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Theorem List for Metamath Proof Explorer - 12701-12800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdensq 12701 Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.)
denom denom

Theoremqden1elz 12702 A rational is an integer iff it has denominator 1. (Contributed by Stefan O'Rear, 15-Sep-2014.)
denom

Theoremzsqrelqelz 12703 If an integer has a rational square root, that root is must be an integer. (Contributed by Stefan O'Rear, 15-Sep-2014.)

Theoremnonsq 12704 Any integer strictly between two adjacent squares has an irrational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.)

6.2.3  Euler's theorem

Syntaxcodz 12705 Extend class notation with the order function on the class of integers mod N.

Syntaxcphi 12706 Extend class notation with the Euler phi function.

Definitiondf-odz 12707* Define the order function on the class of integers mod N. (Contributed by Mario Carneiro, 23-Feb-2014.)

Definitiondf-phi 12708* Define the Euler phi function, which counts the number of integers less than and coprime to it. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremphival 12709* Value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremphicl2 12710 Bounds and closure for the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremphicl 12711 Closure for the value of the Euler function. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremphibndlem 12712* Lemma for phibnd 12713. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremphibnd 12713 A slightly tighter bound on the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremphicld 12714 Closure for the value of the Euler function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremphi1 12715 Value of the Euler function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremdfphi2 12716* Alternate definition of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 2-May-2016.)
..^

Theoremhashdvds 12717* The number of numbers in a given residue class in a finite set of integers. (Contributed by Mario Carneiro, 12-Mar-2014.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)

Theoremphiprmpw 12718 Value of the Euler function at a prime power. (Contributed by Mario Carneiro, 24-Feb-2014.)

Theoremphiprm 12719 Value of the Euler function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremcrt 12720* The Chinese Remainder Theorem: the function that maps to its remainder classes and is 1-1 and onto when and are coprime. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-May-2016.)
..^        ..^ ..^

Theoremphimullem 12721* Lemma for phimul 12722. (Contributed by Mario Carneiro, 24-Feb-2014.)
..^        ..^ ..^                     ..^        ..^

Theoremphimul 12722 The Euler function is a multiplicative function, meaning that it distributes over multiplication at relatively prime arguments. (Contributed by Mario Carneiro, 24-Feb-2014.)

Theoremeulerthlem1 12723* Lemma for eulerth 12725. (Contributed by Mario Carneiro, 8-May-2015.)
..^

Theoremeulerthlem2 12724* Lemma for eulerth 12725. (Contributed by Mario Carneiro, 28-Feb-2014.)
..^

Theoremeulerth 12725 Euler's theorem, a generalization of Fermat's little theorem. If and are coprime, then , mod . (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremfermltl 12726 Fermat's little theorem. When is prime, , mod for any . (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremprmdiv 12727 Show an explicit expression for the modular inverse of . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremprmdiveq 12728 The modular inverse of is unique. (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremprmdivdiv 12729 The (modular) inverse of the inverse of a number is itself. (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremodzval 12730* Value of the order function. This is a function of functions; the inner argument selects the base (i.e. mod for some , often prime) and the outer argument selects the integer or equivalence class (if you want to think about it that way) from the integers mod . In order to ensure the supremum is well-defined, we only define the expression when and are coprime. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremodzcllem 12731 - Lemma for odzcl 12732, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzcl 12732 The order of a group element is an integer. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzid 12733 Any element raised to the power of its order is . (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzdvds 12734 The only powers of that are congruent to are the multiples of the order of . (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzphi 12735 The order of any group element is a divisor of the Euler function. (Contributed by Mario Carneiro, 28-Feb-2014.)

