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Theorem List for Intuitionistic Logic Explorer - 9701-9800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremabs3dif 9701 Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006.)

Theoremabs2dif 9702 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2dif2 9703 Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremabs2difabs 9704 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremrecan 9705* Cancellation law involving the real part of a complex number. (Contributed by NM, 12-May-2005.)

Theoremabsf 9706 Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremabs3lem 9707 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)

Theoremfzomaxdiflem 9708 Lemma for fzomaxdif 9709. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^ ..^

Theoremfzomaxdif 9709 A bound on the separation of two points in a half-open range. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^ ..^

Theoremcau3lem 9710* Lemma for cau3 9711. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremcau3 9711* Convert between three-quantifier and four-quantifier versions of the Cauchy criterion. (In particular, the four-quantifier version has no occurrence of in the assertion, so it can be used with rexanuz 9587 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremcau4 9712* Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremcaubnd2 9713* A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.)

Theoremamgm2 9714 Arithmetic-geometric mean inequality for . (Contributed by Mario Carneiro, 2-Jul-2014.)

Theoremsqrtthi 9715 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtcli 9716 The square root of a nonnegative real is a real. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtgt0i 9717 The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtmsqi 9718 Square root of square. (Contributed by NM, 2-Aug-1999.)

Theoremsqrtsqi 9719 Square root of square. (Contributed by NM, 11-Aug-1999.)

Theoremsqsqrti 9720 Square of square root. (Contributed by NM, 11-Aug-1999.)

Theoremsqrtge0i 9721 The square root of a nonnegative real is nonnegative. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremabsidi 9722 A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsnidi 9723 A negative number is the negative of its own absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremleabsi 9724 A real number is less than or equal to its absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsrei 9725 Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)

Theoremsqrtpclii 9726 The square root of a positive real is a real. (Contributed by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtgt0ii 9727 The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrt11i 9728 The square root function is one-to-one. (Contributed by NM, 27-Jul-1999.)

Theoremsqrtmuli 9729 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)

Theoremsqrtmulii 9730 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)

Theoremsqrtmsq2i 9731 Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.)

Theoremsqrtlei 9732 Square root is monotonic. (Contributed by NM, 3-Aug-1999.)

Theoremsqrtlti 9733 Square root is strictly monotonic. (Contributed by Roy F. Longton, 8-Aug-2005.)

Theoremabslti 9734 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.)

Theoremabslei 9735 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.)

Theoremabsvalsqi 9736 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)

Theoremabsvalsq2i 9737 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)

Theoremabscli 9738 Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsge0i 9739 Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)

Theoremabsval2i 9740 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremabs00i 9741 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.)

Theoremabsgt0api 9742 The absolute value of a nonzero number is positive. Remark in [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)
#

Theoremabsnegi 9743 Absolute value of negative. (Contributed by NM, 2-Aug-1999.)

Theoremabscji 9744 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 9745 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 9746 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)

Theoremabsmuli 9747 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 1-Oct-1999.)

Theoremsqabsaddi 9748 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremsqabssubi 9749 Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007.)

Theoremabsdivapzi 9750 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremabstrii 9751 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. This is Metamath 100 proof #91. (Contributed by NM, 2-Oct-1999.)

Theoremabs3difi 9752 Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999.)

Theoremabs3lemi 9753 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)

Theoremrpsqrtcld 9754 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtgt0d 9755 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsnidd 9756 A negative number is the negative of its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremleabsd 9757 A real number is less than or equal to its absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsred 9758 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremresqrtcld 9759 The square root of a nonnegative real is a real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtmsqd 9760 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtsqd 9761 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtge0d 9762 The square root of a nonnegative real is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsidd 9763 A nonnegative number is its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtdivd 9764 Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtmuld 9765 Square root distributes over multiplication. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtsq2d 9766 Relationship between square root and squares. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtled 9767 Square root is monotonic. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtltd 9768 Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqr11d 9769 The square root function is one-to-one. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsltd 9770 Absolute value and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsled 9771 Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssubge0d 9772 Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssuble0d 9773 Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdifltd 9774 The absolute value of a difference and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdifled 9775 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremicodiamlt 9776 Two elements in a half-open interval have separation strictly less than the difference between the endpoints. (Contributed by Stefan O'Rear, 12-Sep-2014.)

Theoremabscld 9777 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsvalsqd 9778 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsvalsq2d 9779 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsge0d 9780 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsval2d 9781 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs00d 9782 The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsne0d 9783 The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsrpclapd 9784 The absolute value of a complex number apart from zero is a positive real. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremabsnegd 9785 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabscjd 9786 The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreleabsd 9787 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 Mario Carneiro, 29-May-2016.)

Theoremabsexpd 9788 Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssubd 9789 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsmuld 9790 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdivapd 9791 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremabstrid 9792 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs2difd 9793 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs2dif2d 9794 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs2difabsd 9795 Absolute value of difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs3difd 9796 Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs3lemd 9797 Lemma involving absolute value of differences. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremqdenre 9798* The rational numbers are dense in : any real number can be approximated with arbitrary precision by a rational number. For order theoretic density, see qbtwnre 9111. (Contributed by BJ, 15-Oct-2021.)

3.8  Elementary limits and convergence

3.8.1  Limits

Syntaxcli 9799 Extend class notation with convergence relation for limits.

Definitiondf-clim 9800* Define the limit relation for complex number sequences. See clim 9802 for its relational expression. (Contributed by NM, 28-Aug-2005.)

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