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Theorem List for Intuitionistic Logic Explorer - 10301-10400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremzexpcl 10301 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℤ)

Theoremqexpcl 10302 Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℚ)

Theoremreexpcl 10303 Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℝ)

Theoremexpcl 10304 Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℂ)

Theoremrpexpcl 10305 Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
((𝐴 ∈ ℝ+𝑁 ∈ ℤ) → (𝐴𝑁) ∈ ℝ+)

Theoremreexpclzap 10306 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ ℝ)

Theoremqexpclz 10307 Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ ℚ)

Theoremm1expcl2 10308 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1})

Theoremm1expcl 10309 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℤ)

Theoremexpclzaplem 10310* Closure law for integer exponentiation. Lemma for expclzap 10311 and expap0i 10318. (Contributed by Jim Kingdon, 9-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})

Theoremexpclzap 10311 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ ℂ)

Theoremnn0expcli 10312 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0    &   𝑁 ∈ ℕ0       (𝐴𝑁) ∈ ℕ0

Theoremnn0sqcl 10313 The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(𝐴 ∈ ℕ0 → (𝐴↑2) ∈ ℕ0)

Theoremexpm1t 10314 Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴))

Theorem1exp 10315 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)

Theoremexpap0 10316 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 10317 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) → ((𝐴𝑁) # 0 ↔ 𝐴 # 0))

Theoremexpeq0 10317 Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) = 0 ↔ 𝐴 = 0))

Theoremexpap0i 10318 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) → (𝐴𝑁) # 0)

Theoremexpgt0 10319 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 < (𝐴𝑁))

Theoremexpnegzap 10320 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 / (𝐴𝑁)))

Theorem0exp 10321 Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
(𝑁 ∈ ℕ → (0↑𝑁) = 0)

Theoremexpge0 10322 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 ≤ (𝐴𝑁))

Theoremexpge1 10323 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 ≤ (𝐴𝑁))

Theoremexpgt1 10324 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 < (𝐴𝑁))

Theoremmulexp 10325 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) = ((𝐴𝑁) · (𝐵𝑁)))

Theoremmulexpzap 10326 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) = ((𝐴𝑁) · (𝐵𝑁)))

Theoremexprecap 10327 Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴𝑁)))

Theoremexpadd 10328 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 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴𝑀) · (𝐴𝑁)))

(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℝ ∧ -𝑀 ∈ ℕ) ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴𝑀) · (𝐴𝑁)))

Theoremexpaddzap 10330 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)) = ((𝐴𝑀) · (𝐴𝑁)))

Theoremexpmul 10331 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 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐴↑(𝑀 · 𝑁)) = ((𝐴𝑀)↑𝑁))

Theoremexpmulzap 10332 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)) = ((𝐴𝑀)↑𝑁))

Theoremm1expeven 10333 Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)
(𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1)

Theoremexpsubap 10334 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀𝑁)) = ((𝐴𝑀) / (𝐴𝑁)))

Theoremexpp1zap 10335 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)) = ((𝐴𝑁) · 𝐴))

Theoremexpm1ap 10336 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)) = ((𝐴𝑁) / 𝐴))

Theoremexpdivap 10337 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝑁 ∈ ℕ0) → ((𝐴 / 𝐵)↑𝑁) = ((𝐴𝑁) / (𝐵𝑁)))

Theoremltexp2a 10338 Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 < 𝐴𝑀 < 𝑁)) → (𝐴𝑀) < (𝐴𝑁))

Theoremleexp2a 10339 Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)
((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴𝑁 ∈ (ℤ𝑀)) → (𝐴𝑀) ≤ (𝐴𝑁))

Theoremleexp2r 10340 Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑁) ≤ (𝐴𝑀))

Theoremleexp1a 10341 Weak mantissa ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑁 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴𝐵)) → (𝐴𝑁) ≤ (𝐵𝑁))

Theoremexple1 10342 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 ≤ 𝐴𝐴 ≤ 1) ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ≤ 1)

Theoremexpubnd 10343 An upper bound on 𝐴𝑁 when 2 ≤ 𝐴. (Contributed by NM, 19-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 2 ≤ 𝐴) → (𝐴𝑁) ≤ ((2↑𝑁) · ((𝐴 − 1)↑𝑁)))

Theoremsqval 10344 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
(𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴))

Theoremsqneg 10345 The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
(𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2))

Theoremsqsubswap 10346 Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵)↑2) = ((𝐵𝐴)↑2))

Theoremsqcl 10347 Closure of square. (Contributed by NM, 10-Aug-1999.)
(𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ)

Theoremsqmul 10348 Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2)))

Theoremsqeq0 10349 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
(𝐴 ∈ ℂ → ((𝐴↑2) = 0 ↔ 𝐴 = 0))

Theoremsqdivap 10350 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)))

