Type  Label  Description 
Statement 

Theorem  numclwlk1lem2 25501* 
There is a bijection between the set of closed walks (having a fixed
length greater than 2 and starting at a fixed vertex) with the last
but 2 vertex identical with the first (and therefore last) vertex and
the set of closed walks (having a fixed length less by 2 and starting
at the same vertex) and the neighbors of this vertex. (Contributed by
Alexander van der Vekens, 6Jul2018.)

ClWWalksN
USGrph
Neighbors 

Theorem  numclwwlk1 25502* 
Statement 9 in [Huneke] p. 2: "If n >
1, then the number of closed
nwalks v(0) ... v(n2) v(n1) v(n) from v = v(0) = v(n) with v(n2) =
v is kf(n2)". Since is kregular, the vertex v(n2)
= v has k neighbors v(n1), so there are k walks from v(n2) = v to
v(n) = v (via each of v's neighbors) completing each of the f(n2)
walks from v=v(0) to v(n2)=v. This theorem holds even for k=0, but
only for finite graphs! (Contributed by Alexander van der Vekens,
26Sep2018.)

ClWWalksN
RegUSGrph


Theorem  numclwwlkovq 25503* 
Value of operation Q, mapping a vertex v and a nonnegative integer n
to the not closed walks v(0) ... v(n) of length n from a fixed vertex
v = v(0). "Not closed" means v(n) =/= v(0). (Contributed
by
Alexander van der Vekens, 27Sep2018.)

ClWWalksN
WWalksN
lastS
WWalksN
lastS 

Theorem  numclwwlkqhash 25504* 
In a kregular graph, the size of the set of walks of length n
starting with a fixed vertex and ending not at this vertex is the
difference between k to the power of n and the size of the set of
walks of length n starting with this vertex and ending at this
vertex. (Contributed by Alexander van der Vekens, 30Sep2018.)

ClWWalksN
WWalksN
lastS RegUSGrph


Theorem  numclwwlkovh 25505* 
Value of operation H, mapping a vertex v and a nonnegative integer n
to the "closed nwalks v(0) ... v(n2) v(n1) v(n) from v = v(0)
=
v(n) ... with v(n2) =/= v" according to definition 7 in [Huneke]
p. 2. (Contributed by Alexander van der Vekens, 26Aug2018.)

ClWWalksN
WWalksN
lastS


Theorem  numclwwlk2lem1 25506* 
In a friendship graph, for each walk of length n starting with a fixed
vertex and ending not at this vertex, there is a unique vertex so that
the walk extended by an edge to this vertex and an edge from this
vertex to the first vertex of the walk is a value of operation H. If
the walk is represented as a word, it is sufficient to add one vertex
to the word to obtain the closed walk contained in the value of
operation H, since in a word representing a closed walk the starting
vertex is not repeated at the end. This theorem only generally holds
for Friendship Graphs, because these guarantee that for the first and
last vertex there is a third vertex "in between".
(Contributed by
Alexander van der Vekens, 3Oct2018.)

ClWWalksN
WWalksN
lastS
FriendGrph
++ 

Theorem  numclwlk2lem2f 25507* 
R is a function. (Contributed by Alexander van der Vekens,
5Oct2018.)

ClWWalksN
WWalksN
lastS
substr FriendGrph


Theorem  numclwlk2lem2fv 25508* 
Value of the function R. (Contributed by Alexander van der Vekens,
6Oct2018.)

ClWWalksN
WWalksN
lastS
substr
substr 

Theorem  numclwlk2lem2f1o 25509* 
R is a 11 onto function. (Contributed by Alexander van der Vekens,
6Oct2018.)

ClWWalksN
WWalksN
lastS
substr FriendGrph


Theorem  numclwwlk2lem3 25510* 
In a friendship graph, the size of the set of walks of length n
starting with a fixed vertex and ending not at this vertex equals the
size of the set of all closed walks of length (n+2) starting with this
vertex and not having this vertex as last but 2 vertex. (Contributed
by Alexander van der Vekens, 6Oct2018.)

