Type  Label  Description 
Statement 

Theorem  zeo 8401 
An integer is even or odd. (Contributed by NM, 1Jan2006.)



Theorem  zeo2 8402 
An integer is even or odd but not both. (Contributed by Mario Carneiro,
12Sep2015.)



Theorem  peano2uz2 8403* 
Second Peano postulate for upper integers. (Contributed by NM,
3Oct2004.)



Theorem  peano5uzti 8404* 
Peano's inductive postulate for upper integers. (Contributed by NM,
6Jul2005.) (Revised by Mario Carneiro, 25Jul2013.)



Theorem  peano5uzi 8405* 
Peano's inductive postulate for upper integers. (Contributed by NM,
6Jul2005.) (Revised by Mario Carneiro, 3May2014.)



Theorem  dfuzi 8406* 
An expression for the upper integers that start at that is
analogous to dfnn2 7991 for positive integers. (Contributed by NM,
6Jul2005.) (Proof shortened by Mario Carneiro, 3May2014.)



Theorem  uzind 8407* 
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, 5Jul2005.)



Theorem  uzind2 8408* 
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,
25Jul2005.)



Theorem  uzind3 8409* 
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,
26Jul2005.)



Theorem  nn0ind 8410* 
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, 13May2004.)



Theorem  fzind 8411* 
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,
31Mar2011.)



Theorem  fnn0ind 8412* 
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,
31Mar2011.)



Theorem  nn0indraph 8413* 
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,
10Apr2004.)



Theorem  zindd 8414* 
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, 17Apr2009.) (Proof shortened by Mario
Carneiro, 4Jan2017.)



Theorem  btwnz 8415* 
Any real number can be sandwiched between two integers. Exercise 2 of
[Apostol] p. 28. (Contributed by NM,
10Nov2004.)



Theorem  nn0zd 8416 
A positive integer is an integer. (Contributed by Mario Carneiro,
28May2016.)



Theorem  nnzd 8417 
A nonnegative integer is an integer. (Contributed by Mario Carneiro,
28May2016.)



Theorem  zred 8418 
An integer is a real number. (Contributed by Mario Carneiro,
28May2016.)



Theorem  zcnd 8419 
An integer is a complex number. (Contributed by Mario Carneiro,
28May2016.)



Theorem  znegcld 8420 
Closure law for negative integers. (Contributed by Mario Carneiro,
28May2016.)



Theorem  peano2zd 8421 
Deduction from second Peano postulate generalized to integers.
(Contributed by Mario Carneiro, 28May2016.)



Theorem  zaddcld 8422 
Closure of addition of integers. (Contributed by Mario Carneiro,
28May2016.)



Theorem  zsubcld 8423 
Closure of subtraction of integers. (Contributed by Mario Carneiro,
28May2016.)



Theorem  zmulcld 8424 
Closure of multiplication of integers. (Contributed by Mario Carneiro,
28May2016.)



Theorem  zadd2cl 8425 
Increasing an integer by 2 results in an integer. (Contributed by
Alexander van der Vekens, 16Sep2018.)



3.4.9 Decimal arithmetic


Syntax  cdc 8426 
Constant used for decimal constructor.

; 

Definition  dfdec 8427 
Define the "decimal constructor", which is used to build up
"decimal
integers" or "numeric terms" in base . For example,
;;; ;;; ;;; 1kp2ke3k 10250.
(Contributed by Mario Carneiro, 17Apr2015.) (Revised by AV,
1Aug2021.)

; 

Theorem  9p1e10 8428 
9 + 1 = 10. (Contributed by Mario Carneiro, 18Apr2015.) (Revised by
Stanislas Polu, 7Apr2020.) (Revised by AV, 1Aug2021.)

; 

Theorem  dfdec10 8429 
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, 1Aug2021.)

; ;


Theorem  deceq1 8430 
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17Apr2015.) (Revised by AV, 6Sep2021.)

