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Theorem List for Metamath Proof Explorer - 21701-21800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremex-rn 21701 Example for df-rn 4848. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-res 21702 Example for df-res 4849. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-ima 21703 Example for df-ima 4850. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-fv 21704 Example for df-fv 5421. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-1st 21705 Example for df-1st 6308. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremex-2nd 21706 Example for df-2nd 6309. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theorem1kp2ke3k 21707 Example for df-dec 10339, 1000 + 2000 = 3000.

This proof disproves (by counter-example) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with , commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 10339 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

;;; ;;; ;;;

Theoremex-fl 21708 Example for df-fl 11157. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremex-dvds 21709 3 divides into 6. A demonstration of df-dvds 12808. (Contributed by David A. Wheeler, 19-May-2015.)

15.2  Humor

15.2.1  April Fool's theorem

Theoremavril1 21710 Poisson d'Avril's Theorem. This theorem is noted for its Selbstdokumentieren property, which means, literally, "self-documenting" and recalls the principle of quidquid german dictum sit, altum viditur, often used in set theory. Starting with the seemingly simple yet profound fact that any object equals itself (proved by Tarski in 1965; see Lemma 6 of [Tarski] p. 68), we demonstrate that the power set of the real numbers, as a relation on the value of the imaginary unit, does not conjoin with an empty relation on the product of the additive and multiplicative identity elements, leading to this startling conclusion that has left even seasoned professional mathematicians scratching their heads. (Contributed by Prof. Loof Lirpa, 1-Apr-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

A reply to skeptics can be found at http://us.metamath.org/mpeuni/mmnotes.txt, under the 1-Apr-2006 entry.

Theorem2bornot2b 21711 The law of excluded middle. Act III, Theorem 1 of Shakespeare, Hamlet, Prince of Denmark (1602). Its author leaves its proof as an exercise for the reader - "To be, or not to be: that is the question" - starting a trend that has become standard in modern-day textbooks, serving to make the frustrated reader feel inferior, or in some cases to mask the fact that the author does not know its solution. (Contributed by Prof. Loof Lirpa, 1-Apr-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremhelloworld 21712 The classic "Hello world" benchmark has been translated into 314 computer programming languages - see http://www.roesler-ac.de/wolfram/hello.htm. However, for many years it eluded a proof that it is more than just a conjecture, even though a wily mathematician once claimed, "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." Using an IBM 709 mainframe, a team of mathematicians led by Prof. Loof Lirpa, at the New College of Tahiti, were finally able put it rest with a remarkably short proof only 4 lines long. (Contributed by Prof. Loof Lirpa, 1-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem1p1e2apr1 21713 One plus one equals two. Using proof-shortening techniques pioneered by Mr. Mel O'Cat, along with the latest supercomputer technology, Prof. Loof Lirpa and colleagues were able to shorten Whitehead and Russell's 360-page proof that 1+1=2 in Principia Mathematica to this remarkable proof only two steps long, thus establishing a new world's record for this famous theorem. (Contributed by Prof. Loof Lirpa, 1-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremeqid1 21714 Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.

This law is thought to have originated with Aristotle (Metaphysics, Book VII, Part 17). It is one of the three axioms of Ayn Rand's philosophy (Atlas Shrugged, Part Three, Chapter VII). While some have proposed extending Rand's axiomatization to include Compassion and Kindness, others fear that such an extension may flirt with logical inconsistency. (Contributed by Stefan Allan, 1-Apr-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem1div0apr 21715 Division by zero is forbidden! If we try, we encounter the DO NOT ENTER sign, which in mathematics means it is foolhardy to venture any further, possibly putting the underlying fabric of reality at risk. Based on a dare by David A. Wheeler. (Contributed by Mario Carneiro, 1-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

15.3  (Future - to be reviewed and classified)

15.3.1  Planar incidence geometry

Syntaxcplig 21716 Extend class notation with the class of all planar incidence geometries.

Definitiondf-plig 21717* Planar incidence geometry. I use Hilbert's "axioms" adapted to planar geometry. is the incidence relation. I could take a generic incidence relation but I'm lazy and I'm not sure the gain is worth the extra work. Much of what follows is directly borrowed from Aitken. http://public.csusm.edu/aitken_html/m410/betweenness.08.pdf (Contributed by FL, 2-Aug-2009.)

Theoremisplig 21718* The predicate "is a planar incidence geometry". (Contributed by FL, 2-Aug-2009.)

Theoremtncp 21719* There exist three non colinear points. (Contributed by FL, 3-Aug-2009.)

Theoremlpni 21720* For any line, there exists a point not on the line. (Contributed by Jeff Hankins, 15-Aug-2009.)

15.3.2  Algebra preliminaries

Syntaxcrpm 21721 Ring primes.
RPrime

Definitiondf-rprm 21722* Define the set of prime elements in a ring. A prime element is a nonzero non-unit that satisfies an equivalent of Euclid's lemma euclemma 13063. (Contributed by Mario Carneiro, 17-Feb-2015.)
RPrime Unit r

15.3.3  Transitive closure

Syntaxctcl 21723 Extend class notation to include the transitive closure symbol.

Syntaxcrtcl 21724 Extend class notation with transitive closure.

Definitiondf-trcl 21725* Transitive closure of a relation. Experimental. (Contributed by FL, 27-Jun-2011.)

Definitiondf-rtrcl 21726* Reflexive-transitive closure of a relation. Experimental. (Contributed by FL, 27-Jun-2011.)

PART 16  ADDITIONAL MATERIAL ON GROUPS, RINGS, AND FIELDS (DEPRECATED)

This part contains an earlier development of groups, rings, and fields that was defined before extensible structures were introduced.

