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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nn0negzi 12601 | The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
β’ π β β0 β β’ -π β β€ | ||
Theorem | zaddcl 12602 | Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
β’ ((π β β€ β§ π β β€) β (π + π) β β€) | ||
Theorem | peano2z 12603 | Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.) |
β’ (π β β€ β (π + 1) β β€) | ||
Theorem | zsubcl 12604 | Closure of subtraction of integers. (Contributed by NM, 11-May-2004.) |
β’ ((π β β€ β§ π β β€) β (π β π) β β€) | ||
Theorem | peano2zm 12605 | "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.) |
β’ (π β β€ β (π β 1) β β€) | ||
Theorem | zletr 12606 | Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.) |
β’ ((π½ β β€ β§ πΎ β β€ β§ πΏ β β€) β ((π½ β€ πΎ β§ πΎ β€ πΏ) β π½ β€ πΏ)) | ||
Theorem | zrevaddcl 12607 | Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.) |
β’ (π β β€ β ((π β β β§ (π + π) β β€) β π β β€)) | ||
Theorem | znnsub 12608 | The positive difference of unequal integers is a positive integer. (Generalization of nnsub 12256.) (Contributed by NM, 11-May-2004.) |
β’ ((π β β€ β§ π β β€) β (π < π β (π β π) β β)) | ||
Theorem | znn0sub 12609 | The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 12522.) (Contributed by NM, 14-Jul-2005.) |
β’ ((π β β€ β§ π β β€) β (π β€ π β (π β π) β β0)) | ||
Theorem | nzadd 12610 | The sum of a real number not being an integer and an integer is not an integer. (Contributed by AV, 19-Jul-2021.) |
β’ ((π΄ β (β β β€) β§ π΅ β β€) β (π΄ + π΅) β (β β β€)) | ||
Theorem | zmulcl 12611 | Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.) |
β’ ((π β β€ β§ π β β€) β (π Β· π) β β€) | ||
Theorem | zltp1le 12612 | Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
β’ ((π β β€ β§ π β β€) β (π < π β (π + 1) β€ π)) | ||
Theorem | zleltp1 12613 | Integer ordering relation. (Contributed by NM, 10-May-2004.) |
β’ ((π β β€ β§ π β β€) β (π β€ π β π < (π + 1))) | ||
Theorem | zlem1lt 12614 | Integer ordering relation. (Contributed by NM, 13-Nov-2004.) |
β’ ((π β β€ β§ π β β€) β (π β€ π β (π β 1) < π)) | ||
Theorem | zltlem1 12615 | Integer ordering relation. (Contributed by NM, 13-Nov-2004.) |
β’ ((π β β€ β§ π β β€) β (π < π β π β€ (π β 1))) | ||
Theorem | zgt0ge1 12616 | An integer greater than 0 is greater than or equal to 1. (Contributed by AV, 14-Oct-2018.) |
β’ (π β β€ β (0 < π β 1 β€ π)) | ||
Theorem | nnleltp1 12617 | Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ β€ π΅ β π΄ < (π΅ + 1))) | ||
Theorem | nnltp1le 12618 | Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ < π΅ β (π΄ + 1) β€ π΅)) | ||
Theorem | nnaddm1cl 12619 | Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
β’ ((π΄ β β β§ π΅ β β) β ((π΄ + π΅) β 1) β β) | ||
Theorem | nn0ltp1le 12620 | Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
β’ ((π β β0 β§ π β β0) β (π < π β (π + 1) β€ π)) | ||
Theorem | nn0leltp1 12621 | Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.) |
β’ ((π β β0 β§ π β β0) β (π β€ π β π < (π + 1))) | ||
Theorem | nn0ltlem1 12622 | Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
β’ ((π β β0 β§ π β β0) β (π < π β π β€ (π β 1))) | ||
Theorem | nn0sub2 12623 | Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.) |
β’ ((π β β0 β§ π β β0 β§ π β€ π) β (π β π) β β0) | ||