ClWWalksN
WWalksN
lastS
FriendGrph


Theorem  numclwwlk2 25511* 
Statement 10 in [Huneke] p. 2: "If n >
1, then the number of closed
nwalks v(0) ... v(n2) v(n1) v(n) from v = v(0) = v(n) ... with
v(n2) =/= v is k^(n2)  f(n2)." According to rusgranumwlkg 25362, we
have k^(n2) different walks of length (n2): v(0) ... v(n2). From
this number, the number of closed walks of length (n2), which is
f(n2) per definition, must be subtracted, because for these walks
v(n2) =/= v(0) = v would hold. Because of the friendship condition,
there is exactly one vertex v(n1) which is a neighbor of v(n2) as
well as of v(n)=v=v(0), because v(n2) and v(n)=v are different, so
the number of walks v(0) ... v(n2) is identical with the number of
walks v(0) ... v(n), that means each (not closed) walk v(0) ... v(n2)
can be extended by two edges to a closed walk v(0) ... v(n)=v=v(0) in
exactly one way. (Contributed by Alexander van der Vekens,
6Oct2018.)

ClWWalksN
WWalksN
lastS
RegUSGrph FriendGrph


Theorem  numclwwlk3lem 25512* 
Lemma for numclwwlk3 25513. (Contributed by Alexander van der Vekens,
6Oct2018.)

ClWWalksN
WWalksN
lastS
USGrph


Theorem  numclwwlk3 25513* 
Statement 12 in [Huneke] p. 2: "Thus f(n)
= (k  1)f(n  2) +
k^(n2)."  the number of the closed walks v(0) ... v(n2) v(n1)
v(n)
is the sum of the number of the closed walks v(0) ... v(n2) v(n1)
v(n) with v(n2) = v(n) (see numclwwlk1 25502) and with v(n2) =/= v(n)
( see numclwwlk2 25511): f(n) = kf(n2) + k^(n2)  f(n2) = (k 
1)f(n 
2) + k^(n2) (Contributed by Alexander van der Vekens,
26Aug2018.)

ClWWalksN
WWalksN
lastS
RegUSGrph FriendGrph


Theorem  numclwwlk4 25514* 
The total number of closed walks in a finite undirected simple graph is
the sum of the numbers of closed walks starting at each of its
vertices. (Contributed by Alexander van der Vekens, 7Oct2018.)

ClWWalksN
USGrph


Theorem  numclwwlk5lem 25515* 
Lemma for numclwwlk5 25516. (Contributed by Alexander van der Vekens,
7Oct2018.)

ClWWalksN
RegUSGrph


Theorem  numclwwlk5 25516* 
Statement 13 in [Huneke] p. 2: "Let p be
a prime divisor of k1; then
f(p) = 1 (mod p) [for each vertex v]". (Contributed by Alexander
van
der Vekens, 7Oct2018.)

ClWWalksN
RegUSGrph FriendGrph


Theorem  numclwwlk6 25517* 
For a prime divisor p of k1, the total number of closed walks of length
p in an undirected simple graph with m vertices mod p is equal to the
number of vertices mod p. (Contributed by Alexander van der Vekens,
7Oct2018.)

ClWWalksN
RegUSGrph FriendGrph


Theorem  numclwwlk7 25518 
Statement 14 in [Huneke] p. 2: "The total
number of closed walks of
length p [in a friendship graph] is (k(k1)+1)f(p)=1 (mod p)",
since the
number of vertices in a friendship graph is (k(k1)+1), see
frgregordn0 25474 or frrusgraord 25475, and p divides (k1), i.e. (k1) mod p
= 0 => k(k1) mod p = 0 => k(k1)+1 mod p = 1. Since the empty
graph is
a friendship graph, see frgra0 25398, as well as kregular (for any k),
see
0vgrargra 25341, but has no closed walk, see clwlk0 25166, this theorem would
be false: , so this case
must be excluded. ( (Contributed by Alexander van der Vekens,
1Sep2018.)

