Home | Intuitionistic Logic Explorer Theorem List (p. 118 of 131) | < Previous Next > |
Browser slow? Try the
Unicode version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nprmi 11701 | An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.) |
Theorem | dvdsnprmd 11702 | If a number is divisible by an integer greater than 1 and less then the number, the number is not prime. (Contributed by AV, 24-Jul-2021.) |
Theorem | prm2orodd 11703 | A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.) |
Theorem | 2prm 11704 | 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.) |
Theorem | 3prm 11705 | 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | 4nprm 11706 | 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.) |
Theorem | prmuz2 11707 | A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | prmgt1 11708 | A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.) |
Theorem | prmm2nn0 11709 | Subtracting 2 from a prime number results in a nonnegative integer. (Contributed by Alexander van der Vekens, 30-Aug-2018.) |
Theorem | oddprmgt2 11710 | An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.) |
Theorem | oddprmge3 11711 | An odd prime is greater than or equal to 3. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 20-Aug-2021.) |
Theorem | sqnprm 11712 | A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | dvdsprm 11713 | An integer greater than or equal to 2 divides a prime number iff it is equal to it. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | exprmfct 11714* | Every integer greater than or equal to 2 has a prime factor. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 20-Jun-2015.) |
Theorem | prmdvdsfz 11715* | Each integer greater than 1 and less then or equal to a fixed number is divisible by a prime less then or equal to this fixed number. (Contributed by AV, 15-Aug-2020.) |
Theorem | nprmdvds1 11716 | No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.) |
Theorem | divgcdodd 11717 | Either is odd or is odd. (Contributed by Scott Fenton, 19-Apr-2014.) |
This section is about coprimality with respect to primes, and a special version of Euclid's lemma for primes is provided, see euclemma 11720. | ||
Theorem | coprm 11718 | A prime number either divides an integer or is coprime to it, but not both. Theorem 1.8 in [ApostolNT] p. 17. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | prmrp 11719 | Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | euclemma 11720 | Euclid's lemma. A prime number divides the product of two integers iff it divides at least one of them. Theorem 1.9 in [ApostolNT] p. 17. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | isprm6 11721* | A number is prime iff it satisfies Euclid's lemma euclemma 11720. (Contributed by Mario Carneiro, 6-Sep-2015.) |
Theorem | prmdvdsexp 11722 | A prime divides a positive power of an integer iff it divides the integer. (Contributed by Mario Carneiro, 24-Feb-2014.) (Revised by Mario Carneiro, 17-Jul-2014.) |
Theorem | prmdvdsexpb 11723 | A prime divides a positive power of another iff they are equal. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 24-Feb-2014.) |
Theorem | prmdvdsexpr 11724 | If a prime divides a nonnegative power of another, then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Theorem | prmexpb 11725 | Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.) |
Theorem | prmfac1 11726 | The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.) |
Theorem | rpexp 11727 | If two numbers and are relatively prime, then they are still relatively prime if raised to a power. (Contributed by Mario Carneiro, 24-Feb-2014.) |
Theorem | rpexp1i 11728 | Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.) |
Theorem | rpexp12i 11729 | Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.) |
Theorem | prmndvdsfaclt 11730 | A prime number does not divide the factorial of a nonnegative integer less than the prime number. (Contributed by AV, 13-Jul-2021.) |
Theorem | cncongrprm 11731 | Corollary 2 of Cancellability of Congruences: Two products with a common factor are congruent modulo a prime number not dividing the common factor iff the other factors are congruent modulo the prime number. (Contributed by AV, 13-Jul-2021.) |
Theorem | isevengcd2 11732 | The predicate "is an even number". An even number and 2 have 2 as greatest common divisor. (Contributed by AV, 1-Jul-2020.) (Revised by AV, 8-Aug-2021.) |
Theorem | isoddgcd1 11733 | The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020.) (Revised by AV, 8-Aug-2021.) |
Theorem | 3lcm2e6 11734 | The least common multiple of three and two is six. The operands are unequal primes and thus coprime, so the result is (the absolute value of) their product. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 27-Aug-2020.) |
lcm | ||
Theorem | sqrt2irrlem 11735 | Lemma for sqrt2irr 11736. This is the core of the proof: - if , then and are even, so and are smaller representatives, which is absurd by the method of infinite descent (here implemented by strong induction). (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.) |
Theorem | sqrt2irr 11736 |
The square root of 2 is not rational. That is, for any rational number,
does not equal it. However, if we were to say "the
square root of 2 is irrational" that would mean something stronger:
"for any rational number, is apart from it" (the two
statements are equivalent given excluded middle). See sqrt2irrap 11753 for
the proof that the square root of two is irrational.
