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

Theoremelgz 13301 Elementhood in the gaussian integers. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzcn 13302 A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremzgz 13303 An integer is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremigz 13304 is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgznegcl 13305 The gaussian integers are closed under negation. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzcjcl 13306 The gaussian integers are closed under conjugation. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzaddcl 13307 The gaussian integers are closed under addition. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzmulcl 13308 The gaussian integers are closed under multiplication. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzreim 13309 Construct a gaussian integer from real and imaginary parts. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theoremgzsubcl 13310 The gaussian integers are closed under subtraction. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzabssqcl 13311 The squared norm of a gaussian integer is an integer. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem5 13312 Lemma for 4sq 13334. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem6 13313 Lemma for 4sq 13334. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem7 13314 Lemma for 4sq 13334. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem8 13315 Lemma for 4sq 13334. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem9 13316 Lemma for 4sq 13334. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem10 13317 Lemma for 4sq 13334. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem1 13318* Lemma for 4sq 13334. The set is the set of all numbers that are expressible as a sum of four squares. Our goal is to show that ; here we show one subset direction. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theorem4sqlem2 13319* Lemma for 4sq 13334. Change bound variables in . (Contributed by Mario Carneiro, 14-Jul-2014.)

Theorem4sqlem3 13320* Lemma for 4sq 13334. Sufficient condition to be in . (Contributed by Mario Carneiro, 14-Jul-2014.)

Theorem4sqlem4a 13321* Lemma for 4sqlem4 13322. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theorem4sqlem4 13322* Lemma for 4sq 13334. We can express the four-square property more compactly in terms of gaussian integers, because the norms of gaussian integers are exactly sums of two squares. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremmul4sqlem 13323* Lemma for mul4sq 13324: algebraic manipulations. The extra assumptions involving are for a part of 4sqlem17 13331 which needs to know not just that the product is a sum of squares, but also that it preserves divisibility by . (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremmul4sq 13324* Euler's four-square identity: The product of two sums of four squares is also a sum of four squares. This is usually quoted as an explicit formula involving eight real variables; we save some time by working with complex numbers (gaussian integers) instead, so that we only have to work with four variables, and also hiding the actual formula for the product in the proof of mul4sqlem 13323. (For the curious, the explicit formula that is used is .) (Contributed by Mario Carneiro, 14-Jul-2014.)

Theorem4sqlem11 13325* Lemma for 4sq 13334. Use the pigeonhole principle to show that the sets and have a common element, . (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem12 13326* Lemma for 4sq 13334. For any odd prime , there is a such that is a sum of two squares. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem13 13327* Lemma for 4sq 13334. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem14 13328* Lemma for 4sq 13334. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem15 13329* Lemma for 4sq 13334. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem16 13330* Lemma for 4sq 13334. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem17 13331* Lemma for 4sq 13334. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem18 13332* Lemma for 4sq 13334. Inductive step, odd prime case. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem19 13333* Lemma for 4sq 13334. The proof is by strong induction - we show that if all the integers less than are in , then is as well. In this part of the proof we do the induction argument and dispense with all the cases except the odd prime case, which is sent to 4sqlem18 13332. If is , we show directly; otherwise if is composite, is the product of two numbers less than it (and hence in by assumption), so by mul4sq 13324 . (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)

Theorem4sq 13334* Lagrange's four-square theorem, or Bachet's conjecture: every nonnegative integer is expressible as a sum of four squares. (Contributed by Mario Carneiro, 16-Jul-2014.)

6.2.11  Van der Waerden's theorem

Syntaxcvdwa 13335 The arithmetic progression function.
AP

Syntaxcvdwm 13336 The monochromatic arithmetic progression predicate.
MonoAP

Syntaxcvdwp 13337 The polychromatic arithmetic progression predicate.
PolyAP

Definitiondf-vdwap 13338* Define the arithmetic progression function, which takes as input a length , a start point , and a step and outputs the set of points in this progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Definitiondf-vdwmc 13339* Define the "contains a monochromatic AP" predicate. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP AP

Definitiondf-vdwpc 13340* Define the "contains a polychromatic colleciton of APs" predicate. See vdwpc 13350 for more information. (Contributed by Mario Carneiro, 18-Aug-2014.)
PolyAP AP

Theoremvdwapfval 13341* Define the arithmetic progression function, which takes as input a length , a start point , and a step and outputs the set of points in this progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwapf 13342 The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwapval 13343* Value of the arithmetic progression function. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwapun 13344 Remove the first element of an arithmetic progression. (Contributed by Mario Carneiro, 11-Sep-2014.)
AP AP

