Type  Label  Description 
Statement 

Theorem  frgrareg 25901 
If a finite friendship graph is kregular, then k must be 2 (or 0).
(Contributed by Alexander van der Vekens, 9Oct2018.)

FriendGrph
RegUSGrph 

Theorem  frgraregord013 25902 
If a finite friendship graph is kregular, then it must have order 0, 1
or 3. (Contributed by Alexander van der Vekens, 9Oct2018.)

FriendGrph
RegUSGrph


Theorem  frgraregord13 25903 
If a nonempty finite friendship graph is kregular, then it must have
order 1 or 3. Special case of frgraregord013 25902. (Contributed by
Alexander van der Vekens, 9Oct2018.)

FriendGrph
RegUSGrph


Theorem  frgraogt3nreg 25904* 
If a finite friendship graph has an order greater than 3, it cannot be
kregular for any k. (Contributed by Alexander van der Vekens,
9Oct2018.)

FriendGrph
RegUSGrph 

Theorem  friendshipgt3 25905* 
The friendship theorem for big graphs: In every finite friendship graph
with order greater than 3 there is a vertex which is adjacent to all
other vertices. (Contributed by Alexander van der Vekens,
9Oct2018.)

FriendGrph


Theorem  friendship 25906* 
The friendship theorem: In every finite (nonempty) friendship graph
there is a vertex which is adjacent to all other vertices. This is
Metamath 100 proof #83. (Contributed by Alexander van der Vekens,
9Oct2018.)

FriendGrph


PART 17 GUIDES AND
MISCELLANEA


17.1 Guides (conventions, explanations, and
examples)


17.1.1 Conventions
This section describes the conventions we use.
These conventions often refer to existing mathematical practices, which are
discussed in more detail in other references.
For the general conventions, see conventions 25907, and for conventions related
to labels, see conventionslabel 25908.
Logic and set theory provide a foundation for all of mathematics. To learn
about them, you should study one or more of the references listed below.
We indicate references using square brackets. The textbooks provide a
motivation for what we are doing, whereas Metamath lets you see in detail all
hidden and implicit steps. Most standard theorems are accompanied by
citations. Some closely followed texts include the following:
 Axioms of propositional calculus  [Margaris].
 Axioms of predicate calculus  [Megill] (System S3' in the article
referenced).
 Theorems of propositional calculus  [WhiteheadRussell].
 Theorems of pure predicate calculus  [Margaris].
 Theorems of equality and substitution  [Monk2], [Tarski], [Megill].
 Axioms of set theory  [BellMachover].
 Development of set theory  [TakeutiZaring]. (The first part of [Quine]
has a good explanation of the powerful device of "virtual" or
class abstractions, which is essential to our development.)
 Construction of real and complex numbers  [Gleason]
 Theorems about real numbers  [Apostol]


