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Theorem List for Metamath Proof Explorer - 13301-13400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremvdwmc 13301* The predicate " The -coloring contains a monochromatic AP of length ". (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP AP

Theoremvdwmc2 13302* Expand out the definition of an arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP

Theoremvdwpc 13303* The predicate " The coloring contains a polychromatic -tuple of AP's of length ". A polychromatic -tuple of AP's is a set of AP's with the same base point but different step lengths, such that each individual AP is monochromatic, but the AP's all have mutually distinct colors. (The common basepoint is not required to have the same color as any of the AP's.) (Contributed by Mario Carneiro, 18-Aug-2014.)
PolyAP AP

Theoremvdwlem1 13304* Lemma for vdw 13317. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP                      MonoAP

Theoremvdwlem2 13305* Lemma for vdw 13317. (Contributed by Mario Carneiro, 12-Sep-2014.)
MonoAP MonoAP

Theoremvdwlem3 13306 Lemma for vdw 13317. (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwlem4 13307* Lemma for vdw 13317. (Contributed by Mario Carneiro, 12-Sep-2014.)

Theoremvdwlem5 13308* Lemma for vdw 13317. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP                      AP

Theoremvdwlem6 13309* Lemma for vdw 13317. (Contributed by Mario Carneiro, 13-Sep-2014.)
AP                      AP                                    PolyAP MonoAP

Theoremvdwlem7 13310* Lemma for vdw 13317. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP        PolyAP PolyAP MonoAP

Theoremvdwlem8 13311* Lemma for vdw 13317. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP               PolyAP

Theoremvdwlem9 13312* Lemma for vdw 13317. (Contributed by Mario Carneiro, 12-Sep-2014.)
MonoAP                      PolyAP MonoAP               MonoAP                      PolyAP MonoAP

Theoremvdwlem10 13313* Lemma for vdw 13317. Set up secondary induction on . (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP               PolyAP MonoAP

Theoremvdwlem11 13314* Lemma for vdw 13317. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP        MonoAP

Theoremvdwlem12 13315 Lemma for vdw 13317. base case of induction. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP

Theoremvdwlem13 13316* Lemma for vdw 13317. Main induction on ; , base cases. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP

Theoremvdw 13317* Van der Waerden's theorem. For any finite coloring and integer , there is an such that every coloring function from to contains a monochromatic arithmetic progression (which written out in full means that there is a color and base, increment values such that all the numbers lie in the preimage of , i.e. they are all in and evaluated at each one yields ). (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwnnlem1 13318* Corollary of vdw 13317, and lemma for vdwnn 13321. If is a coloring of the integers, then there are arbitrarily long monochromatic APs in . (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwnnlem2 13319* Lemma for vdwnn 13321. The set of all "bad" for the theorem is upwards-closed, because a long AP implies a short AP. (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwnnlem3 13320* Lemma for vdwnn 13321. (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwnn 13321* Van der Waerden's theorem, infinitary version. For any finite coloring of the natural numbers, there is a color that contains arbitrarily long arithmetic progressions. (Contributed by Mario Carneiro, 13-Sep-2014.)

6.2.12  Ramsey's theorem

Syntaxcram 13322 Extend class notation with the Ramsey number function.
Ramsey

Theoremramtlecl 13323* The set of numbers with the Ramsey number property is upward-closed. (Contributed by Mario Carneiro, 21-Apr-2015.)

Definitiondf-ram 13324* Define the Ramsey number function. The input is a number for the size of the edges of the hypergraph, and a tuple from the finite color set to lower bounds for each color. The Ramsey number Ramsey is the smallest number such that for any set with Ramsey and any coloring of the set of -element subsets of (with color set ), there is a color and a subset such that and all the hyperedges of (that is, subsets of of size ) have color . (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremhashbcval 13325* Value of the "binomial set", the set of all -element subsets of . (Contributed by Mario Carneiro, 20-Apr-2015.)

Theoremhashbccl 13326* The binomial set is a finite set. (Contributed by Mario Carneiro, 20-Apr-2015.)

Theoremhashbcss 13327* Subset relation for the binomial set. (Contributed by Mario Carneiro, 20-Apr-2015.)