RegUSGrph FriendGrph
ClWWalksN 

Theorem  numclwwlk8 25519 
The size of the set of closed walks of length p, p prime, is divisible by
p. This corresponds to statement 9 in [Huneke] p. 2: "It follows that,
if p is a prime number, then the number of closed walks of length p is
divisible by p", see also clwlkndivn 25257. (Contributed by Alexander van
der Vekens, 7Oct2018.)

USGrph
ClWWalksN 

Theorem  frgrareggt1 25520 
If a finite friendship graph is kregular with k > 1, then k must be 2.
(Contributed by Alexander van der Vekens, 7Oct2018.)

FriendGrph
RegUSGrph 

Theorem  frgrareg 25521 
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 25522 
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 25523 
If a nonempty finite friendship graph is kregular, then it must have
order 1 or 3. Special case of frgraregord013 25522. (Contributed by
Alexander van der Vekens, 9Oct2018.)

FriendGrph
RegUSGrph


Theorem  frgraogt3nreg 25524* 
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 25525* 
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 25526* 
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.
However, these conventions often refer to existing mathematical
practices, which are discussed in more detail in other references.
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 25527 
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're identical):
 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 13656) which denotes that index
variable ranges over when evaluating . Thus,
means 1/2 + 1/4 + 1/8 + ...
= 1 (geoihalfsum 13841).
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 185, dfcleq 2394, dfclel 2397, dfclab 2388)
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 9579,
proven by the preceding theorem ax1cn 9555.
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 9579 (not ax1cn 9555) and ax1ne0 9590 (not ax1ne0 9566), as these
are proven axioms for complex arithmetic. Thus, both
ax1cn 9555 and ax1ne0 9566 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 8528) 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 7674)
does not use the axiom of choice.
In some cases, the weaker axiom of countable choice (axcc 8846)
or axiom of dependent choice (axdc 8857) 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.
 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 15993) and
connective symbols such as for some group addition operation.
(See prdsplusgval 15085 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 208 or mpbir 209. 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 194 are available, in most cases there is already a theorem that
combines it with your theorem of choice, like mpbir2an 921, sylbir 213,
or 3imtr4i 266.
 Substitution.
" " should be read "the wff that results from the
proper substitution of for in wff ." See dfsb 1764
and the related dfsbc 3277 and dfcsb 3373.
 Isa set.
" " 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 3060 and isset 3062.
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 3067 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 4830.
This can be used to define a subset, e.g., dftan 14014 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 14092). 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 5576.
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 6280).
For example, the in ; see 2p2e4 10693.
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 368, dfan 369, and dfbi 185 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 1843),
equality (see dfcleq 2394),
subset (see dfss 3427), and
lessthan (see dflt 9534). For the general definition
of a binary relation in the form , see dfbr 4395.
For example, (see 0lt1 10114) 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 9841.
 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 14007).
The function is then defined labelled dfNAME; definitions
are typically given using the mapsto notation (e.g., dfcos 14013).
Typically, there are other proofs such as its
closure labelled NAMEcl (e.g., coscl 14069), its
function application form labelled NAMEval (e.g., cosval 14065),
and at least one simple value (e.g., cos0 14092).
 Factorial.
The factorial function is traditionally a postfix operation,
but we treat it as a normal function applied in prefix form, e.g.,
; (dffac 12396 and fac4 12403).
 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 9528), while
"" is the group identity element (df0g 15054),
"" is the poset zero (dfp0 15991),
"" is the zero polynomial (df0p 22367),
"" is the zero vector in a normed complex vector space
(df0v 25891), and
"" is a class variable for use as a connective symbol
(this is used, for example, in p0val 15993).
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).
 Natural numbers.
There are different definitions of "natural" numbers in the literature.
We use (dfnn 10576) for the set of positive integers starting
from 1, and (dfn0 10836) 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 11019, e.g.,
;;; (see 4001prm 14834 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 3616). 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 22707) 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 25, unssd 3618.
