Home Intuitionistic Logic ExplorerTheorem List (p. 100 of 106) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremleabs 9901 A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.)

Theoremqabsor 9902 The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.)

Theoremqabsord 9903 The absolute value of a rational number is either that number or its negative. (Contributed by Jim Kingdon, 8-Nov-2021.)

Theoremabsre 9904 Absolute value of a real number. (Contributed by NM, 17-Mar-2005.)

Theoremabsresq 9905 Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.)

Theoremabsexp 9906 Absolute value of positive integer exponentiation. (Contributed by NM, 5-Jan-2006.)

Theoremabsexpzap 9907 Absolute value of integer exponentiation. (Contributed by Jim Kingdon, 11-Aug-2021.)
#

Theoremabssq 9908 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 9909 The squares of two reals are equal iff their absolute values are equal. (Contributed by NM, 6-Mar-2009.)

Theoremabsrele 9910 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 9911 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 9912 The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.)

Theoremzabscl 9913 The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremltabs 9914 A number which is less than its absolute value is negative. (Contributed by Jim Kingdon, 12-Aug-2021.)

Theoremabslt 9915 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsle 9916 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabssubap0 9917 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 9918 If the absolute value of a complex number is less than a real, its difference from the real is nonzero. See also abssubap0 9917 which is the same with not equal changed to apart. (Contributed by NM, 2-Nov-2007.)

Theoremabsdiflt 9919 The absolute value of a difference and 'less than' relation. (Contributed by Paul Chapman, 18-Sep-2007.)

Theoremabsdifle 9920 The absolute value of a difference and 'less than or equal to' relation. (Contributed by Paul Chapman, 18-Sep-2007.)

Theoremelicc4abs 9921 Membership in a symmetric closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremlenegsq 9922 Comparison to a nonnegative number based on comparison to squares. (Contributed by NM, 16-Jan-2006.)

Theoremreleabs 9923 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 9924 Reciprocal expressed with a real denominator. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremabsidm 9925 The absolute value function is idempotent. (Contributed by NM, 20-Nov-2004.)

Theoremabsgt0ap 9926 The absolute value of a number apart from zero is positive. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremnnabscl 9927 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 9928 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 9929 Absolute value of a nonnegative difference. (Contributed by NM, 14-Feb-2008.)

Theoremabssuble0 9930 Absolute value of a nonpositive difference. (Contributed by FL, 3-Jan-2008.)

Theoremabstri 9931 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 9932 Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006.)

Theoremabs2dif 9933 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremabs2dif2 9934 Difference of absolute values. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremabs2difabs 9935 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremrecan 9936* Cancellation law involving the real part of a complex number. (Contributed by NM, 12-May-2005.)

Theoremabsf 9937 Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremabs3lem 9938 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)

Theoremfzomaxdiflem 9939 Lemma for fzomaxdif 9940. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^ ..^

Theoremfzomaxdif 9940 A bound on the separation of two points in a half-open range. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^ ..^ ..^

Theoremcau3lem 9941* Lemma for cau3 9942. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 1-May-2014.)

Theoremcau3 9942* 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 9815 and friends.) (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremcau4 9943* Change the base of a Cauchy criterion. (Contributed by Mario Carneiro, 18-Mar-2014.)

Theoremcaubnd2 9944* A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.)

Theoremamgm2 9945 Arithmetic-geometric mean inequality for . (Contributed by Mario Carneiro, 2-Jul-2014.)

Theoremsqrtthi 9946 Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtcli 9947 The square root of a nonnegative real is a real. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtgt0i 9948 The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtmsqi 9949 Square root of square. (Contributed by NM, 2-Aug-1999.)

Theoremsqrtsqi 9950 Square root of square. (Contributed by NM, 11-Aug-1999.)

Theoremsqsqrti 9951 Square of square root. (Contributed by NM, 11-Aug-1999.)

Theoremsqrtge0i 9952 The square root of a nonnegative real is nonnegative. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremabsidi 9953 A nonnegative number is its own absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsnidi 9954 A negative number is the negative of its own absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremleabsi 9955 A real number is less than or equal to its absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsrei 9956 Absolute value of a real number. (Contributed by NM, 3-Aug-1999.)

Theoremsqrtpclii 9957 The square root of a positive real is a real. (Contributed by Mario Carneiro, 6-Sep-2013.)

Theoremsqrtgt0ii 9958 The square root of a positive real is positive. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrt11i 9959 The square root function is one-to-one. (Contributed by NM, 27-Jul-1999.)

Theoremsqrtmuli 9960 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)

Theoremsqrtmulii 9961 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.)

Theoremsqrtmsq2i 9962 Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.)

Theoremsqrtlei 9963 Square root is monotonic. (Contributed by NM, 3-Aug-1999.)

Theoremsqrtlti 9964 Square root is strictly monotonic. (Contributed by Roy F. Longton, 8-Aug-2005.)

Theoremabslti 9965 Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.)

Theoremabslei 9966 Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005.)

Theoremabsvalsqi 9967 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)

Theoremabsvalsq2i 9968 Square of value of absolute value function. (Contributed by NM, 2-Oct-1999.)

Theoremabscli 9969 Real closure of absolute value. (Contributed by NM, 2-Aug-1999.)

Theoremabsge0i 9970 Absolute value is nonnegative. (Contributed by NM, 2-Aug-1999.)

Theoremabsval2i 9971 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremabs00i 9972 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 9973 The absolute value of a nonzero number is positive. Remark in [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)
#

Theoremabsnegi 9974 Absolute value of negative. (Contributed by NM, 2-Aug-1999.)

Theoremabscji 9975 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 9976 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 9977 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by NM, 1-Oct-1999.)

Theoremabsmuli 9978 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 1-Oct-1999.)

Theoremsqabsaddi 9979 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)

Theoremsqabssubi 9980 Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007.)

Theoremabsdivapzi 9981 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremabstrii 9982 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 9983 Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999.)

Theoremabs3lemi 9984 Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)

Theoremrpsqrtcld 9985 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtgt0d 9986 The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsnidd 9987 A negative number is the negative of its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremleabsd 9988 A real number is less than or equal to its absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsred 9989 Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremresqrtcld 9990 The square root of a nonnegative real is a real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtmsqd 9991 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtsqd 9992 Square root of square. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtge0d 9993 The square root of a nonnegative real is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremabsidd 9994 A nonnegative number is its own absolute value. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtdivd 9995 Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtmuld 9996 Square root distributes over multiplication. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtsq2d 9997 Relationship between square root and squares. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtled 9998 Square root is monotonic. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqrtltd 9999 Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremsqr11d 10000 The square root function is one-to-one. (Contributed by Mario Carneiro, 29-May-2016.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10511
 Copyright terms: Public domain < Previous  Next >