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Theorem List for Intuitionistic Logic Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremzmulcl 8901 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)

Theoremzltp1le 8902 Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremzleltp1 8903 Integer ordering relation. (Contributed by NM, 10-May-2004.)

Theoremzlem1lt 8904 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)

Theoremzltlem1 8905 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)

Theoremzgt0ge1 8906 An integer greater than is greater than or equal to . (Contributed by AV, 14-Oct-2018.)

Theoremnnleltp1 8907 Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnnltp1le 8908 Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)

Theoremnnaddm1cl 8909 Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0ltp1le 8910 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0leltp1 8911 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremnn0ltlem1 8912 Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremznn0sub 8913 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 8914.) (Contributed by NM, 14-Jul-2005.)

Theoremnn0sub 8914 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)

Theoremnn0n0n1ge2 8915 A nonnegative integer which is neither 0 nor 1 is greater than or equal to 2. (Contributed by Alexander van der Vekens, 6-Dec-2017.)

Theoremelz2 8916* Membership in the set of integers. Commonly used in constructions of the integers as equivalence classes under subtraction of the positive integers. (Contributed by Mario Carneiro, 16-May-2014.)

Theoremdfz2 8917 Alternate definition of the integers, based on elz2 8916. (Contributed by Mario Carneiro, 16-May-2014.)

Theoremnn0sub2 8918 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)

Theoremzapne 8919 Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
#

Theoremzdceq 8920 Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
DECID

Theoremzdcle 8921 Integer is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
DECID

Theoremzdclt 8922 Integer is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
DECID

Theoremzltlen 8923 Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8204 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)

Theoremnn0n0n1ge2b 8924 A nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.)

Theoremnn0lt10b 8925 A nonnegative integer less than is . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremnn0lt2 8926 A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)

Theoremnn0le2is012 8927 A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.)

Theoremnn0lem1lt 8928 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremnnlem1lt 8929 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremnnltlem1 8930 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremnnm1ge0 8931 A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.)

Theoremnn0ge0div 8932 Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)

Theoremzdiv 8933* Two ways to express " divides . (Contributed by NM, 3-Oct-2008.)

Theoremzdivadd 8934 Property of divisibility: if divides and then it divides . (Contributed by NM, 3-Oct-2008.)

Theoremzdivmul 8935 Property of divisibility: if divides then it divides . (Contributed by NM, 3-Oct-2008.)

Theoremzextle 8936* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)

Theoremzextlt 8937* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)

Theoremrecnz 8938 The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)

Theorembtwnnz 8939 A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)

Theoremgtndiv 8940 A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)

Theoremhalfnz 8941 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)

Theorem3halfnz 8942 Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)

Theoremsuprzclex 8943* The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)

Theoremprime 8944* Two ways to express " is a prime number (or 1)." (Contributed by NM, 4-May-2005.)

Theoremmsqznn 8945 The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)

Theoremzneo 8946 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)

Theoremnneoor 8947 A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)

Theoremnneo 8948 A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)

Theoremnneoi 8949 A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.)

Theoremzeo 8950 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)

Theoremzeo2 8951 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theorempeano2uz2 8952* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)

Theorempeano5uzti 8953* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)

Theorempeano5uzi 8954* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)

Theoremdfuzi 8955* An expression for the upper integers that start at that is analogous to dfnn2 8522 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)

Theoremuzind 8956* Induction on the upper integers that start at . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 5-Jul-2005.)

Theoremuzind2 8957* Induction on the upper integers that start after an integer . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 25-Jul-2005.)

Theoremuzind3 8958* Induction on the upper integers that start at an integer . The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 26-Jul-2005.)

Theoremnn0ind 8959* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 13-May-2004.)

Theoremfzind 8960* Induction on the integers from to inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremfnn0ind 8961* Induction on the integers from to inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremnn0ind-raph 8962* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.)

Theoremzindd 8963* Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.)

Theorembtwnz 8964* Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)

Theoremnn0zd 8965 A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnzd 8966 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzred 8967 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzcnd 8968 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremznegcld 8969 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theorempeano2zd 8970 Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzaddcld 8971 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzsubcld 8972 Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzmulcld 8973 Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzadd2cl 8974 Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)

Theorembtwnapz 8975 A number between an integer and its successor is apart from any integer. (Contributed by Jim Kingdon, 6-Jan-2023.)
#

3.4.10  Decimal arithmetic

Syntaxcdc 8976 Constant used for decimal constructor.
;

Definitiondf-dec 8977 Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base . For example, ;;; ;;; ;;; 1kp2ke3k 12363. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
;

Theorem9p1e10 8978 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.)
;

Theoremdfdec10 8979 Version of the definition of the "decimal constructor" using ; instead of the symbol 10. Of course, this statement cannot be used as definition, because it uses the "decimal constructor". (Contributed by AV, 1-Aug-2021.)
; ;

Theoremdeceq1 8980 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theoremdeceq2 8981 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theoremdeceq1i 8982 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq2i 8983 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq12i 8984 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremnumnncl 8985 Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0u 8986 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0h 8987 Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumcl 8988 Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumsuc 8989 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdeccl 8990 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem10nn 8991 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
;

Theorem10pos 8992 The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
;

Theorem10nn0 8993 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem10re 8994 The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
;

Theoremdecnncl 8995 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theoremdec0u 8996 Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theoremdec0h 8997 Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theoremnumnncl2 8998 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)

Theoremdecnncl2 8999 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theoremnumlt 9000 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)

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