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Theorem List for Metamath Proof Explorer - 13901-14000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremleexp2r 13901 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 13902 Weak base ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑁 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴𝐵)) → (𝐴𝑁) ≤ (𝐵𝑁))
 
Theoremexple1 13903 A real between 0 and 1 inclusive raised to a nonnegative integer 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 13904 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)))
 
Theoremsumsqeq0 13905 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 13906 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (𝐴↑2) = (𝐴 · 𝐴)
 
Theoremsqcli 13907 Closure of square. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℂ       (𝐴↑2) ∈ ℂ
 
Theoremsqeq0i 13908 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ       ((𝐴↑2) = 0 ↔ 𝐴 = 0)
 
Theoremsqrecii 13909 Square of reciprocal. (Contributed by NM, 17-Sep-1999.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       ((1 / 𝐴)↑2) = (1 / (𝐴↑2))
 
Theoremsqmuli 13910 Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))
 
Theoremsqdivi 13911 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 13912 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ       (𝐴↑2) ∈ ℝ
 
Theoremsqgt0i 13913 The square of a nonzero real is positive. (Contributed by NM, 17-Sep-1999.)
𝐴 ∈ ℝ       (𝐴 ≠ 0 → 0 < (𝐴↑2))
 
Theoremsqge0i 13914 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
𝐴 ∈ ℝ       0 ≤ (𝐴↑2)
 
Theoremlt2sqi 13915 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 13916 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 13917 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 13918 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
(0↑2) = 0
 
Theoremsq0i 13919 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
(𝐴 = 0 → (𝐴↑2) = 0)
 
Theoremsq0id 13920 If a number is zero, its square is zero. Deduction form of sq0i 13919. Converse of sqeq0d 13872. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 = 0)       (𝜑 → (𝐴↑2) = 0)
 
Theoremsq1 13921 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
(1↑2) = 1
 
Theoremneg1sqe1 13922 -1 squared is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1↑2) = 1
 
Theoremsq2 13923 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
(2↑2) = 4
 
Theoremsq3 13924 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
(3↑2) = 9
 
Theoremsq4e2t8 13925 The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)
(4↑2) = (2 · 8)
 
Theoremcu2 13926 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
(2↑3) = 8
 
Theoremirec 13927 The reciprocal of i. (Contributed by NM, 11-Oct-1999.)
(1 / i) = -i
 
Theoremi2 13928 i squared. (Contributed by NM, 6-May-1999.)
(i↑2) = -1
 
Theoremi3 13929 i cubed. (Contributed by NM, 31-Jan-2007.)
(i↑3) = -i
 
Theoremi4 13930 i to the fourth power. (Contributed by NM, 31-Jan-2007.)
(i↑4) = 1
 
Theoremnnlesq 13931 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 13932 Taking i to the 𝐾-th power is the same as using the 𝐾 mod 4 -th power instead, by i4 13930. (Contributed by Mario Carneiro, 7-Jul-2014.)
(𝐾 ∈ ℤ → (i↑(𝐾 mod 4)) = (i↑𝐾))
 
Theoremexpnass 13933 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 13934 Cancel one factor of a square in a comparison. Unlike lemul1 11836, 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 13935 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴𝐵)))
 
Theoremsubsq2 13936 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 13937 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 13938 Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴𝐵))
 
Theoremsqeqori 13939 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 13940 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 13941 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 13942 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 13943 Special case of binom2 13942 where 𝐵 = 1. (Contributed by Scott Fenton, 11-May-2014.)
(𝐴 ∈ ℂ → ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · 𝐴)) + 1))
 
Theorembinom2sub 13944 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 13945 Special case of binom2sub 13944 where 𝐵 = 1. (Contributed by AV, 2-Aug-2021.)
(𝐴 ∈ ℂ → ((𝐴 − 1)↑2) = (((𝐴↑2) − (2 · 𝐴)) + 1))
 
Theorembinom2subi 13946 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 13947 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 13948 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 13949 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 13950 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 13951 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 13952 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 13953 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 13954 Variation of Bernoulli's inequality bernneq 13953. (Contributed by NM, 18-Oct-2007.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴𝑁))
 
Theorembernneq3 13955 A corollary of bernneq 13953. (Contributed by Mario Carneiro, 11-Mar-2014.)
((𝑃 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (𝑃𝑁))
 
Theoremexpnbnd 13956* Exponentiation with a base greater than 1 has no upper bound. (Contributed by NM, 20-Oct-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵) → ∃𝑘 ∈ ℕ 𝐴 < (𝐵𝑘))
 
Theoremexpnlbnd 13957* The reciprocal of exponentiation with a base greater than 1 has no positive lower bound. (Contributed by NM, 18-Jul-2008.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ ∧ 1 < 𝐵) → ∃𝑘 ∈ ℕ (1 / (𝐵𝑘)) < 𝐴)
 
Theoremexpnlbnd2 13958* The reciprocal of exponentiation with a base greater than 1 has no positive 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 13959* Exponentiation with a base 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 13960 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 13961 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 13962 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 13963* 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 13964* 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)
 
Theoremexpnngt1 13965 If an integer power with a positive integer base is greater than 1, then the exponent is positive. (Contributed by AV, 28-Dec-2022.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ 1 < (𝐴𝐵)) → 𝐵 ∈ ℕ)
 
Theoremexpnngt1b 13966 An integer power with an integer base greater than 1 is greater than 1 iff the exponent is positive. (Contributed by AV, 28-Dec-2022.)
((𝐴 ∈ (ℤ‘2) ∧ 𝐵 ∈ ℤ) → (1 < (𝐴𝐵) ↔ 𝐵 ∈ ℕ))
 