Theoremvdwapid1 13345 The first element of an arithmetic progression. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP

Theoremvdwap0 13346 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwap1 13347 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwmc 13348* The predicate " The -coloring contains a monochromatic AP of length ". (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP AP

Theoremvdwmc2 13349* Expand out the definition of an arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP

Theoremvdwpc 13350* 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 13351* Lemma for vdw 13364. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP                      MonoAP

Theoremvdwlem2 13352* Lemma for vdw 13364. (Contributed by Mario Carneiro, 12-Sep-2014.)
MonoAP MonoAP

Theoremvdwlem3 13353 Lemma for vdw 13364. (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwlem4 13354* Lemma for vdw 13364. (Contributed by Mario Carneiro, 12-Sep-2014.)

Theoremvdwlem5 13355* Lemma for vdw 13364. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP                      AP

Theoremvdwlem6 13356* Lemma for vdw 13364. (Contributed by Mario Carneiro, 13-Sep-2014.)
AP                      AP                                    PolyAP MonoAP

Theoremvdwlem7 13357* Lemma for vdw 13364. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP        PolyAP PolyAP MonoAP

Theoremvdwlem8 13358* Lemma for vdw 13364. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP               PolyAP

Theoremvdwlem9 13359* Lemma for vdw 13364. (Contributed by Mario Carneiro, 12-Sep-2014.)
MonoAP                      PolyAP MonoAP               MonoAP                      PolyAP MonoAP

Theoremvdwlem10 13360* Lemma for vdw 13364. Set up secondary induction on . (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP               PolyAP MonoAP

Theoremvdwlem11 13361* Lemma for vdw 13364. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP        MonoAP

Theoremvdwlem12 13362 Lemma for vdw 13364. base case of induction. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP

Theoremvdwlem13 13363* Lemma for vdw 13364. Main induction on ; , base cases. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP

Theoremvdw 13364* 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 13365* Corollary of vdw 13364, and lemma for vdwnn 13368. If is a coloring of the integers, then there are arbitrarily long monochromatic APs in . (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwnnlem2 13366* Lemma for vdwnn 13368. 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 13367* Lemma for vdwnn 13368. (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwnn 13368* 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 13369 Extend class notation with the Ramsey number function.
Ramsey

Theoremramtlecl 13370* The set of numbers with the Ramsey number property is upward-closed. (Contributed by Mario Carneiro, 21-Apr-2015.)

Definitiondf-ram 13371* 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 13372* Value of the "binomial set", the set of all -element subsets of . (Contributed by Mario Carneiro, 20-Apr-2015.)

Theoremhashbccl 13373* The binomial set is a finite set. (Contributed by Mario Carneiro, 20-Apr-2015.)

Theoremhashbcss 13374* Subset relation for the binomial set. (Contributed by Mario Carneiro, 20-Apr-2015.)

Theoremhashbc0 13375* The set of subsets of size zero is the singleton of the empty set. (Contributed by Mario Carneiro, 22-Apr-2015.)

Theoremhashbc2 13376* The size of the binomial set is the binomial coefficient. (Contributed by Mario Carneiro, 20-Apr-2015.)

Theorem0hashbc 13377* There are no subsets of the empty set with size greater than zero. (Contributed by Mario Carneiro, 22-Apr-2015.)

Theoremramval 13378* The value of the Ramsey number function. (Contributed by Mario Carneiro, 21-Apr-2015.)
Ramsey

Theoremramcl2lem 13379* Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramtcl 13380* The Ramsey number has the Ramsey number property if any number does. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramtcl2 13381* 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 13382* 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 13383* 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 13384* 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 13385* The defining property of a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey               Ramsey

Theoremramcl2 13386 The Ramsey number is either a nonnegative integer or plus infinity. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramxrcl 13387 The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 13399.) (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramubcl 13388 If the Ramsey number is upper bounded, then it is an integer. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey Ramsey

Theoremramlb 13389* Establish a lower bound on a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theorem0ram 13390* The Ramsey number when . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theorem0ram2 13391 The Ramsey number when . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theoremram0 13392 The Ramsey number when . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theorem0ramcl 13393 Lemma for ramcl 13399: Existence of the Ramsey number when . (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey

Theoremramz2 13394 The Ramsey number when has value zero for some color . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theoremramz 13395 The Ramsey number when is the zero function. (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theoremramub1lem1 13396* Lemma for ramub1 13398. (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey               Ramsey                      Ramsey

Theoremramub1lem2 13397* Lemma for ramub1 13398. (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey               Ramsey                      Ramsey

Theoremramub1 13398* 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 13399 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 13400* 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.)

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