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Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsubdiri 7201 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 8-May-1999.)
 CC   &     CC   &     C  CC   =>     -  x.  C  x.  C  -  x.  C
 
Theoremmuladdi 7202 Product of two sums. (Contributed by NM, 17-May-1999.)
 CC   &     CC   &     C  CC   &     D  CC   =>     +  x.  C  +  D  x.  C  +  D  x.  +  x.  D  +  C  x.
 
Theoremmulm1d 7203 Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     -u 1  x.  -u
 
Theoremmulneg1d 7204 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  x.  -u  x.
 
Theoremmulneg2d 7205 Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     x.  -u  -u  x.
 
Theoremmul2negd 7206 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  x.  -u  x.
 
Theoremsubdid 7207 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     x.  -  C  x.  -  x.  C
 
Theoremsubdird 7208 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  x.  C  x.  C  -  x.  C
 
Theoremmuladdd 7209 Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     D  CC   =>     +  x.  C  +  D  x.  C  +  D  x.  +  x.  D  +  C  x.
 
Theoremmulsubd 7210 Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     D  CC   =>     -  x.  C  -  D  x.  C  +  D  x. 
 -  x.  D  +  C  x.
 
Theoremmulsubfacd 7211 Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.)
 CC   &     CC   =>     x.  -  -  1  x.
 
3.3.4  Ordering on reals (cont.)
 
Theoremltadd2 7212 Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  <  C  +  <  C  +
 
Theoremltadd2i 7213 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.)
 RR   &     RR   &     C  RR   =>     <  C  +  <  C  +
 
Theoremltadd2d 7214 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     <  C  +  <  C  +
 
Theoremltadd2dd 7215 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR   &     <    =>     C  +  <  C  +
 
Theoremltletrd 7216 Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)
 RR   &     RR   &     C  RR   &     <    &     <_  C   =>     <  C
 
Theoremgt0ne0 7217 Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  0  <  =/=  0
 
Theoremlt0ne0 7218 A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 RR  <  0  =/=  0
 
Theoremltadd1 7219 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.)
 RR  RR  C  RR  <  +  C  <  +  C
 
Theoremleadd1 7220 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.)
 RR  RR  C  RR  <_  +  C  <_  +  C
 
Theoremleadd2 7221 Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)
 RR  RR  C  RR  <_  C  +  <_  C  +
 
Theoremltsubadd 7222 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  -  <  C  <  C  +
 
Theoremltsubadd2 7223 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
 RR  RR  C  RR  -  <  C  <  +  C
 
Theoremlesubadd 7224 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  -  <_  C  <_  C  +
 
Theoremlesubadd2 7225 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)
 RR  RR  C  RR  -  <_  C  <_  +  C
 
Theoremltaddsub 7226 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
 RR  RR  C  RR  +  <  C  <  C  -
 
Theoremltaddsub2 7227 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
 RR  RR  C  RR  +  <  C  <  C  -
 
Theoremleaddsub 7228 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)
 RR  RR  C  RR  +  <_  C  <_  C  -
 
Theoremleaddsub2 7229 'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)
 RR  RR  C  RR  +  <_  C  <_  C  -
 
Theoremsuble 7230 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
 RR  RR  C  RR  -  <_  C 
 -  C  <_
 
Theoremlesub 7231 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  RR  C  RR  <_  -  C  C  <_  -
 
Theoremltsub23 7232 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)
 RR  RR  C  RR  -  <  C 
 -  C  <
 
Theoremltsub13 7233 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)
 RR  RR  C  RR  <  -  C  C  <  -
 
Theoremle2add 7234 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.)
 RR  RR  C  RR  D  RR  <_  C  <_  D  +  <_  C  +  D
 
Theoremlt2add 7235 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.)
 RR  RR  C  RR  D  RR  <  C  <  D  +  <  C  +  D
 
Theoremltleadd 7236 Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)
 RR  RR  C  RR  D  RR  <  C  <_  D  +  <  C  +  D
 
Theoremleltadd 7237 Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
 RR  RR  C  RR  D  RR  <_  C  <  D  +  <  C  +  D
 
Theoremaddgt0 7238 The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  RR 
 0  <  0  <  0  <  +
 
