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Theorem List for Metamath Proof Explorer - 14001-14100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremabsvalsq 14001 Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)
(𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))

Theoremabsvalsq2 14002 Square of value of absolute value function. (Contributed by NM, 1-Feb-2007.)
(𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2)))

Theoremsqabsadd 14003 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘(𝐴 + 𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) + (2 · (ℜ‘(𝐴 · (∗‘𝐵))))))

Theoremsqabssub 14004 Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘(𝐴𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) − (2 · (ℜ‘(𝐴 · (∗‘𝐵))))))

Theoremabsval2 14005 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.)
(𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))))

Theoremabs0 14006 The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)
(abs‘0) = 0

Theoremabsi 14007 The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
(abs‘i) = 1

Theoremabsge0 14008 Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)
(𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))

Theoremabsrpcl 14009 The absolute value of a nonzero number is a positive real. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ+)

Theoremabs00 14010 The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by NM, 26-Sep-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
(𝐴 ∈ ℂ → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))

Theoremabs00ad 14011 A complex number is zero iff its absolute value is zero. Deduction form of abs00 14010. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))

Theoremabs00bd 14012 If a complex number is zero, its absolute value is zero. Converse of abs00d 14166. One-way deduction form of abs00 14010. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 = 0)       (𝜑 → (abs‘𝐴) = 0)

Theoremabsreimsq 14013 Square of the absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 1-Feb-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘(𝐴 + (i · 𝐵)))↑2) = ((𝐴↑2) + (𝐵↑2)))

Theoremabsreim 14014 Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(𝐴 + (i · 𝐵))) = (√‘((𝐴↑2) + (𝐵↑2))))

Theoremabsmul 14015 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (abs‘𝐵)))

Theoremabsdiv 14016 Absolute value distributes over division. (Contributed by NM, 27-Apr-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))

Theoremabsid 14017 A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (abs‘𝐴) = 𝐴)

Theoremabs1 14018 The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
(abs‘1) = 1

Theoremabsnid 14019 A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.)
((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → (abs‘𝐴) = -𝐴)

Theoremleabs 14020 A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.)
(𝐴 ∈ ℝ → 𝐴 ≤ (abs‘𝐴))

Theoremabsor 14021 The absolute value of a real number is either that number or its negative. (Contributed by NM, 27-Feb-2005.)
(𝐴 ∈ ℝ → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴))

Theoremabsre 14022 Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)
(𝐴 ∈ ℝ → (abs‘𝐴) = (√‘(𝐴↑2)))

Theoremabsresq 14023 Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)
(𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴↑2))

Theoremabsmod0 14024 𝐴 is divisible by 𝐵 iff its absolute value is. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 mod 𝐵) = 0 ↔ ((abs‘𝐴) mod 𝐵) = 0))

Theoremabsexp 14025 Absolute value of positive integer exponentiation. (Contributed by NM, 5-Jan-2006.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (abs‘(𝐴𝑁)) = ((abs‘𝐴)↑𝑁))

Theoremabsexpz 14026 Absolute value of integer exponentiation. (Contributed by Mario Carneiro, 6-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴𝑁)) = ((abs‘𝐴)↑𝑁))

Theoremabssq 14027 Square can be moved in and out of absolute value. (Contributed by Scott Fenton, 18-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)
(𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (abs‘(𝐴↑2)))

Theoremsqabs 14028 The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴↑2) = (𝐵↑2) ↔ (abs‘𝐴) = (abs‘𝐵)))

Theoremabsrele 14029 The absolute value of a complex number is greater than or equal to the absolute value of its real part. (Contributed by NM, 1-Apr-2005.)
(𝐴 ∈ ℂ → (abs‘(ℜ‘𝐴)) ≤ (abs‘𝐴))

Theoremabsimle 14030 The absolute value of a complex number is greater than or equal to the absolute value of its imaginary part. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
(𝐴 ∈ ℂ → (abs‘(ℑ‘𝐴)) ≤ (abs‘𝐴))

Theoremmax0add 14031 The sum of the positive and negative part functions is the absolute value function over the reals. (Contributed by Mario Carneiro, 24-Aug-2014.)
(𝐴 ∈ ℝ → (if(0 ≤ 𝐴, 𝐴, 0) + if(0 ≤ -𝐴, -𝐴, 0)) = (abs‘𝐴))

Theoremabsz 14032 A real number is an integer iff its absolute value is an integer. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 29-May-2016.)
(𝐴 ∈ ℝ → (𝐴 ∈ ℤ ↔ (abs‘𝐴) ∈ ℤ))

Theoremnn0abscl 14033 The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
(𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℕ0)

Theoremzabscl 14034 The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)
(𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℤ)