Theorem | nn0lt10b 12624 | A nonnegative integer less than 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by OpenAI, 25-Mar-2020.) |
β’ (π β β0 β (π < 1 β π = 0)) | ||
Theorem | nn0lt2 12625 | A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.) |
β’ ((π β β0 β§ π < 2) β (π = 0 β¨ π = 1)) | ||
Theorem | nn0le2is012 12626 | A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.) |
β’ ((π β β0 β§ π β€ 2) β (π = 0 β¨ π = 1 β¨ π = 2)) | ||
Theorem | nn0lem1lt 12627 | Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.) |
β’ ((π β β0 β§ π β β0) β (π β€ π β (π β 1) < π)) | ||
Theorem | nnlem1lt 12628 | Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.) |
β’ ((π β β β§ π β β) β (π β€ π β (π β 1) < π)) | ||
Theorem | nnltlem1 12629 | Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.) |
β’ ((π β β β§ π β β) β (π < π β π β€ (π β 1))) | ||
Theorem | nnm1ge0 12630 | A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.) |
β’ (π β β β 0 β€ (π β 1)) | ||
Theorem | nn0ge0div 12631 | Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
β’ ((πΎ β β0 β§ πΏ β β) β 0 β€ (πΎ / πΏ)) | ||
Theorem | zdiv 12632* | Two ways to express "π divides π. (Contributed by NM, 3-Oct-2008.) |
β’ ((π β β β§ π β β€) β (βπ β β€ (π Β· π) = π β (π / π) β β€)) | ||
Theorem | zdivadd 12633 | Property of divisibility: if π· divides π΄ and π΅ then it divides π΄ + π΅. (Contributed by NM, 3-Oct-2008.) |
β’ (((π· β β β§ π΄ β β€ β§ π΅ β β€) β§ ((π΄ / π·) β β€ β§ (π΅ / π·) β β€)) β ((π΄ + π΅) / π·) β β€) | ||
Theorem | zdivmul 12634 | Property of divisibility: if π· divides π΄ then it divides π΅ Β· π΄. (Contributed by NM, 3-Oct-2008.) |
β’ (((π· β β β§ π΄ β β€ β§ π΅ β β€) β§ (π΄ / π·) β β€) β ((π΅ Β· π΄) / π·) β β€) | ||
Theorem | zextle 12635* | An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.) |
β’ ((π β β€ β§ π β β€ β§ βπ β β€ (π β€ π β π β€ π)) β π = π) | ||
Theorem | zextlt 12636* | An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.) |
β’ ((π β β€ β§ π β β€ β§ βπ β β€ (π < π β π < π)) β π = π) | ||
Theorem | recnz 12637 | The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.) |
β’ ((π΄ β β β§ 1 < π΄) β Β¬ (1 / π΄) β β€) | ||
Theorem | btwnnz 12638 | A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.) |
β’ ((π΄ β β€ β§ π΄ < π΅ β§ π΅ < (π΄ + 1)) β Β¬ π΅ β β€) | ||
Theorem | gtndiv 12639 | A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.) |
β’ ((π΄ β β β§ π΅ β β β§ π΅ < π΄) β Β¬ (π΅ / π΄) β β€) | ||
Theorem | halfnz 12640 | One-half is not an integer. (Contributed by NM, 31-Jul-2004.) |
β’ Β¬ (1 / 2) β β€ | ||
Theorem | 3halfnz 12641 | Three halves is not an integer. (Contributed by AV, 2-Jun-2020.) |
β’ Β¬ (3 / 2) β β€ | ||
Theorem | suprzcl 12642* | 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.) |
β’ ((π΄ β β€ β§ π΄ β β β§ βπ₯ β β βπ¦ β π΄ π¦ β€ π₯) β sup(π΄, β, < ) β π΄) | ||
Theorem | prime 12643* | Two ways to express "π΄ is a prime number (or 1)". See also isprm 16610. (Contributed by NM, 4-May-2005.) |
β’ (π΄ β β β (βπ₯ β β ((π΄ / π₯) β β β (π₯ = 1 β¨ π₯ = π΄)) β βπ₯ β β ((1 < π₯ β§ π₯ β€ π΄ β§ (π΄ / π₯) β β) β π₯ = π΄))) | ||
Theorem | msqznn 12644 | The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.) |
β’ ((π΄ β β€ β§ π΄ β 0) β (π΄ Β· π΄) β β) | ||
Theorem | zneo 12645 | 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.) |