Theorem  conventions 25907 
Here are some of the conventions we use in the
Metamath Proof Explorer (aka "set.mm"), and how they correspond to
typical textbook language (skipping the many cases
where they are identical).
For conventions related to labels, see conventionslabel 25908.
 Notation.
Where possible, the notation attempts to conform to modern
conventions, with variations due to our choice of the axiom system
or to make proofs shorter. However, our notation is strictly
sequential (lefttoright). For example, summation is written in the
form (dfsum 13808) which denotes that index
variable ranges over when evaluating . Thus,
means 1/2 + 1/4 + 1/8 + ...
= 1 (geoihalfsum 13993).
The notation is usually explained in more detail when first introduced.
 Axiomatic assertions ($a).
All axiomatic assertions ($a statements)
starting with " " have labels starting
with "ax" (axioms) or "df" (definitions). A statement with a
label starting with "ax" corresponds to what is traditionally
called an axiom. A statement with a label starting with "df"
introduces new symbols or a new relationship among symbols
that can be eliminated; they always extend the definition of
a wff or class. Metamath blindly treats $a statements as new
given facts but does not try to justify them. The mmj2 program
will justify the definitions as sound as discussed below,
except for 4 definitions (dfbi 190, dfcleq 2455, dfclel 2458, dfclab 2449)
that require a more complex metalogical justification by hand.
 Proven axioms.
In some cases we wish to treat an expression as an axiom in
later theorems, even though it can be proved. For example,
we derive the postulates or axioms of complex arithmetic as
theorems of ZFC set theory. For convenience, after deriving
the postulates, we reintroduce them as new axioms on
top of set theory. This lets us easily identify which axioms
are needed for a particular complex number proof, without the
obfuscation of the set theory used to derive them. For more, see
http://us.metamath.org/mpeuni/mmcomplex.html. When we wish
to use a previouslyproven assertion as an axiom, our convention
is that we use the
regular "axNAME" label naming convention to define the axiom,
but we precede it with a proof of the same statement with the label
"axNAME" . An example is complex arithmetic axiom ax1cn 9628,
proven by the preceding theorem ax1cn 9604.
The metamath.exe program will warn if an axiom does not match the
preceding theorem that justifies it if the names match in this way.
 Definitions (df...).
We encourage definitions to include hypertext links to proven examples.
 Statements with hypotheses. Many theorems and some axioms,
such as axmp 5, have hypotheses that must be satisfied in order for
the conclusion to hold, in this case min and maj. When presented in
summarized form such as in the Theorem List (click on "Nearby theorems"
on the axmp 5 page), the hypotheses are connected with an ampersand and
separated from the conclusion with a big arrow, such as in "
& => ". These symbols are _not_
part of the Metamath language but are just informal notation meaning
"and" and "implies".
 Discouraged use and modification.
If something should only be used in limited ways, it is marked with
"(New usage is discouraged.)". This is used, for example, when something
can be constructed in more than one way, and we do not want later
theorems to depend on that specific construction.
This marking is also used if we want later proofs to use proven axioms.
For example, we want later proofs to
use ax1cn 9628 (not ax1cn 9604) and ax1ne0 9639 (not ax1ne0 9615), as these
are proven axioms for complex arithmetic. Thus, both
ax1cn 9604 and ax1ne0 9615 are marked as "(New usage is discouraged.)".
In some cases a proof should not normally be changed, e.g., when it
demonstrates some specific technique.
These are marked with "(Proof modification is discouraged.)".
 New definitions infrequent.
Typically, we are minimalist when introducing new definitions; they are
introduced only when a clear advantage becomes apparent for reducing
the number of symbols, shortening proofs, etc. We generally avoid
the introduction of gratuitous definitions because each one requires
associated theorems and additional elimination steps in proofs.
For example, we use and for inequality expressions, and
use instead of sinh
for the hyperbolic sine.
 Minimizing axioms and the axiom of choice.
We prefer proofs that depend on fewer and/or weaker axioms,
even if the proofs are longer. In particular, we prefer proofs that do
not use the axiom of choice (dfac 8578) where such proofs can be found.
The axiom of choice is widely accepted, and ZFC is the most
commonlyaccepted fundamental set of axioms for mathematics.
However, there have been and still are some lingering controversies
about the Axiom of Choice. Therefore, where a proof
does not require the axiom of choice, we prefer that proof instead.
E.g., our proof of the SchroederBernstein Theorem (sbth 7723)
does not use the axiom of choice.
In some cases, the weaker axiom of countable choice (axcc 8896)
or axiom of dependent choice (axdc 8907) can be used instead.
 Alternative (ALT) proofs.
If a different proof is significantly shorter or clearer but
uses more or stronger axioms, we prefer to make that proof an
"alternative" proof (marked with an ALT label suffix), even if
this alternative proof was formalized first.
We then make the proof that requires fewer axioms the main proof.
This has the effect of reducing (over time)
the number and strength of axioms used by any particular proof.
There can be multiple alternatives if it makes sense to do so.
Alternative (*ALT) theorems should have "(Proof modification is
discouraged.) (New usage is discouraged.)" in their comment and should
follow the main statement, so that people reading the text in order will
see the main statement first. The alternative and main statement
comments should use hyperlinks to refer to each other (so that a reader
of one will become easily aware of the other).
 Alternative (ALTV) versions.
If a theorem or definition is an alternative/variant of an already
existing theorem resp. definition, its label should have the same name
with suffix ALTV. Such alternatives should be temporary only, until it
is decided which alternative should be used in the future. Alternative
(*ALTV) theorems or definitions are usually contained in mathboxes.
Their comments need not to contain "(Proof modification is discouraged.)
(New usage is discouraged.)". Alternative statements should follow the
main statement, so that people reading the text in order will see the
main statement first.
 Old (OLD) versions or proofs.
If a proof, definition, axiom, or theorem is going to be removed,
we often stage that change by first renaming its
label with an OLD suffix (to make it clear that it is going to
be removed). Old (*OLD) statements should have "(Proof modification is
discouraged.) (New usage is discouraged.)" and "Obsolete version of
~ xxx as of ddmmmyyyy." (not enclosed in parentheses) in the comment.
An old statement should follow the main statement, so that people
reading the text in order will see the main statement first.
This typically happens when a shorter proof to an existing theorem is
found: the existing theorem is kept as an *OLD statement for one year.
When a proof is shortened automatically (using Metamath's minimize_with
command), then it is not necessary to keep the old proof, nor to add
credit for the shortening.
 Variables.
Propositional variables (variables for wellformed formulas or wffs) are
represented with lowercase Greek letters and are normally used
in this order:
= phi, = psi, = chi, = theta,
= tau, = eta, = zeta, and = sigma.
Individual setvar variables are represented with lowercase Latin letters
and are normally used in this order:
, , , , , , and .
Variables that represent classes are often represented by
uppercase Latin letters:
, , , , , and so on.
There are other symbols that also represent class variables and suggest
specific purposes, e.g., for poset zero (see p0val 16342) and
connective symbols such as for some group addition operation.
(See prdsplusgval 15426 for an example of the use of ).
Class variables are selected in alphabetical order starting
from if there is no reason to do otherwise, but many
assertions select different class variables or a different order
to make their intended meaning clearer.
 Turnstile.
"", meaning "It is provable that," is the first token
of all assertions
and hypotheses that aren't syntax constructions. This is a standard
convention in logic. For us, it also prevents any ambiguity with
statements that are syntax constructions, such as "wff ".
 Biconditional ().
There are basically two ways to maximize the effectiveness of
biconditionals ():
you can either have onedirectional simplifications of all theorems
that produce biconditionals, or you can have onedirectional
simplifications of theorems that consume biconditionals.
Some tools (like Lean) follow the first approach, but set.mm follows
the second approach. Practically, this means that in set.mm, for
every theorem that uses an implication in the hypothesis, like
axmp 5, there is a corresponding version with a biconditional or a
reversed biconditional, like mpbi 213 or mpbir 214. We prefer this
second approach because the number of duplications in the second
approach is bounded by the size of the propositional calculus section,
which is much smaller than the number of possible theorems in all later
sections that produce biconditionals. So although theorems like
biimpi 199 are available, in most cases there is already a theorem that
combines it with your theorem of choice, like mpbir2an 936, sylbir 218,
or 3imtr4i 274.
 Substitution.
" " should be read "the wff that results from the
proper substitution of for in wff ." See dfsb 1809
and the related dfsbc 3280 and dfcsb 3376.
 Isaset.
" " should be read "Class is a set (i.e. exists)."
This is a convention based on
Definition 2.9 of [Quine] p. 19. See dfv 3059 and isset 3061.
However, instead of using in the antecedent of
a theorem for some variable , we now prefer to
use (or another variable if is not
available) to make it more general. That way we can often avoid
needing extra uses of elex 3066 and syl 17 in the common case where
is already a member of something.
 Converse.
"" should be read "converse of (relation) "
and is the same as the more standard notation R^{1}
(the standard notation is ambiguous). See dfcnv 4864.
This can be used to define a subset, e.g., dftan 14180 notates
"the set of values whose cosine is a nonzero complex number" as
.
 Function application.
"()" should be read "the value
of function at " and has the same meaning as the more
familiar but ambiguous notation F(x). For example,
(see cos0 14259). The left apostrophe notation
originated with Peano and was adopted in Definition *30.01 of
[WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and
Definition 6.11 of [TakeutiZaring] p. 26. See dffv 5613.
In the ASCII (input) representation there are spaces around the grave
accent; there is a single accent when it is used directly,
and it is doubled within comments.
 Infix and parentheses.
When a function that takes two classes and produces a class
is applied as part of an infix expression, the expression is always
surrounded by parentheses (see dfov 6323).
For example, the in ; see 2p2e4 10761.
Function application is itself an example of this.
Similarly, predicate expressions
in infix form that take two or three wffs and produce a wff
are also always surrounded by parentheses, such as
, , , and
(see wi 4, dfor 376, dfan 377, and dfbi 190 respectively).
In contrast, a binary relation (which compares two _classes_ and
produces a _wff_) applied in an infix expression is _not_
surrounded by parentheses.
This includes set membership (see wel 1899),
equality (see dfcleq 2455),
subset (see dfss 3430), and
lessthan (see dflt 9583). For the general definition
of a binary relation in the form , see dfbr 4419.
For example, (see 0lt1 10169) does not use parentheses.
 Unary minus.
The symbol is used to indicate a unary minus, e.g., .
It is specially defined because it is so commonly used.
See cneg 9892.
 Function definition.
Functions are typically defined by first defining the constant symbol
(using $c) and declaring that its symbol is a class with the
label cNAME (e.g., ccos 14172).
The function is then defined labeled dfNAME; definitions
are typically given using the mapsto notation (e.g., dfcos 14179).
Typically, there are other proofs such as its
closure labeled NAMEcl (e.g., coscl 14236), its
function application form labeled NAMEval (e.g., cosval 14232),
and at least one simple value (e.g., cos0 14259).
 Factorial.
The factorial function is traditionally a postfix operation,
but we treat it as a normal function applied in prefix form, e.g.,
; (dffac 12498 and fac4 12505).
 Unambiguous symbols.
A given symbol has a single unambiguous meaning in general.
Thus, where the literature might use the same symbol with different
meanings, here we use different (variant) symbols for different
meanings. These variant symbols often have suffixes, subscripts,
or underlines to distinguish them. For example, here
"" always means the value zero (df0 9577), while
"" is the group identity element (df0g 15395),
"" is the poset zero (dfp0 16340),
"" is the zero polynomial (df0p 22684),
"" is the zero vector in a normed complex vector space
(df0v 26273), and
"" is a class variable for use as a connective symbol
(this is used, for example, in p0val 16342).
There are other class variables used as connective symbols
where traditional notation would use ambiguous symbols, including
"", "", "", and "".
These symbols are very similar to traditional notation, but because
they are different symbols they eliminate ambiguity.
 ASCII representation of symbols.
We must have an ASCII representation for each symbol.
We generally choose short sequences, ideally digraphs, and generally
choose sequences that vaguely resemble the mathematical symbol.
Here are some of the conventions we use when selecting an
ASCII representation.
We generally do not include parentheses inside a symbol because
that confuses text editors (such as emacs).
Greek letters for wff variables always use the first two letters
of their English names, making them easy to type and easy to remember.
Symbols that almost look like letters, such as ,
are often represented by that letter followed by a period.
For example, "A." is used to represent ,
"e." is used to represent , and
"E." is used to represent .
Single letters are now always variable names, so constants that are
often shown as single letters are now typically preceded with "_"
in their ASCII representation, for example,
"_i" is the ASCII representation for the imaginary unit .
A script font constant is often the letter
preceded by "~" meaning "curly", such as "~P" to represent
the power class .
Originally, all setvar and class variables used only single letters
az and AZ, respectively. A big change in recent years was to
allow the use of certain symbols as variable names to make formulas
more readable, such as a variable representing an additive group
operation. The convention is to take the original constant token
(in this case "+" which means complex number addition) and put
a period in front of it to result in the ASCII representation of the
variable ".+", shown as , that can
be used instead of say the letter "P" that had to be used before.
Choosing tokens for more advanced concepts that have no standard
symbols but are represented by words in books, is hard. A few are
reasonably obvious, like "Grp" for group and "Top" for topology,
but often they seem to end up being either too long or too
cryptic. It would be nice if the math community came up with
standardized short abbreviations for English math terminology,
like they have more or less done with symbols, but that probably
won't happen any time soon.
Another informal convention that we've somewhat followed, that is also
not uncommon in the literature, is to start tokens with a
capital letter for collectionlike objects and lower case for
functionlike objects. For example, we have the collections On
(ordinal numbers), Fin, Prime, Grp, and we have the functions sin,
tan, log, sup. Predicates like Ord and Lim also tend to start
with upper case, but in a sense they are really collectionlike,
e.g. Lim indirectly represents the collection of limit ordinals,
but it can't be an actual class since not all limit ordinals
are sets.
This initial capital vs. lower case letter convention is sometimes
ambiguous. In the past there's been a debate about whether
domain and range are collectionlike or functionlike, thus whether
we should use Dom, Ran or dom, ran. Both are used in the literature.
In the end dom, ran won out for aesthetic reasons
(Norm Megill simply just felt they looked nicer).
 Typography conventions.
Class symbols for functions (e.g., , )
should usually not have leading or trailing blanks in their
HTML/Latex representation.
This is in contrast to class symbols for operations
(e.g., , sadd, eval), which usually do
include leading and trailing blanks in their representation.
If a class symbol is used for a function as well as an operation
(according to the definition dfov 6323, each operation value can be
written as function value of an ordered pair), the convention for its
primary usage should be used, e.g. iEdg versus
iEdg for the edges of a graph .
 Number construction independence.
There are many ways to model complex numbers.
After deriving the complex number postulates we
reintroduce them as new axioms on top of set theory.
This lets us easily identify which axioms are needed
for a particular complex number proof, without the obfuscation
of the set theory used to derive them.
This also lets us be independent of the specific construction,
which we believe is valuable.
See mmcomplex for details.
Thus, for example, we don't allow the use of ,
as handy as that would be, because that would be
constructionspecific. We want proofs about to be independent
of whether or not .
 Minimize hypotheses
(except for construction independence and number theorem domains).
In most cases we try to minimize hypotheses, that is,
we eliminate or reduce what must be true to prove something, so that
the proof is more general and easier to use.
There are exceptions. For example, we intentionally add hypotheses
if they help make proofs independent of a particular construction
(e.g., the contruction of complex numbers ).
We also intentionally add hypotheses for many real and complex
number theorems to expressly state their domains even when they
aren't strictly needed. For example, we could show that
without any other hypotheses, but in
practice we also require proving at least some domains
(e.g., see ltnei 9789). Here are the reasons as discussed in
https://groups.google.com/g/metamath/c/2AW7T3d2YiQ/m/iSN7g87t3ikJ :
 Having the hypotheses immediately shows the intended domain of
applicability (is it , , , or something else?),
without having to trace back to definitions.
 Having the hypotheses forces its use in the intended
domain, which generally is desirable.
 The behavior is dependent on accidental behavior of definitions
outside of their domains, so the theorems are nonportable and
"brittle".
 Only a few theorems can have their hypotheses removed
in this fashion due to happy coincidences for our particular
settheoretical definitions. The poor user (especially a
novice learning real number arithmetic) is going to be
confused not knowing when hypotheses are needed and when
they are not. For someone who hasn't traced back the
settheoretical foundations of the definitions, it is
seemingly random and isn't intuitive at all.
 The consensus of opinion of people on this group seemed to be
against doing this.
 Natural numbers.
There are different definitions of "natural" numbers in the literature.
We use (dfnn 10643) for the set of positive integers starting
from 1, and (dfn0 10904) for the set of nonnegative integers
starting at zero.
 Decimal numbers.
Numbers larger than ten are often expressed in base 10 using the decimal
constructor dfdec 11086, e.g.,
;;; (see 4001prm 15171 for a proof that 4001 is prime).
 Theorem forms.
We will use the following descriptive terms to categorize theorems:
 A theorem is in "closed form" if it has no $e hypotheses
(e.g., unss 3620). The term "tautology" is also used, especially in
propositional calculus. This form was formerly called "theorem form"
or "closed theorem form".
 A theorem is in "deduction form" (or is a "deduction") if it
has one or more $e hypotheses, and the hypotheses and the conclusion are
implications that share the same antecedent. More precisely, the
conclusion is an implication with a wff variable as the antecedent
(usually ), and every hypothesis ($e statement) is either:
 an implication with the same antecedent as the conclusion, or
 a definition. A definition can be for a class variable (this is a
class variable followed by , e.g. the definition of in
lhop 23024) or a wff variable (this is a wff variable followed by
); class variable definitions are more common.
In practice, a proof of a theorem in deduction form will also contain
many steps that are implications where the antecedent is either that
wff variable (usually ) or is a conjunction
including that wff variable (). E.g. a1d 26, unssd 3622.
 A theorem is in "inference form" (or is an "inference") if
it has one or more $e hypotheses, but is not in deduction form,
i.e. there is no common antecedent (e.g., unssi 3621).
Any theorem whose conclusion is an implication has an associated
inference, whose hypotheses are the hypotheses of that theorem
together with the antecedent of its conclusion, and whose conclusion is
the consequent of that conclusion. When both theorems are in set.mm,
then the associated inference is often labeled by adding the suffix "i"
to the label of the original theorem (for instance, con3i 142 is the
inference associated with con3 141). The inference associated with a
theorem is easily derivable from that theorem by a simple use of
axmp 5. The other direction is the subject of the Deduction Theorem
discussed below. We may also use the term "associated inference" when
the above process is iterated. For instance, syl 17 is an
inference associated with imim1 79 because it is the inference
associated with imim1i 60 which is itself the inference
associated with imim1 79.
"Deduction form" is the preferred form for theorems because this form
allows us to easily use the theorem in places where (in traditional
textbook formalizations) the standard Deduction Theorem (see below)
would be used. We call this approach "deduction style".
In contrast, we usually avoid theorems in "inference form" when that
would end up requiring us to use the deduction theorem.
Deductions have a label suffix of "d", especially if there are other
forms of the same theorem (e.g., pm2.43d 50). The labels for inferences
usually have the suffix "i" (e.g., pm2.43i 49). The labels of theorems
in "closed form" would have no special suffix (e.g., pm2.43 53). When
an inference is converted to a theorem by eliminating an "is a set"
hypothesis, we sometimes suffix the closed form with "g" (for "more
general") as in uniex 6619 vs. uniexg 6620.
 Deduction theorem.
The Deduction Theorem is a metalogical theorem that
provides an algorithm for constructing a proof of a theorem from
the proof of its corresponding deduction (its associated inference).
See for instance Theorem 3 in [Margaris] p. 56.
In ordinary mathematics, no one actually carries out the algorithm,
because (in its most basic form) it involves an exponential
explosion of the number of proof steps as more hypotheses are
eliminated. Instead, in ordinary mathematics the Deduction Theorem
is invoked simply to claim that something can be done in principle,
without actually doing it. For more details, see
http://us.metamath.org/mpeuni/mmdeduction.html.
The Deduction Theorem is a metalogical theorem that cannot be
applied directly in metamath, and the explosion of steps would
be a problem anyway, so alternatives are used. One alternative
we use sometimes is the "weak deduction theorem" dedth 3944,
which works in certain cases in set theory. We also
sometimes use dedhb 3220.
However, the primary mechanism we use today for
emulating the deduction theorem is to write proofs in deduction
form (aka "deduction style") as described earlier; the
prefixed mimics the context in a deduction proof system.
In practice this mechanism works very well.
This approach is described in the
deduction form
and natural deduction page; a list of translations
for common natural deduction rules is given in natded 25909.
 Recursion.
We define recursive functions using various "recursion constructors".
These allow us to define, with compact direct definitions, functions
that are usually defined in textbooks with indirect selfreferencing
recursive definitions.
This produces compact definition and much simpler proofs, and
greatly reduces the risk of creating unsound definitions.
Examples of recursion constructors include recs in dfrecs 7121,
in dfrdg 7159, seq_{𝜔} in dfseqom 7196, and
in dfseq 12252.
These have characteristic function and initial value .
(_{g} in dfgsum 15396 isn't really designed for arbitrary recursion,
but you could do it with the right magma.)
The logically primary one is dfrecs 7121, but for the "average user"
the most useful one is probably dfseq 12252 provided that
a countable sequence is sufficient for the recursion.
 Extensible structures.
Mathematics includes many structures such as
ring, group, poset, etc. We define an "extensible structure"
which is then used to define group, ring, poset, etc.
This allows theorems from more general structures (groups)
to be reused for more specialized structures (rings) without
having to reprove them. See dfstruct 15178.
 Undefined results and "junk theorems".
Some expressions are only expected to be meaningful
in certain contexts.
For example, consider Russell's definition description binder iota,
where is meant to be "the such that "
(where typically depends on x).
What should that expression produce when there is no such ?
In set.mm we primarily use one of two approaches.
One approach is to make the expression evaluate to the empty set
whenever the expression is being used outside of its expected context.
While not perfect, it makes it a bit more clear when something
is undefined, and it has the advantage that it makes more
things equal outside their domain which can remove hypotheses when
you feel like exploiting these socalled junk theorems.
Note that Quine does this with iota (his definition of iota
evaluates to the empty set when there is no unique value of ).
Quine has no problem with that and we don't see why we should,
so we define iota exactly the same way that Quine does.
The main place where you see this being systematically exploited is in
"reverse closure" theorems like ,
which is useful when is a family of sets. (by this we
mean it's a set set even in a type theoretic interpretation.)
The second approach uses "(New usage is discouraged.)" to prevent
unintentional uses of certain properties.
For example, you could define some construct dfNAME whose
usage is discouraged, and prove only the specific properties
you wish to use (and add those proofs to the list of permitted uses
of "discouraged" information). From then on, you can only use
those specific properties without a warning.
Other approaches often have hidden problems.
For example, you could try to "not define undefined terms"
by creating definitions like ${ $d $. $d $.
dfiota $a $. $}.
This will be rejected by the definition checker, but the bigger
theoretical reason to reject this axiom is that it breaks equality 
the metatheorem P(x) P(y) fails
to hold if definitions don't unfold without some assumptions.
(That is, iotabidv 5590 is no longer provable and must be added
as an axiom.) It is important for every syntax constructor to
satisfy equality theorems *unconditionally*, e.g., expressions
like should not be rejected.
This is forced on us by the context free term
language, and anything else requires a lot more infrastructure
(e.g., a type checker) to support without making everything else
more painful to use.
Another approach would be to try to make nonsensical
statements syntactically invalid, but that can create its own
complexities; in some cases that would make parsing itself undecidable.
In practice this does not seem to be a serious issue.
No one does these things deliberately in "real" situations,
and some knowledgeable people (such as Mario Carneiro)
have never seen this happen accidentally.
Norman Megill doesn't agree that these "junk" consequences are
necessarily bad anyway, and they can significantly shorten proofs
in some cases. This database would be much larger if, for example,
we had to condition fvex 5902 on the argument being in the domain
of the function. It is impossible to derive a contradiction
from sound definitions (i.e. that pass the definition check),
assuming ZFC is consistent, and he doesn't see the point of all the
extra busy work and huge increase in set.mm size that would result
from restricting *all* definitions.
So instead of implementing a complex system to counter a
problem that does not appear to occur in practice, we use
a significantly simpler set of approaches.
 Organizing proofs.
Humans have trouble understanding long proofs.
It is often preferable to break longer proofs into
smaller parts (just as with traditional proofs). In Metamath
this is done by creating separate proofs of the separate parts.
A proof with the sole purpose of supporting a final proof is a
lemma; the naming convention for a lemma is the final proof's name
followed by "lem", and a number if there is more than one. E.g.,
sbthlem1 7713 is the first lemma for sbth 7723. Also, consider proving
reusable results separately, so that others will be able to easily
reuse that part of your work.
 Limit proof size.
It is often preferable to break longer proofs into
smaller parts, just as you would do with traditional proofs.
One reason is that humans have trouble understanding long proofs.
Another reason is that it's generally best to prove
reusable results separately,
so that others will be able to easily reuse them.
Finally, the "minimize" routine can take much longer with
very long proofs.
We encourage proofs to be no more than 200 essential steps, and
generally no more than 500 essential steps,
though these are simply guidelines and not hardandfast rules.
Much smaller proofs are fine!
We also acknowledge that some proofs, especially autogenerated ones,
should sometimes not be broken up (e.g., because
breaking them up might be useless and inefficient due to many
interconnections and reused terms within the proof).
In Metamath, breaking up longer proofs is done by creating multiple
separate proofs of separate parts.
A proof with the sole purpose of supporting a final proof is a
lemma; the naming convention for a lemma is the final proof's name
followed by "lem", and a number if there is more than one. E.g.,
sbthlem1 7713 is the first lemma for sbth 7723.
 Hypertext links.
We strongly encourage comments to have many links to related material,
with accompanying text that explains the relationship. These can help
readers understand the context. Links to other statements, or to
HTTP/HTTPS URLs, can be inserted in ASCII source text by prepending a
spaceseparated tilde (e.g., " ~ dfprm " results in " dfprm 14678").
When metamath.exe is used to generate HTML it automatically inserts
hypertext links for syntax used (e.g., every symbol used), every axiom
and definition depended on, the justification for each step in a proof,
and to both the next and previous assertion.
 Hypertext links to section headers.
Some section headers have text under them that describes or explains the
section. However, they are not part of the description of axioms or
theorems, and there is no way to link to them directly. To provide for
this, section headers with accompanying text (indicated with "*"
prefixed to mmtheorems.html#mmdtoc entries) have an anchor in
mmtheorems.html whose name is the first $a or $p statement that
follows the header. For example there is a glossary under the section
heading called GRAPH THEORY. The first $a or $p statement that follows
is cuhg 25073, which you can see two lines down. To reference it we link
to the anchor using a spaceseparated tilde followed by the
spaceseparated link mmtheorems.html#cuhg, which will become the
hyperlink mmtheorems.html#cuhg. Note that no theorem in set.mm is
allowed to begin with "mm" (enforced by "verify markup" in the metamath
program).
Whenever the software sees a tilde reference beginning with "http:",
"https:", or "mm", the reference is assumed to be a link to something
other than a statement label, and the tilde reference is used as is.
This can also be useful for relative links to other pages such as
mmcomplex.html.
 Bibliography references.
Please include a bibliographic reference to any external material used.
A name in square brackets in a comment indicates a
bibliographic reference. The full reference must be of the form
KEYWORD IDENTIFIER? NOISEWORD(S)* [AUTHOR(S)] p. NUMBER 
note that this is a very specific form that requires a page number.
There should be no comma between the author reference and the
"p." (a constant indicator).
Whitespace, comma, period, or semicolon should follow NUMBER.
An example is Theorem 3.1 of [Monk1] p. 22,
The KEYWORD, which is not casesensitive,
must be one of the following: Axiom, Chapter, Compare, Condition,
Corollary, Definition, Equation, Example, Exercise, Figure, Item,
Lemma, Lemmas, Line, Lines, Notation, Part, Postulate, Problem,
Property, Proposition, Remark, Rule, Scheme, Section, or Theorem.
The IDENTIFIER is optional, as in for example
"Remark in [Monk1] p. 22".
The NOISEWORDS(S) are zero or more from the list: from, in, of, on.
The AUTHOR(S) must be present in the file identified with the
htmlbibliography assignment (e.g., mmset.html) as a named anchor
(NAME=). If there is more than one document by the same author(s),
add a numeric suffix (as shown here).
The NUMBER is a page number, and may be any alphanumeric string such as
an integer or Roman numeral.
Note that we _require_ page numbers in comments for individual
$a or $p statements. We allow names in square brackets without
page numbers (a reference to an entire document) in
heading comments.
If this is a new reference, please also add it to the
"Bibliography" section of mmset.html.
(The file mmbiblio.html is automatically rebuilt, e.g.,
using the metamath.exe "write bibliography" command.)
 Acceptable shorter proofs
Shorter proofs are welcome, and any shorter proof we accept
will be acknowledged in the theorem's description. However,
in some cases a proof may be "shorter" or not depending on
how it is formatted. This section provides general guidelines.
Usually we automatically accept shorter proofs that (1)
shorten the set.mm file (with compressed proofs), (2) reduce
the size of the HTML file generated with SHOW STATEMENT xx
/ HTML, (3) use only existing, unmodified theorems in the
database (the order of theorems may be changed, though), and
(4) use no additional axioms.
Usually we will also automatically accept a _new_ theorem
that is used to shorten multiple proofs, if the total size
of set.mm (including the comment of the new theorem, not
including the acknowledgment) decreases as a result.
In borderline cases, we typically place more importance on
the number of compressed proof steps and less on the length
of the label section (since the names are in principle
arbitrary). If two proofs have the same number of compressed
proof steps, we will typically give preference to the one
with the smaller number of different labels, or if these
numbers are the same, the proof with the fewest number of
characters that the proofs happen to have by chance when
label lengths are included.
A few theorems have a longer proof than necessary in order
to avoid the use of certain axioms, for pedagogical purposes,
and for other reasons. These theorems will (or should) have
a "(Proof modification is discouraged.)" tag in their
description. For example, idALT 23 shows a proof directly from
axioms. Shorter proofs for such cases won't be accepted,
of course, unless the criteria described continues to be
satisfied.
 Input format.
The input is in ASCII with twospace indents. Tab characters are not
allowed. Use embedded math comments or HTML entities for nonASCII
characters (e.g., "é" for "é").
 Information on syntax, axioms, and definitions.
For a hyperlinked list of syntax, axioms, and definitions, see
http://us.metamath.org/mpeuni/mmdefinitions.html.
If you have questions about a specific symbol or axiom, it is best
to go directly to its definition to learn more about it.
The generated HTML for each theorem and axiom includes hypertext
links to each symbol's definition.
 Reserved symbols: 'LETTER.
Some symbols are reserved for potential future use.
Symbols with the pattern 'LETTER are reserved for possibly
representing characters (this is somewhat similar to Lisp).
We would expect '\n to represent newline, 'sp for space, and perhaps
'\x24 for the dollar character.
 Language and spelling.
It is preferred to use American English for comments and symbols, e.g.
we use "neighborhood" instead of the British English "neighbourhood".
An exception is the word "analog", which can be either a noun or an
adjective. Furthermore, "analog" has the confounding meaning "not
digital", whereas "analogue" is often used in the sense something that
bears analogy to something else also in American English. Therefore,
"analogue" is used for the noun and "analogous" for the adjective in
set.mm.
The challenge of varying mathematical conventions
We try to follow mathematical conventions, but in many cases
different texts use different conventions.
In those cases we pick some reasonably common convention and stick to
it.
We have already mentioned that the term "natural number" has
varying definitions (some start from 0, others start from 1), but
that is not the only such case.
A useful example is the set of metavariables used to represent
arbitrary wellformed formulas (wffs).
We use an open phi, φ, to represent the first arbitrary wff in an
assertion with one or more wffs; this is a common convention and
this symbol is easily distinguished from the empty set symbol.
That said, it is impossible to please everyone or simply "follow
the literature" because there are many different conventions for
a variable that represents any arbitrary wff.
To demonstrate the point,
here are some conventions for variables that represent an arbitrary
wff and some texts that use each convention:
 open phi φ (and so on): Tarski's papers,
Rasiowa & Sikorski's
The Mathematics of Metamathematics (1963),
Monk's Introduction to Set Theory (1969),
Enderton's Elements of Set Theory (1977),
Bell & Machover's A Course in Mathematical Logic (1977),
Jech's Set Theory (1978),
Takeuti & Zaring's
Introduction to Axiomatic Set Theory (1982).
 closed phi ϕ (and so on):
Levy's Basic Set Theory (1979),
Kunen's Set Theory (1980),
Paulson's Isabelle: A Generic Theorem Prover (1994),
Huth and Ryan's Logic in Computer Science (2004/2006).
 Greek α, β, γ:
Duffy's Principles of Automated Theorem Proving (1991).
 Roman A, B, C:
Kleene's Introduction to Metamathematics (1974),
Smullyan's FirstOrder Logic (1968/1995).
 script A, B, C:
Hamilton's Logic for Mathematicians (1988).
 italic A, B, C:
Mendelson's Introduction to Mathematical Logic (1997).
 italic P, Q, R:
Suppes's Axiomatic Set Theory (1972),
Gries and Schneider's A Logical Approach to Discrete Math
(1993/1994),
Rosser's Logic for Mathematicians (2008).
 italic p, q, r:
Quine's Set Theory and Its Logic (1969),
Kuratowski & Mostowski's Set Theory (1976).
 italic X, Y, Z:
Dijkstra and Scholten's
Predicate Calculus and Program Semantics (1990).
 Fraktur letters:
Fraenkel et. al's Foundations of Set Theory (1973).
Distinctness or freeness
Here are some conventions that address distinctness or freeness of a
variable:
 is read " is not free in (wff) ";
see dfnf 1679 (whose description has some important technical
details). Similarly, is read is not free in (class)
, see dfnfc 2592.
 "$d x y $." should be read "Assume x and y are distinct
variables."
 "$d x $." should be read "Assume x does not occur in phi $."
Sometimes a theorem is proved using
(dfnf 1679) in place of
"$d $." when a more general result is desired;
ax5 1769 can be used to derive the $d version. For an example of
how to get from the $d version back to the $e version, see the
proof of euf 2318 from dfeu 2314.
 "$d x A $." should be read "Assume x is not a variable occurring in
class A."
 "$d x A $. $d x ps $. $e  $."
is an idiom
often used instead of explicit substitution, meaning "Assume psi results
from the proper substitution of A for x in phi."
 " " occurs early in some cases, and
should be read "If x and y are distinct
variables, then..." This antecedent provides us with a technical
device (called a "distinctor" in Section 7 of [Megill] p. 444)
to avoid the need for the
$d statement early in our development of predicate calculus, permitting
unrestricted substitutions as conceptually simple as those in
propositional calculus. However, the $d eventually becomes a
requirement, and after that this device is rarely used.
There is a general technique to replace a $d x A or
$d x ph condition in a theorem with the corresponding
or ; here it is.
T[x, A] where ,
and you wish to prove T[x, A].
You apply the theorem substituting for and for ,
where is a new dummy variable, so that
$d y A is satisfied.
You obtain T[y, A], and apply chvar to obtain
T[x, A] (or just use mpbir 214 if T[x, A] binds ).
The side goal is T[y, A] T[x, A] ,
where you can use equality theorems, except
that when you get to a bound variable you use a nondv bound variable
renamer theorem like cbval 2125. The section
mmtheorems32.html#mm3146s also describes the
metatheorem that underlies this.
Standard Metamath verifiers do not distinguish between axioms and
definitions (both are $a statements).
In practice, we require that definitions (1) be conservative
(a definition should not allow an expression
that previously qualified as a wff but was not provable
to become provable) and be eliminable
(there should exist an algorithmic method for converting any
expression using the definition into
a logically equivalent expression that previously qualified as a wff).
To ensure this, we have additional rules on almost all definitions
($a statements with a label that does not begin with ax).
These additional rules are not applied in a few cases where they
are too strict (dfbi 190, dfclab 2449, dfcleq 2455, and dfclel 2458);
see those definitions for more information.
These additional rules for definitions are checked by at least
mmj2's definition check (see
mmj2 master file mmj2jar/macros/definitionCheck.js).
This definition check relies on the database being very much like
set.mm, down to the names of certain constants and types, so it
cannot apply to all Metamath databases... but it is useful in set.mm.
In this definition check, a $astatement with a given label and
typecode passes the test if and only if it
respects the following rules (these rules require that we have
an unambiguous tree parse, which is checked separately):
 The expression must be a biconditional or an equality (i.e. its
rootsymbol must be or ).
If the proposed definition passes this first rule, we then
define its definiendum as its left hand side (LHS) and
its definiens as its right hand side (RHS).
We define the *defined symbol* as the rootsymbol of the LHS.
We define a *dummy variable* as a variable occurring
in the RHS but not in the LHS.
Note that the "rootsymbol" is the root of the considered tree;
it need not correspond to a single token in the database
(e.g., see w3o 990 or wsb 1808).
 The defined expression must not appear in any statement
between its syntax axiom () and its definition,
and the defined expression must be not be used in its definiens.
See df3an 993 for an example where the same symbol is used in
different ways (this is allowed).
 No two variables occurring in the LHS may share a
disjoint variable (DV) condition.
 All dummy variables are required to be disjoint from any
other (dummy or not) variable occurring in this labeled expression.
 Either
(a) there must be no nonsetvar dummy variables, or
(b) there must be a justification theorem.
The justification theorem must be of form
definiens rootsymbol definiens'
where definiens' is definiens but the dummy variables are all
replaced with other unused dummy variables of the same type.
Note that rootsymbol is or , and that setvar
variables are simply variables with the typecode.
 One of the following must be true:
(a) there must be no setvar dummy variables,
(b) there must be a justification theorem as described in rule 5, or
(c) if there are setvar dummy variables, every one must not be free.
That is, it must be true that
for each setvar dummy variable
where is the definiens.
We use two different tests for nonfreeness; one must succeed
for each setvar dummy variable .
The first test requires that the setvar dummy variable
be syntactically bound
(this is sometimes called the "fast" test, and this implies
that we must track binding operators).
The second test requires a successful
search for the directlystated proof of
Part c of this rule is how most setvar dummy variables
are handled.
Rule 3 may seem unnecessary, but it is needed.
Without this rule, you can define something like
cbar $a wff Foo x y $.
${ $d x y $. dffoo $a  ( Foo x y <> x = y ) $. $}
and now "Foo x x" is not eliminable;
there is no way to prove that it means anything in particular,
because the definitional theorem that is supposed to be
responsible for connecting it to the original language wants
nothing to do with this expression, even though it is well formed.
A justification theorem for a definition (if used this way)
must be proven before the definition that depends on it.
One example of a justification theorem is vjust 3058.
The definition dfv 3059 is justified
by the justification theorem vjust 3058
.
Another example of a justification theorem is trujust 1457;
the definition dftru 1458
is justified by trujust 1457 .
Here is more information about our processes for checking and
contributing to this work:
 Multiple verifiers.
This entire file is verified by multiple independentlyimplemented
verifiers when it is checked in, giving us extremely high
confidence that all proofs follow from the assumptions.
The checkers also check for various other problems such as
overly long lines.
 Maximum text line length is 79 characters.
You can fix comment line length by running the commands scripts/rewrap
or metamath 'read set.mm' 'save proof */c/f'
'write source set.mm/rewrap' quit .
As a general rule, a math string in a comment should be surrounded
by backquotes on the same line, and if it is too long it should
be broken into multiple adjacent mathstrings on multiple lines.
Those commands don't modify the math content of statements.
In statements we try to break before the outermost important connective
(not including the typecode and perhaps not the antecedent).
For examples, see sqrtmulii 13504 and absmax 13447.
 Discouraged information.
A separate file named "discouraged" lists all
discouraged statements and uses of them, and this file is checked.
If you change the use of discouraged things, you will need to change
this file.
This makes it obvious when there is a change to anything discouraged
(triggering further review).
 LRParser check.
Metamath verifiers ensure that $p statements follow from previous
$a and $p statements.
However, by itself the Metamath language permits certain kinds of
syntactic ambiguity that we choose to avoid in this database.
Thus, we require that this database unambiguously parse
using the "LRParser" check (implemented by at least mmj2).
(For details, see mmj2 master file src/mmj/verify/LRParser.java).
This check
counters, for example, a devious ambiguous construct
developed by saueran at oregonstate dot edu
posted on Mon, 11 Feb 2019 17:32:32 0800 (PST)
based on creating definitions with mismatched parentheses.
 Proposing specific changes.
Please propose specific changes as pull requests (PRs) against the
"develop" branch of set.mm, at:
https://github.com/metamath/set.mm/tree/develop
 Community.
We encourage anyone interested in Metamath to join our mailing list:
https://groups.google.com/forum/#!forum/metamath.
(Contributed by DAW, 27Dec2016.)



