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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | prmex 11001 | The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.) |
Theorem | 1nprm 11002 | 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.) |
Theorem | 1idssfct 11003* | The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | isprm2lem 11004* | Lemma for isprm2 11005. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | isprm2 11005* | The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only positive divisors are 1 and itself. Definition in [ApostolNT] p. 16. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | isprm3 11006* | The predicate "is a prime number". A prime number is an integer greater than or equal to 2 with no divisors strictly between 1 and itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | isprm4 11007* | The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only divisor greater than or equal to 2 is itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | prmind2 11008* | A variation on prmind 11009 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | prmind 11009* | Perform induction over the multiplicative structure of . If a property holds for the primes and and is preserved under multiplication, then it holds for every positive integer. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | dvdsprime 11010 | If divides a prime, then is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.) |
Theorem | nprm 11011 | A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | nprmi 11012 | An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.) |
Theorem | dvdsnprmd 11013 | 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 11014 | A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.) |
Theorem | 2prm 11015 | 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.) |
Theorem | 3prm 11016 | 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | 4nprm 11017 | 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.) |
Theorem | prmuz2 11018 | A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | prmgt1 11019 | A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.) |
Theorem | prmm2nn0 11020 | Subtracting 2 from a prime number results in a nonnegative integer. (Contributed by Alexander van der Vekens, 30-Aug-2018.) |
Theorem | oddprmgt2 11021 | An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.) |
Theorem | oddprmge3 11022 | 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 11023 | A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | dvdsprm 11024 | 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 11025* | 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 11026* | 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 11027 | No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.) |
Theorem | divgcdodd 11028 | 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 11031. | ||
Theorem | coprm 11029 | 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 11030 | Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | euclemma 11031 | 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 11032* | A number is prime iff it satisfies Euclid's lemma euclemma 11031. (Contributed by Mario Carneiro, 6-Sep-2015.) |
Theorem | prmdvdsexp 11033 | 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 11034 | 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 11035 | If a prime divides a nonnegative power of another, then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Theorem | prmexpb 11036 | Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.) |
Theorem | prmfac1 11037 | The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.) |
Theorem | rpexp 11038 | 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 11039 | Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.) |
Theorem | rpexp12i 11040 | Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.) |
Theorem | prmndvdsfaclt 11041 | 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 11042 | 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 11043 | 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 11044 | 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 11045 | 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 11046 | Lemma for sqrt2irr 11047. 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 11047 |
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 11064 for
the proof that the square root of two is irrational.
The proof's core is proven in sqrt2irrlem 11046, 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 11048 | The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.) |
Theorem | pw2dvdslemn 11049* | Lemma for pw2dvds 11050. 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 11050* | A natural number has a highest power of two which divides it. (Contributed by Jim Kingdon, 14-Nov-2021.) |
Theorem | pw2dvdseulemle 11051 | Lemma for pw2dvdseu 11052. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | pw2dvdseu 11052* | A natural number has a unique highest power of two which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdclemxy 11053* | Lemma for oddpwdc 11058. 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 11054* | Lemma for oddpwdc 11058. A natural number is divisible by the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | oddpwdclemndvds 11055* | Lemma for oddpwdc 11058. 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 11056* | Lemma for oddpwdc 11058. Removing the powers of two from a natural number produces an odd number. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdclemdc 11057* | Lemma for oddpwdc 11058. Decomposing a number into odd and even parts. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdc 11058* | 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 11059* | 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 11060* | 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 11061 | 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 11062 | The absolute value of the difference between two unequal integers is at least one. (Contributed by Jim Kingdon, 31-Jan-2022.) |
Theorem | sqrt2irraplemnn 11063 | Lemma for sqrt2irrap 11064. 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 11064 | 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 11047. (Contributed by Jim Kingdon, 2-Oct-2021.) |
# | ||
Syntax | cnumer 11065 | Extend class notation to include canonical numerator function. |
numer | ||
Syntax | cdenom 11066 | Extend class notation to include canonical denominator function. |
denom | ||
Definition | df-numer 11067* | 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 11068* | 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 11069* | Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | qdenval 11070* | Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumdencl 11071 | Lemma for qnumcl 11072 and qdencl 11073. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qnumcl 11072 | The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | qdencl 11073 | The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | fnum 11074 | Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | fden 11075 | Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumdenbi 11076 | 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 11077 | The canonical representation of a rational is fully reduced. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qeqnumdivden 11078 | Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qmuldeneqnum 11079 | Multiplying a rational by its denominator results in an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom numer | ||
Theorem | divnumden 11080 | Calculate the reduced form of a quotient using . (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | divdenle 11081 | Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumgt0 11082 | A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer | ||
Theorem | qgt0numnn 11083 | A rational is positive iff its canonical numerator is a positive integer. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer | ||
Theorem | nn0gcdsq 11084 | Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Theorem | zgcdsq 11085 | nn0gcdsq 11084 extended to integers by symmetry. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Theorem | numdensq 11086 | Squaring a rational squares its canonical components. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer numer denom denom | ||
Theorem | numsq 11087 | Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer numer | ||
Theorem | densq 11088 | Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
denom denom | ||
Theorem | qden1elz 11089 | A rational is an integer iff it has denominator 1. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
denom | ||
Theorem | nn0sqrtelqelz 11090 | If a nonnegative integer has a rational square root, that root must be an integer. (Contributed by Jim Kingdon, 24-May-2022.) |
Theorem | nonsq 11091 | Any integer strictly between two adjacent squares has a non-rational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Syntax | cphi 11092 | Extend class notation with the Euler phi function. |
Definition | df-phi 11093* | 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 11094* | Finiteness of an expression used to define the Euler function. (Contributed by Jim Kingon, 28-May-2022.) |
Theorem | phival 11095* | Value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
♯ | ||
Theorem | phicl2 11096 | Bounds and closure for the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phicl 11097 | Closure for the value of the Euler function. (Contributed by Mario Carneiro, 28-Feb-2014.) |
Theorem | phibndlem 11098* | Lemma for phibnd 11099. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phibnd 11099 | A slightly tighter bound on the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phicld 11100 | Closure for the value of the Euler function. (Contributed by Mario Carneiro, 29-May-2016.) |
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