 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 3617).
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 135 is the
inference associated with con3 134). 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 76 because it is the inference
associated with imim1i 57 which is itself the inference
associated with imim1 76.
"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 47). The labels for inferences
usually have the suffix "i" (e.g. pm2.43i 46). The labels of theorems in
"closed form" would have no special suffix (e.g. pm2.43 50). 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 6577 vs. uniexg 6578.
 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).
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 3935,
which works in certain cases in set theory. We also
sometimes use dedhb 3218.
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 25528.
 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 7074,
in dfrdg 7112, seq_{𝜔} in dfseqom 7149, and
in dfseq 12150.
These have characteristic function and initial value .
(_{g} in dfgsum 15055 isn't really designed for arbitrary recursion,
but you could do it with the right magma.)
The logically primary one is dfrecs 7074, but for the "average user"
the most useful one is probably dfseq 12150 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 14841.
 Junk/undefined results.
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 5553 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 5858 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 7664 is the first lemma for sbth 7674. Also, consider proving
reusable results separately, so that others will be able to easily
reuse that part of your work.
 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.
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 24694, 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.)
 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.
Label naming conventions
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 2763"rgen.1 $e  ( x e. A > ph ) $."
or letters corresponding to the (main) class variable used in the
hypothesis, e.g. for mdet0 19398: "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 2341 and stirling 37220.
 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 37.
 19.x series of theorems.
Similar to the conventions for the theorems from Principia Mathematica,
theorems from section 19 of [Margaris] 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 named
19.38 1683, and the restricted quantifier version of theorem 21 from
section 19 of [Margaris] p. 90 is named r19.21 2802.
 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 names of theorems from Principia Mathematica and the
19.x series of theorems from [Margaris] (see above) and 0.999... 13840.
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 labelled "dfNAME",
and normally NAME is the syntax label fragment.
For example, the difference construct
is defined in dfdif 3416, and thus its syntax label fragment is "dif".
Similarly, the subclass (subset) relation
has syntax label fragment "ss",
because it is defined in dfss 3427.
Most theorem names follow from these fragments, for example,
theorem proving
involves a difference ("dif")
of a subset ("ss"), and thus is named difss 3569.
There are many other syntax label fragments, e.g.,
singleton construct has syntax label fragment "sn"
(because it is defined in dfsn 3972), and
the pair construct has fragment "pr" ( from dfpr 3974).
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 difference ("dif") of a singleton ("sn"), it is named eldifsn 4096.
An "n" is often used for negation (), e.g., nan 578.
 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, label fragment
"cn" represents complex numbers (even though its definition
is in dfc 9527) and "re" represents real numbers .
The empty set often uses fragment 0, even though it is defined
in dfnul 3738.
Syntax construct usually uses the fragment "add"
(which is consistent with dfadd 9532),
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 named 2p2e4 10693.
 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 14092 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 named "NAMEval" and its closure named "NAMEcl".
E.g., for cosine (dfcos 14013) we have
value cosval 14065 and closure coscl 14069.
 Special cases.
Sometimes syntax and related markings are
insufficient to distinguish different theorems.
For example, there are over 100 different
implicationexclusive theorems.
These are then 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's 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 25528 for a list),
and that's about all you need for most things. As for the rest,
you can just assume that if it involves three or less connectives
we probably already have a proof, and searching for it will give
you the name.
 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 1933 via the use of
distinct variable conditions combined with nfv 1728. If two (or three)
such hypotheses are eliminated, the suffix "vv" resp. "vvv" is used,