Theoremhashbc0 13328* The set of subsets of size zero is the singleton of the empty set. (Contributed by Mario Carneiro, 22-Apr-2015.)

Theoremhashbc2 13329* The size of the binomial set is the binomial coefficient. (Contributed by Mario Carneiro, 20-Apr-2015.)

Theorem0hashbc 13330* There are no subsets of the empty set with size greater than zero. (Contributed by Mario Carneiro, 22-Apr-2015.)

Theoremramval 13331* The value of the Ramsey number function. (Contributed by Mario Carneiro, 21-Apr-2015.)
Ramsey

Theoremramcl2lem 13332* Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramtcl 13333* The Ramsey number has the Ramsey number property if any number does. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramtcl2 13334* 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 13335* 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 13336* 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 13337* 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 13338* The defining property of a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey               Ramsey

Theoremramcl2 13339 The Ramsey number is either a nonnegative integer or plus infinity. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramxrcl 13340 The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 13352.) (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramubcl 13341 If the Ramsey number is upper bounded, then it is an integer. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey Ramsey

Theoremramlb 13342* Establish a lower bound on a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theorem0ram 13343* The Ramsey number when . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theorem0ram2 13344 The Ramsey number when . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theoremram0 13345 The Ramsey number when . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theorem0ramcl 13346 Lemma for ramcl 13352: Existence of the Ramsey number when . (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey

Theoremramz2 13347 The Ramsey number when has value zero for some color . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theoremramz 13348 The Ramsey number when is the zero function. (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theoremramub1lem1 13349* Lemma for ramub1 13351. (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey               Ramsey                      Ramsey

Theoremramub1lem2 13350* Lemma for ramub1 13351. (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey               Ramsey                      Ramsey

Theoremramub1 13351* 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 13352 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 13353* 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 13354 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec5dvds 13355 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec5dvds2 13356 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec5nprm 13357 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec2nprm 13358 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremmodxai 13359 Add exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) (Revised by Mario Carneiro, 5-Feb-2015.)

Theoremmod2xi 13360 Double exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)

Theoremmodxp1i 13361 Add one to an exponent in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)

Theoremmod2xnegi 13362 Version of mod2xi 13360 with a negaive mod value. (Contributed by Mario Carneiro, 21-Feb-2014.)

Theoremmodsubi 13363 Subtract from within a mod calculation. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremgcdi 13364 Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)

Theoremgcdmodi 13365 Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)

Theoremdecexp2 13366 Calculate a power of two. (Contributed by Mario Carneiro, 19-Feb-2014.)

Theoremnumexp0 13367 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremnumexp1 13368 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremnumexpp1 13369 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremnumexp2x 13370 Double an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremdecsplit0b 13371 Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.)

Theoremdecsplit0 13372 Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.)

Theoremdecsplit1 13373 Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.)
;

Theoremdecsplit 13374 Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.)
; ;

Theoremkaratsuba 13375 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 10385. (Contributed by Mario Carneiro, 16-Jul-2015.)

Theorem2exp4 13376 Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
;

Theorem2exp6 13377 Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.)
;

Theorem2exp8 13378 Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
;;

Theorem2exp16 13379 Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
; ;;;;

Theorem3exp3 13380 Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.)
;

Theorem2expltfac 13381 The factorial grows faster than two to the power . (Contributed by Mario Carneiro, 15-Sep-2016.)

6.2.14  Specific prime numbers

Theorem4nprm 13382 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.)

Theoremprmlem0 13383* Lemma for prmlem1 13385 and prmlem2 13397. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremprmlem1a 13384* 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 13385 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 13386 5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)

Theorem6nprm 13387 6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem7prm 13388 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)

Theorem8nprm 13389 8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem9nprm 13390 9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem10nprm 13391 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem11prm 13392 11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theorem13prm 13393 13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theorem17prm 13394 17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theorem19prm 13395 19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theorem23prm 13396 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theoremprmlem2 13397 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 13410).

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 13398 37 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;

Theorem43prm 13399 43 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;

Theorem83prm 13400 83 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;

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