Theorem List for Intuitionistic Logic Explorer - 8801-8900 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | mvllmulapd 8801 |
Move LHS left multiplication to RHS. (Contributed by Jim Kingdon,
10-Jun-2020.)
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Theorem | rerecapb 8802* |
A real number has a multiplicative inverse if and only if it is apart
from zero. Theorem 11.2.4 of [HoTT], p.
(varies). (Contributed by Jim
Kingdon, 18-Jan-2025.)
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4.3.9 Ordering on reals (cont.)
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Theorem | ltp1 8803 |
A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
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Theorem | lep1 8804 |
A number is less than or equal to itself plus 1. (Contributed by NM,
5-Jan-2006.)
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Theorem | ltm1 8805 |
A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
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Theorem | lem1 8806 |
A number minus 1 is less than or equal to itself. (Contributed by Mario
Carneiro, 2-Oct-2015.)
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Theorem | letrp1 8807 |
A transitive property of 'less than or equal' and plus 1. (Contributed by
NM, 5-Aug-2005.)
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Theorem | p1le 8808 |
A transitive property of plus 1 and 'less than or equal'. (Contributed by
NM, 16-Aug-2005.)
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Theorem | recgt0 8809 |
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.)
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Theorem | prodgt0gt0 8810 |
Infer that a multiplicand is positive from a positive multiplier and
positive product. See prodgt0 8811 for the same theorem with
replaced by the weaker condition
. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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Theorem | prodgt0 8811 |
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.)
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Theorem | prodgt02 8812 |
Infer that a multiplier is positive from a nonnegative multiplicand and
positive product. (Contributed by NM, 24-Apr-2005.)
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Theorem | prodge0 8813 |
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.)
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Theorem | prodge02 8814 |
Infer that a multiplier is nonnegative from a positive multiplicand and
nonnegative product. (Contributed by NM, 2-Jul-2005.)
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Theorem | ltmul2 8815 |
Multiplication of both sides of 'less than' by a positive number. Theorem
I.19 of [Apostol] p. 20. (Contributed by
NM, 13-Feb-2005.)
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Theorem | lemul2 8816 |
Multiplication of both sides of 'less than or equal to' by a positive
number. (Contributed by NM, 16-Mar-2005.)
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Theorem | lemul1a 8817 |
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.)
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Theorem | lemul2a 8818 |
Multiplication of both sides of 'less than or equal to' by a nonnegative
number. (Contributed by Paul Chapman, 7-Sep-2007.)
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Theorem | ltmul12a 8819 |
Comparison of product of two positive numbers. (Contributed by NM,
30-Dec-2005.)
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Theorem | lemul12b 8820 |
Comparison of product of two nonnegative numbers. (Contributed by NM,
22-Feb-2008.)
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Theorem | lemul12a 8821 |
Comparison of product of two nonnegative numbers. (Contributed by NM,
22-Feb-2008.)
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Theorem | mulgt1 8822 |
The product of two numbers greater than 1 is greater than 1. (Contributed
by NM, 13-Feb-2005.)
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Theorem | ltmulgt11 8823 |
Multiplication by a number greater than 1. (Contributed by NM,
24-Dec-2005.)
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Theorem | ltmulgt12 8824 |
Multiplication by a number greater than 1. (Contributed by NM,
24-Dec-2005.)
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Theorem | lemulge11 8825 |
Multiplication by a number greater than or equal to 1. (Contributed by
NM, 17-Dec-2005.)
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Theorem | lemulge12 8826 |
Multiplication by a number greater than or equal to 1. (Contributed by
Paul Chapman, 21-Mar-2011.)
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Theorem | ltdiv1 8827 |
Division of both sides of 'less than' by a positive number. (Contributed
by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
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Theorem | lediv1 8828 |
Division of both sides of a less than or equal to relation by a positive
number. (Contributed by NM, 18-Nov-2004.)
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Theorem | gt0div 8829 |
Division of a positive number by a positive number. (Contributed by NM,
28-Sep-2005.)
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Theorem | ge0div 8830 |
Division of a nonnegative number by a positive number. (Contributed by
NM, 28-Sep-2005.)
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Theorem | divgt0 8831 |
The ratio of two positive numbers is positive. (Contributed by NM,
12-Oct-1999.)
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Theorem | divge0 8832 |
The ratio of nonnegative and positive numbers is nonnegative.
(Contributed by NM, 27-Sep-1999.)
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Theorem | ltmuldiv 8833 |
'Less than' relationship between division and multiplication.
(Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro,
27-May-2016.)
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Theorem | ltmuldiv2 8834 |
'Less than' relationship between division and multiplication.
(Contributed by NM, 18-Nov-2004.)
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Theorem | ltdivmul 8835 |
'Less than' relationship between division and multiplication.
(Contributed by NM, 18-Nov-2004.)
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Theorem | ledivmul 8836 |
'Less than or equal to' relationship between division and multiplication.
(Contributed by NM, 9-Dec-2005.)
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Theorem | ltdivmul2 8837 |
'Less than' relationship between division and multiplication.
(Contributed by NM, 24-Feb-2005.)
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Theorem | lt2mul2div 8838 |
'Less than' relationship between division and multiplication.
(Contributed by NM, 8-Jan-2006.)
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Theorem | ledivmul2 8839 |
'Less than or equal to' relationship between division and multiplication.
(Contributed by NM, 9-Dec-2005.)
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Theorem | lemuldiv 8840 |
'Less than or equal' relationship between division and multiplication.
(Contributed by NM, 10-Mar-2006.)
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Theorem | lemuldiv2 8841 |
'Less than or equal' relationship between division and multiplication.
(Contributed by NM, 10-Mar-2006.)
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Theorem | ltrec 8842 |
The reciprocal of both sides of 'less than'. (Contributed by NM,
26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
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Theorem | lerec 8843 |
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.)
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Theorem | lt2msq1 8844 |
Lemma for lt2msq 8845. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | lt2msq 8845 |
Two nonnegative numbers compare the same as their squares. (Contributed
by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro,
27-May-2016.)
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Theorem | ltdiv2 8846 |
Division of a positive number by both sides of 'less than'. (Contributed
by NM, 27-Apr-2005.)
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Theorem | ltrec1 8847 |
Reciprocal swap in a 'less than' relation. (Contributed by NM,
24-Feb-2005.)
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Theorem | lerec2 8848 |
Reciprocal swap in a 'less than or equal to' relation. (Contributed by
NM, 24-Feb-2005.)
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Theorem | ledivdiv 8849 |
Invert ratios of positive numbers and swap their ordering. (Contributed
by NM, 9-Jan-2006.)
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Theorem | lediv2 8850 |
Division of a positive number by both sides of 'less than or equal to'.
(Contributed by NM, 10-Jan-2006.)
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Theorem | ltdiv23 8851 |
Swap denominator with other side of 'less than'. (Contributed by NM,
3-Oct-1999.)
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Theorem | lediv23 8852 |
Swap denominator with other side of 'less than or equal to'. (Contributed
by NM, 30-May-2005.)
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Theorem | lediv12a 8853 |
Comparison of ratio of two nonnegative numbers. (Contributed by NM,
31-Dec-2005.)
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Theorem | lediv2a 8854 |
Division of both sides of 'less than or equal to' into a nonnegative
number. (Contributed by Paul Chapman, 7-Sep-2007.)
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Theorem | reclt1 8855 |
The reciprocal of a positive number less than 1 is greater than 1.
(Contributed by NM, 23-Feb-2005.)
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Theorem | recgt1 8856 |
The reciprocal of a positive number greater than 1 is less than 1.
(Contributed by NM, 28-Dec-2005.)
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Theorem | recgt1i 8857 |
The reciprocal of a number greater than 1 is positive and less than 1.
(Contributed by NM, 23-Feb-2005.)
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Theorem | recp1lt1 8858 |
Construct a number less than 1 from any nonnegative number. (Contributed
by NM, 30-Dec-2005.)
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Theorem | recreclt 8859 |
Given a positive number , construct a new positive number less than
both and 1.
(Contributed by NM, 28-Dec-2005.)
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Theorem | le2msq 8860 |
The square function on nonnegative reals is monotonic. (Contributed by
NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
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Theorem | msq11 8861 |
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.)
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Theorem | ledivp1 8862 |
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.)
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Theorem | squeeze0 8863* |
If a nonnegative number is less than any positive number, it is zero.
(Contributed by NM, 11-Feb-2006.)
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Theorem | ltp1i 8864 |
A number is less than itself plus 1. (Contributed by NM,
20-Aug-2001.)
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Theorem | recgt0i 8865 |
The reciprocal of a positive number is positive. Exercise 4 of
[Apostol] p. 21. (Contributed by NM,
15-May-1999.)
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Theorem | recgt0ii 8866 |
The reciprocal of a positive number is positive. Exercise 4 of
[Apostol] p. 21. (Contributed by NM,
15-May-1999.)
