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Theorem List for Metamath Proof Explorer - 13101-13200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremphibnd 13101 A slightly tighter bound on the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremphicld 13102 Closure for the value of the Euler function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremphi1 13103 Value of the Euler function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremdfphi2 13104* Alternate definition of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 2-May-2016.)
..^

Theoremhashdvds 13105* 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 13106 Value of the Euler function at a prime power. (Contributed by Mario Carneiro, 24-Feb-2014.)

Theoremphiprm 13107 Value of the Euler function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremcrt 13108* 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 13109* Lemma for phimul 13110. (Contributed by Mario Carneiro, 24-Feb-2014.)
..^        ..^ ..^                     ..^        ..^

Theoremphimul 13110 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 13111* Lemma for eulerth 13113. (Contributed by Mario Carneiro, 8-May-2015.)
..^

Theoremeulerthlem2 13112* Lemma for eulerth 13113. (Contributed by Mario Carneiro, 28-Feb-2014.)
..^

Theoremeulerth 13113 Euler's theorem, a generalization of Fermat's little theorem. If and are coprime, then , mod . (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremfermltl 13114 Fermat's little theorem. When is prime, , mod for any . (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremprmdiv 13115 Show an explicit expression for the modular inverse of . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremprmdiveq 13116 The modular inverse of is unique. (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremprmdivdiv 13117 The (modular) inverse of the inverse of a number is itself. (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremodzval 13118* 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 13119 - Lemma for odzcl 13120, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzcl 13120 The order of a group element is an integer. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzid 13121 Any element raised to the power of its order is . (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzdvds 13122 The only powers of that are congruent to are the multiples of the order of . (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzphi 13123 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 13124 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 13125 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 13126 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremomoe 13127 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremopeo 13128 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 13129 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 13130 A prime not equal to is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theorempythagtriplem1 13131* Lemma for pythagtrip 13149. 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 13132* Lemma for pythagtrip 13149. 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 13133 Lemma for pythagtrip 13149. Show that and are relatively prime under some conditions. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem4 13134 Lemma for pythagtrip 13149. Show that and are relatively prime. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem10 13135 Lemma for pythagtrip 13149. Show that is positive. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem6 13136 Lemma for pythagtrip 13149. Calculate . (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem7 13137 Lemma for pythagtrip 13149. Calculate . (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem8 13138 Lemma for pythagtrip 13149. Show that is a natural number (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem9 13139 Lemma for pythagtrip 13149. Show that is a natural number (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem11 13140 Lemma for pythagtrip 13149. 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 13141 Lemma for pythagtrip 13149. Calculate the square of . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem13 13142 Lemma for pythagtrip 13149. 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 13143 Lemma for pythagtrip 13149. Calculate the square of . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem15 13144 Lemma for pythagtrip 13149. Show the relationship between , , and . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem16 13145 Lemma for pythagtrip 13149. Show the relationship between , , and . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem17 13146 Lemma for pythagtrip 13149. Show the relationship between , , and . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem18 13147* Lemma for pythagtrip 13149. Wrap the previous and up in quanitifers. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem19 13148* Lemma for pythagtrip 13149. Introduce and remove the relative primality requirement. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtrip 13149* 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 13150* Collect the odd terms in a sequence. (Contributed by Mario Carneiro, 7-Apr-2015.)

6.2.5  The prime count function

Syntaxcpc 13151 Extend class notation with the prime count function.

Definitiondf-pc 13152* 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 13153* - Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcprecl 13154* Closure of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcprendvds 13155* Non-divisibility property of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcprendvds2 13156* Non-divisibility property of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcpre1 13157* Value of the prime power pre-function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 26-Apr-2016.)

Theorempcpremul 13158* 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 13159* The value of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)

Theorempceulem 13160* Lemma for pceu 13161. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempceu 13161* Uniqueness for the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempczpre 13162* 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 13163 Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempccl 13164 Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempccld 13165 Closure of the prime power function. (Contributed by Mario Carneiro, 29-May-2016.)

Theorempcmul 13166 Multiplication property of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcdiv 13167 Division property of the prime power function. (Contributed by Mario Carneiro, 1-Mar-2014.)

Theorempcqmul 13168 Multiplication property of the prime power function. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempc0 13169 The value of the prime power function at zero. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempc1 13170 Value of the prime count function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcqcl 13171 Closure of the general prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcqdiv 13172 Division property of the prime power function. (Contributed by Mario Carneiro, 10-Aug-2015.)

Theorempcrec 13173 Prime power of a reciprocal. (Contributed by Mario Carneiro, 10-Aug-2015.)

Theorempcexp 13174 Prime power of an exponential. (Contributed by Mario Carneiro, 10-Aug-2015.)

Theorempcxcl 13175 Extended real closure of the general prime count function. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcge0 13176 The prime count of an integer is greater or equal to zero. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempczdvds 13177 Defining property of the prime count function. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcdvds 13178 Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempczndvds 13179 Defining property of the prime count function. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcndvds 13180 Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempczndvds2 13181 The remainder after dividing out all factors of is not divisible by . (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcndvds2 13182 The remainder after dividing out all factors of is not divisible by . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcdvdsb 13183 divides if and only if is at most the count of . (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcelnn 13184 There are a positive number of powers of a prime in iff divides . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempceq0 13185 There are zero powers of a prime in iff does not divide . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcidlem 13186 The prime count of a prime power. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorempcid 13187 The prime count of a prime power. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcneg 13188 The prime count of a negative number. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorempcabs 13189 The prime count of an absolute value. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorempcdvdstr 13190 The prime count increases under the divisibility relation. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorempcgcd1 13191 The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcgcd 13192 The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempc2dvds 13193* A characterization of divisibility in terms of prime count. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)

Theorempc11 13194* The prime count function, viewed as a function from to , is one-to-one. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcz 13195* The prime count function can be used as an indicator that a given rational number is an integer. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcprmpw2 13196* Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theorempcprmpw 13197* Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theorempcaddlem 13198 Lemma for pcadd 13199. The original numbers and have been decomposed using the prime count function as where are both not divisible by and , and similarly for . (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcadd 13199 An inequality for the prime count of a sum. This is the source of the ultrametric inequality for the p-adic metric. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcadd2 13200 The inequality of pcadd 13199 becomes an equality when one of the factors has prime count strictly less than the other. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)

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