e.g. exlimivv 1744.
Conversely, we sometimes suffix with "f" the label of a theorem
introducing such a hypothesis to eliminate the need for the distinct
variable condition; e.g. euf 2248 derived from dfeu 2242. 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 4639.
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 name is suffixed with "ALT" if it provides an
alternative lesspreferred proof of a theorem (e.g., the theorem
is clearer but uses more axioms than the preferred theorem).
The "ALT" may be further suffixed with a number if there is more
than one alternate theorem.
Furthermore, a theorem name is suffixed with "OLD" if there is a new
version of it and the OLD version is obsolete (and will be removed
soon).
Finally, it should be mentioned that suffixes can be combined, for
example in cbvaldva 2058 (cbval 2048 in deduction form "d" with a not free
variable replaced by a distinct variable condition "v" with a
conjunction as antecedent "a").
In the following, a list of common suffixes is provided:
 a : theorem having a conjunction as antecedent
 b : theorem expressing a logical equivalence
 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
 v : theorem with one (main) distinct variable
 vv : theorem with two (main) distinct variables
 w : weak(er) form of a theorem
 ALT : alternative proof for a theorem
 ALTV : alternative for another theorem/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 commonlyused abbreviations in labels,
in alphabetical order.
For each abbreviation we provide a mnenomic to help you remember it,
the source theorem/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 482, rexlimiva 2891 
abl  Abelian group  dfabl 17123 
 Yes  ablgrp 17125, zringabl 18810 
abs  absorption    No 
ressabs 14905 
abs  absolute value (of a complex number) 
dfabs 13216   Yes 
absval 13218, absneg 13257, abs1 13277 
ad  adding  
 No  adantr 463, ad2antlr 725 
add  add (see "p")  dfadd 9532 
 Yes 
addcl 9603, addcom 9799, addass 9608 
al  "for all"  
 No  alim 1653, alex 1668 
ALT  alternative/less preferred (suffix)  
 No  aevALT 2089 
an  and  dfan 369 
 Yes 
anor 487, iman 422, imnan 420 
ant  antecedent  
 No  adantr 463 
ass  associative  
 No  biass 357, orass 522, mulass 9609 
asym  asymmetric, antisymmetric  
 No  intasym 5202, asymref 5203, posasymb 15904 
ax  axiom  
 No  ax6dgen 1848, ax1cn 9555 
bas, base 
base (set of an extensible structure)  dfbase 14844 
 Yes 
baseval 14886, ressbas 14896, cnfldbas 18742 
b, bi  biconditional ("iff", "if and only if")
 dfbi 185   Yes 
impbid 191, sspwb 4639 
br  binary relation  dfbr 4395 
 Yes  brab1 4439, brun 4442 
cbv  change bound variable   
No  cbvalivw 1813, cbvrex 3030 
cl  closure    No 
ifclda 3916, ovrcl 6310, zaddcl 10944 
cn  complex numbers  dfc 9527 
 Yes  nnsscn 10580, nncn 10583 
cnfld  field of complex numbers  dfcnfld 18739 
ℂ_{fld}  Yes  cnfldbas 18742, cnfldinv 18767 
cntz  centralizer  dfcntz 16677 
Cntz  Yes 
cntzfval 16680, dprdfcntz 17367 
cnv  converse  dfcnv 4830 
 Yes  opelcnvg 5002, f1ocnv 5810 
co  composition  dfco 4831 
 Yes  cnvco 5008, fmptco 6042 
com  commutative  
 No  orcom 385, bicomi 202, eqcomi 2415 
con  contradiction, contraposition  
 No  condan 795, con2d 115 
csb  class substitution  dfcsb 3373 
 Yes 
csbid 3380, csbie2g 3403 
cyg  cyclic group  dfcyg 17203 
CycGrp  Yes 
iscyg 17204, zringcyg 18824 
d  deduction form (suffix)  
 No  idd 24, impbid 191 
df  (alternate) definition (prefix)  
 No  dfrel2 5273, dffn2 5714 
di, distr  distributive  
 No 
andi 868, imdi 361, ordi 865, difindi 3703, ndmovdistr 6444 
dif  difference  dfdif 3416 
 Yes 
difss 3569, difindi 3703 
div  division  dfdiv 10247 
 Yes 
divcl 10253, divval 10249, divmul 10250 
dm  domain  dfdm 4832 
 Yes  dmmpt 5317, iswrddm0 12615 
e, eq, equ  equals  dfcleq 2394 
 Yes 
2p2e4 10693, uneqri 3584, equtr 1820 
el  element of  
 Yes 
eldif 3423, eldifsn 4096, elssuni 4219 
eu  "there exists exactly one"  dfeu 2242 
 Yes  euex 2264, euabsn 4043 
ex  exists (i.e. is a set)  
 No  brrelex 4861, 0ex 4525 
ex  "there exists (at least one)"  dfex 1634 
 Yes  exim 1675, alex 1668 
exp  export  
 No  expt 156, expcom 433 
f  "not free in" (suffix)  
 No  equs45f 2115, sbf 2145 
f  function  dff 5572 
 Yes  fssxp 5725, opelf 5729 
fal  false  dffal 1411 
 Yes  bifal 1418, falantru 1431 
fi  finite intersection  dffi 7904 
 Yes  fival 7905, inelfi 7911 
fi, fin  finite  dffin 7557 
 Yes 
isfi 7576, snfi 7633, onfin 7745 
fld  field (Note: there is an alternative
definition of a field, see dffld 25816)  dffield 17717 
Field  Yes  isfld 17723, fldidom 18272 
fn  function with domain  dffn 5571 
 Yes  ffn 5713, fndm 5660 
frgp  free group  dffrgp 17050 
freeGrp  Yes 
frgpval 17098, frgpadd 17103 
fsupp  finitely supported function 
dffsupp 7863  finSupp  Yes 
isfsupp 7866, fdmfisuppfi 7871, fsuppco 7894 
fun  function  dffun 5570 
 Yes  funrel 5585, ffun 5715 
fv  function value  dffv 5576 
 Yes  fvres 5862, swrdfv 12703 
fz  finite set of sequential integers 
dffz 11725 
 Yes  fzval 11726, eluzfz 11735 
fz0  finite set of sequential nonnegative integers 