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Theorem | prodgt0i 8867 |
Infer that a multiplicand is positive from a nonnegative multiplier and
positive product. (Contributed by NM, 15-May-1999.)
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Theorem | prodge0i 8868 |
Infer that a multiplicand is nonnegative from a positive multiplier and
nonnegative product. (Contributed by NM, 2-Jul-2005.)
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Theorem | divgt0i 8869 |
The ratio of two positive numbers is positive. (Contributed by NM,
16-May-1999.)
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Theorem | divge0i 8870 |
The ratio of nonnegative and positive numbers is nonnegative.
(Contributed by NM, 12-Aug-1999.)
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Theorem | ltreci 8871 |
The reciprocal of both sides of 'less than'. (Contributed by NM,
15-Sep-1999.)
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Theorem | lereci 8872 |
The reciprocal of both sides of 'less than or equal to'. (Contributed
by NM, 16-Sep-1999.)
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Theorem | lt2msqi 8873 |
The square function on nonnegative reals is strictly monotonic.
(Contributed by NM, 3-Aug-1999.)
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Theorem | le2msqi 8874 |
The square function on nonnegative reals is monotonic. (Contributed by
NM, 2-Aug-1999.)
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Theorem | msq11i 8875 |
The square of a nonnegative number is a one-to-one function.
(Contributed by NM, 29-Jul-1999.)
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Theorem | divgt0i2i 8876 |
The ratio of two positive numbers is positive. (Contributed by NM,
16-May-1999.)
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Theorem | ltrecii 8877 |
The reciprocal of both sides of 'less than'. (Contributed by NM,
15-Sep-1999.)
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Theorem | divgt0ii 8878 |
The ratio of two positive numbers is positive. (Contributed by NM,
18-May-1999.)
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Theorem | ltmul1i 8879 |
Multiplication of both sides of 'less than' by a positive number.
Theorem I.19 of [Apostol] p. 20.
(Contributed by NM, 16-May-1999.)
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Theorem | ltdiv1i 8880 |
Division of both sides of 'less than' by a positive number.
(Contributed by NM, 16-May-1999.)
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Theorem | ltmuldivi 8881 |
'Less than' relationship between division and multiplication.
(Contributed by NM, 12-Oct-1999.)
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Theorem | ltmul2i 8882 |
Multiplication of both sides of 'less than' by a positive number.
Theorem I.19 of [Apostol] p. 20.
(Contributed by NM, 16-May-1999.)
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Theorem | lemul1i 8883 |
Multiplication of both sides of 'less than or equal to' by a positive
number. (Contributed by NM, 2-Aug-1999.)
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Theorem | lemul2i 8884 |
Multiplication of both sides of 'less than or equal to' by a positive
number. (Contributed by NM, 1-Aug-1999.)
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Theorem | ltdiv23i 8885 |
Swap denominator with other side of 'less than'. (Contributed by NM,
26-Sep-1999.)
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Theorem | ltdiv23ii 8886 |
Swap denominator with other side of 'less than'. (Contributed by NM,
26-Sep-1999.)
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Theorem | ltmul1ii 8887 |
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.)
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Theorem | ltdiv1ii 8888 |
Division of both sides of 'less than' by a positive number.
(Contributed by NM, 16-May-1999.)
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Theorem | ltp1d 8889 |
A number is less than itself plus 1. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | lep1d 8890 |
A number is less than or equal to itself plus 1. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | ltm1d 8891 |
A number minus 1 is less than itself. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | lem1d 8892 |
A number minus 1 is less than or equal to itself. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | recgt0d 8893 |
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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Theorem | divgt0d 8894 |
The ratio of two positive numbers is positive. (Contributed by Mario
Carneiro, 28-May-2016.)
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Theorem | mulgt1d 8895 |
The product of two numbers greater than 1 is greater than 1.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | lemulge11d 8896 |
Multiplication by a number greater than or equal to 1. (Contributed
by Mario Carneiro, 28-May-2016.)
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Theorem | lemulge12d 8897 |
Multiplication by a number greater than or equal to 1. (Contributed
by Mario Carneiro, 28-May-2016.)
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Theorem | lemul1ad 8898 |
Multiplication of both sides of 'less than or equal to' by a
nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | lemul2ad 8899 |
Multiplication of both sides of 'less than or equal to' by a
nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | ltmul12ad 8900 |
Comparison of product of two positive numbers. (Contributed by Mario
Carneiro, 28-May-2016.)
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