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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | prm2orodd 11201 | A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.) |
Theorem | 2prm 11202 | 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.) |
Theorem | 3prm 11203 | 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | 4nprm 11204 | 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.) |
Theorem | prmuz2 11205 | A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | prmgt1 11206 | A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.) |
Theorem | prmm2nn0 11207 | Subtracting 2 from a prime number results in a nonnegative integer. (Contributed by Alexander van der Vekens, 30-Aug-2018.) |
Theorem | oddprmgt2 11208 | An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.) |
Theorem | oddprmge3 11209 | 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 11210 | A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | dvdsprm 11211 | 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 11212* | 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 11213* | 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 11214 | No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.) |
Theorem | divgcdodd 11215 | 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 11218. | ||
Theorem | coprm 11216 | 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 11217 | Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | euclemma 11218 | 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 11219* | A number is prime iff it satisfies Euclid's lemma euclemma 11218. (Contributed by Mario Carneiro, 6-Sep-2015.) |
Theorem | prmdvdsexp 11220 | 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 11221 | 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 11222 | If a prime divides a nonnegative power of another, then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Theorem | prmexpb 11223 | Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.) |
Theorem | prmfac1 11224 | The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.) |
Theorem | rpexp 11225 | 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 11226 | Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.) |
Theorem | rpexp12i 11227 | Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.) |
Theorem | prmndvdsfaclt 11228 | 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 11229 | 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 11230 | 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 11231 | 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 11232 | 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 11233 | Lemma for sqrt2irr 11234. 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 11234 |
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 11251 for
the proof that the square root of two is irrational.
The proof's core is proven in sqrt2irrlem 11233, 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 11235 | The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.) |
Theorem | pw2dvdslemn 11236* | Lemma for pw2dvds 11237. 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 11237* | A natural number has a highest power of two which divides it. (Contributed by Jim Kingdon, 14-Nov-2021.) |
Theorem | pw2dvdseulemle 11238 | Lemma for pw2dvdseu 11239. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | pw2dvdseu 11239* | A natural number has a unique highest power of two which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdclemxy 11240* | Lemma for oddpwdc 11245. 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 11241* | Lemma for oddpwdc 11245. A natural number is divisible by the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | oddpwdclemndvds 11242* | Lemma for oddpwdc 11245. 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 11243* | Lemma for oddpwdc 11245. Removing the powers of two from a natural number produces an odd number. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdclemdc 11244* | Lemma for oddpwdc 11245. Decomposing a number into odd and even parts. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdc 11245* | 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 11246* | 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 11247* | 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 11248 | 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 11249 | The absolute value of the difference between two unequal integers is at least one. (Contributed by Jim Kingdon, 31-Jan-2022.) |
Theorem | sqrt2irraplemnn 11250 | Lemma for sqrt2irrap 11251. 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 11251 | 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 11234. (Contributed by Jim Kingdon, 2-Oct-2021.) |
# | ||
Syntax | cnumer 11252 | Extend class notation to include canonical numerator function. |
numer | ||
Syntax | cdenom 11253 | Extend class notation to include canonical denominator function. |
denom | ||
Definition | df-numer 11254* | 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 11255* | 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 11256* | Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | qdenval 11257* | Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumdencl 11258 | Lemma for qnumcl 11259 and qdencl 11260. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qnumcl 11259 | The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | qdencl 11260 | The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | fnum 11261 | Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | fden 11262 | Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumdenbi 11263 | 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 11264 | The canonical representation of a rational is fully reduced. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qeqnumdivden 11265 | Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qmuldeneqnum 11266 | Multiplying a rational by its denominator results in an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom numer | ||
Theorem | divnumden 11267 | Calculate the reduced form of a quotient using . (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | divdenle 11268 | Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumgt0 11269 | A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer | ||
Theorem | qgt0numnn 11270 | A rational is positive iff its canonical numerator is a positive integer. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer | ||
Theorem | nn0gcdsq 11271 | Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Theorem | zgcdsq 11272 | nn0gcdsq 11271 extended to integers by symmetry. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Theorem | numdensq 11273 | Squaring a rational squares its canonical components. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer numer denom denom | ||
Theorem | numsq 11274 | Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer numer | ||
Theorem | densq 11275 | Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
denom denom | ||
Theorem | qden1elz 11276 | A rational is an integer iff it has denominator 1. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
denom | ||
Theorem | nn0sqrtelqelz 11277 | If a nonnegative integer has a rational square root, that root must be an integer. (Contributed by Jim Kingdon, 24-May-2022.) |
Theorem | nonsq 11278 | Any integer strictly between two adjacent squares has a non-rational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Syntax | cphi 11279 | Extend class notation with the Euler phi function. |
Definition | df-phi 11280* | 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 11281* | Finiteness of an expression used to define the Euler function. (Contributed by Jim Kingon, 28-May-2022.) |
Theorem | phival 11282* | Value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
♯ | ||
Theorem | phicl2 11283 | Bounds and closure for the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phicl 11284 | Closure for the value of the Euler function. (Contributed by Mario Carneiro, 28-Feb-2014.) |
Theorem | phibndlem 11285* | Lemma for phibnd 11286. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phibnd 11286 | A slightly tighter bound on the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phicld 11287 | Closure for the value of the Euler function. (Contributed by Mario Carneiro, 29-May-2016.) |
Theorem | phi1 11288 | Value of the Euler function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | dfphi2 11289* | Alternate definition of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 2-May-2016.) |
♯ ..^ | ||
Theorem | hashdvds 11290* | 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 11291 | 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 11292 | Value of the Euler function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.) |
Theorem | crth 11293* | 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 11294* | Lemma for phimul 11295. (Contributed by Mario Carneiro, 24-Feb-2014.) |
..^ ..^ ..^ ..^ ..^ | ||
Theorem | phimul 11295 | 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 11296* | A correspondence between elements of specific GCD and relative primes in a smaller ring. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
..^ ..^ | ||
Theorem | hashgcdeq 11297* | Number of initial positive integers with specified divisors. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
♯ ..^ | ||
Theorem | oddennn 11298 | There are as many odd positive integers as there are positive integers. (Contributed by Jim Kingdon, 11-May-2022.) |
Theorem | evenennn 11299 | There are as many even positive integers as there are positive integers. (Contributed by Jim Kingdon, 12-May-2022.) |
Theorem | xpnnen 11300 | The Cartesian product of the set of positive integers with itself is equinumerous to the set of positive integers. (Contributed by NM, 1-Aug-2004.) |
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