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Theorem List for Intuitionistic Logic Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremabsidm 10901 The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)

Theoremabsgt0ap 10902 The absolute value of a number apart from zero is positive. (Contributed by Jim Kingdon, 13-Aug-2021.)
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Theoremnnabscl 10903 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 10904 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 10905 Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008.)

Theoremabssuble0 10906 Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)

Theoremabstri 10907 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 10908 Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006.)

Theoremabs2dif 10909 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2dif2 10910 Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremabs2difabs 10911 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremrecan 10912* Cancellation law involving the real part of a complex number. (Contributed by NM, 12-May-2005.)

Theoremabsf 10913 Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremabs3lem 10914 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)

Theoremfzomaxdiflem 10915 Lemma for fzomaxdif 10916. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^ ..^

Theoremfzomaxdif 10916 A bound on the separation of two points in a half-open range. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^ ..^

Theoremcau3lem 10917* Lemma for cau3 10918. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremcau3 10918* 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 10791 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremcau4 10919* Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremcaubnd2 10920* A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.)

Theoremamgm2 10921 Arithmetic-geometric mean inequality for . (Contributed by Mario Carneiro, 2-Jul-2014.)

Theoremsqrtthi 10922 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtcli 10923 The square root of a nonnegative real is a real. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtgt0i 10924 The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtmsqi 10925 Square root of square. (Contributed by NM, 2-Aug-1999.)

Theoremsqrtsqi 10926 Square root of square. (Contributed by NM, 11-Aug-1999.)

Theoremsqsqrti 10927 Square of square root. (Contributed by NM, 11-Aug-1999.)

Theoremsqrtge0i 10928 The square root of a nonnegative real is nonnegative. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremabsidi 10929 A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsnidi 10930 A negative number is the negative of its own absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremleabsi 10931 A real number is less than or equal to its absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsrei 10932 Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)

Theoremsqrtpclii 10933 The square root of a positive real is a real. (Contributed by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtgt0ii 10934 The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrt11i 10935 The square root function is one-to-one. (Contributed by NM, 27-Jul-1999.)

Theoremsqrtmuli 10936 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)

Theoremsqrtmulii 10937 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)

Theoremsqrtmsq2i 10938 Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.)

Theoremsqrtlei 10939 Square root is monotonic. (Contributed by NM, 3-Aug-1999.)

Theoremsqrtlti 10940 Square root is strictly monotonic. (Contributed by Roy F. Longton, 8-Aug-2005.)

Theoremabslti 10941 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.)

Theoremabslei 10942 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.)

Theoremabsvalsqi 10943 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)

Theoremabsvalsq2i 10944 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)

Theoremabscli 10945 Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsge0i 10946 Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)

Theoremabsval2i 10947 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremabs00i 10948 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 10949 The absolute value of a nonzero number is positive. Remark in [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)
#

Theoremabsnegi 10950 Absolute value of negative. (Contributed by NM, 2-Aug-1999.)

Theoremabscji 10951 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 10952 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 10953 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)

Theoremabsmuli 10954 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 1-Oct-1999.)

Theoremsqabsaddi 10955 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremsqabssubi 10956 Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007.)

Theoremabsdivapzi 10957 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremabstrii 10958 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 10959 Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999.)

Theoremabs3lemi 10960 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)

Theoremrpsqrtcld 10961 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtgt0d 10962 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsnidd 10963 A negative number is the negative of its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremleabsd 10964 A real number is less than or equal to its absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsred 10965 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremresqrtcld 10966 The square root of a nonnegative real is a real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtmsqd 10967 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtsqd 10968 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtge0d 10969 The square root of a nonnegative real is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsidd 10970 A nonnegative number is its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtdivd 10971 Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtmuld 10972 Square root distributes over multiplication. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtsq2d 10973 Relationship between square root and squares. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtled 10974 Square root is monotonic. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtltd 10975 Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqr11d 10976 The square root function is one-to-one. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsltd 10977 Absolute value and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsled 10978 Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssubge0d 10979 Absolute value of a nonnegative difference. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssuble0d 10980 Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdifltd 10981 The absolute value of a difference and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdifled 10982 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremicodiamlt 10983 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 10984 Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsvalsqd 10985 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsvalsq2d 10986 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsge0d 10987 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsval2d 10988 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs00d 10989 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 10990 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 10991 The absolute value of a complex number apart from zero is a positive real. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremabsnegd 10992 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabscjd 10993 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 10994 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 10995 Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabssubd 10996 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsmuld 10997 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsdivapd 10998 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremabstrid 10999 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabs2difd 11000 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)

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