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Theorem List for Intuitionistic Logic Explorer - 10801-10900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremresqrexlemnmsq 10801* Lemma for resqrex 10810. The difference between the squares of two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 30-Jul-2021.)

Theoremresqrexlemnm 10802* Lemma for resqrex 10810. The difference between two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 31-Jul-2021.)

Theoremresqrexlemcvg 10803* Lemma for resqrex 10810. The sequence has a limit. (Contributed by Jim Kingdon, 6-Aug-2021.)

Theoremresqrexlemgt0 10804* Lemma for resqrex 10810. A limit is nonnegative. (Contributed by Jim Kingdon, 7-Aug-2021.)

Theoremresqrexlemoverl 10805* Lemma for resqrex 10810. Every term in the sequence is an overestimate compared with the limit . Although this theorem is stated in terms of a particular sequence the proof could be adapted for any decreasing convergent sequence. (Contributed by Jim Kingdon, 9-Aug-2021.)

Theoremresqrexlemglsq 10806* Lemma for resqrex 10810. The sequence formed by squaring each term of converges to . (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)

Theoremresqrexlemga 10807* Lemma for resqrex 10810. The sequence formed by squaring each term of converges to . (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)

Theoremresqrexlemsqa 10808* Lemma for resqrex 10810. The square of a limit is . (Contributed by Jim Kingdon, 7-Aug-2021.)

Theoremresqrexlemex 10809* Lemma for resqrex 10810. Existence of square root given a sequence which converges to the square root. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)

Theoremresqrex 10810* Existence of a square root for positive reals. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremrsqrmo 10811* Uniqueness for the square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)

Theoremrersqreu 10812* Existence and uniqueness for the real square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)

Theoremresqrtcl 10813 Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremrersqrtthlem 10814 Lemma for resqrtth 10815. (Contributed by Jim Kingdon, 10-Aug-2021.)

Theoremresqrtth 10815 Square root theorem over the reals. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremremsqsqrt 10816 Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrtge0 10817 The square root function is nonnegative for nonnegative input. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 9-Jul-2013.)

Theoremsqrtgt0 10818 The square root function is positive for positive input. (Contributed by Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtmul 10819 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrtle 10820 Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremsqrtlt 10821 Square root is strictly monotonic. Closed form of sqrtlti 10921. (Contributed by Scott Fenton, 17-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremsqrt11ap 10822 Analogue to sqrt11 10823 but for apartness. (Contributed by Jim Kingdon, 11-Aug-2021.)
# #

Theoremsqrt11 10823 The square root function is one-to-one. Also see sqrt11ap 10822 which would follow easily from this given excluded middle, but which is proved another way without it. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremsqrt00 10824 A square root is zero iff its argument is 0. (Contributed by NM, 27-Jul-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremrpsqrtcl 10825 The square root of a positive real is a positive real. (Contributed by NM, 22-Feb-2008.)

Theoremsqrtdiv 10826 Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.)

Theoremsqrtsq2 10827 Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrtsq 10828 Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrtmsq 10829 Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrt1 10830 The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)

Theoremsqrt4 10831 The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)

Theoremsqrt9 10832 The square root of 9 is 3. (Contributed by NM, 11-May-2004.)

Theoremsqrt2gt1lt2 10833 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.)

Theoremabsneg 10834 Absolute value of negative. (Contributed by NM, 27-Feb-2005.)

Theoremabscl 10835 Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)

Theoremabscj 10836 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 10837 Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)

Theoremabsvalsq2 10838 Square of value of absolute value function. (Contributed by NM, 1-Feb-2007.)

Theoremsqabsadd 10839 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.)

Theoremsqabssub 10840 Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.)

Theoremabsval2 10841 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.)

Theoremabs0 10842 The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsi 10843 The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)

Theoremabsge0 10844 Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsrpclap 10845 The absolute value of a number apart from zero is a positive real. (Contributed by Jim Kingdon, 11-Aug-2021.)
#

Theoremabs00ap 10846 The absolute value of a number is apart from zero iff the number is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
# #

Theoremabsext 10847 Strong extensionality for absolute value. (Contributed by Jim Kingdon, 12-Aug-2021.)
# #

Theoremabs00 10848 The absolute value of a number is zero iff the number is zero. Also see abs00ap 10846 which is similar but for apartness. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by NM, 26-Sep-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremabs00ad 10849 A complex number is zero iff its absolute value is zero. Deduction form of abs00 10848. (Contributed by David Moews, 28-Feb-2017.)

Theoremabs00bd 10850 If a complex number is zero, its absolute value is zero. (Contributed by David Moews, 28-Feb-2017.)

Theoremabsreimsq 10851 Square of the absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 1-Feb-2007.)

Theoremabsreim 10852 Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006.)

Theoremabsmul 10853 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.)

Theoremabsdivap 10854 Absolute value distributes over division. (Contributed by Jim Kingdon, 11-Aug-2021.)
#

Theoremabsid 10855 A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabs1 10856 The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)

Theoremabsnid 10857 A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.)

Theoremleabs 10858 A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.)

Theoremqabsor 10859 The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.)

Theoremqabsord 10860 The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.)

Theoremabsre 10861 Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)

Theoremabsresq 10862 Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)

Theoremabsexp 10863 Absolute value of positive integer exponentiation. (Contributed by NM, 5-Jan-2006.)

Theoremabsexpzap 10864 Absolute value of integer exponentiation. (Contributed by Jim Kingdon, 11-Aug-2021.)
#

Theoremabssq 10865 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 10866 The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.)

Theoremabsrele 10867 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 10868 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.)

Theoremnn0abscl 10869 The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.)

Theoremzabscl 10870 The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremltabs 10871 A number which is less than its absolute value is negative. (Contributed by Jim Kingdon, 12-Aug-2021.)

Theoremabslt 10872 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsle 10873 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabssubap0 10874 If the absolute value of a complex number is less than a real, its difference from the real is apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
#

Theoremabssubne0 10875 If the absolute value of a complex number is less than a real, its difference from the real is nonzero. See also abssubap0 10874 which is the same with not equal changed to apart. (Contributed by NM, 2-Nov-2007.)

Theoremabsdiflt 10876 The absolute value of a difference and 'less than' relation. (Contributed by Paul Chapman, 18-Sep-2007.)

Theoremabsdifle 10877 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Paul Chapman, 18-Sep-2007.)

Theoremelicc4abs 10878 Membership in a symmetric closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremlenegsq 10879 Comparison to a nonnegative number based on comparison to squares. (Contributed by NM, 16-Jan-2006.)

Theoremreleabs 10880 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.)

Theoremrecvalap 10881 Reciprocal expressed with a real denominator. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremabsidm 10882 The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)

Theoremabsgt0ap 10883 The absolute value of a number apart from zero is positive. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremnnabscl 10884 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 10885 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 10886 Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008.)

Theoremabssuble0 10887 Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)

Theoremabstri 10888 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 10889 Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006.)

Theoremabs2dif 10890 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2dif2 10891 Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremabs2difabs 10892 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremrecan 10893* Cancellation law involving the real part of a complex number. (Contributed by NM, 12-May-2005.)

Theoremabsf 10894 Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremabs3lem 10895 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)

Theoremfzomaxdiflem 10896 Lemma for fzomaxdif 10897. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^ ..^

Theoremfzomaxdif 10897 A bound on the separation of two points in a half-open range. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^ ..^

Theoremcau3lem 10898* Lemma for cau3 10899. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremcau3 10899* 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 10772 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremcau4 10900* Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.)

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