6.2.4  Pythagorean Triples

Theoremcoprimeprodsq 12736 If three numbers are coprime, and the square of one is the product of the other two, then there is a formula for the other two in terms of and square. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremcoprimeprodsq2 12737 If three numbers are coprime, and the square of one is the product of the other two, then there is a formula for the other two in terms of and square. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremopoe 12738 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremomoe 12739 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremopeo 12740 The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremomeo 12741 The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremoddprm 12742 A prime not equal to is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theorempythagtriplem1 12743* Lemma for pythagtrip 12761. Prove a weaker version of one direction of the theorem. (Contributed by Scott Fenton, 28-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem2 12744* Lemma for pythagtrip 12761. Prove the full version of one direction of the theorem. (Contributed by Scott Fenton, 28-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem3 12745 Lemma for pythagtrip 12761. Show that and are relatively prime under some conditions. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem4 12746 Lemma for pythagtrip 12761. Show that and are relatively prime. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem10 12747 Lemma for pythagtrip 12761. Show that is positive. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem6 12748 Lemma for pythagtrip 12761. Calculate . (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem7 12749 Lemma for pythagtrip 12761. Calculate . (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem8 12750 Lemma for pythagtrip 12761. Show that is a natural number (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem9 12751 Lemma for pythagtrip 12761. Show that is a natural number (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem11 12752 Lemma for pythagtrip 12761. Show that (which will eventually be closely related to the in the final statement) is a natural. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem12 12753 Lemma for pythagtrip 12761. Calculate the square of . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem13 12754 Lemma for pythagtrip 12761. Show that (which will eventually be closely related to the in the final statement) is a natural. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem14 12755 Lemma for pythagtrip 12761. Calculate the square of . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem15 12756 Lemma for pythagtrip 12761. Show the relationship between , , and . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem16 12757 Lemma for pythagtrip 12761. Show the relationship between , , and . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem17 12758 Lemma for pythagtrip 12761. Show the relationship between , , and . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem18 12759* Lemma for pythagtrip 12761. Wrap the previous and up in quanitifers. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem19 12760* Lemma for pythagtrip 12761. Introduce and remove the relative primality requirement. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtrip 12761* Parameterize the Pythagorean triples. If , , and are naturals, then they obey the Pythagorean triple formula iff they are parameterized by three naturals. This proof follows the Isabelle proof at http://afp.sourceforge.net/entries/Fermat3_4.shtml. (Contributed by Scott Fenton, 19-Apr-2014.)

Theoremiserodd 12762* Collect the odd terms in a sequence. (Contributed by Mario Carneiro, 7-Apr-2015.)

6.2.5  The prime count function

Syntaxcpc 12763 Extend class notation with the prime count function.

Definitiondf-pc 12764* Define the prime count function, which returns the largest exponent of a given prime (or other natural number) that divides the number. For rational numbers, it returns negative values according to the power of a prime in the denominator. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempclem 12765* - Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcprecl 12766* Closure of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcprendvds 12767* Non-divisibility property of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcprendvds2 12768* Non-divisibility property of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcpre1 12769* Value of the prime power pre-function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 26-Apr-2016.)

Theorempcpremul 12770* Multiplicative property of the prime count pre-function. Note that the primality of is essential for this property; but . Since this is needed to show uniqueness for the real prime count function (over ), we don't bother to define it off the primes. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcval 12771* The value of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)

Theorempceulem 12772* Lemma for pceu 12773. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempceu 12773* Uniqueness for the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempczpre 12774* Connect the prime count pre-function to the actual prime count function, when restricted to the integers. (Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Theorempczcl 12775 Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempccl 12776 Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempccld 12777 Closure of the prime power function. (Contributed by Mario Carneiro, 29-May-2016.)

Theorempcmul 12778 Multiplication property of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcdiv 12779 Division property of the prime power function. (Contributed by Mario Carneiro, 1-Mar-2014.)

Theorempcqmul 12780 Multiplication property of the prime power function. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempc0 12781 The value of the prime power function at zero. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempc1 12782 Value of the prime count function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcqcl 12783 Closure of the general prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcqdiv 12784 Division property of the prime power function. (Contributed by Mario Carneiro, 10-Aug-2015.)

Theorempcrec 12785 Prime power of a reciprocal. (Contributed by Mario Carneiro, 10-Aug-2015.)

Theorempcexp 12786 Prime power of an exponential. (Contributed by Mario Carneiro, 10-Aug-2015.)

Theorempcxcl 12787 Extended real closure of the general prime count function. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcge0 12788 The prime count of an integer is greater or equal to zero. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempczdvds 12789 Defining property of the prime count function. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcdvds 12790 Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempczndvds 12791 Defining property of the prime count function. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcndvds 12792 Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempczndvds2 12793 The remainder after dividing out all factors of is not divisible by . (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcndvds2 12794 The remainder after dividing out all factors of is not divisible by . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcdvdsb 12795 divides if and only if is at most the count of . (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcelnn 12796 There are a positive number of powers of a prime in iff divides . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempceq0 12797 There are zero powers of a prime in iff does not divide . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcidlem 12798 The prime count of a prime power. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorempcid 12799 The prime count of a prime power. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcneg 12800 The prime count of a negative number. (Contributed by Mario Carneiro, 13-Mar-2014.)

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