Theorem  conventionslabel 25908 
The following explains some of the label conventions in use
in the Metamath Proof Explorer ("set.mm").
For the general conventions, see conventions 25907.
Every statement has a unique identifying label, which serves the
same purpose as an equation number in a book.
We use various label naming conventions to provide
easytoremember hints about their contents.
Labels are not a 1to1 mapping, because that would create
long names that would be difficult to remember and tedious to type.
Instead, label names are relatively short while
suggesting their purpose.
Names are occasionally changed to make them more consistent or
as we find better ways to name them.
Here are a few of the label naming conventions:
 Axioms, definitions, and wff syntax.
As noted earlier, axioms are named "axNAME",
proofs of proven axioms are named "axNAME", and
definitions are named "dfNAME".
Wff syntax declarations have labels beginning with "w"
followed by short fragment suggesting its purpose.
 Hypotheses.
Hypotheses have the name of the final axiom or theorem, followed by
".", followed by a unique id (these ids are usually consecutive integers
starting with 1, e.g. for rgen 2759"rgen.1 $e  ( x e. A > ph ) $."
or letters corresponding to the (main) class variable used in the
hypothesis, e.g. for mdet0 19686: "mdet0.d $e  D = ( N maDet R ) $.").
 Common names.
If a theorem has a wellknown name, that name (or a short version of it)
is sometimes used directly. Examples include
barbara 2403 and stirling 38052.
 Principia Mathematica.
Proofs of theorems from Principia Mathematica often use a special
naming convention: "pm" followed by its identifier.
For example, Theorem *2.27 of [WhiteheadRussell] p. 104 is named
pm2.27 40.
 19.x series of theorems.
Similar to the conventions for the theorems from Principia Mathematica,
theorems from Section 19 of [Margaris] p. 90 often use a special naming
convention: "19." resp. "r19." (for corresponding restricted quantifier
versions) followed by its identifier.
For example, Theorem 38 from Section 19 of [Margaris] p. 90 is labeled
19.38 1723, and the restricted quantifier version of Theorem 21 from
Section 19 of [Margaris] p. 90 is labeled r19.21 2799.
 Characters to be used for labels
Although the specification of Metamath allows for dots/periods "." in
any label, it is usually used only in labels for hypotheses (see above).
Exceptions are the labels of theorems from Principia Mathematica and the
19.x series of theorems from Section 19 of [Margaris] p. 90 (see above)
and 0.999... 13992. Furthermore, the underscore "_" should not be used.
 Syntax label fragments.
Most theorems are named using a concatenation of syntax label fragments
(omitting variables) that represent the important part of the theorem's
main conclusion. Almost every syntactic construct has a definition
labeled "dfNAME", and normally NAME is the syntax label fragment. For
example, the class difference construct is defined in
dfdif 3419, and thus its syntax label fragment is "dif". Similarly, the
subclass relation has syntax label fragment "ss"
because it is defined in dfss 3430. Most theorem names follow from
these fragments, for example, the theorem proving
involves a class difference ("dif") of a subset ("ss"), and thus is
labeled difss 3572. There are many other syntax label fragments, e.g.,
singleton construct has syntax label fragment "sn" (because it
is defined in dfsn 3981), and the pair construct has
fragment "pr" ( from dfpr 3983). Digits are used to represent
themselves. Suffixes (e.g., with numbers) are sometimes used to
distinguish multiple theorems that would otherwise produce the same
label.
 Phantom definitions.
In some cases there are common label fragments for something that could
be in a definition, but for technical reasons is not. The iselementof
(is member of) construct does not have a dfNAME definition;
in this case its syntax label fragment is "el". Thus, because the
theorem beginning with uses iselementof
("el") of a class difference ("dif") of a singleton ("sn"), it is
labeled eldifsn 4110. An "n" is often used for negation (), e.g.,
nan 588.
 Exceptions.
Sometimes there is a definition dfNAME but the label fragment is not
the NAME part. The definition should note this exception as part of its
definition. In addition, the table below attempts to list all such
cases and marks them in bold. For example, the label fragment "cn"
represents complex numbers (even though its definition is in
dfc 9576) and "re" represents real numbers ( definition dfr 9580).
The empty set often uses fragment 0, even though it is defined
in dfnul 3744. The syntax construct usually uses the
fragment "add" (which is consistent with dfadd 9581), but "p" is used as
the fragment for constant theorems. Equality often uses
"e" as the fragment. As a result, "two plus two equals four" is labeled
2p2e4 10761.
 Other markings.
In labels we sometimes use "com" for "commutative", "ass" for
"associative", "rot" for "rotation", and "di" for "distributive".
 Focus on the important part of the conclusion.
Typically the conclusion is the part the user is most interested in.
So, a rough guideline is that a label typically provides a hint
about only the conclusion; a label rarely says anything about the
hypotheses or antecedents.
If there are multiple theorems with the same conclusion
but different hypotheses/antecedents, then the labels will need
to differ; those label differences should emphasize what is different.
There is no need to always fully describe the conclusion; just
identify the important part. For example,
cos0 14259 is the theorem that provides the value for the cosine of 0;
we would need to look at the theorem itself to see what that value is.
The label "cos0" is concise and we use it instead of "cos0eq1".
There is no need to add the "eq1", because there will never be a case
where we have to disambiguate between different values produced by
the cosine of zero, and we generally prefer shorter labels if
they are unambiguous.
 Closures and values.
As noted above, if a function dfNAME is defined, there is typically a
proof of its value labeled "NAMEval" and of its closure labeld "NAMEcl".
E.g., for cosine (dfcos 14179) we have value cosval 14232 and closure
coscl 14236.
 Special cases.
Sometimes, syntax and related markings are insufficient to distinguish
different theorems. For example, there are over a hundred different
implicationonly theorems. They are grouped in a more adhoc way that
attempts to make their distinctions clearer. These often use
abbreviations such as "mp" for "modus ponens", "syl" for syllogism, and
"id" for "identity". It is especially hard to give good names in the
propositional calculus section because there are so few primitives.
However, in most cases this is not a serious problem. There are a few
very common theorems like axmp 5 and syl 17 that you will have no
trouble remembering, a few theorem series like syl*anc and simp* that
you can use parametrically, and a few other useful glue things for
destructuring 'and's and 'or's (see natded 25909 for a list), and that is
about all you need for most things. As for the rest, you can just
assume that if it involves at most three connectives, then it is
probably already proved in set.mm, and searching for it will give you
the label.
 Suffixes.
Suffixes are used to indicate the form of a theorem (see above).
Additionally, we sometimes suffix with "v" the label of a theorem
eliminating a hypothesis such as in 19.21 1998 via the use of
disjoint variable conditions combined with nfv 1772. If two (or three)
such hypotheses are eliminated, the suffix "vv" resp. "vvv" is used,
e.g. exlimivv 1789.
Conversely, we sometimes suffix with "f" the label of a theorem
introducing such a hypothesis to eliminate the need for the disjoint
variable condition; e.g. euf 2318 derived from dfeu 2314. The "f" stands
for "not free in" which is less restrictive than "does not occur in."
The suffix "b" often means "biconditional" (, "iff" , "if and
only if"), e.g. sspwb 4666.
We sometimes suffix with "s" the label of an inference that manipulates
an antecedent, leaving the consequent unchanged. The "s" means that the
inference eliminates the need for a syllogism (syl 17) type inference
in a proof. A theorem label is suffixed with "ALT" if it provides an
alternate lesspreferred proof of a theorem (e.g., the proof is
clearer but uses more axioms than the preferred version).
The "ALT" may be further suffixed with a number if there is more
than one alternate theorem.
Furthermore, a theorem label is suffixed with "OLD" if there is a new
version of it and the OLD version is obsolete (and will be removed
within one year).
Finally, it should be mentioned that suffixes can be combined, for
example in cbvaldva 2135 (cbval 2125 in deduction form "d" with a not free
variable replaced by a disjoint variable condition "v" with a
conjunction as antecedent "a").
Here is a nonexhaustive list of common suffixes:
 a : theorem having a conjunction as antecedent
 b : theorem expressing a logical equivalence
 c : contraction (e.g., sylc 62, syl2anc 671), commutes
(e.g., biimpac 493)
 d : theorem in deduction form
 f : theorem with a hypothesis such as
 g : theorem in closed form having an "is a set" antecedent
 i : theorem in inference form
 l : theorem concerning something at the left
 r : theorem concerning something at the right
 r : theorem with something reversed (e.g., a biconditional)
 s : inference that manipulates an antecedent ("s" refers to an
application of syl 17 that is eliminated)
 v : theorem with one (main) disjoint variable condition
 vv : theorem with two (main) disjoint variable conditions
 w : weak(er) form of a theorem
 ALT : alternate proof of a theorem
 ALTV : alternate version of a theorem or definition
 OLD : old/obsolete version of a theorem/definition/proof
 Reuse.
When creating a new theorem or axiom, try to reuse abbreviations used
elsewhere. A comment should explain the first use of an abbreviation.
The following table shows some commonly used abbreviations in labels, in
alphabetical order. For each abbreviation we provide a mnenomic, the
source theorem or the assumption defining it, an expression showing what
it looks like, whether or not it is a "syntax fragment" (an abbreviation
that indicates a particular kind of syntax), and hyperlinks to label
examples that use the abbreviation. The abbreviation is bolded if there
is a dfNAME definition but the label fragment is not NAME. This is
not a complete list of abbreviations, though we do want this to
eventually be a complete list of exceptions.
Abbreviation  Mnenomic  Source 
Expression  Syntax?  Example(s) 
a  and (suffix)  
 No  biimpa 491, rexlimiva 2887 
abl  Abelian group  dfabl 17488 
 Yes  ablgrp 17490, zringabl 19098 
abs  absorption    No 
ressabs 15243 
abs  absolute value (of a complex number) 
dfabs 13354   Yes 
absval 13356, absneg 13395, abs1 13415 
ad  adding  
 No  adantr 471, ad2antlr 738 
add  add (see "p")  dfadd 9581 
 Yes 
addcl 9652, addcom 9850, addass 9657 
al  "for all"  
 No  alim 1694, alex 1709 
ALT  alternative/less preferred (suffix)  
 No  aevALT 2166 
an  and  dfan 377 
 Yes 
anor 496, iman 430, imnan 428 
ant  antecedent  
 No  adantr 471 
ass  associative  
 No  biass 365, orass 531, mulass 9658 
asym  asymmetric, antisymmetric  
 No  intasym 5237, asymref 5238, posasymb 16253 
ax  axiom  
 No  ax6dgen 1913, ax1cn 9604 
bas, base 
base (set of an extensible structure)  dfbase 15181 
 Yes 
baseval 15223, ressbas 15234, cnfldbas 19029 
b, bi  biconditional ("iff", "if and only if")
 dfbi 190   Yes 
impbid 195, sspwb 4666 
br  binary relation  dfbr 4419 
 Yes  brab1 4464, brun 4467 
cbv  change bound variable   
No  cbvalivw 1861, cbvrex 3028 
cl  closure    No 
ifclda 3925, ovrcl 6353, zaddcl 11011 
cn  complex numbers  dfc 9576 
 Yes  nnsscn 10647, nncn 10650 
cnfld  field of complex numbers  dfcnfld 19026 
ℂ_{fld}  Yes  cnfldbas 19029, cnfldinv 19054 
cntz  centralizer  dfcntz 17026 
Cntz  Yes 
cntzfval 17029, dprdfcntz 17703 
cnv  converse  dfcnv 4864 
 Yes  opelcnvg 5036, f1ocnv 5853 
co  composition  dfco 4865 
 Yes  cnvco 5042, fmptco 6085 
com  commutative  
 No  orcom 393, bicomi 207, eqcomi 2471 
con  contradiction, contraposition  
 No  condan 808, con2d 120 
csb  class substitution  dfcsb 3376 
 Yes 
csbid 3383, csbie2g 3406 
cyg  cyclic group  dfcyg 17568 
CycGrp  Yes 
iscyg 17569, zringcyg 19115 
d  deduction form (suffix)  
 No  idd 25, impbid 195 
df  (alternate) definition (prefix)  
 No  dfrel2 5308, dffn2 5757 
di, distr  distributive  
 No 
andi 883, imdi 369, ordi 880, difindi 3709, ndmovdistr 6490 
dif  class difference  dfdif 3419 
 Yes 
difss 3572, difindi 3709 
div  division  dfdiv 10303 
 Yes 
divcl 10309, divval 10305, divmul 10306 
dm  domain  dfdm 4866 
 Yes  dmmpt 5353, iswrddm0 12732 
e, eq, equ  equals  dfcleq 2455 
 Yes 
2p2e4 10761, uneqri 3588, equtr 1876 
el  element of  
 Yes 
eldif 3426, eldifsn 4110, elssuni 4241 
eu  "there exists exactly one"  dfeu 2314 
 Yes  euex 2334, euabsn 4057 
ex  exists (i.e. is a set)  
 No  brrelex 4895, 0ex 4551 
ex  "there exists (at least one)"  dfex 1675 
 Yes  exim 1717, alex 1709 
exp  export  
 No  expt 161, expcom 441 
f  "not free in" (suffix)  
 No  equs45f 2192, sbf 2220 
f  function  dff 5609 
 Yes  fssxp 5768, opelf 5772 
fal  false  dffal 1461 
 Yes  bifal 1468, falantru 1479 
fi  finite intersection  dffi 7956 
 Yes  fival 7957, inelfi 7963 
fi, fin  finite  dffin 7604 
 Yes 
isfi 7624, snfi 7681, onfin 7794 
fld  field (Note: there is an alternative
definition of a field, see dffld 26198)  dffield 18033 
Field  Yes  isfld 18039, fldidom 18584 
fn  function with domain  dffn 5608 
 Yes  ffn 5755, fndm 5701 
frgp  free group  dffrgp 17415 
freeGrp  Yes 
frgpval 17463, frgpadd 17468 
fsupp  finitely supported function 
dffsupp 7915  finSupp  Yes 
isfsupp 7918, fdmfisuppfi 7923, fsuppco 7946 
fun  function  dffun 5607 
 Yes  funrel 5622, ffun 5758 
fv  function value  dffv 5613 
 Yes  fvres 5906, swrdfv 12823 
fz  finite set of sequential integers 
dffz 11820 
 Yes  fzval 11821, eluzfz 11830 
fz0  finite set of sequential nonnegative integers 