;
; 

Theorem  deceq2 8431 
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17Apr2015.) (Revised by AV, 6Sep2021.)

;
; 

Theorem  deceq1i 8432 
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17Apr2015.)

; ; 

Theorem  deceq2i 8433 
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17Apr2015.)

; ; 

Theorem  deceq12i 8434 
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17Apr2015.)

; ; 

Theorem  numnncl 8435 
Closure for a numeral (with units place). (Contributed by Mario
Carneiro, 18Feb2014.)



Theorem  num0u 8436 
Add a zero in the units place. (Contributed by Mario Carneiro,
18Feb2014.)



Theorem  num0h 8437 
Add a zero in the higher places. (Contributed by Mario Carneiro,
18Feb2014.)



Theorem  numcl 8438 
Closure for a decimal integer (with units place). (Contributed by Mario
Carneiro, 18Feb2014.)



Theorem  numsuc 8439 
The successor of a decimal integer (no carry). (Contributed by Mario
Carneiro, 18Feb2014.)



Theorem  deccl 8440 
Closure for a numeral. (Contributed by Mario Carneiro, 17Apr2015.)
(Revised by AV, 6Sep2021.)

; 

Theorem  10nn 8441 
10 is a positive integer. (Contributed by NM, 8Nov2012.) (Revised by
AV, 6Sep2021.)

; 

Theorem  10pos 8442 
The number 10 is positive. (Contributed by NM, 5Feb2007.) (Revised by
AV, 8Sep2021.)

; 

Theorem  10nn0 8443 
10 is a nonnegative integer. (Contributed by Mario Carneiro,
19Apr2015.) (Revised by AV, 6Sep2021.)

; 

Theorem  10re 8444 
The number 10 is real. (Contributed by NM, 5Feb2007.) (Revised by AV,
8Sep2021.)

; 

Theorem  decnncl 8445 
Closure for a numeral. (Contributed by Mario Carneiro, 17Apr2015.)
(Revised by AV, 6Sep2021.)

; 

Theorem  dec0u 8446 
Add a zero in the units place. (Contributed by Mario Carneiro,
17Apr2015.) (Revised by AV, 6Sep2021.)

;
; 

Theorem  dec0h 8447 
Add a zero in the higher places. (Contributed by Mario Carneiro,
17Apr2015.) (Revised by AV, 6Sep2021.)

; 

Theorem  numnncl2 8448 
Closure for a decimal integer (zero units place). (Contributed by Mario
Carneiro, 9Mar2015.)



Theorem  decnncl2 8449 
Closure for a decimal integer (zero units place). (Contributed by Mario
Carneiro, 17Apr2015.) (Revised by AV, 6Sep2021.)

; 

Theorem  numlt 8450 
Comparing two decimal integers (equal higher places). (Contributed by
Mario Carneiro, 18Feb2014.)



Theorem  numltc 8451 
Comparing two decimal integers (unequal higher places). (Contributed by
Mario Carneiro, 18Feb2014.)



Theorem  le9lt10 8452 
A "decimal digit" (i.e. a nonnegative integer less than or equal to
9)
is less then 10. (Contributed by AV, 8Sep2021.)

; 

Theorem  declt 8453 
Comparing two decimal integers (equal higher places). (Contributed by
Mario Carneiro, 17Apr2015.) (Revised by AV, 6Sep2021.)

; ; 

Theorem  decltc 8454 
Comparing two decimal integers (unequal higher places). (Contributed
by Mario Carneiro, 18Feb2014.) (Revised by AV, 6Sep2021.)

; ; ; 

Theorem  declth 8455 
Comparing two decimal integers (unequal higher places). (Contributed
by AV, 8Sep2021.)

; ; 

Theorem  decsuc 8456 
The successor of a decimal integer (no carry). (Contributed by Mario
Carneiro, 17Apr2015.) (Revised by AV, 6Sep2021.)

