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

Theoremzeo 8401 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)

Theoremzeo2 8402 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theorempeano2uz2 8403* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)

Theorempeano5uzti 8404* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)

Theorempeano5uzi 8405* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)

Theoremdfuzi 8406* An expression for the upper integers that start at that is analogous to dfnn2 7991 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)

Theoremuzind 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, 5-Jul-2005.)

Theoremuzind2 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, 25-Jul-2005.)

Theoremuzind3 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, 26-Jul-2005.)

Theoremnn0ind 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, 13-May-2004.)

Theoremfzind 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, 31-Mar-2011.)

Theoremfnn0ind 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, 31-Mar-2011.)

Theoremnn0ind-raph 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, 10-Apr-2004.)

Theoremzindd 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, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.)

Theorembtwnz 8415* Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)

Theoremnn0zd 8416 A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnzd 8417 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzred 8418 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzcnd 8419 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremznegcld 8420 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theorempeano2zd 8421 Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzaddcld 8422 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzsubcld 8423 Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzmulcld 8424 Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzadd2cl 8425 Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)

3.4.9  Decimal arithmetic

Syntaxcdc 8426 Constant used for decimal constructor.
;

Definitiondf-dec 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, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
;

Theorem9p1e10 8428 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.)
;

Theoremdfdec10 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, 1-Aug-2021.)
; ;

Theoremdeceq1 8430 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theoremdeceq2 8431 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theoremdeceq1i 8432 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq2i 8433 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq12i 8434 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremnumnncl 8435 Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0u 8436 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0h 8437 Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumcl 8438 Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumsuc 8439 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdeccl 8440 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem10nn 8441 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.)
;

Theorem10pos 8442 The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
;

Theorem10nn0 8443 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem10re 8444 The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.)
;

Theoremdecnncl 8445 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theoremdec0u 8446 Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theoremdec0h 8447 Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theoremnumnncl2 8448 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)

Theoremdecnncl2 8449 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theoremnumlt 8450 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumltc 8451 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremle9lt10 8452 A "decimal digit" (i.e. a nonnegative integer less than or equal to 9) is less then 10. (Contributed by AV, 8-Sep-2021.)
;

Theoremdeclt 8453 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theoremdecltc 8454 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;              ; ;

Theoremdeclth 8455 Comparing two decimal integers (unequal higher places). (Contributed by AV, 8-Sep-2021.)
; ;

Theoremdecsuc 8456 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;       ;

Theorem3declth 8457 Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 8-Sep-2021.)
;; ;;

Theorem3decltc 8458 Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 15-Jun-2021.) (Revised by AV, 6-Sep-2021.)
;       ;       ;; ;;

Theoremdecle 8459 Comparing two decimal integers (equal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
; ;

Theoremdecleh 8460 Comparing two decimal integers (unequal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
; ;

Theoremdeclei 8461 Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.)
;

Theoremnumlti 8462 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdeclti 8463 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;

Theoremdecltdi 8464 Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.)
;

Theoremnumsucc 8465 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecsucc 8466 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;

Theorem1e0p1 8467 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdec10p 8468 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theoremnumma 8469 Perform a multiply-add of two decimal integers and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummac 8470 Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumma2c 8471 Perform a multiply-add of two decimal integers and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumadd 8472 Add two decimal integers and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumaddc 8473 Add two decimal integers and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummul1c 8474 The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnummul2c 8475 The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecma 8476 Perform a multiply-add of two numerals and against a fixed multiplicand (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;                            ;

Theoremdecmac 8477 Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;                                   ;       ;

Theoremdecma2c 8478 Perform a multiply-add of two numerals and against a fixed multiplier (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;                                   ;       ;

Theoremdecadd 8479 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;                     ;

Theoremdecaddc 8480 Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;                     ;       ;

Theoremdecaddc2 8481 Add two numerals and (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;       ;              ;       ;

Theoremdecrmanc 8482 Perform a multiply-add of two numerals and against a fixed multiplicand (no carry). (Contributed by AV, 16-Sep-2021.)
;                            ;

Theoremdecrmac 8483 Perform a multiply-add of two numerals and against a fixed multiplicand (with carry). (Contributed by AV, 16-Sep-2021.)
;                                   ;       ;

Theoremdecaddm10 8484 The sum of two multiples of 10 is a multiple of 10. (Contributed by AV, 30-Jul-2021.)
; ; ;

Theoremdecaddi 8485 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;              ;

Theoremdecaddci 8486 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
;                     ;       ;

Theoremdecaddci2 8487 Add two numerals and (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;              ;       ;

Theoremdecsubi 8488 Difference between a numeral and a nonnegative integer (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
;                     ;

Theoremdecmul1 8489 The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
;                            ;

Theoremdecmul1c 8490 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;                            ;       ;

Theoremdecmul2c 8491 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
;                            ;       ;

Theoremdecmulnc 8492 The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021.)
; ;

Theorem11multnc 8493 The product of 11 (as numeral) with a number (no carry). (Contributed by AV, 15-Jun-2021.)
; ;

Theoremdecmul10add 8494 A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
; ;

Theorem6p5lem 8495 Lemma for 6p5e11 8498 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
;       ;

Theorem5p5e10 8496 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem6p4e10 8497 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

Theorem6p5e11 8498 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

Theorem6p6e12 8499 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem7p3e10 8500 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
;

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