 Yes  nn0fz0 11925, fz0tp 11928 
fzo  halfopen integer range  dffzo 11953 
..^  Yes 
elfzo 11959, elfzofz 11972 
g  more general (suffix); eliminates "is a set"
hypothsis  
 No  uniexg 6620 
gra  graph  
 No  uhgrav 25079, isumgra 25098, usgrares 25152 
grp  group  dfgrp 16728 
 Yes  isgrp 16732, tgpgrp 21148 
gsum  group sum  dfgsum 15396 
_{g}  Yes 
gsumval 16569, gsumwrev 17072 
hash  size (of a set)  dfhash 12554 
 Yes 
hashgval 12556, hashfz1 12567, hashcl 12576 
hb  hypothesis builder (prefix)  
 No  hbxfrbi 1705, hbald 1937, hbequid 32526 
hm  (monoid, group, ring) homomorphism  
 No  ismhm 16639, isghm 16938, isrhm 18004 
i  inference (suffix)  
 No  eleq1i 2531, tcsni 8258 
i  implication (suffix)  
 No  brwdomi 8114, infeq5i 8172 
id  identity  
 No  biid 244 
idm  idempotent  
 No  anidm 654, tpidm13 4087 
im, imp  implication (label often omitted) 
dfim 13219   Yes 
iman 430, imnan 428, impbidd 193 
ima  image  dfima 4869 
 Yes  resima 5159, imaundi 5270 
imp  import  
 No  biimpa 491, impcom 436 
in  intersection  dfin 3423 
 Yes  elin 3629, incom 3637 
inf  infimum  dfinf 7988 
inf  Yes 
fiinfcl 8048, infiso 8054 
is...  is (something a) ...?  
 No  isring 17839 
j  joining, disjoining  
 No  jc 152, jaoi 385 
l  left  
 No  olcd 399, simpl 463 
map  mapping operation or set exponentiation 
dfmap 7505   Yes 
mapvalg 7513, elmapex 7523 
mat  matrix  dfmat 19488 
Mat  Yes 
matval 19491, matring 19523 
mdet  determinant (of a square matrix) 
dfmdet 19665  maDet  Yes 
mdetleib 19667, mdetrlin 19682 
mgm  magma  dfmgm 16543 
 Yes 
mgmidmo 16557, mgmlrid 16564, ismgm 16544 
mgp  multiplicative group  dfmgp 17779 
mulGrp  Yes 
mgpress 17789, ringmgp 17841 
mnd  monoid  dfmnd 16592 
 Yes  mndass 16601, mndodcong 17246 
mo  "there exists at most one"  dfmo 2315 
 Yes  eumo 2339, moim 2359 
mp  modus ponens  axmp 5 
 No  mpd 15, mpi 20 
mpt  modus ponendo tollens  
 No  mptnan 1662, mptxor 1663 
mpt  mapsto notation for a function 
dfmpt 4479   Yes 
fconstmpt 4900, resmpt 5176 
mpt2  mapsto notation for an operation 
dfmpt2 6325   Yes 
mpt2mpt 6420, resmpt2 6426 
mul  multiplication (see "t")  dfmul 9582 
 Yes 
mulcl 9654, divmul 10306, mulcom 9656, mulass 9658 
n, not  not  
 Yes 
nan 588, notnot2 117 
ne  not equal  dfne  
Yes  exmidne 2645, neeqtrd 2705 
nel  not element of  dfnel 