The proof's core is proven in sqrt2irrlem 11735, which shows that if , then and are even, so and are smaller representatives, which is absurd. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
Theorem | sqrt2re 11737 | The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.) |
Theorem | pw2dvdslemn 11738* | Lemma for pw2dvds 11739. If a natural number has some power of two which does not divide it, there is a highest power of two which does divide it. (Contributed by Jim Kingdon, 14-Nov-2021.) |
Theorem | pw2dvds 11739* | A natural number has a highest power of two which divides it. (Contributed by Jim Kingdon, 14-Nov-2021.) |
Theorem | pw2dvdseulemle 11740 | Lemma for pw2dvdseu 11741. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | pw2dvdseu 11741* | A natural number has a unique highest power of two which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdclemxy 11742* | Lemma for oddpwdc 11747. Another way of stating that decomposing a natural number into a power of two and an odd number is unique. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdclemdvds 11743* | Lemma for oddpwdc 11747. A natural number is divisible by the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | oddpwdclemndvds 11744* | Lemma for oddpwdc 11747. A natural number is not divisible by one more than the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | oddpwdclemodd 11745* | Lemma for oddpwdc 11747. Removing the powers of two from a natural number produces an odd number. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdclemdc 11746* | Lemma for oddpwdc 11747. Decomposing a number into odd and even parts. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdc 11747* | The function that decomposes a number into its "odd" and "even" parts, which is to say the largest power of two and largest odd divisor of a number, is a bijection from pairs of a nonnegative integer and an odd number to positive integers. (Contributed by Thierry Arnoux, 15-Aug-2017.) |
Theorem | sqpweven 11748* | The greatest power of two dividing the square of an integer is an even power of two. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | 2sqpwodd 11749* | The greatest power of two dividing twice the square of an integer is an odd power of two. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | sqne2sq 11750 | The square of a natural number can never be equal to two times the square of a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | znege1 11751 | The absolute value of the difference between two unequal integers is at least one. (Contributed by Jim Kingdon, 31-Jan-2022.) |
Theorem | sqrt2irraplemnn 11752 | Lemma for sqrt2irrap 11753. The square root of 2 is apart from a positive rational expressed as a numerator and denominator. (Contributed by Jim Kingdon, 2-Oct-2021.) |
# | ||
Theorem | sqrt2irrap 11753 | The square root of 2 is irrational. That is, for any rational number, is apart from it. In the absence of excluded middle, we can distinguish between this and "the square root of 2 is not rational" which is sqrt2irr 11736. (Contributed by Jim Kingdon, 2-Oct-2021.) |
# | ||
Syntax | cnumer 11754 | Extend class notation to include canonical numerator function. |
numer | ||
Syntax | cdenom 11755 | Extend class notation to include canonical denominator function. |
denom | ||
Definition | df-numer 11756* | The canonical numerator of a rational is the numerator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Definition | df-denom 11757* | The canonical denominator of a rational is the denominator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumval 11758* | Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | qdenval 11759* | Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumdencl 11760 | Lemma for qnumcl 11761 and qdencl 11762. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qnumcl 11761 | The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | qdencl 11762 | The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | fnum 11763 | Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | fden 11764 | Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumdenbi 11765 | Two numbers are the canonical representation of a rational iff they are coprime and have the right quotient. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qnumdencoprm 11766 | The canonical representation of a rational is fully reduced. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qeqnumdivden 11767 | Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qmuldeneqnum 11768 | Multiplying a rational by its denominator results in an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom numer | ||
Theorem | divnumden 11769 | Calculate the reduced form of a quotient using . (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | divdenle 11770 | Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumgt0 11771 | A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer | ||
Theorem | qgt0numnn 11772 | A rational is positive iff its canonical numerator is a positive integer. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer | ||
Theorem | nn0gcdsq 11773 | Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Theorem | zgcdsq 11774 | nn0gcdsq 11773 extended to integers by symmetry. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Theorem | numdensq 11775 | Squaring a rational squares its canonical components. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer numer denom denom | ||
Theorem | numsq 11776 | Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer numer | ||
Theorem | densq 11777 | Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
denom denom | ||
Theorem | qden1elz 11778 | A rational is an integer iff it has denominator 1. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
denom | ||
Theorem | nn0sqrtelqelz 11779 | If a nonnegative integer has a rational square root, that root must be an integer. (Contributed by Jim Kingdon, 24-May-2022.) |
Theorem | nonsq 11780 | Any integer strictly between two adjacent squares has a non-rational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Syntax | cphi 11781 | Extend class notation with the Euler phi function. |
Definition | df-phi 11782* | Define the Euler phi function (also called "Euler totient function"), which counts the number of integers less than and coprime to it, see definition in [ApostolNT] p. 25. (Contributed by Mario Carneiro, 23-Feb-2014.) |
♯ | ||
Theorem | phivalfi 11783* | Finiteness of an expression used to define the Euler function. (Contributed by Jim Kingon, 28-May-2022.) |
Theorem | phival 11784* | Value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
♯ | ||
Theorem | phicl2 11785 | Bounds and closure for the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phicl 11786 | Closure for the value of the Euler function. (Contributed by Mario Carneiro, 28-Feb-2014.) |
Theorem | phibndlem 11787* | Lemma for phibnd 11788. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phibnd 11788 | A slightly tighter bound on the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phicld 11789 | Closure for the value of the Euler function. (Contributed by Mario Carneiro, 29-May-2016.) |
Theorem | phi1 11790 | Value of the Euler function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | dfphi2 11791* | Alternate definition of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 2-May-2016.) |
♯ ..^ | ||
Theorem | hashdvds 11792* | 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.) |
♯ | ||
Theorem | phiprmpw 11793 | Value of the Euler function at a prime power. Theorem 2.5(a) in [ApostolNT] p. 28. (Contributed by Mario Carneiro, 24-Feb-2014.) |
Theorem | phiprm 11794 | Value of the Euler function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.) |
Theorem | crth 11795* | 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.) |
..^ ..^ ..^ | ||
Theorem | phimullem 11796* | Lemma for phimul 11797. (Contributed by Mario Carneiro, 24-Feb-2014.) |
..^ ..^ ..^ ..^ ..^ | ||
Theorem | phimul 11797 | The Euler function is a multiplicative function, meaning that it distributes over multiplication at relatively prime arguments. Theorem 2.5(c) in [ApostolNT] p. 28. (Contributed by Mario Carneiro, 24-Feb-2014.) |
Theorem | hashgcdlem 11798* | A correspondence between elements of specific GCD and relative primes in a smaller ring. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
..^ ..^ | ||
Theorem | hashgcdeq 11799* | Number of initial positive integers with specified divisors. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
♯ ..^ | ||
Theorem | oddennn 11800 | There are as many odd positive integers as there are positive integers. (Contributed by Jim Kingdon, 11-May-2022.) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |