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	ser0 14201	The value of the partial sums in a zero-valued infinite series. (Contributed by Mario Carneiro, 31-Aug-2013.) (Revised by Mario Carneiro, 15-Dec-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑁 ∈ 𝑍 → (seq𝑀( + , (𝑍 × {0}))‘𝑁) = 0)
Theorem	ser0f 14202	A zero-valued infinite series is equal to the constant zero function. (Contributed by Mario Carneiro, 8-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0})) = (𝑍 × {0}))
Theorem	serge0 14203*	A finite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 8-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))
Theorem	serle 14204*	Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ (seq𝑀( + , 𝐺)‘𝑁))
Theorem	ser1const 14205	Value of the partial series sum of a constant function. (Contributed by NM, 8-Aug-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( + , (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴))
Theorem	seqof 14206*	Distribute function operation through a sequence. Note that 𝐺(𝑧) is an implicit function on 𝑧. (Contributed by Mario Carneiro, 3-Mar-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)))    ⇒   ⊢ (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
Theorem	seqof2 14207*	Distribute function operation through a sequence. Maps-to notation version of seqof 14206. (Contributed by Mario Carneiro, 7-Jul-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴)) → 𝑋 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (seq𝑀( ∘f + , (𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋)))‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , (𝑥 ∈ 𝐵 ↦ 𝑋))‘𝑁)))
5.6.7  Integer powers
Syntax	cexp 14208	Extend class notation to include exponentiation of a complex number to an integer power.
class ↑
Definition	df-exp 14209*	Define exponentiation of complex numbers with integer exponents. For example, (5↑2) = 25 (ex-exp 31057). Terminology: In general, "exponentiation" is the operation of raising a "base" 𝑥 to the power of the "exponent" 𝑦, resulting in the "power" (𝑥↑𝑦), also called "x to the power of y". In this case, "integer exponentiation" is the operation of raising a complex "base" 𝑥 to the power of an integer 𝑦, resulting in the "integer power" (𝑥↑𝑦). This definition is not meant to be used directly; instead, exp0 14212 and expp1 14215 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 do not have superscripts. 10-Jun-2005: The definition was extended from positive exponents to nonnegative exponent, so that 0↑0 = 1, following standard convention, for instance Definition 10-4.1 of [Gleason] p. 134 (0exp0e1 14213). 4-Jun-2014: The definition was extended to integer exponents. For example, ( -3↑ -2) = (1 / 9) (ex-exp 31057). The case 𝑥 = 0, 𝑦 < 0 gives the "value" (1 / 0); relying on this should be avoided in applications. For a definition of exponentiation including complex exponents see df-cxp 26891 (complex exponentiation). Both definitions are equivalent for integer exponents, see cxpexpz 27001. (Contributed by Raph Levien, 20-May-2004.) (Revised by NM, 15-Oct-2004.)
⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘ -𝑦)))))
Theorem	expval 14210	Value of exponentiation to integer powers. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘ -𝑁)))))
Theorem	expnnval 14211	Value of exponentiation to positive integer powers. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
Theorem	exp0 14212	Value of a complex number raised to the zeroth power. Under our definition, 0↑0 = 1 (0exp0e1 14213), following standard convention, for instance 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 14213	The zeroth power of zero equals one. See comment of exp0 14212. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (0↑0) = 1
Theorem	exp1 14214	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 14215	Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. When 𝐴 is nonzero, this holds for all integers 𝑁, see expneg 14216. (Contributed by NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴))
Theorem	expneg 14216	Value of a complex number raised to a nonpositive integer power. When 𝐴 = 0 and 𝑁 is nonzero, both sides have the "value" (1 / 0); relying on that should be avoid in applications. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑ -𝑁) = (1 / (𝐴↑𝑁)))
Theorem	expneg2 14217	Value of a complex number raised to a nonpositive integer power. When 𝐴 = 0 and 𝑁 is nonzero, both sides have the "value" (1 / 0); relying on that should be avoid in applications. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ0) → (𝐴↑𝑁) = (1 / (𝐴↑ -𝑁)))
Theorem	expn1 14218	A complex number raised to the negative one power is its reciprocal. When 𝐴 = 0, both sides have the "value" (1 / 0); relying on that should be avoid in applications. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ (𝐴 ∈ ℂ → (𝐴↑ -1) = (1 / 𝐴))
Theorem	expcllem 14219*	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	expcl2lem 14220*	Lemma for proving integer exponentiation closure laws. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)