Theoremsqne0 10351 A number is nonzero iff its square is nonzero. See also sqap0 10352 which is the same but with not equal changed to apart. (Contributed by NM, 11-Mar-2006.)
(𝐴 ∈ ℂ → ((𝐴↑2) ≠ 0 ↔ 𝐴 ≠ 0))

Theoremsqap0 10352 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))

Theoremresqcl 10353 Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
(𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ)

Theoremsqgt0ap 10354 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → 0 < (𝐴↑2))

Theoremnnsqcl 10355 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)

Theoremzsqcl 10356 Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)

Theoremqsqcl 10357 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(𝐴 ∈ ℚ → (𝐴↑2) ∈ ℚ)

Theoremsq11 10358 The square function is one-to-one for nonnegative reals. Also see sq11ap 10451 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))

Theoremlt2sq 10359 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)))

Theoremle2sq 10360 The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴𝐵 ↔ (𝐴↑2) ≤ (𝐵↑2)))

Theoremle2sq2 10361 The square of a 'less than or equal to' ordering. (Contributed by NM, 21-Mar-2008.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐴𝐵)) → (𝐴↑2) ≤ (𝐵↑2))

Theoremsqge0 10362 A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
(𝐴 ∈ ℝ → 0 ≤ (𝐴↑2))

Theoremzsqcl2 10363 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)

Theoremsumsqeq0 10364 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))

Theoremsqvali 10365 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (𝐴↑2) = (𝐴 · 𝐴)

Theoremsqcli 10366 Closure of square. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (𝐴↑2) ∈ ℂ

Theoremsqeq0i 10367 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       ((𝐴↑2) = 0 ↔ 𝐴 = 0)

Theoremsqmuli 10368 Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))

Theoremsqdivapi 10369 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))

Theoremresqcli 10370 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ       (𝐴↑2) ∈ ℝ

Theoremsqgt0api 10371 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
𝐴 ∈ ℝ       (𝐴 # 0 → 0 < (𝐴↑2))

Theoremsqge0i 10372 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
𝐴 ∈ ℝ       0 ≤ (𝐴↑2)

Theoremlt2sqi 10373 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)))

Theoremle2sqi 10374 The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴𝐵 ↔ (𝐴↑2) ≤ (𝐵↑2)))

Theoremsq11i 10375 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))

Theoremsq0 10376 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
(0↑2) = 0

Theoremsq0i 10377 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
(𝐴 = 0 → (𝐴↑2) = 0)

Theoremsq0id 10378 If a number is zero, its square is zero. Deduction form of sq0i 10377. Converse of sqeq0d 10416. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 = 0)       (𝜑 → (𝐴↑2) = 0)

Theoremsq1 10379 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
(1↑2) = 1

Theoremneg1sqe1 10380 -1 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1↑2) = 1

Theoremsq2 10381 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
(2↑2) = 4

Theoremsq3 10382 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
(3↑2) = 9

Theoremsq4e2t8 10383 The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)
(4↑2) = (2 · 8)

Theoremcu2 10384 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
(2↑3) = 8

Theoremirec 10385 The reciprocal of i. (Contributed by NM, 11-Oct-1999.)
(1 / i) = -i

Theoremi2 10386 i squared. (Contributed by NM, 6-May-1999.)
(i↑2) = -1

Theoremi3 10387 i cubed. (Contributed by NM, 31-Jan-2007.)
(i↑3) = -i

Theoremi4 10388 i to the fourth power. (Contributed by NM, 31-Jan-2007.)
(i↑4) = 1

Theoremnnlesq 10389 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))

Theoremiexpcyc 10390 Taking i to the 𝐾-th power is the same as using the 𝐾 mod 4 -th power instead, by i4 10388. (Contributed by Mario Carneiro, 7-Jul-2014.)
(𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾))

Theoremexpnass 10391 A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)
((3↑3)↑3) < (3↑(3↑3))

Theoremsubsq 10392 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴𝐵)))

Theoremsubsq2 10393 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)) = (((𝐴𝐵)↑2) + ((2 · 𝐵) · (𝐴𝐵))))

Theorembinom2i 10394 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))

Theoremsubsqi 10395 Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴𝐵))

Theorembinom2 10396 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)))

Theorembinom21 10397 Special case of binom2 10396 where 𝐵 = 1. (Contributed by Scott Fenton, 11-May-2014.)
(𝐴 ∈ ℂ → ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · 𝐴)) + 1))

Theorembinom2sub 10398 Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵)↑2) = (((𝐴↑2) − (2 · (𝐴 · 𝐵))) + (𝐵↑2)))

Theorembinom2sub1 10399 Special case of binom2sub 10398 where 𝐵 = 1. (Contributed by AV, 2-Aug-2021.)
(𝐴 ∈ ℂ → ((𝐴 − 1)↑2) = (((𝐴↑2) − (2 · 𝐴)) + 1))

Theorembinom2subi 10400 Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴𝐵)↑2) = (((𝐴↑2) − (2 · (𝐴 · 𝐵))) + (𝐵↑2))

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