P. 84 of Theorem List - Intuitionistic Logic Explorer

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

Type	Label	Description
Statement
Theorem	negf1o 8301*	Negation is an isomorphism of a subset of the real numbers to the negated elements of the subset. (Contributed by AV, 9-Aug-2020.)
|- F = ( x e. A |-> -u x )   =>    |- ( A C_ RR -> F : A -1-1-onto-> { n e. RR | -u n e. A } )
4.3.3  Multiplication
Theorem	kcnktkm1cn 8302	k times k minus 1 is a complex number if k is a complex number. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
|- ( K e. CC -> ( K x. ( K - 1 ) ) e. CC )
Theorem	muladd 8303	Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) x. ( C + D ) ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) ) )
Theorem	subdi 8304	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B - C ) ) = ( ( A x. B ) - ( A x. C ) ) )
Theorem	subdir 8305	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 30-Dec-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) ) )
Theorem	mul02 8306	Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 10-Aug-1999.)
|- ( A e. CC -> ( 0 x. A ) = 0 )
Theorem	mul02lem2 8307	Zero times a real is zero. Although we prove it as a corollary of mul02 8306, the name is for consistency with the Metamath Proof Explorer which proves it before mul02 8306. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( A e. RR -> ( 0 x. A ) = 0 )
Theorem	mul01 8308	Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
|- ( A e. CC -> ( A x. 0 ) = 0 )
Theorem	mul02i 8309	Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)
|- A e. CC   =>    |- ( 0 x. A ) = 0
Theorem	mul01i 8310	Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
|- A e. CC   =>    |- ( A x. 0 ) = 0
Theorem	mul02d 8311	Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 0 x. A ) = 0 )
Theorem	mul01d 8312	Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. 0 ) = 0 )
Theorem	ine0 8313	The imaginary unit _i is not zero. (Contributed by NM, 6-May-1999.)
|- _i =/= 0
Theorem	mulneg1 8314	Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> ( -u A x. B ) = -u ( A x. B ) )
Theorem	mulneg2 8315	The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004.)
|- ( ( A e. CC /\ B e. CC ) -> ( A x. -u B ) = -u ( A x. B ) )
Theorem	mulneg12 8316	Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)
|- ( ( A e. CC /\ B e. CC ) -> ( -u A x. B ) = ( A x. -u B ) )
Theorem	mul2neg 8317	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.)
|- ( ( A e. CC /\ B e. CC ) -> ( -u A x. -u B ) = ( A x. B ) )
Theorem	submul2 8318	Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B x. C ) ) = ( A + ( B x. -u C ) ) )
Theorem	mulm1 8319	Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)
|- ( A e. CC -> ( -u 1 x. A ) = -u A )
Theorem	mulsub 8320	Product of two differences. (Contributed by NM, 14-Jan-2006.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) x. ( C - D ) ) = ( ( ( A x. C ) + ( D x. B ) ) - ( ( A x. D ) + ( C x. B ) ) ) )
Theorem	mulsub2 8321	Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) x. ( C - D ) ) = ( ( B - A ) x. ( D - C ) ) )
Theorem	mulm1i 8322	Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)
|- A e. CC   =>    |- ( -u 1 x. A ) = -u A
Theorem	mulneg1i 8323	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.)
