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Theorem List for Metamath Proof Explorer - 10101-10200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremzextle 10101* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)

Theoremzextlt 10102* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)

Theoremrecnz 10103 The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)

Theorembtwnnz 10104 A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)

Theoremgtndiv 10105 A larger number does not divide a smaller natural number. (Contributed by NM, 3-May-2005.)

Theoremhalfnz 10106 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)

Theoremsuprzcl 10107* The supremum of a bounded-above set of integers is a member of the set. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Theoremprime 10108* Two ways to express " is a prime number (or 1)." See also isprm 12776. (Contributed by NM, 4-May-2005.)

Theoremmsqznn 10109 The square of a nonzero integer is a natural number. (Contributed by NM, 2-Aug-2004.)

Theoremzneo 10110 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.)

Theoremnneo 10111 A natural number is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)

Theoremnneoi 10112 A natural number is even or odd but not both. (Contributed by NM, 20-Aug-2001.)

Theoremzeo 10113 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)

Theoremzeo2 10114 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theorempeano2uz2 10115* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)

Theorempeano5uzi 10116* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)

Theorempeano5uzti 10117* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)

Theoremdfuzi 10118* An expression for the upper integers that start at that is analogous to df-nn 9763 for natural numbers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)

Theoremuzind 10119* 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 hypothesis. (Contributed by NM, 5-Jul-2005.)

Theoremuzind2 10120* 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 hypothesis. (Contributed by NM, 25-Jul-2005.)

Theoremuzind3 10121* 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 hypothesis. (Contributed by NM, 26-Jul-2005.)

TheoremuzindOLD 10122* Induction on the upper integers that start at an integer . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis.

Warning: The HTML proof page is 3/4 megabyte in size. An attempt to shorten it is on my to-do list. Anyone is welcome to try. (Contributed by NM, 11-May-2004.) (New usage is discouraged.)

Theoremuzind3OLD 10123* Induction on the set of upper integers that starts at . The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction hypothesis. (Contributed by NM, 9-Nov-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnn0ind 10124* 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 hypothesis. (Contributed by NM, 13-May-2004.)

Theoremnn0indALT 10125* Principle of Mathematical Induction (inference schema) on nonnegative integers. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction hypothesis. Either nn0ind 10124 or nn0indALT 10125 may be used; see comment for nnind 9780. (Contributed by NM, 28-Nov-2005.)

Theoremfzind 10126* 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 hypothesis. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremfnn0ind 10127* 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 hypothesis. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremnn0ind-raph 10128* 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 hypothesis. 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 10129* 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 10130* Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)

Theoremnn0zd 10131 A natural number is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnzd 10132 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzred 10133 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzcnd 10134 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremznegcld 10135 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theorempeano2zd 10136 Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzaddcld 10137 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzsubcld 10138 Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzmulcld 10139 Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)

5.4.8  Decimal arithmetic

Syntaxcdc 10140 Constant used for decimal constructor.
;

Definitiondf-dec 10141 Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, ;;; ;;; ;;; 1kp2ke3k 20849. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdecex 10142 A decimal number is a set. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdeceq1 10143 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq2 10144 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq1i 10145 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq2i 10146 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq12i 10147 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremnumnncl 10148 Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0u 10149 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0h 10150 Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumcl 10151 Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumsuc 10152 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecnncl 10153 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdeccl 10154 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdec0u 10155 Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdec0h 10156 Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremnumnncl2 10157 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)

Theoremdecnncl2 10158 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremnumlt 10159 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumltc 10160 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdeclt 10161 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdecltc 10162 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ;

Theoremdecsuc 10163 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.)
;       ;

Theoremnumlti 10164 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdeclti 10165 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
;

Theoremnumsucc 10166 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecsucc 10167 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;

Theorem1e0p1 10168 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdec10p 10169 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec10 10170 The decimal form of 10. NB: In our presentations of large numbers later on, we will use our symbol for 10 at the highest digits when advantageous, because we can use this theorem to convert back to "long form" (where each digit is in the range 0-9) with no extra effort. However, we cannot do this for lower digits while maintaining the ease of use of the decimal system, since it requires nontrivial number knowledge (more than just equality theorems) to convert back. (Contributed by Mario Carneiro, 18-Feb-2014.)
;

Theoremnumma 10171 Perform a multiply-add of two decimal integers and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummac 10172 Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumma2c 10173 Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumadd 10174 Add two decimal integers and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumaddc 10175 Add two decimal integers and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummul1c 10176 The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummul2c 10177 The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecma 10178 Perform a multiply-add of two numerals and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                            ;

Theoremdecmac 10179 Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                                   ;       ;

Theoremdecma2c 10180 Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                                   ;       ;

Theoremdecadd 10181 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                     ;

Theoremdecaddc 10182 Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                     ;       ;

Theoremdecaddc2 10183 Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                     ;

Theoremdecaddi 10184 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;              ;

Theoremdecaddci 10185 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;                     ;       ;

Theoremdecaddci2 10186 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;                     ;

Theoremdecmul1c 10187 The product of a numeral with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)
;                            ;       ;

Theoremdecmul2c 10188 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;                            ;       ;

Theorem6p5lem 10189 Lemma for 6p5e11 10190 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
;       ;

Theorem6p5e11 10190 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6p6e12 10191 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p4e11 10192 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p5e12 10193 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p6e13 10194 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p7e14 10195 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p3e11 10196 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p4e12 10197 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p5e13 10198 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p6e14 10199 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p7e15 10200 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

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