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝑀( + , 𝐺, 𝑆)‘𝑁))
3.5.9  Integer powers
Syntax	cexp 8908	Extend class notation to include exponentiation of a complex number to an integer power.
class ↑
Definition	df-iexp 8909*	Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 8913 and expp1 8916 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts. 10-Jun-2005: The definition was extended to include zero exponents, so that 0↑0 = 1 per the convention of Definition 10-4.1 of [Gleason] p. 134. 4-Jun-2014: The definition was extended to include negative integer exponents. The case x = 0, y < 0 gives the value (1 / 0), so we will avoid this case in our theorems. (Contributed by Jim Kingdon, 7-Jun-2020.)
⊢ ↑ = (x ∈ ℂ, y ∈ ℤ ↦ if(y = 0, 1, if(0 < y, (seq1( · , (ℕ × {x}), ℂ)‘y), (1 / (seq1( · , (ℕ × {x}), ℂ)‘-y)))))
Theorem	expivallem 8910	Lemma for expival 8911. If we take a complex number apart from zero and raise it to a positive integer power, the result is apart from zero. (Contributed by Jim Kingdon, 7-Jun-2020.)
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℕ) → (seq1( · , (ℕ × {A}), ℂ)‘𝑁) # 0)
Theorem	expival 8911	Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (A # 0 ∨ 0 ≤ 𝑁)) → (A↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {A}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {A}), ℂ)‘-𝑁)))))
Theorem	expinnval 8912	Value of exponentiation to positive integer powers. (Contributed by Jim Kingdon, 8-Jun-2020.)
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ) → (A↑𝑁) = (seq1( · , (ℕ × {A}), ℂ)‘𝑁))
Theorem	exp0 8913	Value of a complex number raised to the 0th power. Note that under our definition, 0↑0 = 1, following the convention used by Gleason. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ (A ∈ ℂ → (A↑0) = 1)
Theorem	0exp0e1 8914	0↑0 = 1 (common case). This is our convention. It follows the convention used by Gleason; see Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (0↑0) = 1
Theorem	exp1 8915	Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.)
⊢ (A ∈ ℂ → (A↑1) = A)
Theorem	expp1 8916	Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.)
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (A↑(𝑁 + 1)) = ((A↑𝑁) · A))
Theorem	expnegap0 8917	Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℕ0) → (A↑-𝑁) = (1 / (A↑𝑁)))
Theorem	expineg2 8918	Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
⊢ (((A ∈ ℂ ∧ A # 0) ∧ (𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ0)) → (A↑𝑁) = (1 / (A↑-𝑁)))
Theorem	expn1ap0 8919	A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.)
⊢ ((A ∈ ℂ ∧ A # 0) → (A↑-1) = (1 / A))
Theorem	expcllem 8920*	Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
⊢ 𝐹 ⊆ ℂ    &   ⊢ ((x ∈ 𝐹 ∧ y ∈ 𝐹) → (x · y) ∈ 𝐹)    &   ⊢ 1 ∈ 𝐹    ⇒   ⊢ ((A ∈ 𝐹 ∧ B ∈ ℕ0) → (A↑B) ∈ 𝐹)
Theorem	expcl2lemap 8921*	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 8922	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
⊢ ((A ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈ ℕ)
Theorem	nn0expcl 8923	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
⊢ ((A ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈ ℕ0)
Theorem	zexpcl 8924	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
⊢ ((A ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈ ℤ)
Theorem	qexpcl 8925	Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
⊢ ((A ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈ ℚ)
Theorem	reexpcl 8926	Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
⊢ ((A ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈ ℝ)
Theorem	expcl 8927	Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (A↑𝑁) ∈ ℂ)
Theorem	rpexpcl 8928	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 8929	Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((A ∈ ℝ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑𝑁) ∈ ℝ)
Theorem	qexpclz 8930	Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
⊢ ((A ∈ ℚ ∧ A ≠ 0 ∧ 𝑁 ∈ ℤ) → (A↑𝑁) ∈ ℚ)
Theorem	m1expcl2 8931	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1})
Theorem	m1expcl 8932	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℤ)
Theorem	expclzaplem 8933*	Closure law for integer exponentiation. Lemma for expclzap 8934 and expap0i 8941. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑𝑁) ∈ {z ∈ ℂ ∣ z # 0})
Theorem	expclzap 8934	Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((A ∈ ℂ ∧ A # 0 ∧ 𝑁 ∈ ℤ) → (A↑𝑁) ∈ ℂ)
Theorem	nn0expcli 8935	Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ A ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (A↑𝑁) ∈ ℕ0
Theorem	nn0sqcl 8936	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 8937	Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
⊢ ((A ∈ ℂ ∧ 𝑁 ∈ ℕ) → (A↑𝑁) = ((A↑(𝑁 − 1)) · A))
