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Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnnsqcl 9201 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 9202 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 9203 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(𝐴 ∈ ℚ → (𝐴↑2) ∈ ℚ)

Theoremsq11 9204 The square function is one-to-one for nonnegative reals. Also see sq11ap 9292 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 9205 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 9206 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 9207 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 9208 A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
(𝐴 ∈ ℝ → 0 ≤ (𝐴↑2))

Theoremzsqcl2 9209 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 9210 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 9211 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (𝐴↑2) = (𝐴 · 𝐴)

Theoremsqcli 9212 Closure of square. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (𝐴↑2) ∈ ℂ

Theoremsqeq0i 9213 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       ((𝐴↑2) = 0 ↔ 𝐴 = 0)

Theoremsqmuli 9214 Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))

Theoremsqdivapi 9215 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 9216 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ       (𝐴↑2) ∈ ℝ

Theoremsqgt0api 9217 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
𝐴 ∈ ℝ       (𝐴 # 0 → 0 < (𝐴↑2))

Theoremsqge0i 9218 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
𝐴 ∈ ℝ       0 ≤ (𝐴↑2)

Theoremlt2sqi 9219 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 9220 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 9221 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 9222 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
(0↑2) = 0

Theoremsq0i 9223 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
(𝐴 = 0 → (𝐴↑2) = 0)

Theoremsq0id 9224 If a number is zero, its square is zero. Deduction form of sq0i 9223. Converse of sqeq0d 9258. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 = 0)       (𝜑 → (𝐴↑2) = 0)

Theoremsq1 9225 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
(1↑2) = 1

Theoremneg1sqe1 9226 -1 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1↑2) = 1

Theoremsq2 9227 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
(2↑2) = 4

Theoremsq3 9228 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
(3↑2) = 9

Theoremcu2 9229 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
(2↑3) = 8

Theoremirec 9230 The reciprocal of i. (Contributed by NM, 11-Oct-1999.)
(1 / i) = -i

Theoremi2 9231 i squared. (Contributed by NM, 6-May-1999.)
(i↑2) = -1

Theoremi3 9232 i cubed. (Contributed by NM, 31-Jan-2007.)
(i↑3) = -i

Theoremi4 9233 i to the fourth power. (Contributed by NM, 31-Jan-2007.)
(i↑4) = 1

Theoremnnlesq 9234 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))

Theoremexpnass 9235 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 9236 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴𝐵)))

Theoremsubsq2 9237 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 9238 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 9239 Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴𝐵))

Theorembinom2 9240 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 9241 Special case of binom2 9240 where 𝐵 = 1. (Contributed by Scott Fenton, 11-May-2014.)
(𝐴 ∈ ℂ → ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · 𝐴)) + 1))

Theorembinom2sub 9242 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)))

Theorembinom2subi 9243 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))

Theorembinom3 9244 The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑3) = (((𝐴↑3) + (3 · ((𝐴↑2) · 𝐵))) + ((3 · (𝐴 · (𝐵↑2))) + (𝐵↑3))))

Theoremzesq 9245 An integer is even iff its square is even. (Contributed by Mario Carneiro, 12-Sep-2015.)
(𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ↔ ((𝑁↑2) / 2) ∈ ℤ))

Theoremnnesq 9246 A positive integer is even iff its square is even. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)
(𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ↔ ((𝑁↑2) / 2) ∈ ℕ))

Theorembernneq 9247 Bernoulli's inequality, due to Johan Bernoulli (1667-1748). (Contributed by NM, 21-Feb-2005.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ -1 ≤ 𝐴) → (1 + (𝐴 · 𝑁)) ≤ ((1 + 𝐴)↑𝑁))

Theorembernneq2 9248 Variation of Bernoulli's inequality bernneq 9247. (Contributed by NM, 18-Oct-2007.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴𝑁))

Theorembernneq3 9249 A corollary of bernneq 9247. (Contributed by Mario Carneiro, 11-Mar-2014.)
((𝑃 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (𝑃𝑁))

Theoremexpnbnd 9250* Exponentiation with a mantissa greater than 1 has no upper bound. (Contributed by NM, 20-Oct-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵) → ∃𝑘 ∈ ℕ 𝐴 < (𝐵𝑘))

Theoremexpnlbnd 9251* The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ ∧ 1 < 𝐵) → ∃𝑘 ∈ ℕ (1 / (𝐵𝑘)) < 𝐴)

Theoremexpnlbnd2 9252* The reciprocal of exponentiation with a mantissa greater than 1 has no lower bound. (Contributed by NM, 18-Jul-2008.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ ∧ 1 < 𝐵) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ𝑗)(1 / (𝐵𝑘)) < 𝐴)

Theoremexp0d 9253 Value of a complex number raised to the 0th power. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴↑0) = 1)

Theoremexp1d 9254 Value of a complex number raised to the first power. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴↑1) = 𝐴)

Theoremexpeq0d 9255 Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐴𝑁) = 0)       (𝜑𝐴 = 0)

Theoremsqvald 9256 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴↑2) = (𝐴 · 𝐴))

Theoremsqcld 9257 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴↑2) ∈ ℂ)

Theoremsqeq0d 9258 A number is zero iff its square is zero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → (𝐴↑2) = 0)       (𝜑𝐴 = 0)

Theoremexpcld 9259 Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑁) ∈ ℂ)

Theoremexpp1d 9260 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.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴𝑁) · 𝐴))

