P. 110 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10901-11000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	exp0 10901	Value of a complex number raised to the 0th power. Note that under our definition, 0↑0 = 1 (0exp0e1 10902) , 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 10902	The zeroth power of zero equals one. See comment of exp0 10901. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (0↑0) = 1
Theorem	exp1 10903	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 10904	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 10905	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 10906	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 10907	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 10908*	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 10909*	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 10910	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ)
Theorem	nn0expcl 10911	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ0)
Theorem	zexpcl 10912	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ)
Theorem	qexpcl 10913	Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℚ)
Theorem	reexpcl 10914	Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℝ)
Theorem	expcl 10915	Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ)
Theorem	rpexpcl 10916	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 10917	Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ)
Theorem	qexpclz 10918	Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℚ)
Theorem	m1expcl2 10919	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1})
Theorem	m1expcl 10920	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℤ)
Theorem	expclzaplem 10921*	Closure law for integer exponentiation. Lemma for expclzap 10922 and expap0i 10929. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})
Theorem	expclzap 10922	Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℂ)
Theorem	nn0expcli 10923	Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝐴↑𝑁) ∈ ℕ0
Theorem	nn0sqcl 10924	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 10925	Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴))
Theorem	1exp 10926	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 10927	Positive integer exponentiation is apart from zero iff its base is apart from zero. That it is easier to prove this first, and then prove expeq0 10928 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." (Remark of [Geuvers], p. 1). (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) # 0 ↔ 𝐴 # 0))
Theorem	expeq0 10928	Positive integer exponentiation is 0 iff its base is 0. (Contributed by NM, 23-Feb-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) = 0 ↔ 𝐴 = 0))
Theorem	expap0i 10929	Integer exponentiation is apart from zero if its base is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) # 0)
Theorem	expgt0 10930	A positive real raised to an integer power 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 10931	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 10932	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
Theorem	expge0 10933	A nonnegative real raised to a nonnegative integer 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 10934	A real greater than or equal to 1 raised to a nonnegative integer 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 10935	A real greater than 1 raised to a positive integer 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 10936	Nonnegative 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 10937	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 10938	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 10939	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 10940	Lemma for expaddzap 10941. (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 10941	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 10942	Product of exponents law for nonnegative 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 10943	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	m1expeven 10944	Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)
⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1)
Theorem	expsubap 10945	Exponent subtraction law for 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 10946	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 10947	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 10948	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 10949	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 10950	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 10951	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 10952	Weak base 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 10953	A real between 0 and 1 inclusive raised to a nonnegative integer 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 10954	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 10955	Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴))
Theorem	sqneg 10956	The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2))
Theorem	sqsubswap 10957	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 10958	Closure of square. (Contributed by NM, 10-Aug-1999.)
⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ)
Theorem	sqmul 10959	Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2)))
Theorem	sqeq0 10960	A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) = 0 ↔ 𝐴 = 0))
Theorem	sqdivap 10961	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	sqdividap 10962	The square of a complex number apart from zero divided by itself equals that number. (Contributed by AV, 19-Jul-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((𝐴↑2) / 𝐴) = 𝐴)
Theorem	sqne0 10963	A number is nonzero iff its square is nonzero. See also sqap0 10964 which is the same but with not equal changed to apart. (Contributed by NM, 11-Mar-2006.)
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) ≠ 0 ↔ 𝐴 ≠ 0))
Theorem	sqap0 10964	A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) # 0 ↔ 𝐴 # 0))
Theorem	resqcl 10965	Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
⊢ (𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ)
Theorem	sqgt0ap 10966	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → 0 < (𝐴↑2))
Theorem	nnsqcl 10967	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 10968	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 10969	The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (𝐴↑2) ∈ ℚ)
Theorem	sq11 10970	The square function is one-to-one for nonnegative reals. Also see sq11ap 11065 which would easily follow from this given excluded middle, but which for us is proved another way. (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 10971	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 10972	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 10973	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 10974	A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴↑2))
Theorem	zsqcl2 10975	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 10976	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 10977	Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴↑2) = (𝐴 · 𝐴)
Theorem	sqcli 10978	Closure of square. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴↑2) ∈ ℂ
Theorem	sqeq0i 10979	A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ ((𝐴↑2) = 0 ↔ 𝐴 = 0)
Theorem	sqmuli 10980	Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))
Theorem	sqdivapi 10981	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 10982	Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (𝐴↑2) ∈ ℝ
Theorem	sqgt0api 10983	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (𝐴 # 0 → 0 < (𝐴↑2))
Theorem	sqge0i 10984	A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 0 ≤ (𝐴↑2)
Theorem	lt2sqi 10985	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 10986	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 10987	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 10988	The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
⊢ (0↑2) = 0
Theorem	sq0i 10989	If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
⊢ (𝐴 = 0 → (𝐴↑2) = 0)
Theorem	sq0id 10990	If a number is zero, its square is zero. Deduction form of sq0i 10989. Converse of sqeq0d 11030. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 = 0)    ⇒   ⊢ (𝜑 → (𝐴↑2) = 0)
Theorem	sq1 10991	The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
⊢ (1↑2) = 1
Theorem	neg1sqe1 10992	-1 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (-1↑2) = 1
Theorem	sq2 10993	The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
⊢ (2↑2) = 4
Theorem	sq3 10994	The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
⊢ (3↑2) = 9
Theorem	sq4e2t8 10995	The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)
⊢ (4↑2) = (2 · 8)
Theorem	cu2 10996	The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
⊢ (2↑3) = 8
Theorem	irec 10997	The reciprocal of i. (Contributed by NM, 11-Oct-1999.)
⊢ (1 / i) = -i
Theorem	i2 10998	i squared. (Contributed by NM, 6-May-1999.)
⊢ (i↑2) = -1
Theorem	i3 10999	i cubed. (Contributed by NM, 31-Jan-2007.)
⊢ (i↑3) = -i
Theorem	i4 11000	i to the fourth power. (Contributed by NM, 31-Jan-2007.)
⊢ (i↑4) = 1