; ; 

Theorem  3declth 8457 
Comparing two decimal integers with three "digits" (unequal higher
places). (Contributed by AV, 8Sep2021.)

;;
;; 

Theorem  3decltc 8458 
Comparing two decimal integers with three "digits" (unequal higher
places). (Contributed by AV, 15Jun2021.) (Revised by AV,
6Sep2021.)

;
; ;; ;; 

Theorem  decle 8459 
Comparing two decimal integers (equal higher places). (Contributed by
AV, 17Aug2021.) (Revised by AV, 8Sep2021.)

; ; 

Theorem  decleh 8460 
Comparing two decimal integers (unequal higher places). (Contributed by
AV, 17Aug2021.) (Revised by AV, 8Sep2021.)

; ; 

Theorem  declei 8461 
Comparing a digit to a decimal integer. (Contributed by AV,
17Aug2021.)

; 

Theorem  numlti 8462 
Comparing a digit to a decimal integer. (Contributed by Mario Carneiro,
18Feb2014.)



Theorem  declti 8463 
Comparing a digit to a decimal integer. (Contributed by Mario
Carneiro, 18Feb2014.) (Revised by AV, 6Sep2021.)

;
; 

Theorem  decltdi 8464 
Comparing a digit to a decimal integer. (Contributed by AV,
8Sep2021.)

; 

Theorem  numsucc 8465 
The successor of a decimal integer (with carry). (Contributed by Mario
Carneiro, 18Feb2014.)



Theorem  decsucc 8466 
The successor of a decimal integer (with carry). (Contributed by Mario
Carneiro, 18Feb2014.) (Revised by AV, 6Sep2021.)

; ; 

Theorem  1e0p1 8467 
The successor of zero. (Contributed by Mario Carneiro, 18Feb2014.)



Theorem  dec10p 8468 
Ten plus an integer. (Contributed by Mario Carneiro, 19Apr2015.)
(Revised by AV, 6Sep2021.)

;
; 

Theorem  numma 8469 
Perform a multiplyadd of two decimal integers and against
a fixed multiplicand (no carry). (Contributed by Mario
Carneiro, 18Feb2014.)



Theorem  nummac 8470 
Perform a multiplyadd of two decimal integers and against
a fixed multiplicand (with carry). (Contributed by Mario
Carneiro, 18Feb2014.)



Theorem  numma2c 8471 
Perform a multiplyadd of two decimal integers and against
a fixed multiplicand (with carry). (Contributed by Mario
Carneiro, 18Feb2014.)



Theorem  numadd 8472 
Add two decimal integers and (no
carry). (Contributed by
Mario Carneiro, 18Feb2014.)



Theorem  numaddc 8473 
Add two decimal integers and (with
carry). (Contributed
by Mario Carneiro, 18Feb2014.)



Theorem  nummul1c 8474 
The product of a decimal integer with a number. (Contributed by Mario
Carneiro, 18Feb2014.)



Theorem  nummul2c 8475 
The product of a decimal integer with a number (with carry).
(Contributed by Mario Carneiro, 18Feb2014.)



Theorem  decma 8476 
Perform a multiplyadd of two numerals and against a fixed
multiplicand
(no carry). (Contributed by Mario Carneiro,
18Feb2014.) (Revised by AV, 6Sep2021.)

; ;
; 

Theorem  decmac 8477 
Perform a multiplyadd of two numerals and against a fixed
multiplicand
(with carry). (Contributed by Mario Carneiro,
18Feb2014.) (Revised by AV, 6Sep2021.)

; ;
;
; 

Theorem  decma2c 8478 
Perform a multiplyadd of two numerals and against a fixed
multiplier
(with carry). (Contributed by Mario Carneiro,
18Feb2014.) (Revised by AV, 6Sep2021.)

; ;
;
; 

Theorem  decadd 8479 
Add two numerals and
(no carry).
(Contributed by Mario
Carneiro, 18Feb2014.) (Revised by AV, 6Sep2021.)

