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

Theoremprmdiv 13101 Show an explicit expression for the modular inverse of . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremprmdiveq 13102 The modular inverse of is unique. (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremprmdivdiv 13103 The (modular) inverse of the inverse of a number is itself. (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremodzval 13104* 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 13105 - Lemma for odzcl 13106, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzcl 13106 The order of a group element is an integer. (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzid 13107 Any element raised to the power of its order is . (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzdvds 13108 The only powers of that are congruent to are the multiples of the order of . (Contributed by Mario Carneiro, 28-Feb-2014.)

Theoremodzphi 13109 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 13110 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 13111 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 13112 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremomoe 13113 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremopeo 13114 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 13115 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 13116 A prime not equal to is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theorempythagtriplem1 13117* Lemma for pythagtrip 13135. 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 13118* Lemma for pythagtrip 13135. 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 13119 Lemma for pythagtrip 13135. Show that and are relatively prime under some conditions. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem4 13120 Lemma for pythagtrip 13135. Show that and are relatively prime. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem10 13121 Lemma for pythagtrip 13135. Show that is positive. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem6 13122 Lemma for pythagtrip 13135. Calculate . (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem7 13123 Lemma for pythagtrip 13135. Calculate . (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem8 13124 Lemma for pythagtrip 13135. Show that is a natural number (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem9 13125 Lemma for pythagtrip 13135. Show that is a natural number (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem11 13126 Lemma for pythagtrip 13135. 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 13127 Lemma for pythagtrip 13135. Calculate the square of . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem13 13128 Lemma for pythagtrip 13135. 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 13129 Lemma for pythagtrip 13135. Calculate the square of . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem15 13130 Lemma for pythagtrip 13135. Show the relationship between , , and . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem16 13131 Lemma for pythagtrip 13135. Show the relationship between , , and . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem17 13132 Lemma for pythagtrip 13135. Show the relationship between , , and . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem18 13133* Lemma for pythagtrip 13135. Wrap the previous and up in quanitifers. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtriplem19 13134* Lemma for pythagtrip 13135. Introduce and remove the relative primality requirement. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorempythagtrip 13135* 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 13136* Collect the odd terms in a sequence. (Contributed by Mario Carneiro, 7-Apr-2015.)

6.2.5  The prime count function

Syntaxcpc 13137 Extend class notation with the prime count function.

Definitiondf-pc 13138* 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 13139* - Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcprecl 13140* Closure of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcprendvds 13141* Non-divisibility property of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcprendvds2 13142* Non-divisibility property of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcpre1 13143* Value of the prime power pre-function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 26-Apr-2016.)

Theorempcpremul 13144* 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 13145* The value of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)

Theorempceulem 13146* Lemma for pceu 13147. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempceu 13147* Uniqueness for the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempczpre 13148* 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 13149 Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempccl 13150 Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempccld 13151 Closure of the prime power function. (Contributed by Mario Carneiro, 29-May-2016.)

Theorempcmul 13152 Multiplication property of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcdiv 13153 Division property of the prime power function. (Contributed by Mario Carneiro, 1-Mar-2014.)

Theorempcqmul 13154 Multiplication property of the prime power function. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempc0 13155 The value of the prime power function at zero. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempc1 13156 Value of the prime count function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcqcl 13157 Closure of the general prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcqdiv 13158 Division property of the prime power function. (Contributed by Mario Carneiro, 10-Aug-2015.)

Theorempcrec 13159 Prime power of a reciprocal. (Contributed by Mario Carneiro, 10-Aug-2015.)

Theorempcexp 13160 Prime power of an exponential. (Contributed by Mario Carneiro, 10-Aug-2015.)

Theorempcxcl 13161 Extended real closure of the general prime count function. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcge0 13162 The prime count of an integer is greater or equal to zero. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempczdvds 13163 Defining property of the prime count function. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcdvds 13164 Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempczndvds 13165 Defining property of the prime count function. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcndvds 13166 Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempczndvds2 13167 The remainder after dividing out all factors of is not divisible by . (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcndvds2 13168 The remainder after dividing out all factors of is not divisible by . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcdvdsb 13169 divides if and only if is at most the count of . (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcelnn 13170 There are a positive number of powers of a prime in iff divides . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempceq0 13171 There are zero powers of a prime in iff does not divide . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcidlem 13172 The prime count of a prime power. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorempcid 13173 The prime count of a prime power. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcneg 13174 The prime count of a negative number. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorempcabs 13175 The prime count of an absolute value. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorempcdvdstr 13176 The prime count increases under the divisibility relation. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorempcgcd1 13177 The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcgcd 13178 The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempc2dvds 13179* A characterization of divisibility in terms of prime count. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)

Theorempc11 13180* The prime count function, viewed as a function from to , is one-to-one. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcz 13181* 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 13182* Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theorempcprmpw 13183* Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theorempcaddlem 13184 Lemma for pcadd 13185. 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 13185 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 13186 The inequality of pcadd 13185 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.)

Theorempcmptcl 13187 Closure for the prime power map. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorempcmpt 13188* Construct a function with given prime count characteristics. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorempcmpt2 13189* Dividing two prime count maps yields a number with all dividing primes confined to an interval. (Contributed by Mario Carneiro, 14-Mar-2014.)

Theorempcmptdvds 13190 The partial products of the prime power map form a divisibility chain. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorempcprod 13191* The product of the primes taken to their respective powers reconstructs the original number. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theoremsumhash 13192* The sum of 1 over a set is the size of the set. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 20-May-2014.)

Theoremfldivp1 13193 The difference between the floors of adjacent fractions is either 1 or 0. (Contributed by Mario Carneiro, 8-Mar-2014.)

Theorempcfaclem 13194 Lemma for pcfac 13195. (Contributed by Mario Carneiro, 20-May-2014.)

Theorempcfac 13195* Calculate the prime count of a factorial. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)

Theorempcbc 13196* Calculate the prime count of a binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)

Theoremqexpz 13197 If a power of a rational number is an integer, then the number is an integer. In other words, all n-th roots are irrational unless they are integers (so that the original number is an n-th power). (Contributed by Mario Carneiro, 10-Aug-2015.)

Theoremexpnprm 13198 A second or higher power of a rational number is not a prime number. Or by contraposition, the n-th root of a prime number is irrational. Suggested by Norm Megill. (Contributed by Mario Carneiro, 10-Aug-2015.)

6.2.6  Pocklington's theorem

Theoremprmpwdvds 13199 A relation involving divisibility by a prime power. (Contributed by Mario Carneiro, 2-Mar-2014.)

Theorempockthlem 13200 Lemma for pockthg 13201. (Contributed by Mario Carneiro, 2-Mar-2014.)

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