 Yes  nn0fz0 11827, fz0tp 11830 
fzo  halfopen integer range  dffzo 11853 
..^  Yes 
elfzo 11859, elfzofz 11872 
g  more general (suffix); eliminates "is a set"
hypothsis  
 No  uniexg 6578 
gra  graph  
 No  uhgrav 24700, isumgra 24719, usgrares 24773 
grp  group  dfgrp 16379 
 Yes  isgrp 16383, tgpgrp 20867 
gsum  group sum  dfgsum 15055 
_{g}  Yes 
gsumval 16220, gsumwrev 16723 
hash  size (of a set)  dfhash 12451 
 Yes 
hashgval 12453, hashfz1 12464, hashcl 12473 
hb  hypothesis builder (prefix)  
 No  hbxfrbi 1664, hbald 1872, hbequid 31912 
hm  (monoid, group, ring) homomorphism  
 No  ismhm 16290, isghm 16589, isrhm 17688 
i  inference (suffix)  
 No  eleq1i 2479, tcsni 8205 
i  implication (suffix)  
 No  brwdomi 8027, infeq5i 8085 
id  identity  
 No  biid 236 
idm  idempotent  
 No  anidm 642, tpidm13 4073 
im, imp  implication (label often omitted) 
dfim 13081   Yes 
iman 422, imnan 420, impbidd 189 
ima  image  dfima 4835 
 Yes  resima 5125, imaundi 5235 
imp  import  
 No  biimpa 482, impcom 428 
in  intersection  dfin 3420 
 Yes  elin 3625, incom 3631 
is...  is (something a) ...?  
 No  isring 17520 
j  joining, disjoining  
 No  jc 147, jaoi 377 
l  left  
 No  olcd 391, simpl 455 
map  mapping operation or set exponentiation 
dfmap 7458   Yes 
mapvalg 7466, elmapex 7476 
mat  matrix  dfmat 19200 
Mat  Yes 
matval 19203, matring 19235 
mdet  determinant (of a square matrix) 
dfmdet 19377  maDet  Yes 
mdetleib 19379, mdetrlin 19394 
mgm  magma  dfmgm 16194 
 Yes 
mgmidmo 16208, mgmlrid 16215, ismgm 16195 
mgp  multiplicative group  dfmgp 17460 
mulGrp  Yes 
mgpress 17470, ringmgp 17522 
mnd  monoid  dfmnd 16243 
 Yes  mndass 16252, mndodcong 16888 
mo  "there exists at most one"  dfmo 2243 
 Yes  eumo 2269, moim 2291 
mp  modus ponens  axmp 5 
 No  mpd 15, mpi 18 
mpt  modus ponendo tollens  
 No  mptnan 1621, mptxor 1622 
mpt  mapsto notation for a function 
dfmpt 4454   Yes 
fconstmpt 4866, resmpt 5142 
mpt2  mapsto notation for an operation 
dfmpt2 6282   Yes 
mpt2mpt 6374, resmpt2 6380 
mul  multiplication (see "t")  dfmul 9533 
 Yes 
mulcl 9605, divmul 10250, mulcom 9607, mulass 9609 
n, not  not  
 Yes 
nan 578, notnot2 112 
ne  not equal  dfne  
Yes  exmidne 2608, neeqtrd 2698 
nel  not element of  dfnel 

