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Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaddge0 7201 The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  RR 
 0  <_  0  <_  0  <_  +
 
Theoremltaddpos 7202 Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
 RR  RR  0  <  <  +
 
Theoremltaddpos2 7203 Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
 RR  RR  0  <  <  +
 
Theoremltsubpos 7204 Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  RR  0  <  -  <
 
Theoremposdif 7205 Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
 RR  RR  <  0  <  -
 
Theoremlesub1 7206 Subtraction from both sides of 'less than or equal to'. (Contributed by NM, 13-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  <_  -  C  <_  -  C
 
Theoremlesub2 7207 Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  <_  C  -  <_  C  -
 
Theoremltsub1 7208 Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  <  -  C  <  -  C
 
Theoremltsub2 7209 Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  <  C  -  <  C  -
 
Theoremlt2sub 7210 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
 RR  RR  C  RR  D  RR  <  C  D  <  -  <  C  -  D
 
Theoremle2sub 7211 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
 RR  RR  C  RR  D  RR  <_  C  D  <_  -  <_  C  -  D
 
Theoremltneg 7212 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.)
 RR  RR  <  -u  <  -u
 
Theoremltnegcon1 7213 Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
 RR  RR  -u  <  -u  <
 
Theoremltnegcon2 7214 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)
 RR  RR  <  -u  <  -u
 
Theoremleneg 7215 Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  <_  -u  <_  -u
 
Theoremlenegcon1 7216 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)
 RR  RR  -u  <_  -u  <_
 
Theoremlenegcon2 7217 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)
 RR  RR  <_  -u  <_  -u
 
Theoremlt0neg1 7218 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
 RR  <  0  0  <  -u
 
Theoremlt0neg2 7219 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
 RR  0  <  -u  <  0
 
Theoremle0neg1 7220 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
 RR  <_  0  0  <_  -u
 
Theoremle0neg2 7221 Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)
 RR  0  <_  -u  <_  0
 
Theoremaddge01 7222 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)
 RR  RR  0  <_  <_  +
 
Theoremaddge02 7223 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)
 RR  RR  0  <_  <_  +
 
Theoremadd20 7224 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.)
 RR  0  <_  RR  0  <_  +  0  0  0
 
Theoremsubge0 7225 Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  0  <_  -  <_
 
Theoremsuble0 7226 Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR 
 -  <_ 
 0  <_
 
Theoremleaddle0 7227 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.)
 RR  RR  +  <_  <_  0
 
Theoremsubge02 7228 Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
 RR  RR  0  <_  -  <_
 
Theoremlesub0 7229 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  0 
 <_  <_  -  0
 
Theoremmullt0 7230 The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)
 RR  <  0  RR  <  0  0  <  x.
 
Theorem0le1 7231 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
 0  <_  1
 
Theoremleidi 7232 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
 RR   =>     <_
 
Theoremgt0ne0i 7233 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
 RR   =>     0  <  =/=  0
 
Theoremgt0ne0ii 7234 Positive implies nonzero. (Contributed by NM, 15-May-1999.)
 RR   &     0  <    =>     =/=  0
 
Theoremaddgt0i 7235 Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR   &     RR   =>     0  <  0  <  0  <  +
 
Theoremaddge0i 7236 Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR   &     RR   =>     0  <_  0  <_  0  <_  +
 
Theoremaddgegt0i 7237 Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     0  <_  0  <  0  <  +
 
Theoremaddgt0ii 7238 Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)
 RR   &     RR   &     0  <    &     0  <    =>     0  <  +
 
Theoremadd20i 7239 Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)
 RR   &     RR   =>     0  <_  0  <_  +  0  0  0
 
Theoremltnegi 7240 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
 RR   &     RR   =>     <  -u  <  -u
 
Theoremlenegi 7241 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)
 RR   &     RR   =>     <_  -u  <_  -u
 
Theoremltnegcon2i 7242 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
 RR   &     RR   =>     <  -u  <  -u
 
Theoremlesub0i 7243 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR   &     RR   =>     0  <_  <_  -  0
 
Theoremltaddposi 7244 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)
 RR   &     RR   =>     0  <  <  +
 
Theoremposdifi 7245 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)
 RR   &     RR   =>     <  0  <  -
 
Theoremltnegcon1i 7246 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
 RR   &     RR   =>     -u  <  -u  <
 
Theoremlenegcon1i 7247 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)
 RR   &     RR   =>     -u  <_  -u  <_
 
Theoremsubge0i 7248 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
 RR   &     RR   =>     0  <_  -  <_
 
Theoremltadd1i 7249 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
 RR   &     RR   &     C  RR   =>     <  +  C  <  +  C
 
