HomeHome Metamath Proof Explorer
Theorem List (p. 93 of 322)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21498)
  Hilbert Space Explorer  Hilbert Space Explorer
(21499-23021)
  Users' Mathboxes  Users' Mathboxes
(23022-32154)
 

Theorem List for Metamath Proof Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsubneintrd 9201 Introducing subtraction on both sides of a statement of nonequality. Contrapositive of subcand 9198. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  ( A  -  B )  =/=  ( A  -  C ) )
 
Theoremsubcan2ad 9202 Cancellation law for subtraction. Deduction form of subcan2 9072. Generalization of subcan2d 9199. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  C )  =  ( B  -  C )  <->  A  =  B ) )
 
Theoremsubneintr2d 9203 Introducing subtraction on both sides of a statement of nonequality. Contrapositive of subcan2d 9199. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A  -  C )  =/=  ( B  -  C ) )
 
Theoremaddsub4d 9204 Rearrangement of 4 terms in a mixed addition and subtraction. (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 )  -  ( C  +  D ) )  =  ( ( A  -  C )  +  ( B  -  D ) ) )
 
Theoremsubadd4d 9205 Rearrangement of 4 terms in a mixed addition and subtraction. (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 )  -  ( C  -  D ) )  =  ( ( A  +  D )  -  ( B  +  C )
 ) )
 
Theoremsub4d 9206 Rearrangement of 4 terms in a subtraction. (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 )  -  ( C  -  D ) )  =  ( ( A  -  C )  -  ( B  -  D ) ) )
 
Theorem2addsubd 9207 Law for subtraction and addition. (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 )  +  C )  -  D )  =  ( ( ( A  +  C )  -  D )  +  B ) )
 
Theoremaddsubeq4d 9208 Relation between sums and 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 )  =  ( C  +  D )  <->  ( C  -  A )  =  ( B  -  D ) ) )
 
Theoremsubeq0bd 9209 If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 9167. Converse of subeq0d 9165. Contrapositive of subne0ad 9168. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  -  B )  =  0 )
 
Theoremrenegcld 9210 Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  -u A  e.  RR )
 
Theoremresubcld 9211 Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  -  B )  e.  RR )
 
5.3.3  Multiplication
 
Theoremmuladd 9212 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 ) ) ) )
 
Theoremsubdi 9213 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 ) ) )
 
Theoremsubdir 9214 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 ) ) )
 
Theoremine0 9215 The imaginary unit  _i is not zero. (Contributed by NM, 6-May-1999.)
 |-  _i  =/=  0
 
Theoremmulneg1 9216 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 ) )
 
Theoremmulneg2 9217 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 ) )
 
Theoremmulneg12 9218 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 ) )
 
Theoremmul2neg 9219 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 ) )
 
Theoremsubmul2 9220 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 ) ) )
 
Theoremmulm1 9221 Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)
 |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
 
Theoremmulsub 9222 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 ) ) ) )
 
Theoremmulsub2 9223 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 ) ) )
 
Theoremmulm1i 9224 Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)
 |-  A  e.  CC   =>    |-  ( -u 1  x.  A )  =  -u A
 
Theoremmulneg1i 9225 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 )
 
Theoremmulneg2i 9226 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 )
 
Theoremmul2negi 9227 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 )
 
Theoremsubdii 9228 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 ) )
 
Theoremsubdiri 9229 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 ) )
 
Theoremmuladdi 9230 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 ) ) )
 
Theoremmulm1d 9231 Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( -u 1  x.  A )  =  -u A )
 
Theoremmulneg1d 9232 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 ) )
 
Theoremmulneg2d 9233 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 ) )
 
Theoremmul2negd 9234 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 ) )
 
Theoremsubdid 9235 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 ) ) )
 
Theoremsubdird 9236 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 ) ) )
 
Theoremmuladdd 9237 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 ) ) ) )
 
Theoremmulsubd 9238 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 ) ) ) )
 
5.3.4  Ordering on reals (cont.)
 
Theoremgt0ne0 9239 Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0
 )
 
Theoremlt0ne0 9240 A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( ( A  e.  RR  /\  A  <  0
 )  ->  A  =/=  0 )
 
Theoremltadd1 9241 Addition to both sides of 'less than'. (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 ) ) )
 
Theoremleadd1 9242 Addition to both sides of 'less than or equal to'. (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 ) ) )
 
Theoremleadd2 9243 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 ) ) )
 
Theoremltsubadd 9244 '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 ) ) )
 
Theoremltsubadd2 9245 '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 ) ) )
 
Theoremlesubadd 9246 '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 ) ) )
 
Theoremlesubadd2 9247 '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 ) ) )
 
Theoremltaddsub 9248 '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 ) ) )
 
Theoremltaddsub2 9249 '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 ) ) )
 
Theoremleaddsub 9250 '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 ) ) )
 
Theoremleaddsub2 9251 '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 ) ) )
 
