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Theorem List for Metamath Proof Explorer - 13701-13800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremresqrtcl 13701 Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ ℝ)
 
Theoremresqrtthlem 13702 Lemma for resqrtth 13703. (Contributed by Mario Carneiro, 9-Jul-2013.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (ℜ‘(√‘𝐴)) ∧ (i · (√‘𝐴)) ∉ ℝ+))
 
Theoremresqrtth 13703 Square root theorem over the reals. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 9-Jul-2013.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√‘𝐴)↑2) = 𝐴)
 
Theoremremsqsqrt 13704 Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√‘𝐴) · (√‘𝐴)) = 𝐴)
 
Theoremsqrtge0 13705 The square root function is nonnegative for nonnegative input. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 9-Jul-2013.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ (√‘𝐴))
 
Theoremsqrtgt0 13706 The square root function is positive for positive input. (Contributed by Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < (√‘𝐴))
 
Theoremsqrtmul 13707 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (√‘(𝐴 · 𝐵)) = ((√‘𝐴) · (√‘𝐵)))
 
Theoremsqrtle 13708 Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴𝐵 ↔ (√‘𝐴) ≤ (√‘𝐵)))
 
Theoremsqrtlt 13709 Square root is strictly monotonic. Closed form of sqrtlti 13836. (Contributed by Scott Fenton, 17-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 < 𝐵 ↔ (√‘𝐴) < (√‘𝐵)))
 
Theoremsqrt11 13710 The square root function is one-to-one. (Contributed by Scott Fenton, 11-Jun-2013.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((√‘𝐴) = (√‘𝐵) ↔ 𝐴 = 𝐵))
 
Theoremsqrt00 13711 A square root is zero iff its argument is 0. (Contributed by NM, 27-Jul-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√‘𝐴) = 0 ↔ 𝐴 = 0))
 
Theoremrpsqrtcl 13712 The square root of a positive real is a positive real. (Contributed by NM, 22-Feb-2008.)
(𝐴 ∈ ℝ+ → (√‘𝐴) ∈ ℝ+)
 
Theoremsqrtdiv 13713 Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (√‘(𝐴 / 𝐵)) = ((√‘𝐴) / (√‘𝐵)))
 
Theoremsqrtneglem 13714 The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (((i · (√‘𝐴))↑2) = -𝐴 ∧ 0 ≤ (ℜ‘(i · (√‘𝐴))) ∧ (i · (i · (√‘𝐴))) ∉ ℝ+))
 
Theoremsqrtneg 13715 The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘-𝐴) = (i · (√‘𝐴)))
 
Theoremsqrtsq2 13716 Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((√‘𝐴) = 𝐵𝐴 = (𝐵↑2)))
 
Theoremsqrtsq 13717 Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘(𝐴↑2)) = 𝐴)
 
Theoremsqrtmsq 13718 Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘(𝐴 · 𝐴)) = 𝐴)
 
Theoremsqrt1 13719 The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)
(√‘1) = 1
 
Theoremsqrt4 13720 The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)
(√‘4) = 2
 
Theoremsqrt9 13721 The square root of 9 is 3. (Contributed by NM, 11-May-2004.)
(√‘9) = 3
 
Theoremsqrt2gt1lt2 13722 The square root of 2 is bounded by 1 and 2. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 6-Sep-2013.)
(1 < (√‘2) ∧ (√‘2) < 2)
 
Theoremsqrtm1 13723 The imaginary unit is the square root of negative 1. A lot of people like to call this the "definition" of i, but the definition of df-sqrt 13682 has already been crafted with i being mentioned explicitly, and in any case it doesn't make too much sense to define a value based on a function evaluated outside its domain. A more appropriate view is to take ax-i2m1 9759 or i2 12695 as the "definition", and simply postulate the existence of a number satisfying this property. This is the approach we take here. (Contributed by Mario Carneiro, 10-Jul-2013.)
i = (√‘-1)
 
Theoremabsneg 13724 Absolute value of negative. (Contributed by NM, 27-Feb-2005.)
(𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴))
 