β’ ((π΄ β β€ β§ π΅ β β€) β (2 Β· π΄) β ((2 Β· π΅) + 1)) | ||
Theorem | nneo 12646 | A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.) |
β’ (π β β β ((π / 2) β β β Β¬ ((π + 1) / 2) β β)) | ||
Theorem | nneoi 12647 | A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.) |
β’ π β β β β’ ((π / 2) β β β Β¬ ((π + 1) / 2) β β) | ||
Theorem | zeo 12648 | An integer is even or odd. (Contributed by NM, 1-Jan-2006.) |
β’ (π β β€ β ((π / 2) β β€ β¨ ((π + 1) / 2) β β€)) | ||
Theorem | zeo2 12649 | An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.) |
β’ (π β β€ β ((π / 2) β β€ β Β¬ ((π + 1) / 2) β β€)) | ||
Theorem | peano2uz2 12650* | Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.) |
β’ ((π΄ β β€ β§ π΅ β {π₯ β β€ β£ π΄ β€ π₯}) β (π΅ + 1) β {π₯ β β€ β£ π΄ β€ π₯}) | ||
Theorem | peano5uzi 12651* | Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.) |
β’ π β β€ β β’ ((π β π΄ β§ βπ₯ β π΄ (π₯ + 1) β π΄) β {π β β€ β£ π β€ π} β π΄) | ||
Theorem | peano5uzti 12652* | Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.) |
β’ (π β β€ β ((π β π΄ β§ βπ₯ β π΄ (π₯ + 1) β π΄) β {π β β€ β£ π β€ π} β π΄)) | ||
Theorem | dfuzi 12653* | An expression for the upper integers that start at π that is analogous to dfnn2 12225 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.) |
β’ π β β€ β β’ {π§ β β€ β£ π β€ π§} = β© {π₯ β£ (π β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)} | ||
Theorem | uzind 12654* | Induction on the upper integers that start at π. 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.) |
β’ (π = π β (π β π)) & β’ (π = π β (π β π)) & β’ (π = (π + 1) β (π β π)) & β’ (π = π β (π β π)) & β’ (π β β€ β π) & β’ ((π β β€ β§ π β β€ β§ π β€ π) β (π β π)) β β’ ((π β β€ β§ π β β€ β§ π β€ π) β π) | ||
Theorem | uzind2 12655* | Induction on the upper integers that start after an integer π. 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.) |
β’ (π = (π + 1) β (π β π)) & β’ (π = π β (π β π)) & β’ (π = (π + 1) β (π β π)) & β’ (π = π β (π β π)) & β’ (π β β€ β π) & β’ ((π β β€ β§ π β β€ β§ π < π) β (π β π)) β β’ ((π β β€ β§ π β β€ β§ π < π) β π) | ||
Theorem | uzind3 12656* | Induction on the upper integers that start at an integer π. 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.) |
β’ (π = π β (π β π)) & β’ (π = π β (π β π)) & β’ (π = (π + 1) β (π β π)) & β’ (π = π β (π β π)) & β’ (π β β€ β π) & β’ ((π β β€ β§ π β {π β β€ β£ π β€ π}) β (π β π)) β β’ ((π β β€ β§ π β {π β β€ β£ π β€ π}) β π) | ||
Theorem | nn0ind 12657* | 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.) |
β’ (π₯ = 0 β (π β π)) & β’ (π₯ = π¦ β (π β π)) & β’ (π₯ = (π¦ + 1) β (π β π)) & β’ (π₯ = π΄ β (π β π)) & β’ π & β’ (π¦ β β0 β (π β π)) β β’ (π΄ β β0 β π) | ||
Theorem | nn0indALT 12658* | 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 step. Either nn0ind 12657 or nn0indALT 12658 may be used; see comment for nnind 12230. (Contributed by NM, 28-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |
β’ (π¦ β β0 β (π β π)) & β’ π & β’ (π₯ = 0 β (π β π)) & β’ (π₯ = π¦ β (π β π)) & β’ (π₯ = (π¦ + 1) β (π β π)) & β’ (π₯ = π΄ β (π β π)) β β’ (π΄ β β0 β π) | ||
Theorem | nn0indd 12659* | Principle of Mathematical Induction (inference schema) on nonnegative integers, a deduction version. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