⊢ 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 14221	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ)
Theorem	nn0expcl 14222	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ0)
Theorem	zexpcl 14223	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ)
Theorem	qexpcl 14224	Closure of exponentiation of rationals. For integer exponents, see qexpclz 14228. (Contributed by NM, 16-Dec-2005.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℚ)
Theorem	reexpcl 14225	Closure of exponentiation of reals. For integer exponents, see reexpclz 14229. (Contributed by NM, 14-Dec-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℝ)
Theorem	expcl 14226	Closure law for nonnegative integer exponentiation. For integer exponents, see expclz 14231. (Contributed by NM, 26-May-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ)
Theorem	rpexpcl 14227	Closure law for integer 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	qexpclz 14228	Closure of integer exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℚ)
Theorem	reexpclz 14229	Closure of integer exponentiation of reals. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ)
Theorem	expclzlem 14230	Lemma for expclz 14231. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ (ℂ ∖ {0}))
Theorem	expclz 14231	Closure law for integer exponentiation of complex numnbers. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℂ)
Theorem	m1expcl2 14232	Closure of integer exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑁 ∈ ℤ → ( -1↑𝑁) ∈ { -1, 1})
Theorem	m1expcl 14233	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑁 ∈ ℤ → ( -1↑𝑁) ∈ ℤ)
Theorem	zexpcld 14234	Closure of exponentiation of integers, deduction form. (Contributed by SN, 15-Sep-2024.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℤ)
Theorem	nn0expcli 14235	Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝐴↑𝑁) ∈ ℕ0
Theorem	nn0sqcl 14236	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 14237	Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴))
Theorem	1exp 14238	Value of 1 raised to an integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1)
Theorem	expeq0 14239	A positive integer power is zero if and only if its base is zero. (Contributed by NM, 23-Feb-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) = 0 ↔ 𝐴 = 0))
Theorem	expne0 14240	A positive integer power is nonzero if and only if its base is nonzero. (Contributed by NM, 6-May-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) ≠ 0 ↔ 𝐴 ≠ 0))
Theorem	expne0i 14241	An integer power is nonzero if its base is nonzero. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ≠ 0)
Theorem	expgt0 14242	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	expnegz 14243	Value of a nonzero complex number raised to the negative of an integer power. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑ -𝑁) = (1 / (𝐴↑𝑁)))
Theorem	0exp 14244	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
Theorem	expge0 14245	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 14246	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 14247	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 14248	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	mulexpz 14249	Integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ 𝑁 ∈ ℤ) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁)))
Theorem	exprec 14250	Integer exponentiation of a reciprocal. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))
Theorem	expadd 14251	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	expaddzlem 14252	Lemma for expaddz 14253. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑀 ∈ ℝ ∧ -𝑀 ∈ ℕ) ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) · (𝐴↑𝑁)))
Theorem	expaddz 14253	Sum of exponents law for integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) · (𝐴↑𝑁)))
Theorem	expmul 14254	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	expmulz 14255	Product of exponents law for integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 7-Jul-2014.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 · 𝑁)) = ((𝐴↑𝑀)↑𝑁))
Theorem	m1expeven 14256	Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)
⊢ (𝑁 ∈ ℤ → ( -1↑(2 · 𝑁)) = 1)
Theorem	expsub 14257	Exponent subtraction law for integer exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 − 𝑁)) = ((𝐴↑𝑀) / (𝐴↑𝑁)))
Theorem	expp1z 14258	Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴))
Theorem	expm1 14259	Value of a nonzero complex number raised to an integer power minus one. (Contributed by NM, 25-Dec-2008.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 − 1)) = ((𝐴↑𝑁) / 𝐴))
Theorem	expdiv 14260	Nonnegative integer exponentiation of a quotient. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ 𝑁 ∈ ℕ0) → ((𝐴 / 𝐵)↑𝑁) = ((𝐴↑𝑁) / (𝐵↑𝑁)))
Theorem	sqval 14261	Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴))
Theorem	sqneg 14262	The square of the negative of a number. (Contributed by NM, 15-Jan-2006.)
