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

Theoremrddif 11701 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 11702 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 11703 Lemma for fzomaxdif 11704. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^ ..^

Theoremfzomaxdif 11704 A bound on the separation of two points in a half-open range. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^ ..^

Theoremuzin2 11705 The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)

Theoremrexanuz 11706* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 25-Dec-2013.)

Theoremrexanre 11707* Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.)

Theoremrexfiuz 11708* Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)

Theoremrexuz3 11709* Rextrict the base of the upper integers set to another upper integers set. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremrexanuz2 11710* Combine two different upper integer properties into one. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremr19.29uz 11711* A version of 19.29 1595 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014.)

Theoremr19.2uz 11712* A version of r19.2z 3449 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremrexuzre 11713* Convert an upper real quantifier to an upper integer quantifier. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremrexico 11714* Rextrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremcau3lem 11715* Lemma for cau3 11716. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremcau3 11716* 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 11706 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremcau4 11717* Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremcaubnd2 11718* A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.)

Theoremcaubnd 11719* A Cauchy sequence of complex numbers is bounded. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 14-Feb-2014.)

Theoremsqreulem 11720 Lemma for sqreu 11721: write a general complex square root in terms of the square root function over nonnegative reals. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqreu 11721* Existence and uniqueness for the square root function in general. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqrcl 11722 Closure of the square root function over the complexes. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrthlem 11723 Lemma for sqrth 11725. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrf 11724 Mapping domain and codomain of the square root function. (Contributed by Mario Carneiro, 13-Sep-2015.)

Theoremsqrth 11725 Square root theorem over the complexes. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrrege0 11726 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 11727 Solve an equation containing a square. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremeqsqrd 11728 A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015.)

Theoremeqsqr2d 11729 A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015.)

Theoremamgm2 11730 Arithmetic-geometric mean inequality for . (Contributed by Mario Carneiro, 2-Jul-2014.)

Theoremsqrthi 11731 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrcli 11732 The square root of a nonnegative real is a real. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrgt0i 11733 The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrmsqi 11734 Square root of square. (Contributed by NM, 2-Aug-1999.)

Theoremsqrsqi 11735 Square root of square. (Contributed by NM, 11-Aug-1999.)

Theoremsqsqri 11736 Square of square root. (Contributed by NM, 11-Aug-1999.)

Theoremsqrge0i 11737 The square root of a nonnegative real is nonnegative. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremabsidi 11738 A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsnidi 11739 A negative number is the negative of its own absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremleabsi 11740 A real number is less than or equal to its absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsori 11741 The absolute value of a real number is either that number or its negative. (Contributed by NM, 30-Sep-1999.)

Theoremabsrei 11742 Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)

Theoremsqrpclii 11743 The square root of a positive real is a real. (Contributed by Mario Carneiro, 6-Sep-2013.)

Theoremsqrgt0ii 11744 The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqr11i 11745 The square root function is one-to-one. (Contributed by NM, 27-Jul-1999.)

Theoremsqrmuli 11746 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)

Theoremsqrmulii 11747 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)

Theoremsqrmsq2i 11748 Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.)

Theoremsqrlei 11749 Square root is monotonic. (Contributed by NM, 3-Aug-1999.)

Theoremsqrlti 11750 Square root is strictly monotonic. (Contributed by Roy F. Longton, 8-Aug-2005.)

Theoremabslti 11751 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.)

Theoremabslei 11752 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.)

Theoremabsvalsqi 11753 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)

Theoremabsvalsq2i 11754 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)

Theoremabscli 11755 Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsge0i 11756 Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)

Theoremabsval2i 11757 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremabs00i 11758 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, 28-Jul-1999.)

Theoremabsgt0i 11759 The absolute value of a nonzero number is positive. Remark in [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)

Theoremabsnegi 11760 Absolute value of negative. (Contributed by NM, 2-Aug-1999.)

Theoremabscji 11761 The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremreleabsi 11762 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, 2-Oct-1999.)

Theoremabssubi 11763 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)

Theoremabsmuli 11764 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 1-Oct-1999.)

Theoremsqabsaddi 11765 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremsqabssubi 11766 Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007.)

Theoremabsdivzi 11767 Absolute value distributes over division. (Contributed by NM, 26-Mar-2005.)

Theoremabstrii 11768 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremabs3difi 11769 Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999.)

Theoremabs3lemi 11770 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)

Theoremrpsqrcld 11771 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrgt0d 11772 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsnidd 11773 A negative number is the negative of its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremleabsd 11774 A real number is less than or equal to its absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsord 11775 The absolute value of a real number is either that number or its negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsred 11776 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremresqrcld 11777 The square root of a nonnegative real is a real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrmsqd 11778 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrsqd 11779 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrge0d 11780 The square root of a nonnegative real is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrnegd 11781 The square root of a negative number. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsidd 11782 A nonnegative number is its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrdivd 11783 Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrmuld 11784 Square root distributes over multiplication. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrsq2d 11785 Relationship between square root and squares. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrled 11786 Square root is monotonic. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrltd 11787 Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqr11d 11788 The square root function is one-to-one. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsltd 11789 Absolute value and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsled 11790 Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssubge0d 11791 Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssuble0d 11792 Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdifltd 11793 The absolute value of a difference and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdifled 11794 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabscld 11795 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrcld 11796 Closure of the square root function over the complexes. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrrege0d 11797 The real part of the square root function is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqsqrd 11798 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremmsqsqrd 11799 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqr00d 11800 A square root is zero iff its argument is 0. (Contributed by Mario Carneiro, 29-May-2016.)

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