Theoremleadd1i 7250 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
 RR   &     RR   &     C  RR   =>     <_  +  C  <_  +  C
 
Theoremleadd2i 7251 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
 RR   &     RR   &     C  RR   =>     <_  C  +  <_  C  +
 
Theoremltsubaddi 7252 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR   &     RR   &     C  RR   =>     -  <  C  <  C  +
 
Theoremlesubaddi 7253 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR   &     RR   &     C  RR   =>     -  <_  C  <_  C  +
 
Theoremltsubadd2i 7254 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
 RR   &     RR   &     C  RR   =>     -  <  C  <  +  C
 
Theoremlesubadd2i 7255 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)
 RR   &     RR   &     C  RR   =>     -  <_  C  <_  +  C
 
Theoremltaddsubi 7256 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
 RR   &     RR   &     C  RR   =>     +  <  C  <  C  -
 
Theoremlt2addi 7257 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
 RR   &     RR   &     C  RR   &     D  RR   =>     <  C  <  D  +  <  C  +  D
 
Theoremle2addi 7258 Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)
 RR   &     RR   &     C  RR   &     D  RR   =>     <_  C  <_  D  +  <_  C  +  D
 
Theoremgt0ne0d 7259 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 0  <    =>     =/=  0
 
Theoremlt0ne0d 7260 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)
 <  0   =>     =/=  0
 
Theoremleidd 7261 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>     <_
 
Theoremlt0neg1d 7262 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>     <  0  0  <  -u
 
Theoremlt0neg2d 7263 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>    
 0  <  -u  <  0
 
Theoremle0neg1d 7264 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>     <_  0  0  <_  -u
 
Theoremle0neg2d 7265 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>    
 0  <_  -u  <_  0
 
Theoremaddgegt0d 7266 Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     0  <_    &     0  <    =>     0  <  +
 
Theoremaddgt0d 7267 Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     0  <    &     0  <    =>     0  <  +
 
Theoremaddge0d 7268 Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     0  <_    &     0  <_    =>     0  <_  +
 
Theoremltnegd 7269 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     <  -u  < 
 -u
 
Theoremlenegd 7270 Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     <_  -u  <_ 
 -u
 
Theoremltnegcon1d 7271 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     -u  <    =>     -u  <
 
Theoremltnegcon2d 7272 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     < 
 -u   =>     < 
 -u
 
Theoremlenegcon1d 7273 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     -u  <_    =>     -u  <_
 
Theoremlenegcon2d 7274 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     <_ 
 -u   =>     <_ 
 -u
 
Theoremltaddposd 7275 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 0  <  <  +
 
Theoremltaddpos2d 7276 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 0  <  <  +
 
Theoremltsubposd 7277 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 0  < 
 -  <
 
Theoremposdifd 7278 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     <  0  <  -
 
Theoremaddge01d 7279 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 0  <_  <_  +
 
Theoremaddge02d 7280 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 0  <_  <_  +
 
Theoremsubge0d 7281 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 0  <_  -  <_
 
Theoremsuble0d 7282 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     -  <_  0  <_
 
Theoremsubge02d 7283 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 0  <_ 
 -  <_
 
Theoremltadd1d 7284 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     <  +  C  <  +  C
 
Theoremleadd1d 7285 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     <_  +  C  <_  +  C
 
Theoremleadd2d 7286 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     <_  C  +  <_  C  +
 
Theoremltsubaddd 7287 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     -  <  C  <  C  +
 
Theoremlesubaddd 7288 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     -  <_  C  <_  C  +
 
Theoremltsubadd2d 7289 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     -  <  C  <  +  C
 
Theoremlesubadd2d 7290 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     -  <_  C  <_  +  C
 
Theoremltaddsubd 7291 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     +  <  C  <  C  -
 
Theoremltaddsub2d 7292 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)
 RR   &     RR   &     C  RR   =>     +  <  C  <  C  -
 
Theoremleaddsub2d 7293 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     +  <_  C  <_  C  -
 
Theoremsubled 7294 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     -  <_  C   =>     -  C  <_
 
Theoremlesubd 7295 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     <_  -  C   =>     C  <_  -
 
Theoremltsub23d 7296 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     -  <  C   =>     -  C  <
 
Theoremltsub13d 7297 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     <  -  C   =>     C  <  -
 
Theoremlesub1d 7298 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     <_ 
 -  C  <_  -  C
 
Theoremlesub2d 7299 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     <_  C  -  <_  C  -
 
Theoremltsub1d 7300 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     < 
 -  C  <  -  C
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