Yes  neli 2738, nnel 2745 
ne0  not equal to zero (see n0)  
 No 
negne0d 10015, ine0 10087, gt0ne0 10112 
nf  "not free in" (prefix)  
 No  nfnd 1995 
ngp  normed group  dfngp 21653 
NrmGrp  Yes  isngp 21665, ngptps 21671 
nm  norm (on a group or ring)  dfnm 21652 
 Yes 
nmval 21659, subgnm 21696 
nn  positive integers  dfnn 10643 
 Yes  nnsscn 10647, nncn 10650 
nn0  nonnegative integers  dfn0 10904 
 Yes  nnnn0 10910, nn0cn 10913 
n0  not the empty set (see ne0)  
 No  n0i 3748, vn0 3751, ssn0 3779 
OLD  old, obsolete (to be removed soon)  
 No  19.43OLD 1757 
op  ordered pair  dfop 3987 
 Yes  dfopif 4177, opth 4693 
or  or  dfor 376 
 Yes 
orcom 393, anor 496 
ot  ordered triple  dfot 3989 
 Yes 
euotd 4719, fnotovb 6361 
ov  operation value  dfov 6323 
 Yes
 fnotovb 6361, fnovrn 6476 
p  plus (see "add"), for allconstant
theorems  dfadd 9581 
 Yes 
3p2e5 10776 
pfx  prefix  dfpfx 39060 
prefix  Yes 
pfxlen 39069, ccatpfx 39087 
pm  Principia Mathematica  
 No  pm2.27 40 
pm  partial mapping (operation)  dfpm 7506 
 Yes  elpmi 7521, pmsspw 7537 
pr  pair  dfpr 3983 
 Yes 
elpr 3998, prcom 4063, prid1g 4091, prnz 4104 
prm, prime  prime (number)  dfprm 14678 
 Yes  1nprm 14684, dvdsprime 14692 
pss  proper subset  dfpss 3432 
 Yes  pssss 3540, sspsstri 3547 
q  rational numbers ("quotients")  dfq 11299 
 Yes  elq 11300 
r  right  
 No  orcd 398, simprl 769 
rab  restricted class abstraction 
dfrab 2758   Yes 
rabswap 2982, dfoprab 6324 
ral  restricted universal quantification 
dfral 2754   Yes 
ralnex 2846, ralrnmpt2 6443 
rcl  reverse closure  
 No  ndmfvrcl 5917, nnarcl 7348 
re  real numbers  dfr 9580 
 Yes  recn 9660, 0re 9674 
rel  relation  dfrel 4863  
Yes  brrelex 4895, relmpt2opab 6910 
res  restriction  dfres 4868 
 Yes 
opelres 5132, f1ores 5855 
reu  restricted existential uniqueness 
dfreu 2756   Yes 
nfreud 2975, reurex 3021 
rex  restricted existential quantification 
dfrex 2755   Yes 
rexnal 2848, rexrnmpt2 6444 
rmo  restricted "at most one" 
dfrmo 2757   Yes 
nfrmod 2976, nrexrmo 3024 
rn  range  dfrn 4867  
Yes  elrng 5048, rncnvcnv 5080 
rng  (unital) ring  dfring 17837 
 Yes 
ringidval 17792, isring 17839, ringgrp 17840 
rot  rotation  
 No  3anrot 996, 3orrot 997 
s  eliminates need for syllogism (suffix) 
  No  ancoms 459 
sb  (proper) substitution (of a set) 
dfsb 1809   Yes 
spsbe 1812, sbimi 1814 
sbc  (proper) substitution of a class 
dfsbc 3280   Yes 
sbc2or 3288, sbcth 3294 
sca  scalar  dfsca 15261 
Scalar  Yes 
resssca 15330, mgpsca 17785 
simp  simple, simplification  
 No  simpl 463, simp3r3 1124 
sn  singleton  dfsn 3981 
 Yes  eldifsn 4110 
sp  specialization  
 No  spsbe 1812, spei 2116 
ss  subset  dfss 3430 
 Yes  difss 3572 
struct  structure  dfstruct 15178 
Struct  Yes  brstruct 15184, structfn 15189 
sub  subtract  dfsub 9893 
 Yes 
subval 9897, subaddi 9993 
sup  supremum  dfsup 7987 
 Yes 
fisupcl 8016, supmo 7997 
supp  support (of a function)  dfsupp 6947 
supp  Yes 
ressuppfi 7940, mptsuppd 6970 
swap  swap (two parts within a theorem) 
  No  rabswap 2982, 2reuswap 3254 
syl  syllogism  syl 17 
 No  3syl 18 
sym  symmetric  
 No  dfsymdif 3675, cnvsym 5236 
symg  symmetric group  dfsymg 17074 
 Yes 
symghash 17081, pgrpsubgsymg 17104 
t 
times (see "mul"), for allconstant theorems 
dfmul 9582 
 Yes 
3t2e6 10795 
th  theorem  
 No  nfth 1687, sbcth 3294, weth 8956 
tp  triple  dftp 3985 
 Yes 
eltpi 4028, tpeq1 4073 
tr  transitive  
 No  bitrd 261, biantr 947 
tru  true  dftru 1458 
 Yes  bitru 1467, truanfal 1478 
un  union  dfun 3421 
 Yes 
uneqri 3588, uncom 3590 
unit  unit (in a ring) 
dfunit 17925  Unit  Yes 
isunit 17940, nzrunit 18546 
v  disjoint variable conditions used when
a notfree hypothesis (suffix) 
  No  spimv 2112 
vv  2 disjoint variables (in a notfree hypothesis)
(suffix)    No  19.23vv 1830 
w  weak (version of a theorem) (suffix)  
 No  ax11w 1915, spnfw 1855 
wrd  word 
dfword 12703  Word  Yes 
iswrdb 12716, wrdfn 12724, ffz0iswrd 12736 
xp  cross product (Cartesian product) 
dfxp 4862   Yes 
elxp 4873, opelxpi 4888, xpundi 4909 
xr  eXtended reals  dfxr 9710 
 Yes  ressxr 9715, rexr 9717, 0xr 9718 
z  integers (from German "Zahlen") 
dfz 10972   Yes 
elz 10973, zcn 10976 
zn  ring of integers  dfzn 19133 
ℤ/nℤ  Yes 
znval 19161, zncrng 19170, znhash 19184 
zring  ring of integers  dfzring 19095 
ℤ_{ring}  Yes  zringbas 19100, zringcrng 19096