Theoremabslt 14035 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) < 𝐵 ↔ (-𝐵 < 𝐴𝐴 < 𝐵)))

Theoremabsle 14036 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘𝐴) ≤ 𝐵 ↔ (-𝐵𝐴𝐴𝐵)))

Theoremabssubne0 14037 If the absolute value of a complex number is less than a real, its difference from the real is nonzero. (Contributed by NM, 2-Nov-2007.) (Proof shortened by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ (abs‘𝐴) < 𝐵) → (𝐵𝐴) ≠ 0)

Theoremabsdiflt 14038 The absolute value of a difference and 'less than' relation. (Contributed by Paul Chapman, 18-Sep-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((abs‘(𝐴𝐵)) < 𝐶 ↔ ((𝐵𝐶) < 𝐴𝐴 < (𝐵 + 𝐶))))

Theoremabsdifle 14039 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Paul Chapman, 18-Sep-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((abs‘(𝐴𝐵)) ≤ 𝐶 ↔ ((𝐵𝐶) ≤ 𝐴𝐴 ≤ (𝐵 + 𝐶))))

Theoremelicc4abs 14040 Membership in a symmetric closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ ((𝐴𝐵)[,](𝐴 + 𝐵)) ↔ (abs‘(𝐶𝐴)) ≤ 𝐵))

Theoremlenegsq 14041 Comparison to a nonnegative number based on comparison to squares. (Contributed by NM, 16-Jan-2006.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) → ((𝐴𝐵 ∧ -𝐴𝐵) ↔ (𝐴↑2) ≤ (𝐵↑2)))

Theoremreleabs 14042 The real part of a number is less than or equal to its absolute value. Proposition 10-3.7(d) of [Gleason] p. 133. (Contributed by NM, 1-Apr-2005.)
(𝐴 ∈ ℂ → (ℜ‘𝐴) ≤ (abs‘𝐴))

Theoremrecval 14043 Reciprocal expressed with a real denominator. (Contributed by Mario Carneiro, 1-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) = ((∗‘𝐴) / ((abs‘𝐴)↑2)))

Theoremabsidm 14044 The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)
(𝐴 ∈ ℂ → (abs‘(abs‘𝐴)) = (abs‘𝐴))

Theoremabsgt0 14045 The absolute value of a nonzero number is positive. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)
(𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ 0 < (abs‘𝐴)))

Theoremnnabscl 14046 The absolute value of a nonzero integer is a positive integer. (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)
((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)

Theoremabssub 14047 Swapping order of subtraction doesn't change the absolute value. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴𝐵)) = (abs‘(𝐵𝐴)))

Theoremabssubge0 14048 Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (abs‘(𝐵𝐴)) = (𝐵𝐴))

Theoremabssuble0 14049 Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (abs‘(𝐴𝐵)) = (𝐵𝐴))

Theoremabsmax 14050 The maximum of two numbers using absolute value. (Contributed by NM, 7-Aug-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴𝐵, 𝐵, 𝐴) = (((𝐴 + 𝐵) + (abs‘(𝐴𝐵))) / 2))

Theoremabstri 14051 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by NM, 7-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 + 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))

Theoremabs3dif 14052 Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (abs‘(𝐴𝐵)) ≤ ((abs‘(𝐴𝐶)) + (abs‘(𝐶𝐵))))

Theoremabs2dif 14053 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴𝐵)))

Theoremabs2dif2 14054 Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))

Theoremabs2difabs 14055 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴𝐵)))

Theoremabs1m 14056* For any complex number, there exists a unit-magnitude multiplier that produces its absolute value. Part of proof of Theorem 13-2.12 of [Gleason] p. 195. (Contributed by NM, 26-Mar-2005.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ ((abs‘𝑥) = 1 ∧ (abs‘𝐴) = (𝑥 · 𝐴)))

Theoremrecan 14057* Cancellation law involving the real part of a complex number. (Contributed by NM, 12-May-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∀𝑥 ∈ ℂ (ℜ‘(𝑥 · 𝐴)) = (ℜ‘(𝑥 · 𝐵)) ↔ 𝐴 = 𝐵))

Theoremabsf 14058 Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro, 7-Nov-2013.)
abs:ℂ⟶ℝ

Theoremabs3lem 14059 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℝ)) → (((abs‘(𝐴𝐶)) < (𝐷 / 2) ∧ (abs‘(𝐶𝐵)) < (𝐷 / 2)) → (abs‘(𝐴𝐵)) < 𝐷))

Theoremabslem2 14060 Lemma involving absolute values. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (((∗‘(𝐴 / (abs‘𝐴))) · 𝐴) + ((𝐴 / (abs‘𝐴)) · (∗‘𝐴))) = (2 · (abs‘𝐴)))