β’ (π₯ = 0 β (π β π)) & β’ (π₯ = π¦ β (π β π)) & β’ (π₯ = (π¦ + 1) β (π β π)) & β’ (π₯ = π΄ β (π β π)) & β’ (π β π) & β’ (((π β§ π¦ β β0) β§ π) β π) β β’ ((π β§ π΄ β β0) β π) | ||
Theorem | fzind 12660* | Induction on the integers from π to π 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.) |
β’ (π₯ = π β (π β π)) & β’ (π₯ = π¦ β (π β π)) & β’ (π₯ = (π¦ + 1) β (π β π)) & β’ (π₯ = πΎ β (π β π)) & β’ ((π β β€ β§ π β β€ β§ π β€ π) β π) & β’ (((π β β€ β§ π β β€) β§ (π¦ β β€ β§ π β€ π¦ β§ π¦ < π)) β (π β π)) β β’ (((π β β€ β§ π β β€) β§ (πΎ β β€ β§ π β€ πΎ β§ πΎ β€ π)) β π) | ||
Theorem | fnn0ind 12661* | Induction on the integers from 0 to π 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.) |
β’ (π₯ = 0 β (π β π)) & β’ (π₯ = π¦ β (π β π)) & β’ (π₯ = (π¦ + 1) β (π β π)) & β’ (π₯ = πΎ β (π β π)) & β’ (π β β0 β π) & β’ ((π β β0 β§ π¦ β β0 β§ π¦ < π) β (π β π)) β β’ ((π β β0 β§ πΎ β β0 β§ πΎ β€ π) β π) | ||
Theorem | nn0ind-raph 12662* | 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.) |
β’ (π₯ = 0 β (π β π)) & β’ (π₯ = π¦ β (π β π)) & β’ (π₯ = (π¦ + 1) β (π β π)) & β’ (π₯ = π΄ β (π β π)) & β’ π & β’ (π¦ β β0 β (π β π)) β β’ (π΄ β β0 β π) | ||
Theorem | zindd 12663* | 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.) |
β’ (π₯ = 0 β (π β π)) & β’ (π₯ = π¦ β (π β π)) & β’ (π₯ = (π¦ + 1) β (π β π)) & β’ (π₯ = -π¦ β (π β π)) & β’ (π₯ = π΄ β (π β π)) & β’ (π β π) & β’ (π β (π¦ β β0 β (π β π))) & β’ (π β (π¦ β β β (π β π))) β β’ (π β (π΄ β β€ β π)) | ||
Theorem | fzindd 12664* | Induction on the integers from M to N inclusive, a deduction version. (Contributed by metakunt, 12-May-2024.) |
β’ (π₯ = π β (π β π)) & β’ (π₯ = π¦ β (π β π)) & β’ (π₯ = (π¦ + 1) β (π β π)) & β’ (π₯ = π΄ β (π β π)) & β’ (π β π) & β’ ((π β§ (π¦ β β€ β§ π β€ π¦ β§ π¦ < π) β§ π) β π) & β’ (π β π β β€) & β’ (π β π β β€) & β’ (π β π β€ π) β β’ ((π β§ (π΄ β β€ β§ π β€ π΄ β§ π΄ β€ π)) β π) | ||
Theorem | btwnz 12665* | Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.) |
β’ (π΄ β β β (βπ₯ β β€ π₯ < π΄ β§ βπ¦ β β€ π΄ < π¦)) | ||
Theorem | zred 12666 | An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.) |
β’ (π β π΄ β β€) β β’ (π β π΄ β β) | ||
Theorem | zcnd 12667 | An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.) |
β’ (π β π΄ β β€) β β’ (π β π΄ β β) | ||
Theorem | znegcld 12668 | Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.) |
β’ (π β π΄ β β€) β β’ (π β -π΄ β β€) | ||
Theorem | peano2zd 12669 | Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.) |
β’ (π β π΄ β β€) β β’ (π β (π΄ + 1) β β€) | ||
Theorem | zaddcld 12670 | Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
β’ (π β π΄ β β€) & β’ (π β π΅ β β€) β β’ (π β (π΄ + π΅) β β€) | ||
Theorem | zsubcld 12671 | Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
β’ (π β π΄ β β€) & β’ (π β π΅ β β€) β β’ (π β (π΄ β π΅) β β€) | ||
Theorem | zmulcld 12672 | Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.) |
β’ (π β π΄ β β€) & β’ (π β π΅ β β€) β β’ (π β (π΄ Β· π΅) β β€) | ||
Theorem | znnn0nn 12673 | The negative of a negative integer, is a natural number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ ((π β β€ β§ Β¬ π β β0) β -π β β) | ||
Theorem | zadd2cl 12674 | Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.) |
β’ (π β β€ β (π + 2) β β€) | ||
Theorem | zriotaneg 12675* | The negative of the unique integer such that π. (Contributed by AV, 1-Dec-2018.) |
