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Theorem List for Intuitionistic Logic Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubdir 8201 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 30-Dec-2005.)

Theoremmul02 8202 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 10-Aug-1999.)

Theoremmul02lem2 8203 Zero times a real is zero. Although we prove it as a corollary of mul02 8202, the name is for consistency with the Metamath Proof Explorer which proves it before mul02 8202. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul01 8204 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremmul02i 8205 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)

Theoremmul01i 8206 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremmul02d 8207 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul01d 8208 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremine0 8209 The imaginary unit is not zero. (Contributed by NM, 6-May-1999.)

Theoremmulneg1 8210 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.)

Theoremmulneg2 8211 The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004.)

Theoremmulneg12 8212 Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)

Theoremmul2neg 8213 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.)

Theoremsubmul2 8214 Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.)

Theoremmulm1 8215 Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)

Theoremmulsub 8216 Product of two differences. (Contributed by NM, 14-Jan-2006.)

Theoremmulsub2 8217 Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremmulm1i 8218 Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)

Theoremmulneg1i 8219 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.)

Theoremmulneg2i 8220 Product with negative is negative of product. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmul2negi 8221 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsubdii 8222 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.)

Theoremsubdiri 8223 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 8-May-1999.)

Theoremmuladdi 8224 Product of two sums. (Contributed by NM, 17-May-1999.)

Theoremmulm1d 8225 Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulneg1d 8226 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulneg2d 8227 Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul2negd 8228 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubdid 8229 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubdird 8230 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmuladdd 8231 Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulsubd 8232 Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulsubfacd 8233 Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.)

4.3.4  Ordering on reals (cont.)

Theoremltadd2 8234 Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremltadd2i 8235 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.)

Theoremltadd2d 8236 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd2dd 8237 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltletrd 8238 Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)

Theoremltaddneg 8239 Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremltaddnegr 8240 Adding a negative number to another number decreases it. (Contributed by AV, 19-Mar-2021.)

Theoremlelttrdi 8241 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.)

Theoremgt0ne0 8242 Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlt0ne0 8243 A number which is less than zero is not zero. See also lt0ap0 8463 which is similar but for apartness. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremltadd1 8244 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.)

Theoremleadd1 8245 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.)

Theoremleadd2 8246 Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)

Theoremltsubadd 8247 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltsubadd2 8248 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)

Theoremlesubadd 8249 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlesubadd2 8250 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)

Theoremltaddsub 8251 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)

Theoremltaddsub2 8252 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)

Theoremleaddsub 8253 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)

Theoremleaddsub2 8254 'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)

Theoremsuble 8255 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)

Theoremlesub 8256 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltsub23 8257 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)

Theoremltsub13 8258 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)

Theoremle2add 8259 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.)

Theoremlt2add 8260 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.)

Theoremltleadd 8261 Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)

Theoremleltadd 8262 Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)

Theoremaddgt0 8263 The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddgegt0 8264 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddgtge0 8265 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddge0 8266 The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltaddpos 8267 Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)

Theoremltaddpos2 8268 Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)

Theoremltsubpos 8269 Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremposdif 8270 Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)

Theoremlesub1 8271 Subtraction from both sides of 'less than or equal to'. (Contributed by NM, 13-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlesub2 8272 Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremltsub1 8273 Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltsub2 8274 Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlt2sub 8275 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremle2sub 8276 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremltneg 8277 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.)

Theoremltnegcon1 8278 Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)

Theoremltnegcon2 8279 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremleneg 8280 Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlenegcon1 8281 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)

Theoremlenegcon2 8282 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)

Theoremlt0neg1 8283 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)

Theoremlt0neg2 8284 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)

Theoremle0neg1 8285 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)

Theoremle0neg2 8286 Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)

Theoremaddge01 8287 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)

Theoremaddge02 8288 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)

Theoremadd20 8289 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.)

Theoremsubge0 8290 Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremsuble0 8291 Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremleaddle0 8292 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.)

Theoremsubge02 8293 Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)

Theoremlesub0 8294 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremmullt0 8295 The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)

Theorem0le1 8296 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremltordlem 8297* Lemma for eqord1 8298. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremeqord1 8298* A strictly increasing real function on a subset of is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by Jim Kingdon, 20-Dec-2022.)

Theoremeqord2 8299* A strictly decreasing real function on a subset of is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremleidi 8300 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)

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