Theoremexpaddd 9261 Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℕ0)       (𝜑 → (𝐴↑(𝑀 + 𝑁)) = ((𝐴𝑀) · (𝐴𝑁)))

Theoremexpmuld 9262 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.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℕ0)       (𝜑 → (𝐴↑(𝑀 · 𝑁)) = ((𝐴𝑀)↑𝑁))

Theoremsqrecapd 9263 Square of reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → ((1 / 𝐴)↑2) = (1 / (𝐴↑2)))

Theoremexpclzapd 9264 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴𝑁) ∈ ℂ)

Theoremexpap0d 9265 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴𝑁) # 0)

Theoremexpnegapd 9266 Value of a complex number raised to a negative power. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴↑-𝑁) = (1 / (𝐴𝑁)))

Theoremexprecapd 9267 Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → ((1 / 𝐴)↑𝑁) = (1 / (𝐴𝑁)))

Theoremexpp1zapd 9268 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴𝑁) · 𝐴))

Theoremexpm1apd 9269 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴↑(𝑁 − 1)) = ((𝐴𝑁) / 𝐴))

Theoremexpsubapd 9270 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐴↑(𝑀𝑁)) = ((𝐴𝑀) / (𝐴𝑁)))

Theoremsqmuld 9271 Distribution of square over multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2)))

Theoremsqdivapd 9272 Distribution of square over division. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))

Theoremexpdivapd 9273 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝐴 / 𝐵)↑𝑁) = ((𝐴𝑁) / (𝐵𝑁)))

Theoremmulexpd 9274 Positive integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝐴 · 𝐵)↑𝑁) = ((𝐴𝑁) · (𝐵𝑁)))

Theorem0expd 9275 Value of zero raised to a positive integer power. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → (0↑𝑁) = 0)

Theoremreexpcld 9276 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑁) ∈ ℝ)

Theoremexpge0d 9277 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → 0 ≤ (𝐴𝑁))

Theoremexpge1d 9278 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → 1 ≤ 𝐴)       (𝜑 → 1 ≤ (𝐴𝑁))

Theoremnnsqcld 9279 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → (𝐴↑2) ∈ ℕ)

Theoremnnexpcld 9280 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑁) ∈ ℕ)

Theoremnn0expcld 9281 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑁) ∈ ℕ0)

Theoremrpexpcld 9282 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴𝑁) ∈ ℝ+)

Theoremreexpclzapd 9283 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴𝑁) ∈ ℝ)

Theoremresqcld 9284 Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴↑2) ∈ ℝ)

Theoremsqge0d 9285 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → 0 ≤ (𝐴↑2))

Theoremsqgt0apd 9286 The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 # 0)       (𝜑 → 0 < (𝐴↑2))

Theoremleexp2ad 9287 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 ≤ 𝐴)    &   (𝜑𝑁 ∈ (ℤ𝑀))       (𝜑 → (𝐴𝑀) ≤ (𝐴𝑁))

Theoremleexp2rd 9288 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 ≤ 1)       (𝜑 → (𝐴𝑁) ≤ (𝐴𝑀))

Theoremlt2sqd 9289 The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐴↑2) < (𝐵↑2)))

Theoremle2sqd 9290 The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → (𝐴𝐵 ↔ (𝐴↑2) ≤ (𝐵↑2)))

Theoremsq11d 9291 The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 ≤ 𝐵)    &   (𝜑 → (𝐴↑2) = (𝐵↑2))       (𝜑𝐴 = 𝐵)

Theoremsq11ap 9292 Analogue to sq11 9204 but for apartness. (Contributed by Jim Kingdon, 12-Aug-2021.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) # (𝐵↑2) ↔ 𝐴 # 𝐵))

3.6  Elementary real and complex functions

3.6.1  The "shift" operation

Syntaxcshi 9293 Extend class notation with function shifter.
class shift

Definitiondf-shft 9294* Define a function shifter. This operation offsets the value argument of a function (ordinarily on a subset of ) and produces a new function on . See shftval 9304 for its value. (Contributed by NM, 20-Jul-2005.)
shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ℂ ∧ (𝑦𝑥)𝑓𝑧)})

Theoremshftlem 9295* Two ways to write a shifted set (𝐵 + 𝐴). (Contributed by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ⊆ ℂ) → {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ 𝐵} = {𝑥 ∣ ∃𝑦𝐵 𝑥 = (𝑦 + 𝐴)})

Theoremshftuz 9296* A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝑥 ∈ ℂ ∣ (𝑥𝐴) ∈ (ℤ𝐵)} = (ℤ‘(𝐵 + 𝐴)))

Theoremshftfvalg 9297* The value of the sequence shifter operation is a function on . 𝐴 is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℂ ∧ 𝐹𝑉) → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})

Theoremovshftex 9298 Existence of the result of applying shift. (Contributed by Jim Kingdon, 15-Aug-2021.)
((𝐹𝑉𝐴 ∈ ℂ) → (𝐹 shift 𝐴) ∈ V)

Theoremshftfibg 9299 Value of a fiber of the relation 𝐹. (Contributed by Jim Kingdon, 15-Aug-2021.)
((𝐹𝑉𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) “ {𝐵}) = (𝐹 “ {(𝐵𝐴)}))

Theoremshftfval 9300* The value of the sequence shifter operation is a function on . 𝐴 is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
𝐹 ∈ V       (𝐴 ∈ ℂ → (𝐹 shift 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ (𝑥𝐴)𝐹𝑦)})

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