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Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdiv12apd 7801 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdiv23apd 7802 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdivdirapd 7803 Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdivsubdirapd 7804 Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdiv11apd 7805 One-to-one relationship for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
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Theoremdivmuldivapd 7806 Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.)
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Theoremrerecclapd 7807 Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremredivclapd 7808 Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.)
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Theoremmvllmulapd 7809 Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.)
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3.3.9  Ordering on reals (cont.)

Theoremltp1 7810 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)

Theoremlep1 7811 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)

Theoremltm1 7812 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)

Theoremlem1 7813 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremletrp1 7814 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)

Theoremp1le 7815 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)

Theoremrecgt0 7816 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremprodgt0gt0 7817 Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 7818 for the same theorem with replaced by the weaker condition . (Contributed by Jim Kingdon, 29-Feb-2020.)

Theoremprodgt0 7818 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremprodgt02 7819 Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.)

Theoremprodge0 7820 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremprodge02 7821 Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)

Theoremltmul2 7822 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)

Theoremlemul2 7823 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)

Theoremlemul1a 7824 Multiplication of both sides of 'less than or equal to' by a nonnegative number. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 21-Feb-2005.)

Theoremlemul2a 7825 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremltmul12a 7826 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)

Theoremlemul12b 7827 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)

Theoremlemul12a 7828 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)

Theoremmulgt1 7829 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)

Theoremltmulgt11 7830 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)

Theoremltmulgt12 7831 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)

Theoremlemulge11 7832 Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)

Theoremlemulge12 7833 Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremltdiv1 7834 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 7835 Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)

Theoremgt0div 7836 Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)

Theoremge0div 7837 Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)

Theoremdivgt0 7838 The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)

Theoremdivge0 7839 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)

Theoremltmuldiv 7840 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltmuldiv2 7841 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)

Theoremltdivmul 7842 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)

Theoremledivmul 7843 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)

Theoremltdivmul2 7844 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)

Theoremlt2mul2div 7845 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)

Theoremledivmul2 7846 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)

Theoremlemuldiv 7847 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)

Theoremlemuldiv2 7848 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)

Theoremltrec 7849 The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremlerec 7850 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 7851 Lemma for lt2msq 7852. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2msq 7852 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 7853 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)

Theoremltrec1 7854 Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)

Theoremlerec2 7855 Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)

Theoremledivdiv 7856 Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)

Theoremlediv2 7857 Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)

Theoremltdiv23 7858 Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)

Theoremlediv23 7859 Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)

Theoremlediv12a 7860 Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)

Theoremlediv2a 7861 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremreclt1 7862 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)

Theoremrecgt1 7863 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)

Theoremrecgt1i 7864 The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)

Theoremrecp1lt1 7865 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)

Theoremrecreclt 7866 Given a positive number , construct a new positive number less than both and 1. (Contributed by NM, 28-Dec-2005.)

Theoremle2msq 7867 The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremmsq11 7868 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 7869 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 7870* If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)

Theoremltp1i 7871 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)

Theoremrecgt0i 7872 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)

Theoremrecgt0ii 7873 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)

Theoremprodgt0i 7874 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.)

Theoremprodge0i 7875 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)

Theoremdivgt0i 7876 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremdivge0i 7877 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)

Theoremltreci 7878 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)

Theoremlereci 7879 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)

Theoremlt2msqi 7880 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)

Theoremle2msqi 7881 The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)

Theoremmsq11i 7882 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)

Theoremdivgt0i2i 7883 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremltrecii 7884 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)

Theoremdivgt0ii 7885 The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremltmul1i 7886 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 7887 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)

Theoremltmuldivi 7888 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)

Theoremltmul2i 7889 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 7890 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)

Theoremlemul2i 7891 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)

Theoremltdiv23i 7892 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)

Theoremltdiv23ii 7893 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)

Theoremltmul1ii 7894 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 7895 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)

Theoremltp1d 7896 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlep1d 7897 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltm1d 7898 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlem1d 7899 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrecgt0d 7900 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)

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