Theoremaddgegt0 7239 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  RR 
 0  <_  0  <  0  <  +
 
Theoremaddgtge0 7240 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  RR 
 0  <  0  <_  0  <  +
 
Theoremaddge0 7241 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 7242 Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
 RR  RR  0  <  <  +
 
Theoremltaddpos2 7243 Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
 RR  RR  0  <  <  +
 
Theoremltsubpos 7244 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 7245 Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
 RR  RR  <  0  <  -
 
Theoremlesub1 7246 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 7247 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 7248 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 7249 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 7250 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 7251 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 7252 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 7253 Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
 RR  RR  -u  <  -u  <
 
Theoremltnegcon2 7254 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)
 RR  RR  <  -u  <  -u
 
Theoremleneg 7255 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 7256 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)
 RR  RR  -u  <_  -u  <_
 
Theoremlenegcon2 7257 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)
 RR  RR  <_  -u  <_  -u
 
Theoremlt0neg1 7258 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 7259 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
 RR  0  <  -u  <  0
 
Theoremle0neg1 7260 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
 RR  <_  0  0  <_  -u
 
Theoremle0neg2 7261 Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)
 RR  0  <_  -u  <_  0
 
Theoremaddge01 7262 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)
 RR  RR  0  <_  <_  +
 
Theoremaddge02 7263 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)
 RR  RR  0  <_  <_  +
 
Theoremadd20 7264 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 7265 Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  0  <_  -  <_
 
Theoremsuble0 7266 Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR 
 -  <_ 
 0  <_
 
Theoremleaddle0 7267 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 7268 Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
 RR  RR  0  <_  -  <_
 
Theoremlesub0 7269 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 7270 The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)
 RR  <  0  RR  <  0  0  <  x.
 
Theorem0le1 7271 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
 0  <_  1
 
Theoremleidi 7272 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
 RR   =>     <_
 
Theoremgt0ne0i 7273 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
 RR   =>     0  <  =/=  0
 
Theoremgt0ne0ii 7274 Positive implies nonzero. (Contributed by NM, 15-May-1999.)
 RR   &     0  <    =>     =/=  0
 
Theoremaddgt0i 7275 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 7276 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 7277 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 7278 Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)
 RR   &     RR   &     0  <    &     0  <    =>     0  <  +
 
Theoremadd20i 7279 Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)
 RR   &     RR   =>     0  <_  0  <_  +  0  0  0
 
Theoremltnegi 7280 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 7281 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)
 RR   &     RR   =>     <_  -u  <_  -u
 
Theoremltnegcon2i 7282 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
 RR   &     RR   =>     <  -u  <  -u
 
Theoremlesub0i 7283 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 7284 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)
 RR   &     RR   =>     0  <  <  +
 
Theoremposdifi 7285 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)
 RR   &     RR   =>     <  0  <  -
 
Theoremltnegcon1i 7286 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
 RR   &     RR   =>     -u  <  -u  <
 
Theoremlenegcon1i 7287 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)
 RR   &     RR   =>     -u  <_  -u  <_
 
Theoremsubge0i 7288 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
 RR   &     RR   =>     0  <_  -  <_
 
Theoremltadd1i 7289 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 7290 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
 RR   &     RR   &     C  RR   =>     <_  +  C  <_  +  C
 
Theoremleadd2i 7291 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
 RR   &     RR   &     C  RR   =>     <_  C  +  <_  C  +
 
Theoremltsubaddi 7292 '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 7293 '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 7294 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
 RR   &     RR   &     C  RR   =>     -  <  C  <  +  C
 
Theoremlesubadd2i 7295 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)
 RR   &     RR   &     C  RR   =>     -  <_  C  <_  +  C
 
Theoremltaddsubi 7296 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
 RR   &     RR   &     C  RR   =>     +  <  C  <  C  -
 
Theoremlt2addi 7297 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 7298 Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)
 RR   &     RR   &     C  RR   &     D  RR   =>     <_  C  <_  D  +  <_  C  +  D
 
Theoremgt0ne0d 7299 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 0  <    =>     =/=  0
 
Theoremlt0ne0d 7300 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)
 <  0   =>     =/=  0
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