0, z 
slashed zero (empty set) (see n0)  dfnul 3744 
 Yes 
n0i 3748, vn0 3751; snnz 4103, prnz 4104 
(Contributed by DAW, 27Dec2016.)



17.1.2 Natural deduction


Theorem  natded 25909 
Here are typical natural deduction (ND) rules in the style of Gentzen
and Jaśkowski, along with MPE translations of them.
This also shows the recommended theorems when you find yourself
needing these rules (the recommendations encourage a slightly
different proof style that works more naturally with metamath).
A decent list of the standard rules of natural deduction can be
found beginning with definition /\I in [Pfenning] p. 18.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
Many more citations could be added.
Name  Natural Deduction Rule  Translation 
Recommendation  Comments 
IT 
=> 
idi 2 
nothing  Reiteration is always redundant in Metamath.
Definition "new rule" in [Pfenning] p. 18,
definition IT in [Clemente] p. 10. 
I 
& => 
jca 539 
jca 539, pm3.2i 461 
Definition I in [Pfenning] p. 18,
definition Im,n in [Clemente] p. 10, and
definition I in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) 
E_{L} 
=> 
simpld 465 
simpld 465, adantr 471 
Definition E_{L} in [Pfenning] p. 18,
definition E(1) in [Clemente] p. 11, and
definition E in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) 
E_{R} 
=> 
simprd 469 
simpr 467, adantl 472 
Definition E_{R} in [Pfenning] p. 18,
definition E(2) in [Clemente] p. 11, and
definition E in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) 
I 
=> 
ex 440  ex 440 
Definition I in [Pfenning] p. 18,
definition I=>m,n in [Clemente] p. 11, and
definition I in [Indrzejczak] p. 33. 
E 
& => 
mpd 15  axmp 5, mpd 15, mpdan 679, imp 435 
Definition E in [Pfenning] p. 18,
definition E=>m,n in [Clemente] p. 11, and
definition E in [Indrzejczak] p. 33. 
I_{L}  =>

olcd 399 
olc 390, olci 397, olcd 399 
Definition I in [Pfenning] p. 18,
definition In(1) in [Clemente] p. 12 
I_{R}  =>

orcd 398 
orc 391, orci 396, orcd 398 
Definition I_{R} in [Pfenning] p. 18,
definition In(2) in [Clemente] p. 12. 
E  & &
=> 
mpjaodan 800 
mpjaodan 800, jaodan 799, jaod 386 
Definition E in [Pfenning] p. 18,
definition Em,n,p in [Clemente] p. 12. 
I  => 
inegd 1474  pm2.01d 174 

