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Theorem List for Metamath Proof Explorer - 13401-13500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhashbcval 13401* Value of the "binomial set", the set of all -element subsets of . (Contributed by Mario Carneiro, 20-Apr-2015.)

Theoremhashbccl 13402* The binomial set is a finite set. (Contributed by Mario Carneiro, 20-Apr-2015.)

Theoremhashbcss 13403* Subset relation for the binomial set. (Contributed by Mario Carneiro, 20-Apr-2015.)

Theoremhashbc0 13404* The set of subsets of size zero is the singleton of the empty set. (Contributed by Mario Carneiro, 22-Apr-2015.)

Theoremhashbc2 13405* The size of the binomial set is the binomial coefficient. (Contributed by Mario Carneiro, 20-Apr-2015.)

Theorem0hashbc 13406* There are no subsets of the empty set with size greater than zero. (Contributed by Mario Carneiro, 22-Apr-2015.)

Theoremramval 13407* The value of the Ramsey number function. (Contributed by Mario Carneiro, 21-Apr-2015.)
Ramsey

Theoremramcl2lem 13408* Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramtcl 13409* The Ramsey number has the Ramsey number property if any number does. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramtcl2 13410* The Ramsey number is an integer iff there is a number with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramtub 13411* The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramub 13412* The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theoremramub2 13413* It is sufficient to check the Ramsey property on finite sets of size equal to the upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey

Theoremrami 13414* The defining property of a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey               Ramsey

Theoremramcl2 13415 The Ramsey number is either a nonnegative integer or plus infinity. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramxrcl 13416 The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 13428.) (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramubcl 13417 If the Ramsey number is upper bounded, then it is an integer. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey Ramsey

Theoremramlb 13418* Establish a lower bound on a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theorem0ram 13419* The Ramsey number when . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theorem0ram2 13420 The Ramsey number when . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theoremram0 13421 The Ramsey number when . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theorem0ramcl 13422 Lemma for ramcl 13428: Existence of the Ramsey number when . (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey

Theoremramz2 13423 The Ramsey number when has value zero for some color . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theoremramz 13424 The Ramsey number when is the zero function. (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theoremramub1lem1 13425* Lemma for ramub1 13427. (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey               Ramsey                      Ramsey

Theoremramub1lem2 13426* Lemma for ramub1 13427. (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey               Ramsey                      Ramsey

Theoremramub1 13427* Inductive step for Ramsey's theorem, in the form of an explicit upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey               Ramsey        Ramsey Ramsey

Theoremramcl 13428 Ramsey's theorem: the Ramsey number is an integer for every finite coloring and set of upper bounds. (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey

Theoremramsey 13429* Ramsey's theorem with the definition Ramsey eliminated. If is an integer, is a specified finite set of colors, and is a set of lower bounds for each color, then there is an such that for every set of size greater than and every coloring of the set of all -element subsets of , there is a color and a subset such that is larger than and the -element subsets of are monochromatic with color . This is the hypergraph version of Ramsey's theorem; the version for simple graphs is the case . (Contributed by Mario Carneiro, 23-Apr-2015.)

6.2.13  Decimal arithmetic (cont.)

Theoremdec2dvds 13430 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec5dvds 13431 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec5dvds2 13432 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec5nprm 13433 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec2nprm 13434 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremmodxai 13435 Add exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) (Revised by Mario Carneiro, 5-Feb-2015.)

Theoremmod2xi 13436 Double exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)

Theoremmodxp1i 13437 Add one to an exponent in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)

Theoremmod2xnegi 13438 Version of mod2xi 13436 with a negaive mod value. (Contributed by Mario Carneiro, 21-Feb-2014.)

Theoremmodsubi 13439 Subtract from within a mod calculation. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremgcdi 13440 Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)

Theoremgcdmodi 13441 Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)

Theoremdecexp2 13442 Calculate a power of two. (Contributed by Mario Carneiro, 19-Feb-2014.)

Theoremnumexp0 13443 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremnumexp1 13444 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremnumexpp1 13445 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremnumexp2x 13446 Double an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremdecsplit0b 13447 Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.)

Theoremdecsplit0 13448 Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.)

Theoremdecsplit1 13449 Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.)
;

Theoremdecsplit 13450 Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.)
; ;

Theoremkaratsuba 13451 The Karatsuba multiplication algorithm. If and are decomposed into two groups of digits of length (only the lower group is known to be this size but the algorithm is most efficient when the partition is chosen near the middle of the digit string), then can be written in three groups of digits, where each group needs only one multiplication. Thus, we can halve both inputs with only three multiplications on the smaller operands, yielding an asymptotic improvement of n^(log2 3) instead of n^2 for the "naive" algorithm decmul1c 10460. (Contributed by Mario Carneiro, 16-Jul-2015.)

