P. 76 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 7501-7600 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	mulneg12 7501	Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)
$/\$

Theorem	mul2neg 7502	Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 30-Jul-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	submul2 7503	Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.)
$/\$ $/\$

Theorem	mulm1 7504	Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)


Theorem	mulsub 7505	Product of two differences. (Contributed by NM, 14-Jan-2006.)
$/\$ $/\$ $/\$

Theorem	mulsub2 7506	Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)
$/\$ $/\$ $/\$

Theorem	mulm1i 7507	Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)


Theorem	mulneg1i 7508	Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 10-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)


Theorem	mulneg2i 7509	Product with negative is negative of product. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)


Theorem	mul2negi 7510	Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)


Theorem	subdii 7511	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.)


Theorem	subdiri 7512	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 8-May-1999.)


Theorem	muladdi 7513	Product of two sums. (Contributed by NM, 17-May-1999.)


Theorem	mulm1d 7514	Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	mulneg1d 7515	Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	mulneg2d 7516	Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	mul2negd 7517	Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	subdid 7518	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	subdird 7519	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	muladdd 7520	Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	mulsubd 7521	Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	mulsubfacd 7522	Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.)


3.3.4 Ordering on reals (cont.)

Theorem	ltadd2 7523	Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	ltadd2i 7524	Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.)


Theorem	ltadd2d 7525	Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)


Theorem	ltadd2dd 7526	Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)


Theorem	ltletrd 7527	Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)


Theorem	ltaddneg 7528	Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
$/\$

Theorem	ltaddnegr 7529	Adding a negative number to another number decreases it. (Contributed by AV, 19-Mar-2021.)
$/\$

Theorem	lelttrdi 7530	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 7531	Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$

Theorem	lt0ne0 7532	A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.)
$/\$

Theorem	ltadd1 7533	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 7534	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 7535	Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)
$/\$ $/\$

Theorem	ltsubadd 7536	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	ltsubadd2 7537	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
$/\$ $/\$

Theorem	lesubadd 7538	'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 7539	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)
$/\$ $/\$

Theorem	ltaddsub 7540	'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
$/\$ $/\$

Theorem	ltaddsub2 7541	'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
$/\$ $/\$

Theorem	leaddsub 7542	'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)
$/\$ $/\$

Theorem	leaddsub2 7543	'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)
$/\$ $/\$

Theorem	suble 7544	Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
$/\$ $/\$

Theorem	lesub 7545	Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$ $/\$

Theorem	ltsub23 7546	'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)
$/\$ $/\$

Theorem	ltsub13 7547	'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)
$/\$ $/\$

Theorem	le2add 7548	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 7549	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 7550	Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)
$/\$ $/\$ $/\$ $/\$

Theorem	leltadd 7551	Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
$/\$ $/\$ $/\$ $/\$

Theorem	addgt0 7552	The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$ $/\$ $/\$

Theorem	addgegt0 7553	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 7554	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 7555	The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$ $/\$ $/\$

Theorem	ltaddpos 7556	Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
$/\$

Theorem	ltaddpos2 7557	Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
$/\$

Theorem	ltsubpos 7558	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 7559	Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
$/\$

Theorem	lesub1 7560	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 7561	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 7562	Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	ltsub2 7563	Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	lt2sub 7564	Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
$/\$ $/\$ $/\$ $/\$

Theorem	le2sub 7565	Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
$/\$ $/\$ $/\$ $/\$

Theorem	ltneg 7566	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 7567	Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
$/\$

Theorem	ltnegcon2 7568	Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)
$/\$

Theorem	leneg 7569	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 7570	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)
$/\$

Theorem	lenegcon2 7571	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)
$/\$

Theorem	lt0neg1 7572	Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)


Theorem	lt0neg2 7573	Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)


Theorem	le0neg1 7574	Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)


Theorem	le0neg2 7575	Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)


Theorem	addge01 7576	A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)
$/\$

Theorem	addge02 7577	A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)
$/\$

Theorem	add20 7578	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 7579	Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$

Theorem	suble0 7580	Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$

Theorem	leaddle0 7581	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 7582	Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
$/\$

Theorem	lesub0 7583	Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
$/\$ $/\$

Theorem	mullt0 7584	The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)
$/\$ $/\$ $/\$

Theorem	0le1 7585	0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)


Theorem	leidi 7586	'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)


Theorem	gt0ne0i 7587	Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)


Theorem	gt0ne0ii 7588	Positive implies nonzero. (Contributed by NM, 15-May-1999.)


Theorem	addgt0i 7589	Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	addge0i 7590	Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	addgegt0i 7591	Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
$/\$

Theorem	addgt0ii 7592	Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)


Theorem	add20i 7593	Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)
$/\$ $/\$

Theorem	ltnegi 7594	Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)


Theorem	lenegi 7595	Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)


Theorem	ltnegcon2i 7596	Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)


Theorem	lesub0i 7597	Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
$/\$

Theorem	ltaddposi 7598	Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)


Theorem	posdifi 7599	Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)


Theorem	ltnegcon1i 7600	Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)

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