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Theorem List for Metamath Proof Explorer - 12701-12800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsqlecan 12701 Cancel one factor of a square in a comparison. Unlike lemul1 10624, 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 12702 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴𝐵)))

Theoremsubsq2 12703 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 12704 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 12705 Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴𝐵))

Theoremsqeqori 12706 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 12707 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 12708 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 12709 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 12710 Special case of binom2 12709 where 𝐵 = 1. (Contributed by Scott Fenton, 11-May-2014.)
(𝐴 ∈ ℂ → ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · 𝐴)) + 1))

Theorembinom2sub 12711 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 12712 Special case of binom2sub 12711 where 𝐵 = 1. (Contributed by AV, 2-Aug-2021.)
(𝐴 ∈ ℂ → ((𝐴 − 1)↑2) = (((𝐴↑2) − (2 · 𝐴)) + 1))

Theorembinom2subi 12713 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 12714 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 12715 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 12716 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 12717 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 12718 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 12719 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 12720 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 12721 Variation of Bernoulli's inequality bernneq 12720. (Contributed by NM, 18-Oct-2007.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴𝑁))

Theorembernneq3 12722 A corollary of bernneq 12720. (Contributed by Mario Carneiro, 11-Mar-2014.)
((𝑃 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (𝑃𝑁))

Theoremexpnbnd 12723* 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 12724* 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 12725* 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 12726* 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 12727 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 12728 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 12729 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 12730* 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 12731* 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 12732 Value of a complex number raised to the 0th power. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴↑0) = 1)

Theoremexp1d 12733 Value of a complex number raised to the first power. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴↑1) = 𝐴)

Theoremexpeq0d 12734 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 12735 Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴↑2) = (𝐴 · 𝐴))

Theoremsqcld 12736 Closure of square. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴↑2) ∈ ℂ)

Theoremsqeq0d 12737 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 12738 Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑁) ∈ ℂ)

Theoremexpp1d 12739 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 12740 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 12741 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 12742 Square of reciprocal. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → ((1 / 𝐴)↑2) = (1 / (𝐴↑2)))

Theoremexpclzd 12743 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴𝑁) ∈ ℂ)

Theoremexpne0d 12744 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 12745 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 12746 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 12747 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))

Theoremexpm1d 12748 Value of a complex number raised to an integer power minus one. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴↑(𝑁 − 1)) = ((𝐴𝑁) / 𝐴))

Theoremexpsubd 12749 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐴↑(𝑀𝑁)) = ((𝐴𝑀) / (𝐴𝑁)))

Theoremsqmuld 12750 Distribution of square over multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2)))

Theoremsqdivd 12751 Distribution of square over division. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))

Theoremexpdivd 12752 Nonnegative integer exponentiation of a quotient. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝐴 / 𝐵)↑𝑁) = ((𝐴𝑁) / (𝐵𝑁)))

Theoremmulexpd 12753 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 12754 Value of zero raised to a positive integer power. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → (0↑𝑁) = 0)

Theoremreexpcld 12755 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑁) ∈ ℝ)

Theoremexpge0d 12756 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 12757 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))

Theoremsqoddm1div8 12758 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 12759 The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ)       (𝜑 → (𝐴↑2) ∈ ℕ)

Theoremnnexpcld 12760 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑁) ∈ ℕ)

Theoremnn0expcld 12761 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑁) ∈ ℕ0)

Theoremrpexpcld 12762 Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴𝑁) ∈ ℝ+)

Theoremltexp2rd 12763 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 12764 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐴𝑁) ∈ ℝ)

Theoremresqcld 12765 Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴↑2) ∈ ℝ)

Theoremsqge0d 12766 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → 0 ≤ (𝐴↑2))

Theoremsqgt0d 12767 The square of a nonzero real is positive. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → 0 < (𝐴↑2))

Theoremltexp2d 12768 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 12769 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 12770 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 12771 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 ≤ 𝐴)    &   (𝜑𝑁 ∈ (ℤ𝑀))       (𝜑 → (𝐴𝑀) ≤ (𝐴𝑁))

Theoremleexp2rd 12772 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 12773 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 12774 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 12775 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 12776 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 12777 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 12778 The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
(10↑2) = 100

Theoremsq10e99m1 12779 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 12780 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)

Theoremsq10OLD 12781 Old version of sq10 12778. Obsolete as of 1-Aug-2021. (Contributed by AV, 14-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(10↑2) = 100

Theoremsq10e99m1OLD 12782 Old version of sq10e99m1 12779. Obsolete as of 1-Aug-2021. (Contributed by AV, 14-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(10↑2) = (99 + 1)

Theorem3decOLD 12783 Old version of 3dec 12780. Obsolete as of 1-Aug-2021. (Contributed by AV, 14-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       𝐴𝐵𝐶 = ((((10↑2) · 𝐴) + (10 · 𝐵)) + 𝐶)

5.6.8  Ordered pair theorem for nonnegative integers

Theoremnn0le2msqi 12784 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 12785 A rather pretty lemma for nn0opthi 12787. (Contributed by Raph Levien, 10-Dec-2002.)
𝐴 ∈ ℕ0    &   𝐶 ∈ ℕ0       (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶))

Theoremnn0opthlem2 12786 Lemma for nn0opthi 12787. (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 12787 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 4035 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 12788 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthi 12787. (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 12789 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 12787. (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 12790 Extend class notation to include the factorial of nonnegative integers.
class !

Definitiondf-fac 12791 Define the factorial function on nonnegative integers. For example, (!‘5) = 120 because 1 · 2 · 3 · 4 · 5 = 120 (ex-fac 26438). In the literature, the factorial function is written as a postscript exclamation point. (Contributed by NM, 2-Dec-2004.)
! = ({⟨0, 1⟩} ∪ seq1( · , I ))

Theoremfacnn 12792 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 12793 The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
(!‘0) = 1

Theoremfac1 12794 The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
(!‘1) = 1

Theoremfacp1 12795 The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
(𝑁 ∈ ℕ0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1)))

Theoremfac2 12796 The factorial of 2. (Contributed by NM, 17-Mar-2005.)
(!‘2) = 2

Theoremfac3 12797 The factorial of 3. (Contributed by NM, 17-Mar-2005.)
(!‘3) = 6

Theoremfac4 12798 The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
(!‘4) = 24

Theoremfacnn2 12799 Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)
(𝑁 ∈ ℕ → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁))

Theoremfaccl 12800 Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)
(𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ)

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