P. 84 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 8301-8400 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	mulneg2d 8301	Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	mul2negd 8302	Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	subdid 8303	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	subdird 8304	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	muladdd 8305	Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	mulsubd 8306	Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	mulsubfacd 8307	Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.)


4.3.4 Ordering on reals (cont.)

Theorem	ltadd2 8308	Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	ltadd2i 8309	Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.)


Theorem	ltadd2d 8310	Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	ltadd2dd 8311	Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)


Theorem	ltletrd 8312	Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)


Theorem	ltaddneg 8313	Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
$/\$

Theorem	ltaddnegr 8314	Adding a negative number to another number decreases it. (Contributed by AV, 19-Mar-2021.)
$/\$

Theorem	lelttrdi 8315	If a number is less than another number, and the other number is less than or equal to a third number, the first number is less than the third number. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
$/\$ $/\$

Theorem	gt0ne0 8316	Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$

Theorem	lt0ne0 8317	A number which is less than zero is not zero. See also lt0ap0 8537 which is similar but for apartness. (Contributed by Stefan O'Rear, 13-Sep-2014.)
$/\$

Theorem	ltadd1 8318	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 8319	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 8320	Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)
$/\$ $/\$

Theorem	ltsubadd 8321	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	ltsubadd2 8322	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
$/\$ $/\$

Theorem	lesubadd 8323	'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 8324	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)
$/\$ $/\$

Theorem	ltaddsub 8325	'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
$/\$ $/\$

Theorem	ltaddsub2 8326	'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
$/\$ $/\$

Theorem	leaddsub 8327	'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)
$/\$ $/\$

Theorem	leaddsub2 8328	'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)
$/\$ $/\$

Theorem	suble 8329	Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
$/\$ $/\$

Theorem	lesub 8330	Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$ $/\$

Theorem	ltsub23 8331	'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)
$/\$ $/\$

Theorem	ltsub13 8332	'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)
$/\$ $/\$

Theorem	le2add 8333	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 8334	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 8335	Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)
$/\$ $/\$ $/\$ $/\$

Theorem	leltadd 8336	Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
$/\$ $/\$ $/\$ $/\$

Theorem	addgt0 8337	The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$ $/\$ $/\$

Theorem	addgegt0 8338	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 8339	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 8340	The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$ $/\$ $/\$

Theorem	ltaddpos 8341	Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
$/\$

Theorem	ltaddpos2 8342	Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
$/\$

Theorem	ltsubpos 8343	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 8344	Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
$/\$

Theorem	lesub1 8345	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 8346	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 8347	Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	ltsub2 8348	Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	lt2sub 8349	Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
$/\$ $/\$ $/\$ $/\$

Theorem	le2sub 8350	Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltneg 8351	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 8352	Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
$/\$

Theorem	ltnegcon2 8353	Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)
$/\$

Theorem	leneg 8354	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 8355	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)
$/\$

Theorem	lenegcon2 8356	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)
$/\$

Theorem	lt0neg1 8357	Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)


Theorem	lt0neg2 8358	Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)


Theorem	le0neg1 8359	Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)


Theorem	le0neg2 8360	Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)


Theorem	addge01 8361	A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)
$/\$

Theorem	addge02 8362	A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)
$/\$

Theorem	add20 8363	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 8364	Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$

Theorem	suble0 8365	Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$

Theorem	leaddle0 8366	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 8367	Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
$/\$

Theorem	lesub0 8368	Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	mullt0 8369	The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)
$/\$ $/\$ $/\$

Theorem	0le1 8370	0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)


Theorem	ltordlem 8371*	Lemma for eqord1 8372. (Contributed by Mario Carneiro, 14-Jun-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	eqord1 8372*	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 8373*	A strictly decreasing real function on a subset of is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	leidi 8374	'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)


Theorem	gt0ne0i 8375	Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)


Theorem	gt0ne0ii 8376	Positive implies nonzero. (Contributed by NM, 15-May-1999.)


Theorem	addgt0i 8377	Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	addge0i 8378	Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	addgegt0i 8379	Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
$/\$

Theorem	addgt0ii 8380	Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)


Theorem	add20i 8381	Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)
$/\$ $/\$

Theorem	ltnegi 8382	Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)


Theorem	lenegi 8383	Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)


Theorem	ltnegcon2i 8384	Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)


Theorem	lesub0i 8385	Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	ltaddposi 8386	Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)


Theorem	posdifi 8387	Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)


Theorem	ltnegcon1i 8388	Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)


Theorem	lenegcon1i 8389	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)


Theorem	subge0i 8390	Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)


Theorem	ltadd1i 8391	Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)


Theorem	leadd1i 8392	Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)


Theorem	leadd2i 8393	Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)


Theorem	ltsubaddi 8394	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)


Theorem	lesubaddi 8395	'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 8396	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)


Theorem	lesubadd2i 8397	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)


Theorem	ltaddsubi 8398	'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)


Theorem	lt2addi 8399	Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
$/\$

Theorem	le2addi 8400	Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)
$/\$

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