P. 88 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 8701-8800 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	gt0ne0 8701	Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$

Theorem	lt0ne0 8702	A number which is less than zero is not zero. See also lt0ap0 8922 which is similar but for apartness. (Contributed by Stefan O'Rear, 13-Sep-2014.)
$/\$

Theorem	ltadd1 8703	Addition to both sides of 'less than'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 12-Nov-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	leadd1 8704	Addition to both sides of 'less than or equal to'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 18-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	leadd2 8705	Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)
$/\$ $/\$

Theorem	ltsubadd 8706	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	ltsubadd2 8707	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
$/\$ $/\$

Theorem	lesubadd 8708	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	lesubadd2 8709	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)
$/\$ $/\$

Theorem	ltaddsub 8710	'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
$/\$ $/\$

Theorem	ltaddsub2 8711	'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
$/\$ $/\$

Theorem	leaddsub 8712	'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)
$/\$ $/\$

Theorem	leaddsub2 8713	'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)
$/\$ $/\$

Theorem	suble 8714	Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
$/\$ $/\$

Theorem	lesub 8715	Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$ $/\$

Theorem	ltsub23 8716	'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)
$/\$ $/\$

Theorem	ltsub13 8717	'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)
$/\$ $/\$

Theorem	le2add 8718	Adding both sides of two 'less than or equal to' relations. (Contributed by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$ $/\$ $/\$

Theorem	lt2add 8719	Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltleadd 8720	Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)
$/\$ $/\$ $/\$ $/\$

Theorem	leltadd 8721	Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
$/\$ $/\$ $/\$ $/\$

Theorem	addgt0 8722	The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$ $/\$ $/\$

Theorem	addgegt0 8723	The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$ $/\$ $/\$

Theorem	addgtge0 8724	The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$ $/\$ $/\$

Theorem	addge0 8725	The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$ $/\$ $/\$

Theorem	ltaddpos 8726	Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
$/\$

Theorem	ltaddpos2 8727	Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
$/\$

Theorem	ltsubpos 8728	Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	posdif 8729	Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
$/\$

Theorem	lesub1 8730	Subtraction from both sides of 'less than or equal to'. (Contributed by NM, 13-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	lesub2 8731	Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	ltsub1 8732	Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	ltsub2 8733	Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	lt2sub 8734	Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
$/\$ $/\$ $/\$ $/\$

Theorem	le2sub 8735	Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltneg 8736	Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 27-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$

Theorem	ltnegcon1 8737	Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
$/\$

Theorem	ltnegcon2 8738	Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)
$/\$

Theorem	leneg 8739	Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$

Theorem	lenegcon1 8740	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)
$/\$

Theorem	lenegcon2 8741	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)
$/\$

Theorem	lt0neg1 8742	Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)


Theorem	lt0neg2 8743	Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)


Theorem	le0neg1 8744	Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)


Theorem	le0neg2 8745	Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)


Theorem	addge01 8746	A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)
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Theorem	addge02 8747	A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)
$/\$

Theorem	add20 8748	Two nonnegative numbers are zero iff their sum is zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$ $/\$ $/\$

Theorem	subge0 8749	Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$

Theorem	suble0 8750	Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$

Theorem	leaddle0 8751	The sum of a real number and a second real number is less then the real number iff the second real number is negative. (Contributed by Alexander van der Vekens, 30-May-2018.)
$/\$

Theorem	subge02 8752	Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
$/\$

Theorem	lesub0 8753	Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	mullt0 8754	The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)
$/\$ $/\$ $/\$

Theorem	0le1 8755	0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)


Theorem	ltordlem 8756*	Lemma for eqord1 8757. (Contributed by Mario Carneiro, 14-Jun-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	eqord1 8757*	A strictly increasing real function on a subset of is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by Jim Kingdon, 20-Dec-2022.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	eqord2 8758*	A strictly decreasing real function on a subset of is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	leidi 8759	'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)


Theorem	gt0ne0i 8760	Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)


Theorem	gt0ne0ii 8761	Positive implies nonzero. (Contributed by NM, 15-May-1999.)


Theorem	addgt0i 8762	Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	addge0i 8763	Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	addgegt0i 8764	Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
$/\$

Theorem	addgt0ii 8765	Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)


Theorem	add20i 8766	Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)
$/\$ $/\$

Theorem	ltnegi 8767	Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)


Theorem	lenegi 8768	Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)


Theorem	ltnegcon2i 8769	Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)


Theorem	lesub0i 8770	Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	ltaddposi 8771	Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)


Theorem	posdifi 8772	Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)


Theorem	ltnegcon1i 8773	Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)


Theorem	lenegcon1i 8774	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)


Theorem	subge0i 8775	Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)


Theorem	ltadd1i 8776	Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)


Theorem	leadd1i 8777	Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)


Theorem	leadd2i 8778	Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)


Theorem	ltsubaddi 8779	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)


Theorem	lesubaddi 8780	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)


Theorem	ltsubadd2i 8781	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)


Theorem	lesubadd2i 8782	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)


Theorem	ltaddsubi 8783	'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)


Theorem	lt2addi 8784	Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
$/\$

Theorem	le2addi 8785	Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)
$/\$

Theorem	gt0ne0d 8786	Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	lt0ne0d 8787	Something less than zero is not zero. Deduction form. See also lt0ap0d 8923 which is similar but for apartness. (Contributed by David Moews, 28-Feb-2017.)


Theorem	leidd 8788	'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	lt0neg1d 8789	Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	lt0neg2d 8790	Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	le0neg1d 8791	Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	le0neg2d 8792	Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	addgegt0d 8793	Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	addgtge0d 8794	Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021.)


Theorem	addgt0d 8795	Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	addge0d 8796	Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	ltnegd 8797	Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	lenegd 8798	Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	ltnegcon1d 8799	Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	ltnegcon2d 8800	Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

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