Theoremrddif 14061 The difference between a real number and its nearest integer is less than or equal to one half. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Sep-2015.)
(𝐴 ∈ ℝ → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ≤ (1 / 2))

Theoremabsrdbnd 14062 Bound on the absolute value of a real number rounded to the nearest integer. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Sep-2015.)
(𝐴 ∈ ℝ → (abs‘(⌊‘(𝐴 + (1 / 2)))) ≤ ((⌊‘(abs‘𝐴)) + 1))

Theoremfzomaxdiflem 14063 Lemma for fzomaxdif 14064. (Contributed by Stefan O'Rear, 6-Sep-2015.)
(((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) ∧ 𝐴𝐵) → (abs‘(𝐵𝐴)) ∈ (0..^(𝐷𝐶)))

Theoremfzomaxdif 14064 A bound on the separation of two points in a half-open range. (Contributed by Stefan O'Rear, 6-Sep-2015.)
((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → (abs‘(𝐴𝐵)) ∈ (0..^(𝐷𝐶)))

Theoremuzin2 14065 The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)
((𝐴 ∈ ran ℤ𝐵 ∈ ran ℤ) → (𝐴𝐵) ∈ ran ℤ)

Theoremrexanuz 14066* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)
(∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜓))

Theoremrexanre 14067* Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.)
(𝐴 ⊆ ℝ → (∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘 → (𝜑𝜓)) ↔ (∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜑) ∧ ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜓))))

Theoremrexfiuz 14068* Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)
(𝐴 ∈ Fin → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝐴 𝜑 ↔ ∀𝑛𝐴𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))

Theoremrexuz3 14069* Restrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))

Theoremrexanuz2 14070* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝑀)       (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓))

Theoremr19.29uz 14071* A version of 19.29 1799 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)
𝑍 = (ℤ𝑀)       ((∀𝑘𝑍 𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓))

Theoremr19.2uz 14072* A version of r19.2z 4051 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)
𝑍 = (ℤ𝑀)       (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 → ∃𝑘𝑍 𝜑)

Theoremrexuzre 14073* Convert an upper real quantifier to an upper integer quantifier. (Contributed by Mario Carneiro, 7-May-2016.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℝ ∀𝑘𝑍 (𝑗𝑘𝜑)))

Theoremrexico 14074* Restrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.)
((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∃𝑗 ∈ (𝐵[,)+∞)∀𝑘𝐴 (𝑗𝑘𝜑) ↔ ∃𝑗 ∈ ℝ ∀𝑘𝐴 (𝑗𝑘𝜑)))

Theoremcau3lem 14075* Lemma for cau3 14076. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.)
𝑍 ⊆ ℤ    &   (𝜏𝜓)    &   ((𝐹𝑘) = (𝐹𝑗) → (𝜓𝜒))    &   ((𝐹𝑘) = (𝐹𝑚) → (𝜓𝜃))    &   ((𝜑𝜒𝜓) → (𝐺‘((𝐹𝑗)𝐷(𝐹𝑘))) = (𝐺‘((𝐹𝑘)𝐷(𝐹𝑗))))    &   ((𝜑𝜃𝜒) → (𝐺‘((𝐹𝑚)𝐷(𝐹𝑗))) = (𝐺‘((𝐹𝑗)𝐷(𝐹𝑚))))    &   ((𝜑 ∧ (𝜓𝜃) ∧ (𝜒𝑥 ∈ ℝ)) → (((𝐺‘((𝐹𝑘)𝐷(𝐹𝑗))) < (𝑥 / 2) ∧ (𝐺‘((𝐹𝑗)𝐷(𝐹𝑚))) < (𝑥 / 2)) → (𝐺‘((𝐹𝑘)𝐷(𝐹𝑚))) < 𝑥))       (𝜑 → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜏 ∧ (𝐺‘((𝐹𝑘)𝐷(𝐹𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜏 ∧ ∀𝑚 ∈ (ℤ𝑘)(𝐺‘((𝐹𝑘)𝐷(𝐹𝑚))) < 𝑥)))

Theoremcau3 14076* Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurrence of 𝑗 in the assertion, so it can be used with rexanuz 14066 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)
𝑍 = (ℤ𝑀)       (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ ∀𝑚 ∈ (ℤ𝑘)(abs‘((𝐹𝑘) − (𝐹𝑚))) < 𝑥))

Theoremcau4 14077* Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ𝑁)       (𝑁𝑍 → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑗𝑊𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥)))

Theoremcaubnd2 14078* A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.)
𝑍 = (ℤ𝑀)       (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) → ∃𝑦 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑦)

Theoremcaubnd 14079* A Cauchy sequence of complex numbers is bounded. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 14-Feb-2014.)
𝑍 = (ℤ𝑀)       ((∀𝑘𝑍 (𝐹𝑘) ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)

