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Theorem List for Intuitionistic Logic Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcnm2m1cnm3 8401 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 8402 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 8403 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 8404* 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 8405* 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 8406* 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 8407 Extend class notation to include the class of nonnegative integers.

Definitiondf-n0 8408 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremelnn0 8409 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnnssnn0 8410 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0ssre 8411 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0sscn 8412 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)

Theoremnn0ex 8413 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)

Theoremnnnn0 8414 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)

Theoremnnnn0i 8415 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)

Theoremnn0re 8416 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)

Theoremnn0cn 8417 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)

Theoremnn0rei 8418 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)

Theoremnn0cni 8419 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)

Theoremdfn2 8420 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 8421 The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Theorem0nn0 8422 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)

Theorem1nn0 8423 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)

Theorem2nn0 8424 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)

Theorem3nn0 8425 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem4nn0 8426 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem5nn0 8427 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem6nn0 8428 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem7nn0 8429 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem8nn0 8430 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem9nn0 8431 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theoremnn0ge0 8432 A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremnn0nlt0 8433 A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnn0ge0i 8434 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0le0eq0 8435 A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.)

Theoremnn0p1gt0 8436 A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.)

Theoremnnnn0addcl 8437 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 8438 A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.)

Theorem0mnnnnn0 8439 The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.)

Theoremun0addcl 8440 If is closed under addition, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.)

Theoremun0mulcl 8441 If is closed under multiplication, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.)

Theoremnn0addcl 8442 Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)

Theoremnn0mulcl 8443 Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)

Theoremnn0addcli 8444 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0mulcli 8445 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0p1nn 8446 A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)

Theorempeano2nn0 8447 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)

Theoremnnm1nn0 8448 A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremelnn0nn 8449 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 8450 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)

Theoremelnnnn0b 8451 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)

Theoremelnnnn0c 8452 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)

Theoremnn0addge1 8453 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge2 8454 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge1i 8455 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge2i 8456 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0le2xi 8457 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0lele2xi 8458 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0supp 8459 Two ways to write the support of a function on . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremnnnn0d 8460 A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0red 8461 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0cnd 8462 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0ge0d 8463 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0addcld 8464 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0mulcld 8465 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0readdcl 8466 Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)

Theoremnn0ge2m1nn 8467 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 8468 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.)

Theoremnn0nndivcl 8469 Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)

3.4.8  Extended nonnegative integers

The function values of the hash (set size) function are either nonnegative integers or positive infinity. To avoid the need to distinguish between finite and infinite sets (and therefore if the set size is a nonnegative integer or positive infinity), it is useful to provide a definition of the set of nonnegative integers extended by positive infinity, analogously to the extension of the real numbers , see df-xr 7271.

Syntaxcxnn0 8470 The set of extended nonnegative integers.
NN0*

Definitiondf-xnn0 8471 Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers , see df-xr 7271. (Contributed by AV, 10-Dec-2020.)
NN0*

Theoremelxnn0 8472 An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.)
NN0*

Theoremnn0ssxnn0 8473 The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
NN0*

Theoremnn0xnn0 8474 A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
NN0*

Theoremxnn0xr 8475 An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
NN0*

Theorem0xnn0 8476 Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
NN0*

Theorempnf0xnn0 8477 Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
NN0*

Theoremnn0nepnf 8478 No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)

Theoremnn0xnn0d 8479 A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.)
NN0*

Theoremnn0nepnfd 8480 No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)

Theoremxnn0nemnf 8481 No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
NN0*

Theoremxnn0xrnemnf 8482 The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
NN0*

Theoremxnn0nnn0pnf 8483 An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.)
NN0*

3.4.9  Integers (as a subset of complex numbers)

Syntaxcz 8484 Extend class notation to include the class of integers.

Definitiondf-z 8485 Define the set of integers, which are the positive and negative integers together with zero. Definition of integers in [Apostol] p. 22. The letter Z abbreviates the German word Zahlen meaning "numbers." (Contributed by NM, 8-Jan-2002.)

Theoremelz 8486 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)

Theoremnnnegz 8487 The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)

Theoremzre 8488 An integer is a real. (Contributed by NM, 8-Jan-2002.)

Theoremzcn 8489 An integer is a complex number. (Contributed by NM, 9-May-2004.)

Theoremzrei 8490 An integer is a real number. (Contributed by NM, 14-Jul-2005.)

Theoremzssre 8491 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)

Theoremzsscn 8492 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremzex 8493 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremelnnz 8494 Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)

Theorem0z 8495 Zero is an integer. (Contributed by NM, 12-Jan-2002.)

Theorem0zd 8496 Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremelnn0z 8497 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)

Theoremelznn0nn 8498 Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)

Theoremelznn0 8499 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)

Theoremelznn 8500 Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)

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