; ;
; 

Theorem  decaddc 8480 
Add two numerals and
(with carry).
(Contributed by Mario
Carneiro, 18Feb2014.) (Revised by AV, 6Sep2021.)

; ;
;
; 

Theorem  decaddc2 8481 
Add two numerals and
(with carry).
(Contributed by Mario
Carneiro, 18Feb2014.) (Revised by AV, 6Sep2021.)

; ;
;
; 

Theorem  decrmanc 8482 
Perform a multiplyadd of two numerals and against a fixed
multiplicand
(no carry). (Contributed by AV, 16Sep2021.)

;
; 

Theorem  decrmac 8483 
Perform a multiplyadd of two numerals and against a fixed
multiplicand
(with carry). (Contributed by AV, 16Sep2021.)

;
;
; 

Theorem  decaddm10 8484 
The sum of two multiples of 10 is a multiple of 10. (Contributed by AV,
30Jul2021.)

; ; ;


Theorem  decaddi 8485 
Add two numerals and
(no carry).
(Contributed by Mario
Carneiro, 18Feb2014.)

;
; 

Theorem  decaddci 8486 
Add two numerals and
(no carry).
(Contributed by Mario
Carneiro, 18Feb2014.)

;
;
; 

Theorem  decaddci2 8487 
Add two numerals and
(no carry).
(Contributed by Mario
Carneiro, 18Feb2014.) (Revised by AV, 6Sep2021.)

;
;
; 

Theorem  decsubi 8488 
Difference between a numeral and a nonnegative integer (no
underflow). (Contributed by AV, 22Jul2021.) (Revised by AV,
6Sep2021.)

;
; 

Theorem  decmul1 8489 
The product of a numeral with a number (no carry). (Contributed by
AV, 22Jul2021.) (Revised by AV, 6Sep2021.)

;
; 

Theorem  decmul1c 8490 
The product of a numeral with a number (with carry). (Contributed by
Mario Carneiro, 18Feb2014.) (Revised by AV, 6Sep2021.)

;
;
; 

Theorem  decmul2c 8491 
The product of a numeral with a number (with carry). (Contributed by
Mario Carneiro, 18Feb2014.) (Revised by AV, 6Sep2021.)

;
;
; 

Theorem  decmulnc 8492 
The product of a numeral with a number (no carry). (Contributed by AV,
15Jun2021.)

; ; 

Theorem  11multnc 8493 
The product of 11 (as numeral) with a number (no carry). (Contributed
by AV, 15Jun2021.)

; ; 

Theorem  decmul10add 8494 
A multiplication of a number and a numeral expressed as addition with
first summand as multiple of 10. (Contributed by AV, 22Jul2021.)
(Revised by AV, 6Sep2021.)

; ; 

Theorem  6p5lem 8495 
Lemma for 6p5e11 8498 and related theorems. (Contributed by Mario
Carneiro, 19Apr2015.)

;
; 

Theorem  5p5e10 8496 
5 + 5 = 10. (Contributed by NM, 5Feb2007.) (Revised by Stanislas Polu,
7Apr2020.) (Revised by AV, 6Sep2021.)

; 

Theorem  6p4e10 8497 
6 + 4 = 10. (Contributed by NM, 5Feb2007.) (Revised by Stanislas Polu,
7Apr2020.) (Revised by AV, 6Sep2021.)

; 

Theorem  6p5e11 8498 
6 + 5 = 11. (Contributed by Mario Carneiro, 19Apr2015.) (Revised by
AV, 6Sep2021.)

; 

Theorem  6p6e12 8499 
6 + 6 = 12. (Contributed by Mario Carneiro, 19Apr2015.)

; 

Theorem  7p3e10 8500 
7 + 3 = 10. (Contributed by NM, 5Feb2007.) (Revised by Stanislas Polu,
7Apr2020.) (Revised by AV, 6Sep2021.)

; 