|- A e. CC   &    |- B e. CC   =>    |- ( -u A x. B ) = -u ( A x. B )
Theorem	mulneg2i 8324	Product with negative is negative of product. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|- A e. CC   &    |- B e. CC   =>    |- ( A x. -u B ) = -u ( A x. B )
Theorem	mul2negi 8325	Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)
|- A e. CC   &    |- B e. CC   =>    |- ( -u A x. -u B ) = ( A x. B )
Theorem	subdii 8326	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( A x. ( B - C ) ) = ( ( A x. B ) - ( A x. C ) )
Theorem	subdiri 8327	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 8-May-1999.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) )
Theorem	muladdi 8328	Product of two sums. (Contributed by NM, 17-May-1999.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- D e. CC   =>    |- ( ( A + B ) x. ( C + D ) ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) )
Theorem	mulm1d 8329	Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( -u 1 x. A ) = -u A )
Theorem	mulneg1d 8330	Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( -u A x. B ) = -u ( A x. B ) )
Theorem	mulneg2d 8331	Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( A x. -u B ) = -u ( A x. B ) )
Theorem	mul2negd 8332	Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( -u A x. -u B ) = ( A x. B ) )
Theorem	subdid 8333	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( A x. ( B - C ) ) = ( ( A x. B ) - ( A x. C ) ) )
Theorem	subdird 8334	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) ) )
Theorem	muladdd 8335	Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   =>    |- ( ph -> ( ( A + B ) x. ( C + D ) ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) ) )
Theorem	mulsubd 8336	Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   =>    |- ( ph -> ( ( A - B ) x. ( C - D ) ) = ( ( ( A x. C ) + ( D x. B ) ) - ( ( A x. D ) + ( C x. B ) ) ) )
Theorem	mulsubfacd 8337	Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( ( A x. B ) - B ) = ( ( A - 1 ) x. B ) )
4.3.4  Ordering on reals (cont.)
Theorem	ltadd2 8338	Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C + A ) < ( C + B ) ) )
Theorem	ltadd2i 8339	Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( A < B <-> ( C + A ) < ( C + B ) )
Theorem	ltadd2d 8340	Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A < B <-> ( C + A ) < ( C + B ) ) )
Theorem	ltadd2dd 8341	Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> ( C + A ) < ( C + B ) )
Theorem	ltletrd 8342	Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> B <_ C )   =>    |- ( ph -> A < C )
Theorem	ltaddneg 8343	Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < 0 <-> ( B + A ) < B ) )
Theorem	ltaddnegr 8344	Adding a negative number to another number decreases it. (Contributed by AV, 19-Mar-2021.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < 0 <-> ( A + B ) < B ) )
Theorem	lelttrdi 8345	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.)
|- ( ph -> ( A e. RR /\ B e. RR /\ C e. RR ) )   &    |- ( ph -> B <_ C )   =>    |- ( ph -> ( A < B -> A < C ) )
Theorem	gt0ne0 8346	Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ 0 < A ) -> A =/= 0 )
Theorem	lt0ne0 8347	A number which is less than zero is not zero. See also lt0ap0 8567 which is similar but for apartness. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( ( A e. RR /\ A < 0 ) -> A =/= 0 )
Theorem	ltadd1 8348	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.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( A + C ) < ( B + C ) ) )
Theorem	leadd1 8349	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.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( A + C ) <_ ( B + C ) ) )
Theorem	leadd2 8350	Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( C + A ) <_ ( C + B ) ) )
Theorem	ltsubadd 8351	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) < C <-> A < ( C + B ) ) )
Theorem	ltsubadd2 8352	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) < C <-> A < ( B + C ) ) )
Theorem	lesubadd 8353	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) <_ C <-> A <_ ( C + B ) ) )
Theorem	lesubadd2 8354	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) <_ C <-> A <_ ( B + C ) ) )
Theorem	ltaddsub 8355	'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) < C <-> A < ( C - B ) ) )
Theorem	ltaddsub2 8356	'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) < C <-> B < ( C - A ) ) )
Theorem	leaddsub 8357	'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) <_ C <-> A <_ ( C - B ) ) )
Theorem	leaddsub2 8358	'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) <_ C <-> B <_ ( C - A ) ) )
Theorem	suble 8359	Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) <_ C <-> ( A - C ) <_ B ) )
Theorem	lesub 8360	Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ ( B - C ) <-> C <_ ( B - A ) ) )
