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Theorem List for Metamath Proof Explorer - 10301-10400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrecnz 10301 The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)

Theorembtwnnz 10302 A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)

Theoremgtndiv 10303 A larger number does not divide a smaller natural number. (Contributed by NM, 3-May-2005.)

Theoremhalfnz 10304 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)

Theoremsuprzcl 10305* 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 10306* Two ways to express " is a prime number (or 1)." See also isprm 13036. (Contributed by NM, 4-May-2005.)

Theoremmsqznn 10307 The square of a nonzero integer is a natural number. (Contributed by NM, 2-Aug-2004.)

Theoremzneo 10308 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 10309 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 10310 A natural number is even or odd but not both. (Contributed by NM, 20-Aug-2001.)

Theoremzeo 10311 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)

Theoremzeo2 10312 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theorempeano2uz2 10313* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)

Theorempeano5uzi 10314* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)

Theorempeano5uzti 10315* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)

Theoremdfuzi 10316* An expression for the upper integers that start at that is analogous to df-nn 9957 for natural numbers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)

Theoremuzind 10317* 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 10318* 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 10319* 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 10320* 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.) (Proof modification is discouraged.)

Theoremuzind3OLD 10321* 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 10322* 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 10323* 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 10322 or nn0indALT 10323 may be used; see comment for nnind 9974. (Contributed by NM, 28-Nov-2005.)

Theoremfzind 10324* 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 10325* 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 10326* 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 10327* 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 10328* Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)

Theoremnn0zd 10329 A natural number is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnzd 10330 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzred 10331 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzcnd 10332 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremznegcld 10333 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theorempeano2zd 10334 Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzaddcld 10335 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzsubcld 10336 Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzmulcld 10337 Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)

5.4.8  Decimal arithmetic

Syntaxcdc 10338 Constant used for decimal constructor.
;

Definitiondf-dec 10339 Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, ;;; ;;; ;;; 1kp2ke3k 21707. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdecex 10340 A decimal number is a set. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdeceq1 10341 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq2 10342 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq1i 10343 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq2i 10344 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq12i 10345 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremnumnncl 10346 Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0u 10347 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0h 10348 Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumcl 10349 Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumsuc 10350 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecnncl 10351 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdeccl 10352 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdec0u 10353 Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdec0h 10354 Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremnumnncl2 10355 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)

Theoremdecnncl2 10356 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremnumlt 10357 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumltc 10358 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdeclt 10359 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdecltc 10360 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
; ;

Theoremdecsuc 10361 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.)
;       ;

Theoremnumlti 10362 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdeclti 10363 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
;

Theoremnumsucc 10364 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecsucc 10365 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;

Theorem1e0p1 10366 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdec10p 10367 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec10 10368 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 10369 Perform a multiply-add of two decimal integers and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummac 10370 Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumma2c 10371 Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumadd 10372 Add two decimal integers and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumaddc 10373 Add two decimal integers and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummul1c 10374 The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummul2c 10375 The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecma 10376 Perform a multiply-add of two numerals and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                            ;

Theoremdecmac 10377 Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                                   ;       ;

Theoremdecma2c 10378 Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                                   ;       ;

Theoremdecadd 10379 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                     ;

Theoremdecaddc 10380 Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                     ;       ;

Theoremdecaddc2 10381 Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;       ;                     ;

Theoremdecaddi 10382 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;              ;

Theoremdecaddci 10383 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;                     ;       ;

Theoremdecaddci2 10384 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;                     ;

Theoremdecmul1c 10385 The product of a numeral with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)
;                            ;       ;

Theoremdecmul2c 10386 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;                            ;       ;

Theorem6p5lem 10387 Lemma for 6p5e11 10388 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
;       ;

Theorem6p5e11 10388 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem6p6e12 10389 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p4e11 10390 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p5e12 10391 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p6e13 10392 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p7e14 10393 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p3e11 10394 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p4e12 10395 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p5e13 10396 8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p6e14 10397 8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p7e15 10398 8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem8p8e16 10399 8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9p2e11 10400 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

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