Theorem2exp4 13452 Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
;

Theorem2exp6 13453 Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.)
;

Theorem2exp8 13454 Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
;;

Theorem2exp16 13455 Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
; ;;;;

Theorem3exp3 13456 Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.)
;

Theorem2expltfac 13457 The factorial grows faster than two to the power . (Contributed by Mario Carneiro, 15-Sep-2016.)

6.2.14  Specific prime numbers

Theorem4nprm 13458 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.)

Theoremprmlem0 13459* Lemma for prmlem1 13461 and prmlem2 13473. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremprmlem1a 13460* A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremprmlem1 13461 A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.)
;

Theorem5prm 13462 5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)

Theorem6nprm 13463 6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem7prm 13464 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)

Theorem8nprm 13465 8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem9nprm 13466 9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem10nprm 13467 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem11prm 13468 11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theorem13prm 13469 13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theorem17prm 13470 17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theorem19prm 13471 19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theorem23prm 13472 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theoremprmlem2 13473 Our last proving session got as far as 25 because we started with the two "bootstrap" primes 2 and 3, and the next prime is 5, so knowing that 2 and 3 are prime and 4 is not allows us to cover the numbers less than . Additionally, nonprimes are "easy", so we can extend this range of known prime/nonprimes all the way until 29, which is the first prime larger than 25. Thus, in this lemma we extend another blanket out to , from which we can prove even more primes. If we wanted, we could keep doing this, but the goal is Bertrand's postulate, and for that we only need a few large primes - we don't need to find them all, as we have been doing thus far. So after this blanket runs out, we'll have to switch to another method (see 1259prm 13486).

As a side note, you can see the pattern of the primes in the indentation pattern of this lemma! (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

;;                                          ;        ;        ;        ;        ;

Theorem37prm 13474 37 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;

Theorem43prm 13475 43 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;

Theorem83prm 13476 83 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;

Theorem139prm 13477 139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;;

Theorem163prm 13478 163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;;

Theorem317prm 13479 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;;

Theorem631prm 13480 631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;;

6.2.15  Very large primes

Theorem1259lem1 13481 Lemma for 1259prm 13486. Calculate a power mod. In decimal, we calculate and in this lemma. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ; ;;

Theorem1259lem2 13482 Lemma for 1259prm 13486. Calculate a power mod. In decimal, we calculate . (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ; ;;

Theorem1259lem3 13483 Lemma for 1259prm 13486. Calculate a power mod. In decimal, we calculate and . (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ;

Theorem1259lem4 13484 Lemma for 1259prm 13486. Calculate a power mod. In decimal, we calculate , , and finally . (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;

Theorem1259lem5 13485 Lemma for 1259prm 13486. Calculate the GCD of with . (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ;

Theorem1259prm 13486 1259 is a prime number. (Contributed by Mario Carneiro, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;;;

Theorem2503lem1 13487 Lemma for 2503prm 13490. Calculate a power mod. In decimal, we calculate . (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ; ;;;

Theorem2503lem2 13488 Lemma for 2503prm 13490. Calculate a power mod. We calculate , , , , , , , , and finally . (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;

Theorem2503lem3 13489 Lemma for 2503prm 13490. Calculate the GCD of with . (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ;

Theorem2503prm 13490 2503 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;;;

Theorem4001lem1 13491 Lemma for 4001prm 13495. Calculate a power mod. In decimal, we calculate , , , , and . (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ;; ;;

Theorem4001lem2 13492 Lemma for 2503prm 13490. Calculate a power mod. In decimal, we calculate and . (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ;; ;;;

Theorem4001lem3 13493 Lemma for 4001prm 13495. Calculate a power mod. In decimal, we calculate and finally . (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;

Theorem4001lem4 13494 Lemma for 4001prm 13495. Calculate the GCD of with . (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ;;

Theorem4001prm 13495 4001 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;;;

PART 7  BASIC STRUCTURES

7.1  Extensible structures

7.1.1  Basic definitions

An "extensible structure" is a function (set of ordered pairs) on a finite (and not necessarily sequential) subset of , used to define a specific group, ring, poset, etc. The function's argument is the index of a structure component (such as for the base set of a group), and its value is the component (such as the base set). A group will have at least two components (base set and operation), although it can be further specialized by adding other components such as a multiplicative operation for rings (and still remain a group per our definition). Thus, every ring is also a group. This allows theorems from more general structures (groups) to be reused for more specialized structures (rings) without having to reprove them.

Syntaxcstr 13496 Extend class notation with the class of structures with components numbered below .
Struct

Syntaxcnx 13497 Extend class notation with the structure component index extractor.

Syntaxcsts 13498 Set components of a structure.
sSet

Syntaxcslot 13499 Extend class notation with the slot function.
Slot

Syntaxcbs 13500 Extend class notation with the class of all base set extractors.

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