⊢ (𝐴 ∈ ℂ → ( -𝐴↑2) = (𝐴↑2))
Theorem	sqnegd 14263	The square of the negative of a number. (Contributed by Igor Ieskov, 21-Jan-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ( -𝐴↑2) = (𝐴↑2))
Theorem	sqsubswap 14264	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 14265	Closure of square. (Contributed by NM, 10-Aug-1999.)
⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ)
Theorem	sqmul 14266	Distribution of squaring over multiplication. (Contributed by NM, 21-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2)))
Theorem	sqeq0 14267	A complex number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) = 0 ↔ 𝐴 = 0))
Theorem	sqdiv 14268	Distribution of squaring over division. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by Mario Carneiro, 9-Jul-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))
Theorem	sqdivid 14269	The square of a nonzero complex number divided by itself equals that number. (Contributed by AV, 19-Jul-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑2) / 𝐴) = 𝐴)
Theorem	sqne0 14270	A complex number is nonzero if and only if its square is nonzero. (Contributed by NM, 11-Mar-2006.)
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) ≠ 0 ↔ 𝐴 ≠ 0))
Theorem	resqcl 14271	Closure of squaring in reals. (Contributed by NM, 18-Oct-1999.)
⊢ (𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ)
Theorem	resqcld 14272	Closure of squaring in reals, deduction form. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴↑2) ∈ ℝ)
Theorem	sqgt0 14273	The square of a nonzero real is positive. (Contributed by NM, 8-Sep-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → 0 < (𝐴↑2))
Theorem	sqn0rp 14274	The square of a nonzero real is a positive real. (Contributed by AV, 5-Mar-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴↑2) ∈ ℝ+)
Theorem	nnsqcl 14275	The positive 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 14276	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 14277	The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → (𝐴↑2) ∈ ℚ)
Theorem	sq11 14278	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	nn0sq11 14279	The square function is one-to-one for nonnegative integers. (Contributed by AV, 25-Jun-2023.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
Theorem	lt2sq 14280	The square function is increasing on nonnegative reals. (Contributed by NM, 24-Feb-2006.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 < 𝐵 ↔ (𝐴↑2) < (𝐵↑2)))
Theorem	le2sq 14281	The square function is nondecreasing on nonnegative reals. (Contributed by NM, 18-Oct-1999.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 ≤ 𝐵 ↔ (𝐴↑2) ≤ (𝐵↑2)))
Theorem	le2sq2 14282	The square function is nondecreasing on the nonnegative reals. (Contributed by NM, 21-Mar-2008.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵)) → (𝐴↑2) ≤ (𝐵↑2))
Theorem	sqge0 14283	The square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴↑2))
Theorem	sqge0d 14284	The square of a real is nonnegative, deduction form. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴↑2))
Theorem	zsqcl2 14285	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	0expd 14286	Value of zero raised to a positive integer power. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (0↑𝑁) = 0)
Theorem	exp0d 14287	Value of a complex number raised to the zeroth power. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑0) = 1)
Theorem	exp1d 14288	Value of a complex number raised to the first power. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑1) = 𝐴)
Theorem	expeq0d 14289	If a positive integer power is zero, then its base is zero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝐴↑𝑁) = 0)    ⇒   ⊢ (𝜑 → 𝐴 = 0)
Theorem	sqvald 14290	Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑2) = (𝐴 · 𝐴))
Theorem	sqcld 14291	Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑2) ∈ ℂ)
Theorem	sqeq0d 14292	A number is zero iff its square is zero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴↑2) = 0)    ⇒   ⊢ (𝜑 → 𝐴 = 0)
Theorem	expcld 14293	Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ)
Theorem	expp1d 14294	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 Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴))
Theorem	expaddd 14295	Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) · (𝐴↑𝑁)))
Theorem	expmuld 14296	Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴↑(𝑀 · 𝑁)) = ((𝐴↑𝑀)↑𝑁))
Theorem	sqrecd 14297	Square of reciprocal is reciprocal of square. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → ((1 / 𝐴)↑2) = (1 / (𝐴↑2)))
Theorem	expclzd 14298	Closure law for integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ)
Theorem	expne0d 14299	A nonnegative integer power is nonzero if its base is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴↑𝑁) ≠ 0)
Theorem	expnegd 14300	Value of a nonzero complex number raised to the negative of an integer power. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐴↑ -𝑁) = (1 / (𝐴↑𝑁)))