Theoremsuble 9252 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 ) )
 
Theoremlesub 9253 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 ) ) )
 
Theoremltsub23 9254 '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 ) )
 
Theoremltsub13 9255 '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 ) ) )
 
Theoremle2add 9256 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 ) ) )
 
Theoremltleadd 9257 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 ) ) )
 
Theoremleltadd 9258 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 ) ) )
 
Theoremlt2add 9259 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 ) ) )
 
Theoremaddgt0 9260 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 ) )
 
Theoremaddgegt0 9261 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 ) )
 
Theoremaddgtge0 9262 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 ) )
 
Theoremaddge0 9263 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 ) )
 
Theoremltaddpos 9264 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 ) ) )
 
Theoremltaddpos2 9265 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 ) ) )
 
Theoremltsubpos 9266 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 ) )
 
Theoremposdif 9267 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 ) ) )
 
Theoremlesub1 9268 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 ) ) )
 
Theoremlesub2 9269 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 ) ) )
 
Theoremltsub1 9270 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 ) ) )
 
Theoremltsub2 9271 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 ) ) )
 
Theoremlt2sub 9272 Adding 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 ) ) )
 
Theoremle2sub 9273 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 ) ) )
 
Theoremltneg 9274 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 ) )
 
Theoremltnegcon1 9275 Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  <  B  <->  -u B  <  A ) )
 
Theoremltnegcon2 9276 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 ) )
 
Theoremleneg 9277 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 ) )
 
Theoremlenegcon1 9278 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 ) )
 
Theoremlenegcon2 9279 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 ) )
 
Theoremlt0neg1 9280 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 ) )
 
Theoremlt0neg2 9281 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
 |-  ( A  e.  RR  ->  ( 0  <  A  <->  -u A  <  0 ) )
 
Theoremle0neg1 9282 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
 |-  ( A  e.  RR  ->  ( A  <_  0  <->  0 
 <_  -u A ) )
 
Theoremle0neg2 9283 Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)
 |-  ( A  e.  RR  ->  ( 0  <_  A  <->  -u A  <_  0 ) )
 
Theoremaddge01 9284 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 ) ) )
 
Theoremaddge02 9285 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 ) ) )
 
Theoremadd20 9286 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 )
 ) )
 
Theoremsubge0 9287 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 )
 )
 
Theoremsuble0 9288 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 )
 )
 
Theoremsubge02 9289 Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B 
 <->  ( A  -  B )  <_  A ) )
 
Theoremlesub0 9290 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
 ) )
 
Theoremmulge0 9291 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  -> 
 0  <_  ( A  x.  B ) )
 
Theoremmulge0OLD 9292 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 0  <_  A  /\  0  <_  B ) ) 
 ->  0  <_  ( A  x.  B ) )
 
Theoremmullt0 9293 The product of two negative numbers is positive. (Contributed by Jeffrey Hankins, 8-Jun-2009.)
 |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  B  <  0 ) )  -> 
 0  <  ( A  x.  B ) )
 
Theoremmsqgt0 9294 A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by NM, 6-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  0  <  ( A  x.  A ) )
 
Theoremmsqge0 9295 A square is nonnegative. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  RR  ->  0  <_  ( A  x.  A ) )
 
Theorem0lt1 9296 0 is less than 1. Theorem I.21 of [Apostol] p. 20. (Contributed by NM, 17-Jan-1997.)
 |-  0  <  1
 
Theorem0le1 9297 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  0  <_  1
 
Theoremltordlem 9298* Lemma for ltord1 9299. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( x  =  y 
 ->  A  =  B )   &    |-  ( x  =  C  ->  A  =  M )   &    |-  ( x  =  D  ->  A  =  N )   &    |-  S  C_  RR   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  A  e.  RR )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  <  y  ->  A  <  B ) )   =>    |-  ( ( ph  /\  ( C  e.  S  /\  D  e.  S )
 )  ->  ( C  <  D  ->  M  <  N ) )
 
Theoremltord1 9299* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( x  =  y 
 ->  A  =  B )   &    |-  ( x  =  C  ->  A  =  M )   &    |-  ( x  =  D  ->  A  =  N )   &    |-  S  C_  RR   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  A  e.  RR )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  <  y  ->  A  <  B ) )   =>    |-  ( ( ph  /\  ( C  e.  S  /\  D  e.  S )
 )  ->  ( C  <  D  <->  M  <  N ) )
 
Theoremleord1 9300* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)
 |-  ( x  =  y 
 ->  A  =  B )   &    |-  ( x  =  C  ->  A  =  M )   &    |-  ( x  =  D  ->  A  =  N )   &    |-  S  C_  RR   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  A  e.  RR )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  <  y  ->  A  <  B ) )   =>    |-  ( ( ph  /\  ( C  e.  S  /\  D  e.  S )
 )  ->  ( C  <_  D  <->  M  <_  N ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32154
  Copyright terms: Public domain < Previous  Next >