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Theorem List for Metamath Proof Explorer - 12901-13000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremleexp2r 12901 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 12902 Weak mantissa ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑁 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴𝐵)) → (𝐴𝑁) ≤ (𝐵𝑁))

Theoremexple1 12903 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 12904 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 12905 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
(𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴))

Theoremsqneg 12906 The square of the negative of a number. (Contributed by NM, 15-Jan-2006.)
(𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2))

Theoremsqsubswap 12907 Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵)↑2) = ((𝐵𝐴)↑2))

Theoremsqcl 12908 Closure of square. (Contributed by NM, 10-Aug-1999.)
(𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ)

Theoremsqmul 12909 Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2)))

Theoremsqeq0 12910 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
(𝐴 ∈ ℂ → ((𝐴↑2) = 0 ↔ 𝐴 = 0))

Theoremsqdiv 12911 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 12912 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 12913 A number is nonzero iff its square is nonzero. (Contributed by NM, 11-Mar-2006.)
(𝐴 ∈ ℂ → ((𝐴↑2) ≠ 0 ↔ 𝐴 ≠ 0))

Theoremresqcl 12914 Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
(𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ)

Theoremsqgt0 12915 The square of a nonzero real is positive. (Contributed by NM, 8-Sep-2007.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → 0 < (𝐴↑2))

Theoremnnsqcl 12916 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 12917 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 12918 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(𝐴 ∈ ℚ → (𝐴↑2) ∈ ℚ)

Theoremsq11 12919 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 12920 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 12921 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 12922 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 12923 A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
(𝐴 ∈ ℝ → 0 ≤ (𝐴↑2))

Theoremzsqcl2 12924 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 12925 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 12926 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (𝐴↑2) = (𝐴 · 𝐴)

Theoremsqcli 12927 Closure of square. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (𝐴↑2) ∈ ℂ

Theoremsqeq0i 12928 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       ((𝐴↑2) = 0 ↔ 𝐴 = 0)

Theoremsqrecii 12929 Square of reciprocal. (Contributed by NM, 17-Sep-1999.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       ((1 / 𝐴)↑2) = (1 / (𝐴↑2))

Theoremsqmuli 12930 Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))

Theoremsqdivi 12931 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 12932 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ       (𝐴↑2) ∈ ℝ

Theoremsqgt0i 12933 The square of a nonzero real is positive. (Contributed by NM, 17-Sep-1999.)
𝐴 ∈ ℝ       (𝐴 ≠ 0 → 0 < (𝐴↑2))

Theoremsqge0i 12934 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
𝐴 ∈ ℝ       0 ≤ (𝐴↑2)

Theoremlt2sqi 12935 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 12936 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 12937 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 12938 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
(0↑2) = 0

Theoremsq0i 12939 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
(𝐴 = 0 → (𝐴↑2) = 0)

Theoremsq0id 12940 If a number is zero, its square is zero. Deduction form of sq0i 12939. Converse of sqeq0d 12990. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 = 0)       (𝜑 → (𝐴↑2) = 0)

Theoremsq1 12941 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
(1↑2) = 1

Theoremneg1sqe1 12942 -1 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1↑2) = 1

Theoremsq2 12943 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
(2↑2) = 4

Theoremsq3 12944 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
(3↑2) = 9

Theoremsq4e2t8 12945 The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)
(4↑2) = (2 · 8)

Theoremcu2 12946 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
(2↑3) = 8

Theoremirec 12947 The reciprocal of i. (Contributed by NM, 11-Oct-1999.)
(1 / i) = -i

Theoremi2 12948 i squared. (Contributed by NM, 6-May-1999.)
(i↑2) = -1

Theoremi3 12949 i cubed. (Contributed by NM, 31-Jan-2007.)
(i↑3) = -i

Theoremi4 12950 i to the fourth power. (Contributed by NM, 31-Jan-2007.)
(i↑4) = 1

Theoremnnlesq 12951 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 12952 Taking i to the 𝐾-th power is the same as using the 𝐾 mod 4 -th power instead, by i4 12950. (Contributed by Mario Carneiro, 7-Jul-2014.)
(𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾))

Theoremexpnass 12953 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 12954 Cancel one factor of a square in a comparison. Unlike lemul1 10860, 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 12955 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴𝐵)))

Theoremsubsq2 12956 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 12957 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 12958 Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴𝐵))

Theoremsqeqori 12959 The squares of two complex numbers are equal iff one number equals the other or its negative. Lemma 15-4.7 of [Gleason] p. 311 and its converse. (Contributed by NM, 15-Jan-2006.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵))

Theoremsubsq0i 12960 The two solutions to the difference of squares set equal to zero. (Contributed by NM, 25-Apr-2006.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (((𝐴↑2) − (𝐵↑2)) = 0 ↔ (𝐴 = 𝐵𝐴 = -𝐵))

Theoremsqeqor 12961 The squares of two complex numbers are equal iff one number equals the other or its negative. Lemma 15-4.7 of [Gleason] p. 311 and its converse. (Contributed by Paul Chapman, 15-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵𝐴 = -𝐵)))

Theorembinom2 12962 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 12963 Special case of binom2 12962 where 𝐵 = 1. (Contributed by Scott Fenton, 11-May-2014.)
(𝐴 ∈ ℂ → ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · 𝐴)) + 1))

