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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | prmind2 11801* | A variation on prmind 11802 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | prmind 11802* | 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 11803 | If divides a prime, then is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.) |
Theorem | nprm 11804 | A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | nprmi 11805 | An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.) |
Theorem | dvdsnprmd 11806 | 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 11807 | A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.) |
Theorem | 2prm 11808 | 2 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 16-Jun-2016.) |
Theorem | 3prm 11809 | 3 is a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | 4nprm 11810 | 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.) |
Theorem | prmuz2 11811 | A prime number is an integer greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | prmgt1 11812 | A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.) |
Theorem | prmm2nn0 11813 | Subtracting 2 from a prime number results in a nonnegative integer. (Contributed by Alexander van der Vekens, 30-Aug-2018.) |
Theorem | oddprmgt2 11814 | An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.) |
Theorem | oddprmge3 11815 | 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 11816 | A square is never prime. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | dvdsprm 11817 | 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 11818* | 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 11819* | 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 11820 | No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 2-Jul-2015.) |
Theorem | divgcdodd 11821 | 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 11824. | ||
Theorem | coprm 11822 | 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 11823 | Unequal prime numbers are relatively prime. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | euclemma 11824 | 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 11825* | A number is prime iff it satisfies Euclid's lemma euclemma 11824. (Contributed by Mario Carneiro, 6-Sep-2015.) |
Theorem | prmdvdsexp 11826 | 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 11827 | 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 11828 | If a prime divides a nonnegative power of another, then they are equal. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Theorem | prmexpb 11829 | Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.) |
Theorem | prmfac1 11830 | The factorial of a number only contains primes less than the base. (Contributed by Mario Carneiro, 6-Mar-2014.) |
Theorem | rpexp 11831 | 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 11832 | Relative primality passes to asymmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.) |
Theorem | rpexp12i 11833 | Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.) |
Theorem | prmndvdsfaclt 11834 | 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 11835 | 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 11836 | 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 11837 | 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 11838 | 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 11839 | Lemma for sqrt2irr 11840. 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 11840 |
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 11858 for
the proof that the square root of two is irrational.
The proof's core is proven in sqrt2irrlem 11839, 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 11841 | The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.) |
Theorem | sqrt2irr0 11842 | The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.) |
Theorem | pw2dvdslemn 11843* | Lemma for pw2dvds 11844. 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 11844* | A natural number has a highest power of two which divides it. (Contributed by Jim Kingdon, 14-Nov-2021.) |
Theorem | pw2dvdseulemle 11845 | Lemma for pw2dvdseu 11846. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | pw2dvdseu 11846* | A natural number has a unique highest power of two which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdclemxy 11847* | Lemma for oddpwdc 11852. 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 11848* | Lemma for oddpwdc 11852. A natural number is divisible by the highest power of two which divides it. (Contributed by Jim Kingdon, 17-Nov-2021.) |
Theorem | oddpwdclemndvds 11849* | Lemma for oddpwdc 11852. 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 11850* | Lemma for oddpwdc 11852. Removing the powers of two from a natural number produces an odd number. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdclemdc 11851* | Lemma for oddpwdc 11852. Decomposing a number into odd and even parts. (Contributed by Jim Kingdon, 16-Nov-2021.) |
Theorem | oddpwdc 11852* | 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 11853* | 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 11854* | 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 11855 | 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 11856 | The absolute value of the difference between two unequal integers is at least one. (Contributed by Jim Kingdon, 31-Jan-2022.) |
Theorem | sqrt2irraplemnn 11857 | Lemma for sqrt2irrap 11858. 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 11858 | 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 11840. (Contributed by Jim Kingdon, 2-Oct-2021.) |
# | ||
Syntax | cnumer 11859 | Extend class notation to include canonical numerator function. |
numer | ||
Syntax | cdenom 11860 | Extend class notation to include canonical denominator function. |
denom | ||
Definition | df-numer 11861* | 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 11862* | 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 11863* | Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | qdenval 11864* | Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumdencl 11865 | Lemma for qnumcl 11866 and qdencl 11867. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qnumcl 11866 | The canonical numerator of a rational is an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | qdencl 11867 | The canonical denominator is a positive integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | fnum 11868 | Canonical numerator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer | ||
Theorem | fden 11869 | Canonical denominator defines a function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumdenbi 11870 | 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 11871 | The canonical representation of a rational is fully reduced. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qeqnumdivden 11872 | Recover a rational number from its canonical representation. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | qmuldeneqnum 11873 | Multiplying a rational by its denominator results in an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom numer | ||
Theorem | divnumden 11874 | Calculate the reduced form of a quotient using . (Contributed by Stefan O'Rear, 13-Sep-2014.) |
numer denom | ||
Theorem | divdenle 11875 | Reducing a quotient never increases the denominator. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
denom | ||
Theorem | qnumgt0 11876 | A rational is positive iff its canonical numerator is. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer | ||
Theorem | qgt0numnn 11877 | A rational is positive iff its canonical numerator is a positive integer. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer | ||
Theorem | nn0gcdsq 11878 | Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Theorem | zgcdsq 11879 | nn0gcdsq 11878 extended to integers by symmetry. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Theorem | numdensq 11880 | Squaring a rational squares its canonical components. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer numer denom denom | ||
Theorem | numsq 11881 | Square commutes with canonical numerator. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
numer numer | ||
Theorem | densq 11882 | Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
denom denom | ||
Theorem | qden1elz 11883 | A rational is an integer iff it has denominator 1. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
denom | ||
Theorem | nn0sqrtelqelz 11884 | If a nonnegative integer has a rational square root, that root must be an integer. (Contributed by Jim Kingdon, 24-May-2022.) |
Theorem | nonsq 11885 | Any integer strictly between two adjacent squares has a non-rational square root. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Syntax | cphi 11886 | Extend class notation with the Euler phi function. |
Definition | df-phi 11887* | 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 11888* | Finiteness of an expression used to define the Euler function. (Contributed by Jim Kingon, 28-May-2022.) |
Theorem | phival 11889* | Value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
♯ | ||
Theorem | phicl2 11890 | Bounds and closure for the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phicl 11891 | Closure for the value of the Euler function. (Contributed by Mario Carneiro, 28-Feb-2014.) |
Theorem | phibndlem 11892* | Lemma for phibnd 11893. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phibnd 11893 | A slightly tighter bound on the value of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | phicld 11894 | Closure for the value of the Euler function. (Contributed by Mario Carneiro, 29-May-2016.) |
Theorem | phi1 11895 | Value of the Euler function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Theorem | dfphi2 11896* | Alternate definition of the Euler function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 2-May-2016.) |
♯ ..^ | ||
Theorem | hashdvds 11897* | 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 11898 | 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 11899 | Value of the Euler function at a prime. (Contributed by Mario Carneiro, 28-Feb-2014.) |
Theorem | crth 11900* | 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.) |
..^ ..^ ..^ |
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