Theoremabscl 13725 Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)
(𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
 
Theoremabscj 13726 The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 28-Apr-2005.)
(𝐴 ∈ ℂ → (abs‘(∗‘𝐴)) = (abs‘𝐴))
 
Theoremabsvalsq 13727 Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)
(𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
 
Theoremabsvalsq2 13728 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 13729 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 13730 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 13731 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 13732 The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)
(abs‘0) = 0
 
Theoremabsi 13733 The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
(abs‘i) = 1
 
Theoremabsge0 13734 Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)
(𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
 
Theoremabsrpcl 13735 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 13736 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 13737 A complex number is zero iff its absolute value is zero. Deduction form of abs00 13736. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
 
Theoremabs00bd 13738 If a complex number is zero, its absolute value is zero. Converse of abs00d 13892. One-way deduction form of abs00 13736. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 = 0)       (𝜑 → (abs‘𝐴) = 0)
 
Theoremabsreimsq 13739 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 13740 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 13741 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 13742 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 13743 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 13744 The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
(abs‘1) = 1
 
Theoremabsnid 13745 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 13746 A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.)
(𝐴 ∈ ℝ → 𝐴 ≤ (abs‘𝐴))
 
Theoremabsor 13747 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 13748 Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)
(𝐴 ∈ ℝ → (abs‘𝐴) = (√‘(𝐴↑2)))
 
Theoremabsresq 13749 Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)
(𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴↑2))
 
Theoremabsmod0 13750 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 13751 Absolute value of positive integer exponentiation. (Contributed by NM, 5-Jan-2006.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (abs‘(𝐴𝑁)) = ((abs‘𝐴)↑𝑁))
 
Theoremabsexpz 13752 Absolute value of integer exponentiation. (Contributed by Mario Carneiro, 6-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴𝑁)) = ((abs‘𝐴)↑𝑁))
 
Theoremabssq 13753 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 13754 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 13755 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 13756 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 13757 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 13758 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 13759 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 13760 The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)
(𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℤ)
 
Theoremabslt 13761 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 13762 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 13763 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 13764 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 13765 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 13766 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 13767 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 13768 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 13769 Reciprocal expressed with a real denominator. (Contributed by Mario Carneiro, 1-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) = ((∗‘𝐴) / ((abs‘𝐴)↑2)))
 
Theoremabsidm 13770 The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)
(𝐴 ∈ ℂ → (abs‘(abs‘𝐴)) = (abs‘𝐴))
 
Theoremabsgt0 13771 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 13772 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 13773 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 13774 Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (abs‘(𝐵𝐴)) = (𝐵𝐴))
 
Theoremabssuble0 13775 Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (abs‘(𝐴𝐵)) = (𝐵𝐴))
 
Theoremabsmax 13776 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 13777 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 13778 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 13779 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴𝐵)))
 
Theoremabs2dif2 13780 Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))
 
Theoremabs2difabs 13781 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 13782* 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 13783* 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 13784 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 13785 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 13786 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 13787 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 13788 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 13789 Lemma for fzomaxdif 13790. (Contributed by Stefan O'Rear, 6-Sep-2015.)
(((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) ∧ 𝐴𝐵) → (abs‘(𝐵𝐴)) ∈ (0..^(𝐷𝐶)))
 
Theoremfzomaxdif 13790 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 13791 The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)
((𝐴 ∈ ran ℤ𝐵 ∈ ran ℤ) → (𝐴𝐵) ∈ ran ℤ)
 
Theoremrexanuz 13792* 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 13793* 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 13794* 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 13795* 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 13796* 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 13797* A version of 19.29 1770 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)
𝑍 = (ℤ𝑀)       ((∀𝑘𝑍 𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
 
Theoremr19.2uz 13798* A version of r19.2z 3915 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)
𝑍 = (ℤ𝑀)       (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 → ∃𝑘𝑍 𝜑)
 
Theoremrexuzre 13799* 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 13800* 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)))
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