Theorembinom2sub 12964 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 12965 Special case of binom2sub 12964 where 𝐵 = 1. (Contributed by AV, 2-Aug-2021.)
(𝐴 ∈ ℂ → ((𝐴 − 1)↑2) = (((𝐴↑2) − (2 · 𝐴)) + 1))

Theorembinom2subi 12966 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))

Theoremmulbinom2 12967 The square of a binomial with factor. (Contributed by AV, 19-Jul-2021.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐶 · 𝐴) + 𝐵)↑2) = ((((𝐶 · 𝐴)↑2) + ((2 · 𝐶) · (𝐴 · 𝐵))) + (𝐵↑2)))

Theorembinom3 12968 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))))

Theoremsq01 12969 If a complex number equals its square, it must be 0 or 1. (Contributed by NM, 6-Jun-2006.)
(𝐴 ∈ ℂ → ((𝐴↑2) = 𝐴 ↔ (𝐴 = 0 ∨ 𝐴 = 1)))

Theoremzesq 12970 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 12971 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))

Theoremcrreczi 12972 Reciprocal of a complex number in terms of real and imaginary components. Remark in [Apostol] p. 361. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Jeff Hankins, 16-Dec-2013.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) → (1 / (𝐴 + (i · 𝐵))) = ((𝐴 − (i · 𝐵)) / ((𝐴↑2) + (𝐵↑2))))

Theorembernneq 12973 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 12974 Variation of Bernoulli's inequality bernneq 12973. (Contributed by NM, 18-Oct-2007.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴𝑁))

Theorembernneq3 12975 A corollary of bernneq 12973. (Contributed by Mario Carneiro, 11-Mar-2014.)
((𝑃 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (𝑃𝑁))

Theoremexpnbnd 12976* 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 12977* 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 12978* 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)

Theoremexpmulnbnd 12979* Exponentiation with a mantissa greater than 1 is not bounded by any linear function. (Contributed by Mario Carneiro, 31-Mar-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵) → ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝐴 · 𝑘) < (𝐵𝑘))

Theoremdigit2 12980 Two ways to express the 𝐾 th digit in the decimal (when base 𝐵 = 10) expansion of a number 𝐴. 𝐾 = 1 corresponds to the first digit after the decimal point. (Contributed by NM, 25-Dec-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ ∧ 𝐾 ∈ ℕ) → ((⌊‘((𝐵𝐾) · 𝐴)) mod 𝐵) = ((⌊‘((𝐵𝐾) · 𝐴)) − (𝐵 · (⌊‘((𝐵↑(𝐾 − 1)) · 𝐴)))))

Theoremdigit1 12981 Two ways to express the 𝐾 th digit in the decimal expansion of a number 𝐴 (when base 𝐵 = 10). 𝐾 = 1 corresponds to the first digit after the decimal point. (Contributed by NM, 3-Jan-2009.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ ∧ 𝐾 ∈ ℕ) → ((⌊‘((𝐵𝐾) · 𝐴)) mod 𝐵) = (((⌊‘((𝐵𝐾) · 𝐴)) mod (𝐵𝐾)) − ((𝐵 · (⌊‘((𝐵↑(𝐾 − 1)) · 𝐴))) mod (𝐵𝐾))))

Theoremmodexp 12982 Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.)
(((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℕ0𝐷 ∈ ℝ+) ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴𝐶) mod 𝐷) = ((𝐵𝐶) mod 𝐷))

Theoremdiscr1 12983* A nonnegative quadratic form has nonnegative leading coefficient. (Contributed by Mario Carneiro, 4-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑥 ∈ ℝ) → 0 ≤ (((𝐴 · (𝑥↑2)) + (𝐵 · 𝑥)) + 𝐶))    &   𝑋 = if(1 ≤ (((𝐵 + if(0 ≤ 𝐶, 𝐶, 0)) + 1) / -𝐴), (((𝐵 + if(0 ≤ 𝐶, 𝐶, 0)) + 1) / -𝐴), 1)       (𝜑 → 0 ≤ 𝐴)

Theoremdiscr 12984* If a quadratic polynomial with real coefficients is nonnegative for all values, then its discriminant is nonpositive. (Contributed by NM, 10-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑥 ∈ ℝ) → 0 ≤ (((𝐴 · (𝑥↑2)) + (𝐵 · 𝑥)) + 𝐶))       (𝜑 → ((𝐵↑2) − (4 · (𝐴 · 𝐶))) ≤ 0)

Theoremexp0d 12985 Value of a complex number raised to the 0th power. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴↑0) = 1)

Theoremexp1d 12986 Value of a complex number raised to the first power. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴↑1) = 𝐴)

Theoremexpeq0d 12987 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 12988 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴↑2) = (𝐴 · 𝐴))

Theoremsqcld 12989 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴↑2) ∈ ℂ)

Theoremsqeq0d 12990 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 12991 Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑁) ∈ ℂ)

Theoremexpp1d 12992 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 12993 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 12994 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))

Theoremsqrecd 12995 Square of reciprocal. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → ((1 / 𝐴)↑2) = (1 / (𝐴↑2)))

Theoremexpclzd 12996 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴𝑁) ∈ ℂ)

Theoremexpne0d 12997 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴𝑁) ≠ 0)

Theoremexpnegd 12998 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴↑-𝑁) = (1 / (𝐴𝑁)))

Theoremexprecd 12999 Nonnegative integer exponentiation of a reciprocal. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → ((1 / 𝐴)↑𝑁) = (1 / (𝐴𝑁)))

Theoremexpp1zd 13000 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴𝑁) · 𝐴))

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