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Theorem List for Metamath Proof Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrpsqrcl 12101 The square root of a positive real is a postive real. (Contributed by NM, 22-Feb-2008.)

Theoremsqrdiv 12102 Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.)

Theoremsqrneglem 12103 The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqrneg 12104 The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqrsq2 12105 Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrsq 12106 Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrmsq 12107 Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqr1 12108 The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)

Theoremsqr4 12109 The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)

Theoremsqr9 12110 The square root of 9 is 3. (Contributed by NM, 11-May-2004.)

Theoremsqr2gt1lt2 12111 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.)

Theoremsqrm1 12112 The imaginary unit is the square root of negative 1. A lot of people like to call this the "definition" of , but the definition of df-sqr 12071 has already been crafted with 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 9089 or i2 11512 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.)

Theoremabsneg 12113 Absolute value of negative. (Contributed by NM, 27-Feb-2005.)

Theoremabscl 12114 Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)

Theoremabscj 12115 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.)

Theoremabsvalsq 12116 Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)

Theoremabsvalsq2 12117 Square of value of absolute value function. (Contributed by NM, 1-Feb-2007.)

Theoremsqabsadd 12118 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.)

Theoremsqabssub 12119 Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.)

Theoremabsval2 12120 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.)

Theoremabs0 12121 The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsi 12122 The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)

Theoremabsge0 12123 Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsrpcl 12124 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 12125 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 12126 A complex number is zero iff its absolute value is zero. Deduction form of abs00 12125. (Contributed by David Moews, 28-Feb-2017.)

Theoremabs00bd 12127 If a complex number is zero, its absolute value is zero. Converse of abs00d 12279. One-way deduction form of abs00 12125. (Contributed by David Moews, 28-Feb-2017.)

Theoremabsreimsq 12128 Square of the absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 1-Feb-2007.)

Theoremabsreim 12129 Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006.)

Theoremabsmul 12130 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 12131 Absolute value distributes over division. (Contributed by NM, 27-Apr-2005.)

Theoremabsid 12132 A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabs1 12133 The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)

Theoremabsnid 12134 A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.)

Theoremleabs 12135 A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.)

Theoremabsor 12136 The absolute value of a real number is either that number or its negative. (Contributed by NM, 27-Feb-2005.)

Theoremabsre 12137 Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)

Theoremabsresq 12138 Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)

Theoremabsmod0 12139 is divisible by iff its absolute value is. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremabsexp 12140 Absolute value of natural number exponentiation. (Contributed by NM, 5-Jan-2006.)

Theoremabsexpz 12141 Absolute value of integer exponentiation. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremabssq 12142 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 12143 The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.)

Theoremabsrele 12144 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 12145 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 12146 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 12147 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 12148 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 12149 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsle 12150 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabssubne0 12151 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 12152 The absolute value of a difference and 'less than' relation. (Contributed by Paul Chapman, 18-Sep-2007.)

Theoremabsdifle 12153 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Paul Chapman, 18-Sep-2007.)

Theoremelicc4abs 12154 Membership in a symmetric closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremlenegsq 12155 Comparison to a nonnegative number based on comparison to squares. (Contributed by NM, 16-Jan-2006.)

Theoremreleabs 12156 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 12157 Reciprocal expressed with a real denominator. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremabsidm 12158 The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)

Theoremabsgt0 12159 The absolute value of a nonzero number is positive. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremnnabscl 12160 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 12161 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 12162 Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008.)

Theoremabssuble0 12163 Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)

Theoremabsmax 12164 The maximum of two numbers using absolute value. (Contributed by NM, 7-Aug-2008.)

Theoremabstri 12165 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 12166 Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006.)

Theoremabs2dif 12167 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2dif2 12168 Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremabs2difabs 12169 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs1m 12170* 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 12171* Cancellation law involving the real part of a complex number. (Contributed by NM, 12-May-2005.)

Theoremabsf 12172 Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremabs3lem 12173 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)

Theoremabslem2 12174 Lemma involving absolute values. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremrddif 12175 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 12176 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 12177 Lemma for fzomaxdif 12178. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^ ..^

Theoremfzomaxdif 12178 A bound on the separation of two points in a half-open range. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^ ..^

Theoremuzin2 12179 The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)

Theoremrexanuz 12180* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)

Theoremrexanre 12181* Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.)

Theoremrexfiuz 12182* Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)

Theoremrexuz3 12183* Rextrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremrexanuz2 12184* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremr19.29uz 12185* A version of 19.29 1607 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)

Theoremr19.2uz 12186* A version of r19.2z 3741 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremrexuzre 12187* Convert an upper real quantifier to an upper integer quantifier. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremrexico 12188* Rextrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremcau3lem 12189* Lemma for cau3 12190. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremcau3 12190* 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 12180 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremcau4 12191* Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremcaubnd2 12192* A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.)

Theoremcaubnd 12193* A Cauchy sequence of complex numbers is bounded. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 14-Feb-2014.)

Theoremsqreulem 12194 Lemma for sqreu 12195: write a general complex square root in terms of the square root function over nonnegative reals. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqreu 12195* Existence and uniqueness for the square root function in general. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqrcl 12196 Closure of the square root function over the complexes. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrthlem 12197 Lemma for sqrth 12199. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrf 12198 Mapping domain and codomain of the square root function. (Contributed by Mario Carneiro, 13-Sep-2015.)

Theoremsqrth 12199 Square root theorem over the complexes. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrrege0 12200 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.)

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