Theoremsqreulem 14080 Lemma for sqreu 14081: write a general complex square root in terms of the square root function over nonnegative reals. (Contributed by Mario Carneiro, 9-Jul-2013.)
𝐵 = ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))       ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ((𝐵↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝐵) ∧ (i · 𝐵) ∉ ℝ+))

Theoremsqreu 14081* Existence and uniqueness for the square root function in general. (Contributed by Mario Carneiro, 9-Jul-2013.)
(𝐴 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+))

Theoremsqrtcl 14082 Closure of the square root function over the complex numbers. (Contributed by Mario Carneiro, 10-Jul-2013.)
(𝐴 ∈ ℂ → (√‘𝐴) ∈ ℂ)

Theoremsqrtthlem 14083 Lemma for sqrtth 14085. (Contributed by Mario Carneiro, 10-Jul-2013.)
(𝐴 ∈ ℂ → (((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (ℜ‘(√‘𝐴)) ∧ (i · (√‘𝐴)) ∉ ℝ+))

Theoremsqrtf 14084 Mapping domain and codomain of the square root function. (Contributed by Mario Carneiro, 13-Sep-2015.)
√:ℂ⟶ℂ

Theoremsqrtth 14085 Square root theorem over the complex numbers. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 10-Jul-2013.)
(𝐴 ∈ ℂ → ((√‘𝐴)↑2) = 𝐴)

Theoremsqrtrege0 14086 The square root function must make a choice between the two roots, which differ by a sign change. In the general complex case, the choice of "positive" and "negative" is not so clear. The convention we use is to take the root with positive real part, unless 𝐴 is a nonpositive real (in which case both roots have 0 real part); in this case we take the one in the positive imaginary direction. Another way to look at this is that we choose the root that is largest with respect to lexicographic order on the complex numbers (sorting by real part first, then by imaginary part as tie-breaker). (Contributed by Mario Carneiro, 10-Jul-2013.)
(𝐴 ∈ ℂ → 0 ≤ (ℜ‘(√‘𝐴)))

Theoremeqsqrtor 14087 Solve an equation containing a square. (Contributed by Mario Carneiro, 23-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) = 𝐵 ↔ (𝐴 = (√‘𝐵) ∨ 𝐴 = -(√‘𝐵))))

Theoremeqsqrtd 14088 A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴↑2) = 𝐵)    &   (𝜑 → 0 ≤ (ℜ‘𝐴))    &   (𝜑 → ¬ (i · 𝐴) ∈ ℝ+)       (𝜑𝐴 = (√‘𝐵))

Theoremeqsqrt2d 14089 A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴↑2) = 𝐵)    &   (𝜑 → 0 < (ℜ‘𝐴))       (𝜑𝐴 = (√‘𝐵))

Theoremamgm2 14090 Arithmetic-geometric mean inequality for 𝑛 = 2. (Contributed by Mario Carneiro, 2-Jul-2014.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (√‘(𝐴 · 𝐵)) ≤ ((𝐴 + 𝐵) / 2))

Theoremsqrtthi 14091 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
𝐴 ∈ ℝ       (0 ≤ 𝐴 → ((√‘𝐴) · (√‘𝐴)) = 𝐴)

Theoremsqrtcli 14092 The square root of a nonnegative real is a real. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
𝐴 ∈ ℝ       (0 ≤ 𝐴 → (√‘𝐴) ∈ ℝ)

Theoremsqrtgt0i 14093 The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
𝐴 ∈ ℝ       (0 < 𝐴 → 0 < (√‘𝐴))

Theoremsqrtmsqi 14094 Square root of square. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ       (0 ≤ 𝐴 → (√‘(𝐴 · 𝐴)) = 𝐴)

Theoremsqrtsqi 14095 Square root of square. (Contributed by NM, 11-Aug-1999.)
𝐴 ∈ ℝ       (0 ≤ 𝐴 → (√‘(𝐴↑2)) = 𝐴)

Theoremsqsqrti 14096 Square of square root. (Contributed by NM, 11-Aug-1999.)
𝐴 ∈ ℝ       (0 ≤ 𝐴 → ((√‘𝐴)↑2) = 𝐴)

Theoremsqrtge0i 14097 The square root of a nonnegative real is nonnegative. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)
𝐴 ∈ ℝ       (0 ≤ 𝐴 → 0 ≤ (√‘𝐴))

Theoremabsidi 14098 A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ       (0 ≤ 𝐴 → (abs‘𝐴) = 𝐴)

Theoremabsnidi 14099 A negative number is the negative of its own absolute value. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ       (𝐴 ≤ 0 → (abs‘𝐴) = -𝐴)

Theoremleabsi 14100 A real number is less than or equal to its absolute value. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ       𝐴 ≤ (abs‘𝐴)

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