Theorem grpo2grp 21775 shows the relationship between the older group definition and the extensible structure definition.

16.1  Additional material on group theory

16.1.1  Definitions and basic properties for groups

Syntaxcgr 21727 Extend class notation with the class of all group operations.

Syntaxcgi 21728 Extend class notation with a function mapping a group operation to the group's identity element.
GId

Syntaxcgn 21729 Extend class notation with a function mapping a group operation to the inverse function for the group.

Syntaxcgs 21730 Extend class notation with a function mapping a group operation to the division (or subtraction) operation for the group.

Syntaxcgx 21731 Extend class notation with a function mapping a group operation to the power operation for the group.

Definitiondf-grpo 21732* Define the class of all group operations. The base set for a group can be determined from its group operation. Based on the definition in Exercise 28 of [Herstein] p. 54. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Definitiondf-gid 21733* Define a function that maps a group operation to the group's identity element. (Contributed by FL, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Definitiondf-ginv 21734* Define a function that maps a group operation to the group's inverse function. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)
GId

Definitiondf-gdiv 21735* Define a function that maps a group operation to the group's division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Definitiondf-gx 21736* Define a function that maps a group operation to the group's power operation. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
GId

Theoremisgrpo 21737* The predicate "is a group operation." Note that is the base set of the group. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Theoremisgrpo2 21738* The predicate "is a group operation." (Contributed by NM, 23-Oct-2012.) (New usage is discouraged.)

Theoremisgrpoi 21739* Properties that determine a group operation. Read as . (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremgrpofo 21740 A group operation maps onto the group's underlying set. (Contributed by NM, 30-Oct-2006.) (New usage is discouraged.)

Theoremgrpocl 21741 Closure law for a group operation. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Theoremgrpolidinv 21742* A group has a left identity element, and every member has a left inverse. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremgrpon0 21743 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (New usage is discouraged.)

Theoremgrpoass 21744 A group operation is associative. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Theoremgrpoidinvlem1 21745 Lemma for grpoidinv 21749. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Theoremgrpoidinvlem2 21746 Lemma for grpoidinv 21749. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Theoremgrpoidinvlem3 21747* Lemma for grpoidinv 21749. (Contributed by NM, 11-Oct-2006.) (New usage is discouraged.)

Theoremgrpoidinvlem4 21748* Lemma for grpoidinv 21749. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)

Theoremgrpoidinv 21749* A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)

Theoremgrpoideu 21750* The left identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)

Theoremgrporndm 21751 A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)

Theorem0ngrp 21752 The empty set is not a group. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)

Theoremgrporn 21753 The range of a group operation. Useful for satisfying group base set hypotheses of the form . (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Theoremgidval 21754* The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremfngid 21755 GId is a function. (Contributed by FL, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrposn 21756 The group operation for the singleton group. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremgrpoidval 21757* Lemma for grpoidcl 21758 and others. (Contributed by NM, 5-Feb-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrpoidcl 21758 The identity element of a group belongs to the group. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrpoidinv2 21759* A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrpolid 21760 The identity element of a group is a left identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrporid 21761 The identity element of a group is a right identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrporcan 21762 Right cancellation law for groups. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)

Theoremgrpoinveu 21763* The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
GId

Theoremgrpoid 21764 Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GId

Theoremgrpoinvfval 21765* The inverse function of a group. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrpoinvval 21766* The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrpoinvcl 21767 A group element's inverse is a group element. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgrpoinv 21768 The properties of a group element's inverse. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgrpolinv 21769 The left inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
GId

Theoremgrporinv 21770 The right inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
GId

Theoremgrpoinvid1 21771 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
GId

Theoremgrpoinvid2 21772 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
GId

Theoremgrpoinvid 21773 The inverse of the identity element of a group. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
GId

Theoremgrpolcan 21774 Left cancellation law for groups. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Theoremgrpo2grp 21775 Convert a group operation to a group structure. (Contributed by NM, 25-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) (New usage is discouraged.)

Theoremisgrp2d 21776* An alternate way to show a group operation. Exercise 1 of [Herstein] p. 57. (Contributed by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremisgrp2i 21777* An alternate way to show a group operation. Exercise 1 of [Herstein] p. 57. (Contributed by NM, 5-Dec-2006.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremgrpoasscan1 21778 An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)

Theoremgrpoasscan2 21779 An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgrpo2inv 21780 Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Theoremgrpoinvf 21781 Mapping of the inverse function of a group. (Contributed by NM, 29-Mar-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgrpoinvop 21782 The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Theoremgrpodivfval 21783* Group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgrpodivval 21784 Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgrpodivinv 21785 Group division by an inverse. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrpoinvdiv 21786 Inverse of a group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)

Theoremgrpodivf 21787 Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgrpodivcl 21788 Closure of group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrpodivdiv 21789 Double group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)

Theoremgrpomuldivass 21790 Associative-type law for multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrpodivid 21791 Division of a group member by itself. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
GId

Theoremgrpopncan 21792 Cancellation law for group division. (pncan 9267 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrponpcan 21793 Cancellation law for group division. (npcan 9270 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrpopnpcan2 21794 Cancellation law for mixed addition and group division. (pnpcan2 9297 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrponnncan2 21795 Cancellation law for group division. (nnncan2 9294 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremgrponpncan 21796 Cancellation law for group division. (npncan 9279 analog.) (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)

Theoremgrpodiveq 21797 Relationship between group division and group multiplication. (Contributed by Mario Carneiro, 11-Jul-2014.) (New usage is discouraged.)

Theoremgxfval 21798* The value of the group power operator function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgxval 21799 The result of the group power operator. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgxpval 21800 The result of the group power operator when the exponent is positive. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

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