Theoremsqoddm1div8 13967 A squared odd number minus 1 divided by 8 is the odd number multiplied with its successor divided by 2. (Contributed by AV, 19-Jul-2021.)
((𝑁 ∈ ℤ ∧ 𝑀 = ((2 · 𝑁) + 1)) → (((𝑀↑2) − 1) / 8) = ((𝑁 · (𝑁 + 1)) / 2))
 
Theoremnnsqcld 13968 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → (𝐴↑2) ∈ ℕ)
 
Theoremnnexpcld 13969 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑁) ∈ ℕ)
 
Theoremnn0expcld 13970 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑁) ∈ ℕ0)
 
Theoremrpexpcld 13971 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴𝑁) ∈ ℝ+)
 
Theoremltexp2rd 13972 The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 < 1)       (𝜑 → (𝑀 < 𝑁 ↔ (𝐴𝑁) < (𝐴𝑀)))
 
Theoremreexpclzd 13973 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴𝑁) ∈ ℝ)
 
Theoremresqcld 13974 Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴↑2) ∈ ℝ)
 
Theoremsqge0d 13975 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → 0 ≤ (𝐴↑2))
 
Theoremsqgt0d 13976 The square of a nonzero real is positive. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → 0 < (𝐴↑2))
 
Theoremltexp2d 13977 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑 → 1 < 𝐴)       (𝜑 → (𝑀 < 𝑁 ↔ (𝐴𝑀) < (𝐴𝑁)))
 
Theoremleexp2d 13978 Ordering law for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑 → 1 < 𝐴)       (𝜑 → (𝑀𝑁 ↔ (𝐴𝑀) ≤ (𝐴𝑁)))
 
Theoremexpcand 13979 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑 → 1 < 𝐴)    &   (𝜑 → (𝐴𝑀) = (𝐴𝑁))       (𝜑𝑀 = 𝑁)
 
Theoremleexp2ad 13980 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 ≤ 𝐴)    &   (𝜑𝑁 ∈ (ℤ𝑀))       (𝜑 → (𝐴𝑀) ≤ (𝐴𝑁))
 
Theoremleexp2rd 13981 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 13982 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 13983 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 13984 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)
 
Theoremmulsubdivbinom2 13985 The square of a binomial with factor minus a number divided by a nonzero number. (Contributed by AV, 19-Jul-2021.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((((𝐶 · 𝐴) + 𝐵)↑2) − 𝐷) / 𝐶) = (((𝐶 · (𝐴↑2)) + (2 · (𝐴 · 𝐵))) + (((𝐵↑2) − 𝐷) / 𝐶)))
 
Theoremmuldivbinom2 13986 The square of a binomial with factor divided by a nonzero number. (Contributed by AV, 19-Jul-2021.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((((𝐶 · 𝐴) + 𝐵)↑2) / 𝐶) = (((𝐶 · (𝐴↑2)) + (2 · (𝐴 · 𝐵))) + ((𝐵↑2) / 𝐶)))
 
Theoremsq10 13987 The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
(10↑2) = 100
 
Theoremsq10e99m1 13988 The square of 10 is 99 plus 1. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
(10↑2) = (99 + 1)
 
Theorem3dec 13989 A "decimal constructor" which is used to build up "decimal integers" or "numeric terms" in base 10 with 3 "digits". (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       𝐴𝐵𝐶 = ((((10↑2) · 𝐴) + (10 · 𝐵)) + 𝐶)
 
5.6.8  Ordered pair theorem for nonnegative integers
 
Theoremnn0le2msqi 13990 The square function on nonnegative integers is monotonic. (Contributed by Raph Levien, 10-Dec-2002.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       (𝐴𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵))
 
Theoremnn0opthlem1 13991 A rather pretty lemma for nn0opthi 13993. (Contributed by Raph Levien, 10-Dec-2002.)
𝐴 ∈ ℕ0    &   𝐶 ∈ ℕ0       (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶))
 
Theoremnn0opthlem2 13992 Lemma for nn0opthi 13993. (Contributed by Raph Levien, 10-Dec-2002.) (Revised by Scott Fenton, 8-Sep-2010.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0       ((𝐴 + 𝐵) < 𝐶 → ((𝐶 · 𝐶) + 𝐷) ≠ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵))
 
Theoremnn0opthi 13993 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. We can represent an ordered pair of nonnegative integers 𝐴 and 𝐵 by (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵). If two such ordered pairs are equal, their first elements are equal and their second elements are equal. Contrast this ordered pair representation with the standard one df-op 4569 that works for any set. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Scott Fenton, 8-Sep-2010.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0       ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremnn0opth2i 13994 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthi 13993. (Contributed by NM, 22-Jul-2004.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0       ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremnn0opth2 13995 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 13993. (Contributed by NM, 22-Jul-2004.)
(((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) ∧ (𝐶 ∈ ℕ0𝐷 ∈ ℕ0)) → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 
5.6.9  Factorial function
 
Syntaxcfa 13996 Extend class notation to include the factorial of nonnegative integers.
class !
 
Definitiondf-fac 13997 Define the factorial function on nonnegative integers. For example, (!‘5) = 120 because 1 · 2 · 3 · 4 · 5 = 120 (ex-fac 28824). In the literature, the factorial function is written as a postscript exclamation point. (Contributed by NM, 2-Dec-2004.)
! = ({⟨0, 1⟩} ∪ seq1( · , I ))
 
Theoremfacnn 13998 Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
(𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁))
 
Theoremfac0 13999 The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
(!‘0) = 1
 
Theoremfac1 14000 The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
(!‘1) = 1
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