Home Intuitionistic Logic ExplorerTheorem List (p. 80 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 - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltmul12a 7901 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)

Theoremlemul12b 7902 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)

Theoremlemul12a 7903 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)

Theoremmulgt1 7904 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)

Theoremltmulgt11 7905 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)

Theoremltmulgt12 7906 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)

Theoremlemulge11 7907 Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)

Theoremlemulge12 7908 Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremltdiv1 7909 Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremlediv1 7910 Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)

Theoremgt0div 7911 Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)

Theoremge0div 7912 Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)

Theoremdivgt0 7913 The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)

Theoremdivge0 7914 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)

Theoremltmuldiv 7915 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltmuldiv2 7916 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)

Theoremltdivmul 7917 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)

Theoremledivmul 7918 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)

Theoremltdivmul2 7919 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)

Theoremlt2mul2div 7920 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)

Theoremledivmul2 7921 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)

Theoremlemuldiv 7922 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)

Theoremlemuldiv2 7923 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)

Theoremltrec 7924 The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremlerec 7925 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlt2msq1 7926 Lemma for lt2msq 7927. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2msq 7927 Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremltdiv2 7928 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)

Theoremltrec1 7929 Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)

Theoremlerec2 7930 Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)

Theoremledivdiv 7931 Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)

Theoremlediv2 7932 Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)

Theoremltdiv23 7933 Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)

Theoremlediv23 7934 Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)

Theoremlediv12a 7935 Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)

Theoremlediv2a 7936 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremreclt1 7937 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)

Theoremrecgt1 7938 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)

Theoremrecgt1i 7939 The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)

Theoremrecp1lt1 7940 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)

Theoremrecreclt 7941 Given a positive number , construct a new positive number less than both and 1. (Contributed by NM, 28-Dec-2005.)

Theoremle2msq 7942 The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremmsq11 7943 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremledivp1 7944 Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.)

Theoremsqueeze0 7945* If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)

Theoremltp1i 7946 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)

Theoremrecgt0i 7947 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)

Theoremrecgt0ii 7948 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)

Theoremprodgt0i 7949 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.)

Theoremprodge0i 7950 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)

Theoremdivgt0i 7951 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremdivge0i 7952 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)

Theoremltreci 7953 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)

Theoremlereci 7954 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)

Theoremlt2msqi 7955 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)

Theoremle2msqi 7956 The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)

Theoremmsq11i 7957 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)

Theoremdivgt0i2i 7958 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremltrecii 7959 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)

Theoremdivgt0ii 7960 The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremltmul1i 7961 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)

Theoremltdiv1i 7962 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)

Theoremltmuldivi 7963 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)

Theoremltmul2i 7964 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)

Theoremlemul1i 7965 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)

Theoremlemul2i 7966 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)

Theoremltdiv23i 7967 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)

Theoremltdiv23ii 7968 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)

Theoremltmul1ii 7969 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.)

Theoremltdiv1ii 7970 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)

Theoremltp1d 7971 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlep1d 7972 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltm1d 7973 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlem1d 7974 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrecgt0d 7975 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremdivgt0d 7976 The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremmulgt1d 7977 The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemulge11d 7978 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemulge12d 7979 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul1ad 7980 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul2ad 7981 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul12ad 7982 Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul12ad 7983 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul12bd 7984 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremmulle0r 7985 Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.)

3.3.10  Imaginary and complex number properties

Theoremcrap0 7986 The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.)
# # #

Theoremcreur 7987* The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremcreui 7988* The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremcju 7989* The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)

3.4  Integer sets

3.4.1  Positive integers (as a subset of complex numbers)

Syntaxcn 7990 Extend class notation to include the class of positive integers.

Definitiondf-inn 7991* Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 7992 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)

Theoremdfnn2 7992* Definition of the set of positive integers. Another name for df-inn 7991. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)

Theorempeano5nni 7993* Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnssre 7994 The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)

Theoremnnsscn 7995 The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremnnex 7996 The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnre 7997 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)

Theoremnncn 7998 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)

Theoremnnrei 7999 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)

Theoremnncni 8000 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)

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 >