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Theorem List for Metamath Proof Explorer - 12601-12700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnn0expcl 12601 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
((𝐴 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℕ0)
 
Theoremzexpcl 12602 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℤ)
 
Theoremqexpcl 12603 Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℚ)
 
Theoremreexpcl 12604 Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℝ)
 
Theoremexpcl 12605 Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℂ)
 
Theoremrpexpcl 12606 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+)
 
Theoremreexpclz 12607 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ ℝ)
 
Theoremqexpclz 12608 Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ ℚ)
 
Theoremm1expcl2 12609 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1})
 
Theoremm1expcl 12610 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℤ)
 
Theoremexpclzlem 12611 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ (ℂ ∖ {0}))
 
Theoremexpclz 12612 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ ℂ)
 
Theoremnn0expcli 12613 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0    &   𝑁 ∈ ℕ0       (𝐴𝑁) ∈ ℕ0
 
Theoremnn0sqcl 12614 The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(𝐴 ∈ ℕ0 → (𝐴↑2) ∈ ℕ0)
 
Theoremexpm1t 12615 Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴))
 
Theorem1exp 12616 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)
 
Theoremexpeq0 12617 Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) = 0 ↔ 𝐴 = 0))
 
Theoremexpne0 12618 Positive integer exponentiation is nonzero iff its mantissa is nonzero. (Contributed by NM, 6-May-2005.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) ≠ 0 ↔ 𝐴 ≠ 0))
 
Theoremexpne0i 12619 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ≠ 0)
 
Theoremexpgt0 12620 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))
 
Theoremexpnegz 12621 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 12622 Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
(𝑁 ∈ ℕ → (0↑𝑁) = 0)
 
Theoremexpge0 12623 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 12624 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 12625 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 12626 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)))
 
Theoremmulexpz 12627 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) = ((𝐴𝑁) · (𝐵𝑁)))
 
Theoremexprec 12628 Nonnegative 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 / (𝐴𝑁)))
 
Theoremexpadd 12629 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)))
 
Theoremexpaddzlem 12630 Lemma for expaddz 12631. (Contributed by Mario Carneiro, 4-Jun-2014.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑀 ∈ ℝ ∧ -𝑀 ∈ ℕ) ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴𝑀) · (𝐴𝑁)))
 
Theoremexpaddz 12631 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)) = ((𝐴𝑀) · (𝐴𝑁)))
 
Theoremexpmul 12632 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))
 
Theoremexpmulz 12633 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)) = ((𝐴𝑀)↑𝑁))
 
Theoremm1expeven 12634 Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)
(𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1)
 
Theoremexpsub 12635 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀𝑁)) = ((𝐴𝑀) / (𝐴𝑁)))
 
Theoremexpp1z 12636 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)) = ((𝐴𝑁) · 𝐴))
 
Theoremexpm1 12637 Value of a 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)) = ((𝐴𝑁) / 𝐴))
 
Theoremexpdiv 12638 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) = ((𝐴𝑁) / (𝐵𝑁)))
 
Theoremltexp2a 12639 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))
 
Theoremexpcan 12640 Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴𝑀) = (𝐴𝑁) ↔ 𝑀 = 𝑁))
 
Theoremltexp2 12641 Ordering law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴𝑀) < (𝐴𝑁)))
 
Theoremleexp2 12642 Ordering law for exponentiation. (Contributed by Mario Carneiro, 26-Apr-2016.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀𝑁 ↔ (𝐴𝑀) ≤ (𝐴𝑁)))
 
Theoremleexp2a 12643 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))
 
Theoremltexp2r 12644 The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
(((𝐴 ∈ ℝ+𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝑀 < 𝑁 ↔ (𝐴𝑁) < (𝐴𝑀)))
 
Theoremleexp2r 12645 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 12646 Weak mantissa ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑁 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴𝐵)) → (𝐴𝑁) ≤ (𝐵𝑁))
 
Theoremexple1 12647 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 12648 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 12649 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
(𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴))
 
Theoremsqneg 12650 The square of the negative of a number. (Contributed by NM, 15-Jan-2006.)
(𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2))
 
Theoremsqsubswap 12651 Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵)↑2) = ((𝐵𝐴)↑2))
 
Theoremsqcl 12652 Closure of square. (Contributed by NM, 10-Aug-1999.)
(𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ)
 
Theoremsqmul 12653 Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2)))
 