Theorem	ltsub23 8361	'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) < C <-> ( A - C ) < B ) )
Theorem	ltsub13 8362	'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < ( B - C ) <-> C < ( B - A ) ) )
Theorem	le2add 8363	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.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ B <_ D ) -> ( A + B ) <_ ( C + D ) ) )
Theorem	lt2add 8364	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.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A < C /\ B < D ) -> ( A + B ) < ( C + D ) ) )
Theorem	ltleadd 8365	Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A < C /\ B <_ D ) -> ( A + B ) < ( C + D ) ) )
Theorem	leltadd 8366	Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ B < D ) -> ( A + B ) < ( C + D ) ) )
Theorem	addgt0 8367	The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < B ) ) -> 0 < ( A + B ) )
Theorem	addgegt0 8368	The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < B ) ) -> 0 < ( A + B ) )
Theorem	addgtge0 8369	The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 <_ B ) ) -> 0 < ( A + B ) )
Theorem	addge0 8370	The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 <_ B ) ) -> 0 <_ ( A + B ) )
Theorem	ltaddpos 8371	Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
|- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( B + A ) ) )
Theorem	ltaddpos2 8372	Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
|- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( A + B ) ) )
Theorem	ltsubpos 8373	Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> ( B - A ) < B ) )
Theorem	posdif 8374	Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> 0 < ( B - A ) ) )
Theorem	lesub1 8375	Subtraction from both sides of 'less than or equal to'. (Contributed by NM, 13-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( A - C ) <_ ( B - C ) ) )
Theorem	lesub2 8376	Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( C - B ) <_ ( C - A ) ) )
Theorem	ltsub1 8377	Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( A - C ) < ( B - C ) ) )
Theorem	ltsub2 8378	Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C - B ) < ( C - A ) ) )
Theorem	lt2sub 8379	Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A < C /\ D < B ) -> ( A - B ) < ( C - D ) ) )
Theorem	le2sub 8380	Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ D <_ B ) -> ( A - B ) <_ ( C - D ) ) )
Theorem	ltneg 8381	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.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -u B < -u A ) )
Theorem	ltnegcon1 8382	Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
|- ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < A ) )
Theorem	ltnegcon2 8383	Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < -u B <-> B < -u A ) )
Theorem	leneg 8384	Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -u B <_ -u A ) )
Theorem	lenegcon1 8385	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)
|- ( ( A e. RR /\ B e. RR ) -> ( -u A <_ B <-> -u B <_ A ) )
Theorem	lenegcon2 8386	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ -u B <-> B <_ -u A ) )
Theorem	lt0neg1 8387	Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
|- ( A e. RR -> ( A < 0 <-> 0 < -u A ) )
Theorem	lt0neg2 8388	Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
|- ( A e. RR -> ( 0 < A <-> -u A < 0 ) )
Theorem	le0neg1 8389	Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
|- ( A e. RR -> ( A <_ 0 <-> 0 <_ -u A ) )
Theorem	le0neg2 8390	Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)
|- ( A e. RR -> ( 0 <_ A <-> -u A <_ 0 ) )
Theorem	addge01 8391	A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)
|- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ B <-> A <_ ( A + B ) ) )
Theorem	addge02 8392	A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)
|- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ B <-> A <_ ( B + A ) ) )
Theorem	add20 8393	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.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
Theorem	subge0 8394	Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ ( A - B ) <-> B <_ A ) )
Theorem	suble0 8395	Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( A - B ) <_ 0 <-> A <_ B ) )
Theorem	leaddle0 8396	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.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( A + B ) <_ A <-> B <_ 0 ) )
Theorem	subge02 8397	Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
|- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ B <-> ( A - B ) <_ A ) )
Theorem	lesub0 8398	Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <_ A /\ B <_ ( B - A ) ) <-> A = 0 ) )
Theorem	mullt0 8399	The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)
|- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> 0 < ( A x. B ) )
Theorem	0le1 8400	0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
|- 0 <_ 1