I  & =>

mtand 669  mtand 669 
definition Im,n,p in [Clemente] p. 13. 
I  & =>

pm2.65da 584  pm2.65da 584 
Contradiction. 
I 
=> 
pm2.01da 448  pm2.01d 174, pm2.65da 584, pm2.65d 180 
For an alternative falsumfree natural deduction ruleset 
E 
& => 
pm2.21fal 1476 
pm2.21dd 179  
E 
=> 

pm2.21dd 179 
definition E in [Indrzejczak] p. 33. 
E 
& => 
pm2.21dd 179  pm2.21dd 179, pm2.21d 110, pm2.21 112 
For an alternative falsumfree natural deduction ruleset.
Definition E in [Pfenning] p. 18. 
I  
a1tru 1471  tru 1459, a1tru 1471, trud 1464 
Definition I in [Pfenning] p. 18. 
E  
falimd 1470  falim 1469 
Definition E in [Pfenning] p. 18. 
I 
=> 
alrimiv 1784  alrimiv 1784, ralrimiva 2814 
Definition I^{a} in [Pfenning] p. 18,
definition In in [Clemente] p. 32. 
E 
=> 
spsbcd 3293  spcv 3152, rspcv 3158 
Definition E in [Pfenning] p. 18,
definition En,t in [Clemente] p. 32. 
I 
=> 
spesbcd 3362  spcev 3153, rspcev 3162 
Definition I in [Pfenning] p. 18,
definition In,t in [Clemente] p. 32. 
E 
& =>

exlimddv 1792  exlimddv 1792, exlimdd 2081,
exlimdv 1790, rexlimdva 2891 
Definition E^{a,u} in [Pfenning] p. 18,
definition Em,n,p,a in [Clemente] p. 32. 
C 
=> 
efald 1475  efald 1475 
Proof by contradiction (classical logic),
definition C in [Pfenning] p. 17. 
C 
=> 
pm2.18da 449  pm2.18da 449, pm2.18d 116, pm2.18 114 
For an alternative falsumfree natural deduction ruleset 
C 
=> 
notnotrd 118  notnotrd 118, notnot2 117 
Double negation rule (classical logic),
definition NNC in [Pfenning] p. 17,
definition En in [Clemente] p. 14. 
EM  
exmidd 422  exmid 421 
Excluded middle (classical logic),
definition XM in [Pfenning] p. 17,
proof 5.11 in [Clemente] p. 14. 
I  
eqidd 2463  eqid 2462, eqidd 2463 
Introduce equality,
definition =I in [Pfenning] p. 127. 
E  & =>

sbceq1dd 3285  sbceq1d 3284, equality theorems 
Eliminate equality,
definition =E in [Pfenning] p. 127. (Both E1 and E2.) 
Note that MPE uses classical logic, not intuitionist logic.
As is conventional, the "I" rules are introduction rules,
"E" rules are elimination rules, the "C" rules are conversion rules,
and
represents the set of (current) hypotheses.
We use wff variable names beginning with to provide a closer
representation of the Metamath equivalents
(which typically use the antedent to represent
the context ).
Most of this information was developed by
Mario Carneiro and posted on 3Feb2017.
For more information, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
For annotated examples where some traditional ND rules
are directly applied in MPE, see exnatded5.2 25910, exnatded5.3 25913,
exnatded5.5 25916, exnatded5.7 25917, exnatded5.8 25919, exnatded5.13 25921,
exnatded9.20 25923, and exnatded9.26 25925.
(Contributed by DAW, 4Feb2017.) (New usage is discouraged.)



17.1.3 Natural deduction examples
These are examples of how natural deduction rules can be applied
in metamath (both as lineforline translations of ND rules, and as
a way to apply deduction forms without being limited to applying
ND rules). For more information, see natded 25909 and
http://us.metamath.org/mpeuni/mmnatded.html.
Since these examples should not be used within proofs of other theorems,
especially in Mathboxes, they are marked with "(New usage is
discouraged.)".


Theorem  exnatded5.2 25910 
Theorem 5.2 of [Clemente] p. 15, translated line by line using the
interpretation of natural deduction in Metamath.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows:
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  5  

Given 
$e. 
2  2  

Given 
$e. 
3  1  

Given 
$e. 
4  3  

E 2,3 
mpd 15, the MPE equivalent of E, 1,2 
5  4  

I 4,3 
jca 539, the MPE equivalent of I, 3,1 
6  6  

E 1,5 
mpd 15, the MPE equivalent of E, 4,5 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including and uses the Metamath equivalents
of the natural deduction rules.
Below is the final metamath proof (which reorders some steps).
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.22 25911.
A proof without context is shown in exnatded5.2i 25912.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)



Theorem  exnatded5.22 25911 
A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with
exnatded5.2 25910 and exnatded5.2i 25912. (New usage is discouraged.)
(Proof modification is discouraged.) (Contributed by Mario Carneiro,
9Feb2017.)



Theorem  exnatded5.2i 25912 
The same as exnatded5.2 25910 and exnatded5.22 25911 but with no context.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)



Theorem  exnatded5.3 25913 
Theorem 5.3 of [Clemente] p. 16, translated line by line using an
interpretation of natural deduction in Metamath.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.32 25914.
A proof without context is shown in exnatded5.3i 25915.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer
.
The original proof, which uses Fitch style, was written as follows:
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  2;3  

Given 
$e; adantr 471 to move it into the ND hypothesis 
2  5;6  

Given 
$e; adantr 471 to move it into the ND hypothesis 
3  1  ... 

ND hypothesis assumption 
simpr 467, to access the new assumption 
4  4  ... 

E 1,3 
mpd 15, the MPE equivalent of E, 1.3.
adantr 471 was used to transform its dependency
(we could also use imp 435 to get this directly from 1)

5  7  ... 

E 2,4 
mpd 15, the MPE equivalent of E, 4,6.
adantr 471 was used to transform its dependency 
6  8  ... 

I 4,5 
jca 539, the MPE equivalent of I, 4,7 
7  9  

I 3,6 
ex 440, the MPE equivalent of I, 8 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including and uses the Metamath equivalents
of the natural deduction rules.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)



Theorem  exnatded5.32 25914 
A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with
exnatded5.3 25913 and exnatded5.3i 25915. (New usage is discouraged.)
(Proof modification is discouraged.) (Contributed by Mario Carneiro,
9Feb2017.)



Theorem  exnatded5.3i 25915 
The same as exnatded5.3 25913 and exnatded5.32 25914 but with no context.
Identical to jccir 546, which should be used instead.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)



Theorem  exnatded5.5 25916 
Theorem 5.5 of [Clemente] p. 18, translated line by line using the
usual translation of natural deduction (ND) in the
Metamath Proof Explorer (MPE) notation.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  2;3 


Given 
$e; adantr 471 to move it into the ND hypothesis 
2  5  
 Given 
$e; we'll use adantr 471 to move it into the ND hypothesis 
3  1 
...  
ND hypothesis assumption 
simpr 467 
4  4  ... 

E 1,3 
mpd 15 1,3 
5  6  ... 

IT 2 
adantr 471 5 
6  7  

I 3,4,5 
pm2.65da 584 4,6 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 471; simpr 467 is useful when you want to
depend directly on the new assumption).
Below is the final metamath proof (which reorders some steps).
A much more efficient proof is mtod 182;
a proof without context is shown in mto 181.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by David A. Wheeler, 19Feb2017.)



Theorem  exnatded5.7 25917 
Theorem 5.7 of [Clemente] p. 19, translated line by line using the
interpretation of natural deduction in Metamath.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.72 25918.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer
.
The original proof, which uses Fitch style, was written as follows:
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  6 


Given 
$e. No need for adantr 471 because we do not move this
into an ND hypothesis 
2  1  ... 

ND hypothesis assumption (new scope) 
simpr 467 
3  2  ... 

I_{L} 2 
orcd 398, the MPE equivalent of I_{L}, 1 
4  3  ... 

ND hypothesis assumption (new scope) 
simpr 467 
5  4  ... 

E_{L} 4 
simpld 465, the MPE equivalent of E_{L}, 3 
6  6  ... 

I_{R} 5 
olcd 399, the MPE equivalent of I_{R}, 4 
7  7  

E 1,3,6 
mpjaodan 800, the MPE equivalent of E, 2,5,6 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including and uses the Metamath equivalents
of the natural deduction rules.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)



Theorem  exnatded5.72 25918 
A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with
exnatded5.7 25917. (Proof modification is discouraged.)
(New usage is discouraged.) (Contributed by Mario Carneiro,
9Feb2017.)



Theorem  exnatded5.8 25919 
Theorem 5.8 of [Clemente] p. 20, translated line by line using the
usual translation of natural deduction (ND) in the
Metamath Proof Explorer (MPE) notation.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  10;11 


Given 
$e; adantr 471 to move it into the ND hypothesis 
2  3;4  
 Given 
$e; adantr 471 to move it into the ND hypothesis 
3  7;8 
 
Given 
$e; adantr 471 to move it into the ND hypothesis 
4  1;2   
Given 
$e. adantr 471 to move it into the ND hypothesis 
5  6  ... 

ND Hypothesis/Assumption 
simpr 467. New ND hypothesis scope, each reference outside
the scope must change antecedent to . 
6  9  ... 

I 5,3 
jca 539 (I), 6,8 (adantr 471 to bring in scope) 
7  5  ... 

E 1,6 
mpd 15 (E), 2,4 
8  12  ... 

E 2,4 
mpd 15 (E), 9,11;
note the contradiction with ND line 7 (MPE line 5) 
9  13  

I 5,7,8 
pm2.65da 584 (I), 5,12; proof by contradiction.
MPE step 6 (ND#5) does not need a reference here, because
the assumption is embedded in the antecedents 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 471; simpr 467 is useful when you want to
depend directly on the new assumption).
Below is the final metamath proof (which reorders some steps).
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.82 25920.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)



Theorem  exnatded5.82 25920 
A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer
linebyline translation, see exnatded5.8 25919.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)



Theorem  exnatded5.13 25921 
Theorem 5.13 of [Clemente] p. 20, translated line by line using the
interpretation of natural deduction in Metamath.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.132 25922.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  15  

Given 
$e. 
2;3  2  
 Given 
$e. adantr 471 to move it into the ND hypothesis 
3  9  

Given 
$e. ad2antrr 737 to move it into the ND subhypothesis 
4  1  ... 

ND hypothesis assumption 
simpr 467 
5  4  ... 

E 2,4 
mpd 15 1,3 
6  5  ... 

I 5 
orcd 398 4 
7  6  ... 

ND hypothesis assumption 
simpr 467 
8  8  ... ... 

(sub) ND hypothesis assumption 
simpr 467 
9  11  ... ... 

E 3,8 
mpd 15 8,10 
10  7  ... ... 

IT 7 
adantr 471 6 
11  12  ... 

I 8,9,10 
pm2.65da 584 7,11 
12  13  ... 

E 11 
notnotrd 118 12 
13  14  ... 

I 12 
olcd 399 13 
14  16  

E 1,6,13 
mpjaodan 800 5,14,15 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 471; simpr 467 is useful when you want to
depend directly on the new assumption).
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)



Theorem  exnatded5.132 25922 
A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare
with exnatded5.13 25921. (Proof modification is discouraged.)
(New usage is discouraged.) (Contributed by Mario Carneiro,
9Feb2017.)



Theorem  exnatded9.20 25923 
Theorem 9.20 of [Clemente] p. 43, translated line by line using the
usual translation of natural deduction (ND) in the
Metamath Proof Explorer (MPE) notation.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  1 


Given 
$e 
2  2  

E_{L} 1 
simpld 465 1 
3  11 


E_{R} 1 
simprd 469 1 
4  4 
... 

ND hypothesis assumption 
simpr 467 
5  5 
... 

I 2,4 
jca 539 3,4 
6  6 
... 

I_{R} 5 
orcd 398 5 
7  8 
... 

ND hypothesis assumption 
simpr 467 
8  9 
... 

I 2,7 
jca 539 7,8 
9  10 
... 

I_{L} 8 
olcd 399 9 
10  12 


E 3,6,9 
mpjaodan 800 6,10,11 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 471; simpr 467 is useful when you want to
depend directly on the new assumption).
Below is the final metamath proof (which reorders some steps).
A much more efficient proof is exnatded9.202 25924.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by David A. Wheeler, 19Feb2017.)



Theorem  exnatded9.202 25924 
A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare
with exnatded9.20 25923. (Proof modification is discouraged.)
(New usage is discouraged.) (Contributed by David A. Wheeler,
19Feb2017.)



Theorem  exnatded9.26 25925* 
Theorem 9.26 of [Clemente] p. 45, translated line by line using an
interpretation of natural deduction in Metamath. This proof has some
additional complications due to the fact that Metamath's existential
elimination rule does not change bound variables, so we need to verify
that is bound in the conclusion.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  3  

Given 
$e. 
2  6  ... 

ND hypothesis assumption 
simpr 467. Later statements will have this scope. 
3  7;5,4  ... 