Theoremsqeq0 12654 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
(𝐴 ∈ ℂ → ((𝐴↑2) = 0 ↔ 𝐴 = 0))
 
Theoremsqdiv 12655 Distribution of square 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)))
 
Theoremsqdivid 12656 The square of a nonzero number divided by itself yields the number itself. (Contributed by AV, 19-Jul-2021.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑2) / 𝐴) = 𝐴)
 
Theoremsqne0 12657 A number is nonzero iff its square is nonzero. (Contributed by NM, 11-Mar-2006.)
(𝐴 ∈ ℂ → ((𝐴↑2) ≠ 0 ↔ 𝐴 ≠ 0))
 
Theoremresqcl 12658 Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
(𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ)
 
Theoremsqgt0 12659 The square of a nonzero real is positive. (Contributed by NM, 8-Sep-2007.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → 0 < (𝐴↑2))
 
Theoremnnsqcl 12660 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 12661 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 12662 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(𝐴 ∈ ℚ → (𝐴↑2) ∈ ℚ)
 
Theoremsq11 12663 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))
 
Theoremlt2sq 12664 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 12665 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 12666 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 12667 A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
(𝐴 ∈ ℝ → 0 ≤ (𝐴↑2))
 
Theoremzsqcl2 12668 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 12669 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 12670 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (𝐴↑2) = (𝐴 · 𝐴)
 
Theoremsqcli 12671 Closure of square. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (𝐴↑2) ∈ ℂ
 
Theoremsqeq0i 12672 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       ((𝐴↑2) = 0 ↔ 𝐴 = 0)
 
Theoremsqrecii 12673 Square of reciprocal. (Contributed by NM, 17-Sep-1999.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       ((1 / 𝐴)↑2) = (1 / (𝐴↑2))
 
Theoremsqmuli 12674 Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))
 
Theoremsqdivi 12675 Distribution of square over division. (Contributed by NM, 20-Aug-2001.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 ≠ 0       ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))
 
Theoremresqcli 12676 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ       (𝐴↑2) ∈ ℝ
 
Theoremsqgt0i 12677 The square of a nonzero real is positive. (Contributed by NM, 17-Sep-1999.)
𝐴 ∈ ℝ       (𝐴 ≠ 0 → 0 < (𝐴↑2))
 
Theoremsqge0i 12678 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
𝐴 ∈ ℝ       0 ≤ (𝐴↑2)
 
Theoremlt2sqi 12679 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 12680 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 12681 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 12682 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
(0↑2) = 0
 
Theoremsq0i 12683 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
(𝐴 = 0 → (𝐴↑2) = 0)
 
Theoremsq0id 12684 If a number is zero, its square is zero. Deduction form of sq0i 12683. Converse of sqeq0d 12734. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 = 0)       (𝜑 → (𝐴↑2) = 0)
 
Theoremsq1 12685 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
(1↑2) = 1
 
Theoremneg1sqe1 12686 -1 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1↑2) = 1
 
Theoremsq2 12687 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
(2↑2) = 4
 
Theoremsq3 12688 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
(3↑2) = 9
 
Theoremsq4e2t8 12689 The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)
(4↑2) = (2 · 8)
 
Theoremcu2 12690 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
(2↑3) = 8
 
Theoremirec 12691 The reciprocal of i. (Contributed by NM, 11-Oct-1999.)
(1 / i) = -i
 
Theoremi2 12692 i squared. (Contributed by NM, 6-May-1999.)
(i↑2) = -1
 
Theoremi3 12693 i cubed. (Contributed by NM, 31-Jan-2007.)
(i↑3) = -i
 
Theoremi4 12694 i to the fourth power. (Contributed by NM, 31-Jan-2007.)
(i↑4) = 1
 
Theoremnnlesq 12695 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 12696 Taking i to the 𝐾-th power is the same as using the 𝐾 mod 4 -th power instead, by i4 12694. (Contributed by Mario Carneiro, 7-Jul-2014.)
(𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾))
 
Theoremexpnass 12697 A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)
((3↑3)↑3) < (3↑(3↑3))
 
Theoremsqlecan 12698 Cancel one factor of a square in a comparison. Unlike lemul1 10622, the common factor 𝐴 may be zero. (Contributed by NM, 17-Jan-2008.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) ≤ (𝐵 · 𝐴) ↔ 𝐴𝐵))
 
Theoremsubsq 12699 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴𝐵)))
 
Theoremsubsq2 12700 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))))
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