Theorem	1exp 8938	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 8939	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 8940 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 8940	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 8941	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 8942	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 8943	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 8944	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
Theorem	expge0 8945	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 8946	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 8947	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 8948	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 8949	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 8950	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 8951	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 8952	Lemma for expaddzap 8953. (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 8953	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 8954	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 8955	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 8956	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 8957	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 8958	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 8959	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 8960	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 8961	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 8962	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 8963	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 8964	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 8965	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 8966	Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
⊢ (A ∈ ℂ → (A↑2) = (A · A))
Theorem	sqneg 8967	The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
⊢ (A ∈ ℂ → (-A↑2) = (A↑2))
Theorem	sqsubswap 8968	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 8969	Closure of square. (Contributed by NM, 10-Aug-1999.)
⊢ (A ∈ ℂ → (A↑2) ∈ ℂ)
Theorem	sqmul 8970	Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
⊢ ((A ∈ ℂ ∧ B ∈ ℂ) → ((A · B)↑2) = ((A↑2) · (B↑2)))
Theorem	sqeq0 8971	A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
⊢ (A ∈ ℂ → ((A↑2) = 0 ↔ A = 0))
Theorem	sqdivap 8972	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 8973	A number is nonzero iff its square is nonzero. (Contributed by NM, 11-Mar-2006.)
⊢ (A ∈ ℂ → ((A↑2) ≠ 0 ↔ A ≠ 0))
Theorem	resqcl 8974	Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
⊢ (A ∈ ℝ → (A↑2) ∈ ℝ)
Theorem	sqgt0ap 8975	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ ((A ∈ ℝ ∧ A # 0) → 0 < (A↑2))
Theorem	nnsqcl 8976	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 8977	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 8978	The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
⊢ (A ∈ ℚ → (A↑2) ∈ ℚ)
Theorem	sq11 8979	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 8980	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 8981	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 8982	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 8983	A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
⊢ (A ∈ ℝ → 0 ≤ (A↑2))
Theorem	zsqcl2 8984	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 8985	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 8986	Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
⊢ A ∈ ℂ    ⇒   ⊢ (A↑2) = (A · A)
Theorem	sqcli 8987	Closure of square. (Contributed by NM, 2-Aug-1999.)
⊢ A ∈ ℂ    ⇒   ⊢ (A↑2) ∈ ℂ
Theorem	sqeq0i 8988	A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
⊢ A ∈ ℂ    ⇒   ⊢ ((A↑2) = 0 ↔ A = 0)
Theorem	sqmuli 8989	Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)
⊢ A ∈ ℂ    &   ⊢ B ∈ ℂ    ⇒   ⊢ ((A · B)↑2) = ((A↑2) · (B↑2))
Theorem	sqdivapi 8990	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 8991	Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
⊢ A ∈ ℝ    ⇒   ⊢ (A↑2) ∈ ℝ
Theorem	sqgt0api 8992	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ A ∈ ℝ    ⇒   ⊢ (A # 0 → 0 < (A↑2))
Theorem	sqge0i 8993	A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
⊢ A ∈ ℝ    ⇒   ⊢ 0 ≤ (A↑2)
Theorem	lt2sqi 8994	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 8995	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 8996	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 8997	The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
⊢ (0↑2) = 0
Theorem	sq0i 8998	If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
⊢ (A = 0 → (A↑2) = 0)
Theorem	sq0id 8999	If a number is zero, its square is zero. Deduction form of sq0i 8998. Converse of sqeq0d 9033. (Contributed by David Moews, 28-Feb-2017.)
⊢ (φ → A = 0)    ⇒   ⊢ (φ → (A↑2) = 0)
Theorem	sq1 9000	The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
⊢ (1↑2) = 1