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

Theorem8th4div3 8201 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)

Theoremhalfpm6th 8202 One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)

Theoremit0e0 8203 i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem2mulicn 8204 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremiap0 8205 The imaginary unit is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
#

Theorem2muliap0 8206 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
#

Theorem2muline0 8207 . See also 2muliap0 8206. (Contributed by David A. Wheeler, 8-Dec-2018.)

3.4.5  Simple number properties

Theoremhalfcl 8208 Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)

Theoremrehalfcl 8209 Real closure of half. (Contributed by NM, 1-Jan-2006.)

Theoremhalf0 8210 Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)

Theorem2halves 8211 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)

Theoremhalfpos2 8212 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)

Theoremhalfpos 8213 A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremhalfnneg2 8214 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)

Theoremhalfaddsubcl 8215 Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremhalfaddsub 8216 Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremlt2halves 8217 A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)

Theoremaddltmul 8218 Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)

Theoremnominpos 8219* There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)

Theoremavglt1 8220 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremavglt2 8221 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremavgle1 8222 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremavgle2 8223 Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)

Theorem2timesd 8224 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremtimes2d 8225 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremhalfcld 8226 Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)

Theorem2halvesd 8227 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrehalfcld 8228 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2halvesd 8229 A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrehalfcli 8230 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)

Theoremadd1p1 8231 Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)

Theoremsub1m1 8232 Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)

Theoremcnm2m1cnm3 8233 Subtracting 2 and afterwards 1 from a number results in the difference between the number and 3. (Contributed by Alexander van der Vekens, 16-Sep-2018.)

Theoremxp1d2m1eqxm1d2 8234 A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020.)

Theoremdiv4p1lem1div2 8235 An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)

3.4.6  The Archimedean property

Theoremarch 8236* Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)

Theoremnnrecl 8237* There exists a positive integer whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM, 8-Nov-2004.)

Theorembndndx 8238* A bounded real sequence is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)

3.4.7  Nonnegative integers (as a subset of complex numbers)

Syntaxcn0 8239 Extend class notation to include the class of nonnegative integers.

Definitiondf-n0 8240 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremelnn0 8241 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnnssnn0 8242 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0ssre 8243 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0sscn 8244 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)

Theoremnn0ex 8245 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)

Theoremnnnn0 8246 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)

Theoremnnnn0i 8247 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)

Theoremnn0re 8248 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)

Theoremnn0cn 8249 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)

Theoremnn0rei 8250 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)

Theoremnn0cni 8251 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)

Theoremdfn2 8252 The set of positive integers defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)

Theoremelnnne0 8253 The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Theorem0nn0 8254 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)

Theorem1nn0 8255 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)

Theorem2nn0 8256 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)

Theorem3nn0 8257 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem4nn0 8258 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem5nn0 8259 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem6nn0 8260 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem7nn0 8261 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem8nn0 8262 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem9nn0 8263 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theoremnn0ge0 8264 A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremnn0nlt0 8265 A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnn0ge0i 8266 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0le0eq0 8267 A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.)

Theoremnn0p1gt0 8268 A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.)

Theoremnnnn0addcl 8269 A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0nnaddcl 8270 A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.)

Theorem0mnnnnn0 8271 The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.)

Theoremun0addcl 8272 If is closed under addition, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.)

Theoremun0mulcl 8273 If is closed under multiplication, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.)

Theoremnn0addcl 8274 Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)

Theoremnn0mulcl 8275 Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)

Theoremnn0addcli 8276 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0mulcli 8277 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0p1nn 8278 A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)

Theorempeano2nn0 8279 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)

Theoremnnm1nn0 8280 A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremelnn0nn 8281 The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremelnnnn0 8282 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)

Theoremelnnnn0b 8283 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)

Theoremelnnnn0c 8284 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)

Theoremnn0addge1 8285 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge2 8286 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge1i 8287 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge2i 8288 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0le2xi 8289 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0lele2xi 8290 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0supp 8291 Two ways to write the support of a function on . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremnnnn0d 8292 A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0red 8293 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0cnd 8294 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0ge0d 8295 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0addcld 8296 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0mulcld 8297 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0readdcl 8298 Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)

Theoremnn0ge2m1nn 8299 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.)

Theoremnn0ge2m1nn0 8300 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is also a nonnegative integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.)

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