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Theorem List for Metamath Proof Explorer - 11701-11800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremabsi 11701 The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)

Theoremabsge0 11702 Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsrpcl 11703 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.)

Theoremabs00 11704 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.)

Theoremabs00ad 11705 A complex number is zero iff its absolute value is zero. Deduction form of abs00 11704. (Contributed by David Moews, 28-Feb-2017.)

Theoremabs00bd 11706 If a complex number is zero, its absolute value is zero. Converse of abs00d 11858. One-way deduction form of abs00 11704. (Contributed by David Moews, 28-Feb-2017.)

Theoremabsreimsq 11707 Square of the absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 1-Feb-2007.)

Theoremabsreim 11708 Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006.)

Theoremabsmul 11709 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.)

Theoremabsdiv 11710 Absolute value distributes over division. (Contributed by NM, 27-Apr-2005.)

Theoremabsid 11711 A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabs1 11712 The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)

Theoremabsnid 11713 A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.)

Theoremleabs 11714 A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.)

Theoremabsor 11715 The absolute value of a real number is either that number or its negative. (Contributed by NM, 27-Feb-2005.)

Theoremabsre 11716 Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)

Theoremabsresq 11717 Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)

Theoremabsmod0 11718 is divisible by iff its absolute value is. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremabsexp 11719 Absolute value of natural number exponentiation. (Contributed by NM, 5-Jan-2006.)

Theoremabsexpz 11720 Absolute value of integer exponentiation. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremabssq 11721 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.)

Theoremsqabs 11722 The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.)

Theoremabsrele 11723 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.)

Theoremabsimle 11724 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.)

Theoremmax0add 11725 The sum of the positive and negative part functions is the absolute value function over the reals. (Contributed by Mario Carneiro, 24-Aug-2014.)

Theoremabsz 11726 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.)

Theoremnn0abscl 11727 The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremabslt 11728 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsle 11729 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabssubne0 11730 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.)

Theoremabsdiflt 11731 The absolute value of a difference and 'less than' relation. (Contributed by Paul Chapman, 18-Sep-2007.)

Theoremabsdifle 11732 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Paul Chapman, 18-Sep-2007.)

Theoremelicc4abs 11733 Membership in a symmetric closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremlenegsq 11734 Comparison to a nonnegative number based on comparison to squares. (Contributed by NM, 16-Jan-2006.)

Theoremreleabs 11735 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.)

Theoremrecval 11736 Reciprocal expressed with a real denominator. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremabsidm 11737 The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)

Theoremabsgt0 11738 The absolute value of a nonzero number is positive. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremnnabscl 11739 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.)

Theoremabssub 11740 Swapping order of subtraction doesn't change the absolute value. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremabssubge0 11741 Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008.)

Theoremabssuble0 11742 Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)

Theoremabsmax 11743 The maximum of two numbers using absolute value. (Contributed by NM, 7-Aug-2008.)

Theoremabstri 11744 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.)

Theoremabs3dif 11745 Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006.)

Theoremabs2dif 11746 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2dif2 11747 Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremabs2difabs 11748 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs1m 11749* 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.)

Theoremrecan 11750* Cancellation law involving the real part of a complex number. (Contributed by NM, 12-May-2005.)

Theoremabsf 11751 Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremabs3lem 11752 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)

Theoremabslem2 11753 Lemma involving absolute values. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremrddif 11754 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.)

Theoremabsrdbnd 11755 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.)

Theoremfzomaxdiflem 11756 Lemma for fzomaxdif 11757. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^ ..^

Theoremfzomaxdif 11757 A bound on the separation of two points in a half-open range. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^ ..^

Theoremuzin2 11758 The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)

Theoremrexanuz 11759* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)

Theoremrexanre 11760* Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.)

Theoremrexfiuz 11761* Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)

Theoremrexuz3 11762* Rextrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremrexanuz2 11763* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremr19.29uz 11764* A version of 19.29 1595 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)

Theoremr19.2uz 11765* A version of r19.2z 3485 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremrexuzre 11766* Convert an upper real quantifier to an upper integer quantifier. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremrexico 11767* Rextrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremcau3lem 11768* Lemma for cau3 11769. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremcau3 11769* Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurence of in the assertion, so it can be used with rexanuz 11759 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremcau4 11770* Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremcaubnd2 11771* A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.)

Theoremcaubnd 11772* A Cauchy sequence of complex numbers is bounded. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 14-Feb-2014.)

Theoremsqreulem 11773 Lemma for sqreu 11774: write a general complex square root in terms of the square root function over nonnegative reals. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqreu 11774* Existence and uniqueness for the square root function in general. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqrcl 11775 Closure of the square root function over the complexes. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrthlem 11776 Lemma for sqrth 11778. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrf 11777 Mapping domain and codomain of the square root function. (Contributed by Mario Carneiro, 13-Sep-2015.)

Theoremsqrth 11778 Square root theorem over the complexes. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrrege0 11779 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 is a non-positive 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 complexes (sorting by real part first, then by imaginary part as tie-breaker). (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremeqsqror 11780 Solve an equation containing a square. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremeqsqrd 11781 A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015.)

Theoremeqsqr2d 11782 A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015.)

Theoremamgm2 11783 Arithmetic-geometric mean inequality for . (Contributed by Mario Carneiro, 2-Jul-2014.)

Theoremsqrthi 11784 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrcli 11785 The square root of a nonnegative real is a real. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrgt0i 11786 The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrmsqi 11787 Square root of square. (Contributed by NM, 2-Aug-1999.)

Theoremsqrsqi 11788 Square root of square. (Contributed by NM, 11-Aug-1999.)

Theoremsqsqri 11789 Square of square root. (Contributed by NM, 11-Aug-1999.)

Theoremsqrge0i 11790 The square root of a nonnegative real is nonnegative. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremabsidi 11791 A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsnidi 11792 A negative number is the negative of its own absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremleabsi 11793 A real number is less than or equal to its absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsori 11794 The absolute value of a real number is either that number or its negative. (Contributed by NM, 30-Sep-1999.)

Theoremabsrei 11795 Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)

Theoremsqrpclii 11796 The square root of a positive real is a real. (Contributed by Mario Carneiro, 6-Sep-2013.)

Theoremsqrgt0ii 11797 The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqr11i 11798 The square root function is one-to-one. (Contributed by NM, 27-Jul-1999.)

Theoremsqrmuli 11799 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)

Theoremsqrmulii 11800 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)

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