P. 108 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10701-10800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	expn1ap0 10701	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 10702*	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 10703*	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 10704	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ)
Theorem	nn0expcl 10705	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ0)
Theorem	zexpcl 10706	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ)
Theorem	qexpcl 10707	Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℚ)
Theorem	reexpcl 10708	Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℝ)
Theorem	expcl 10709	Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ)
Theorem	rpexpcl 10710	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 10711	Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ)
Theorem	qexpclz 10712	Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℚ)
Theorem	m1expcl2 10713	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1})
Theorem	m1expcl 10714	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℤ)
Theorem	expclzaplem 10715*	Closure law for integer exponentiation. Lemma for expclzap 10716 and expap0i 10723. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})
Theorem	expclzap 10716	Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℂ)
Theorem	nn0expcli 10717	Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝐴↑𝑁) ∈ ℕ0
Theorem	nn0sqcl 10718	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 10719	Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴))
Theorem	1exp 10720	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 10721	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 10722 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 10722	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 10723	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 10724	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 10725	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 10726	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
Theorem	expge0 10727	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 10728	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 10729	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 10730	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 10731	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 10732	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 10733	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 10734	Lemma for expaddzap 10735. (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 10735	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 10736	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 10737	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 10738	Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)
⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1)
Theorem	expsubap 10739	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 10740	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 10741	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 10742	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 10743	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 10744	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 10745	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 10746	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 10747	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 10748	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 10749	Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴))
Theorem	sqneg 10750	The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2))
Theorem	sqsubswap 10751	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 10752	Closure of square. (Contributed by NM, 10-Aug-1999.)
⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ)
Theorem	sqmul 10753	Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2)))
Theorem	sqeq0 10754	A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) = 0 ↔ 𝐴 = 0))
Theorem	sqdivap 10755	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 10756	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 10757	A number is nonzero iff its square is nonzero. See also sqap0 10758 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 10758	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 10759	Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
⊢ (𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ)
Theorem	sqgt0ap 10760	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → 0 < (𝐴↑2))
Theorem	nnsqcl 10761	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 10762	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 10763	The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (𝐴↑2) ∈ ℚ)
Theorem	sq11 10764	The square function is one-to-one for nonnegative reals. Also see sq11ap 10859 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 10765	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 10766	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 10767	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 10768	A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴↑2))
Theorem	zsqcl2 10769	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 10770	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 10771	Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴↑2) = (𝐴 · 𝐴)
Theorem	sqcli 10772	Closure of square. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴↑2) ∈ ℂ
Theorem	sqeq0i 10773	A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ ((𝐴↑2) = 0 ↔ 𝐴 = 0)
Theorem	sqmuli 10774	Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))
Theorem	sqdivapi 10775	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 10776	Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (𝐴↑2) ∈ ℝ
Theorem	sqgt0api 10777	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (𝐴 # 0 → 0 < (𝐴↑2))
Theorem	sqge0i 10778	A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 0 ≤ (𝐴↑2)
Theorem	lt2sqi 10779	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 10780	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 10781	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 10782	The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
⊢ (0↑2) = 0
Theorem	sq0i 10783	If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
⊢ (𝐴 = 0 → (𝐴↑2) = 0)
Theorem	sq0id 10784	If a number is zero, its square is zero. Deduction form of sq0i 10783. Converse of sqeq0d 10824. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 = 0)    ⇒   ⊢ (𝜑 → (𝐴↑2) = 0)
Theorem	sq1 10785	The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
⊢ (1↑2) = 1
Theorem	neg1sqe1 10786	-1 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (-1↑2) = 1
Theorem	sq2 10787	The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
⊢ (2↑2) = 4
Theorem	sq3 10788	The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
⊢ (3↑2) = 9
Theorem	sq4e2t8 10789	The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)
⊢ (4↑2) = (2 · 8)
Theorem	cu2 10790	The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
⊢ (2↑3) = 8
Theorem	irec 10791	The reciprocal of i. (Contributed by NM, 11-Oct-1999.)
⊢ (1 / i) = -i
Theorem	i2 10792	i squared. (Contributed by NM, 6-May-1999.)
⊢ (i↑2) = -1
Theorem	i3 10793	i cubed. (Contributed by NM, 31-Jan-2007.)
⊢ (i↑3) = -i
Theorem	i4 10794	i to the fourth power. (Contributed by NM, 31-Jan-2007.)
⊢ (i↑4) = 1
Theorem	nnlesq 10795	A positive integer is less than or equal to its square. For general integers, see zzlesq 10860. (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.)
⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁↑2))
Theorem	iexpcyc 10796	Taking i to the 𝐾-th power is the same as using the 𝐾 mod 4 -th power instead, by i4 10794. (Contributed by Mario Carneiro, 7-Jul-2014.)
⊢ (𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾))
Theorem	expnass 10797	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 10798	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 10799	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 10800	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))