E 2,y 
spsbcd 3293 (E), 5,6. To use it we need a1i 11 and vex 3060.
This could be immediately done with 19.21bi 1958, but we want to show
the general approach for substitution.

4  12;8,9,10,11  ... 

I 3,a 
spesbcd 3362 (I), 11.
To use it we need sylibr 217, which in turn requires sylib 201 and
two uses of sbcid 3296.
This could be more immediately done using 19.8a 1946, but we want to show
the general approach for substitution.

5  13;1,2  
 E 1,2,4,a 
exlimdd 2081 (E), 1,2,3,12.
We'll need supporting
assertions that the variable is free (not bound),
as provided in nfv 1772 and nfe1 1929 (MPE# 1,2) 
6  14  

I 5 
alrimiv 1784 (I), 13 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including and uses the Metamath equivalents
of the natural deduction rules.
Below is the final metamath proof (which reorders some steps).
Note that in the original proof, has explicit
parameters. In Metamath, these parameters are always implicit, and the
parameters upon which a wff variable can depend are recorded in the
"allowed substitution hints" below.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded9.262 25926.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)
(Revised by David A. Wheeler, 18Feb2017.)



Theorem  exnatded9.262 25926* 
A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare
with exnatded9.26 25925. (Proof modification is discouraged.)
(New usage is discouraged.) (Contributed by Mario Carneiro,
9Feb2017.)



17.1.4 Definitional examples


Theorem  exor 25927 
Example for dfor 376. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 9May2015.)



Theorem  exan 25928 
Example for dfan 377. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 9May2015.)



Theorem  exdif 25929 
Example for dfdif 3419. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)



Theorem  exun 25930 
Example for dfun 3421. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)



Theorem  exin 25931 
Example for dfin 3423. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)



Theorem  exuni 25932 
Example for dfuni 4213. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 2Jul2016.)



Theorem  exss 25933 
Example for dfss 3430. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)



Theorem  expss 25934 
Example for dfpss 3432. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)



Theorem  expw 25935 
Example for dfpw 3965. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 2Jul2016.)



Theorem  expr 25936 
Example for dfpr 3983. (Contributed by Mario Carneiro,
7May2015.)



Theorem  exbr 25937 
Example for dfbr 4419. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)



Theorem  exopab 25938* 
Example for dfopab 4478. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)



Theorem  exeprel 25939 
Example for dfeprel 4767. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)



Theorem  exid 25940 
Example for dfid 4771. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)



Theorem  expo 25941 
Example for dfpo 4777. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)



Theorem  exxp 25942 
Example for dfxp 4862. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)



Theorem  excnv 25943 
Example for dfcnv 4864. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)



Theorem  exco 25944 
Example for dfco 4865. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)



Theorem  exdm 25945 
Example for dfdm 4866. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)



Theorem  exrn 25946 
Example for dfrn 4867. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)



Theorem  exres 25947 
Example for dfres 4868. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 7May2015.)



Theorem  exima 25948 
Example for dfima 4869. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 7May2015.)



Theorem  exfv 25949 
Example for dffv 5613. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)



Theorem  ex1st 25950 
Example for df1st 6825. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)



Theorem  ex2nd 25951 
Example for df2nd 6826. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)



Theorem  1kp2ke3k 25952 
Example for dfdec 11086, 1000 + 2000 = 3000.
This proof disproves (by counterexample) 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 129152, 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 dfdec 11086 developed by
Mario Carneiro in 2015. The underlying Metamath language has an
intentionally very small set of primitives; it doesn't even have a
builtin 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, 29Jun2016.) (Shortened by Mario
Carneiro using the arithmetic algorithm in mmj2, 30Jun2016.)

;;; ;;; ;;; 

Theorem  exfl 25953 
Example for dffl 12066. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)



Theorem  exdvds 25954 
3 divides into 6. A demonstration of dfdvds 14361. (Contributed by David
A. Wheeler, 19May2015.)



17.1.5 Other examples


Theorem  exinddvds 25955 
Example of a proof by induction (divisibility result). (Contributed by
Stanislas Polu, 9Mar2020.) (Revised by BJ, 24Mar2020.)



17.2 Humor


17.2.1 April Fool's theorem


Theorem  avril1 25956 
Poisson d'Avril's Theorem. This theorem is noted for its
Selbstdokumentieren property, which means, literally,
"selfdocumenting" 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, 1Apr2005.) (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 1Apr2006
entry.



Theorem  2bornot2b 25957 
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 modernday 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, 1Apr2006.) (Proof modification is discouraged.)
(New usage is discouraged.)



Theorem  helloworld 25958 
The classic "Hello world" benchmark has been translated into 314
computer
programming languages  see
http://www.roeslerac.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, 1Apr2007.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  1p1e2apr1 25959 
One plus one equals two. Using proofshortening techniques pioneered by
Mr. Mel L. O'Cat, along with the latest supercomputer technology, Prof.
Loof Lirpa and colleagues were able to shorten Whitehead and Russell's
360page 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, 1Apr2008.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  eqid1 25960 
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, 1Apr2009.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  1div0apr 25961 
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, 1Apr2014.)
(Proof modification is discouraged.) (New usage is discouraged.)



Theorem  topnfbey 25962 
Nothing seems to be impossible to Prof. Lirpa. After years of intensive
research, he managed to find a proof that when given a chance to reach
infinity, one could indeed go beyond, thus giving formal soundness to Buzz
Lightyear's motto "To infinity... and beyond!" (Contributed by
Prof.
Loof Lirpa, 1Apr2020.) (Modified by Thierry Arnoux, 2Aug2020.)
(Proof modification is discouraged.) (New usage is discouraged.)



17.3 (Future  to be reviewed and
classified)


17.3.1 Planar incidence geometry


Syntax  cplig 25963 
Extend class notation with the class of all planar incidence
geometries.



Definition  dfplig 25964* 
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, 2Aug2009.)



Theorem  isplig 25965* 
The predicate "is a planar incidence geometry". (Contributed by FL,
2Aug2009.)



Theorem  tncp 25966* 
There exist three non colinear points. (Contributed by FL,
3Aug2009.)



Theorem  lpni 25967* 
For any line, there exists a point not on the line. (Contributed by
Jeff Hankins, 15Aug2009.)



17.3.2 Algebra preliminaries


Syntax  crpm 25968 
Ring primes.

RPrime 

Definition  dfrprm 25969* 
Define the set of prime elements in a ring. A prime element is a
nonzero nonunit that satisfies an equivalent of Euclid's lemma
euclemma 14720. (Contributed by Mario Carneiro,
17Feb2015.)

RPrime Unit
_{r} 

PART 18 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 26018 shows the relationship between the older
group definition
and the extensible structure definition.
The intent is for this deprecated section to be deleted
once its theorems have extensible structure versions
(or are not useful). You can make a list of "terminal" theorems
(i.e. theorems not referenced by anything else) and for each theorem see
if there exists an extensible structure version (or decide it's not useful),
and if so, delete it. Then repeat this recursively.
One way to search for terminal theorems, for example in deprecated
group theory, is to log the output ("open log x.txt") of
"show usage cgr~circgrp" in metamath.exe and search for
"(None)".


18.1 Additional material on group
theory


18.1.1 Definitions and basic properties for
groups


Syntax  cgr 25970 
Extend class notation with the class of all group operations.



Syntax  cgi 25971 
Extend class notation with a function mapping a group operation to the
group's identity element.

GId 

Syntax  cgn 25972 
Extend class notation with a function mapping a group operation to the
inverse function for the group.



Syntax  cgs 25973 
Extend class notation with a function mapping a group operation to the
division (or subtraction) operation for the group.



Syntax  cgx 25974 
Extend class notation with a function mapping a group operation to the
power operation for the group.



Definition  dfgrpo 25975* 
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, 10Oct2006.)
(New usage is discouraged.)



Definition  dfgid 25976* 
Define a function that maps a group operation to the group's identity
element. (Contributed by FL, 5Feb2010.) (Revised by Mario Carneiro,
15Dec2013.) (New usage is discouraged.)

GId 

Definition  dfginv 25977* 
Define a function that maps a group operation to the group's inverse
function. (Contributed by NM, 26Oct2006.)
(New usage is discouraged.)

GId 

Definition  dfgdiv 25978* 
Define a function that maps a group operation to the group's division
(or subtraction) operation. (Contributed by NM, 15Feb2008.)
(New usage is discouraged.)



Definition  dfgx 25979* 
Define a function that maps a group operation to the group's power
operation. (Contributed by Paul Chapman, 17Apr2009.)
(New usage is discouraged.)

GId


Theorem  isgrpo 25980* 
The predicate "is a group operation." Note that is the base set
of the group. (Contributed by NM, 10Oct2006.)
(New usage is discouraged.)



Theorem  isgrpo2 25981* 
The predicate "is a group operation." (Contributed by NM,
23Oct2012.) (New usage is discouraged.)



Theorem  isgrpoi 25982* 
Properties that determine a group operation. Read as
. (Contributed by NM, 4Nov2006.)
(New usage is discouraged.)



Theorem  grpofo 25983 
A group operation maps onto the group's underlying set. (Contributed by
NM, 30Oct2006.) (New usage is discouraged.)



Theorem  grpocl 25984 
Closure law for a group operation. (Contributed by NM, 10Oct2006.)
(New usage is discouraged.)



Theorem  grpolidinv 25985* 
A group has a left identity element, and every member has a left
inverse. (Contributed by NM, 2Nov2006.)
(New usage is discouraged.)



Theorem  grpon0 25986 
The base set of a group is not empty. (Contributed by Szymon
Jaroszewicz, 3Apr2007.) (New usage is discouraged.)



Theorem  grpoass 25987 
A group operation is associative. (Contributed by NM, 10Oct2006.)
(New usage is discouraged.)



Theorem  grpoidinvlem1 25988 
Lemma for grpoidinv 25992. (Contributed by NM, 10Oct2006.)
(New usage is discouraged.)



Theorem  grpoidinvlem2 25989 
Lemma for grpoidinv 25992. (Contributed by NM, 10Oct2006.)
(New usage is discouraged.)



Theorem  grpoidinvlem3 25990* 
Lemma for grpoidinv 25992. (Contributed by NM, 11Oct2006.)
(New usage is discouraged.)



Theorem  grpoidinvlem4 25991* 
Lemma for grpoidinv 25992. (Contributed by NM, 14Oct2006.)
(New usage is discouraged.)



Theorem  grpoidinv 25992* 
A group has a left and right identity element, and every member has a
left and right inverse. (Contributed by NM, 14Oct2006.)
(New usage is discouraged.)



Theorem  grpoideu 25993* 
The left identity element of a group is unique. Lemma 2.2.1(a) of
[Herstein] p. 55. (Contributed by NM,
14Oct2006.)
(New usage is discouraged.)



Theorem  grporndm 25994 
A group's range in terms of its domain. (Contributed by NM, 6Apr2008.)
(New usage is discouraged.)



Theorem  0ngrp 25995 
The empty set is not a group. (Contributed by NM, 25Apr2007.)
(New usage is discouraged.)



Theorem  grporn 25996 
The range of a group operation. Useful for satisfying group base set
hypotheses of the form . (Contributed by NM,
5Nov2006.) (New usage is discouraged.)



Theorem  gidval 25997* 
The value of the identity element of a group. (Contributed by Mario
Carneiro, 15Dec2013.) (New usage is discouraged.)

GId


Theorem  fngid 25998 
GId is a function. (Contributed by FL, 5Feb2010.) (Revised by
Mario Carneiro, 15Dec2013.) (New usage is discouraged.)

GId 

Theorem  grposn 25999 
The group operation for the singleton group. (Contributed by NM,
4Nov2006.) (New usage is discouraged.)



Theorem  grpoidval 26000* 
Lemma for grpoidcl 26001 and others. (Contributed by NM,
5Feb2010.)
(Proof shortened by Mario Carneiro, 15Dec2013.)
(New usage is discouraged.)

GId