β’ (π₯ = -π¦ β (π β π)) β β’ (β!π₯ β β€ π β (β©π₯ β β€ π) = -(β©π¦ β β€ π)) | ||
Theorem | suprfinzcl 12676 | The supremum of a nonempty finite set of integers is a member of the set. (Contributed by AV, 1-Oct-2019.) |
β’ ((π΄ β β€ β§ π΄ β β β§ π΄ β Fin) β sup(π΄, β, < ) β π΄) | ||
Syntax | cdc 12677 | Constant used for decimal constructor. |
class ;π΄π΅ | ||
Definition | df-dec 12678 | Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 29699. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.) |
β’ ;π΄π΅ = (((9 + 1) Β· π΄) + π΅) | ||
Theorem | 9p1e10 12679 | 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.) |
β’ (9 + 1) = ;10 | ||
Theorem | dfdec10 12680 | Version of the definition of the "decimal constructor" using ;10 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.) |
β’ ;π΄π΅ = ((;10 Β· π΄) + π΅) | ||
Theorem | decex 12681 | A decimal number is a set. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
β’ ;π΄π΅ β V | ||
Theorem | deceq1 12682 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
β’ (π΄ = π΅ β ;π΄πΆ = ;π΅πΆ) | ||
Theorem | deceq2 12683 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
β’ (π΄ = π΅ β ;πΆπ΄ = ;πΆπ΅) | ||
Theorem | deceq1i 12684 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) |
β’ π΄ = π΅ β β’ ;π΄πΆ = ;π΅πΆ | ||
Theorem | deceq2i 12685 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) |
β’ π΄ = π΅ β β’ ;πΆπ΄ = ;πΆπ΅ | ||
Theorem | deceq12i 12686 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) |
β’ π΄ = π΅ & β’ πΆ = π· β β’ ;π΄πΆ = ;π΅π· | ||
Theorem | numnncl 12687 | Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.) |
β’ π β β0 & β’ π΄ β β0 & β’ π΅ β β β β’ ((π Β· π΄) + π΅) β β | ||
Theorem | num0u 12688 | Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.) |
β’ π β β0 & β’ π΄ β β0 β β’ (π Β· π΄) = ((π Β· π΄) + 0) | ||
Theorem | num0h 12689 | Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.) |
β’ π β β0 & β’ π΄ β β0 β β’ π΄ = ((π Β· 0) + π΄) | ||
Theorem | numcl 12690 | Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.) |
β’ π β β0 & β’ π΄ β β0 & β’ π΅ β β0 β β’ ((π Β· π΄) + π΅) β β0 | ||
Theorem | numsuc 12691 | The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
β’ π β β0 & β’ π΄ β β0 & β’ π΅ β β0 & β’ (π΅ + 1) = πΆ & β’ π = ((π Β· π΄) + π΅) β β’ (π + 1) = ((π Β· π΄) + πΆ) | ||
Theorem | deccl 12692 | Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
β’ π΄ β β0 & β’ π΅ β β0 β β’ ;π΄π΅ β β0 | ||
Theorem | 10nn 12693 | 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.) |
β’ ;10 β β | ||
Theorem | 10pos 12694 | The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.) |
β’ 0 < ;10 | ||
Theorem | 10nn0 12695 | 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
β’ ;10 β β0 | ||
Theorem | 10re 12696 | The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.) |
β’ ;10 β β | ||
Theorem | decnncl 12697 | Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
β’ π΄ β β0 & β’ π΅ β β β β’ ;π΄π΅ β β | ||
Theorem | dec0u 12698 | Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
β’ π΄ β β0 β β’ (;10 Β· π΄) = ;π΄0 | ||
Theorem | dec0h 12699 | Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
β’ π΄ β β0 β β’ π΄ = ;0π΄ | ||
Theorem | numnncl2 12700 | Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.) |
β’ π β β & β’ π΄ β β β β’ ((π Β· π΄) + 0) β β |
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