Yes  neli 2738, nnel 2748 
ne0  not equal to zero (see n0)  
 No 
negne0d 9964, ine0 10032, gt0ne0 10057 
nf  "not free in" (prefix)  
 No  nfnd 1930 
ngp  normed group  dfngp 21394 
NrmGrp  Yes  isngp 21406, ngptps 21412 
nm  norm (on a group or ring)  dfnm 21393 
 Yes 
nmval 21400, subgnm 21437 
nn  positive integers  dfnn 10576 
 Yes  nnsscn 10580, nncn 10583 
nn0  nonnegative integers  dfn0 10836 
 Yes  nnnn0 10842, nn0cn 10845 
n0  not the empty set (see ne0)  
 No  n0i 3742, vn0 3745, ssn0 3771 
OLD  old, obsolete (to be removed soon)  
 No  19.43OLD 1715 
op  ordered pair  dfop 3978 
 Yes  dfopif 4155, opth 4664 
or  or  dfor 368 
 Yes 
orcom 385, anor 487 
ot  ordered triple  dfot 3980 
 Yes 
euotd 4690, fnotovb 6318 
ov  operation value  dfov 6280 
 Yes
 fnotovb 6318, fnovrn 6430 
p  plus (see "add"), for allconstant
theorems  dfadd 9532 
 Yes 
3p2e5 10708 
pfx  prefix  dfpfx 37850 
prefix  Yes 
pfxlen 37859, ccatpfx 37877 
pm  Principia Mathematica  
 No  pm2.27 37 
pm  partial mapping (operation)  dfpm 7459 
 Yes  elpmi 7474, pmsspw 7490 
pr  pair  dfpr 3974 
 Yes 
elpr 3989, prcom 4049, prid1g 4077, prnz 4090 
prm, prime  prime (number)  dfprm 14425 
 Yes  1nprm 14429, dvdsprime 14437 
pss  proper subset  dfpss 3429 
 Yes  pssss 3537, sspsstri 3544 
q  rational numbers ("quotients")  dfq 11227 
 Yes  elq 11228 
r  right  
 No  orcd 390, simprl 756 
rab  restricted class abstraction 
dfrab 2762   Yes 
rabswap 2986, dfoprab 6281 
ral  restricted universal quantification 
dfral 2758   Yes 
ralnex 2849, ralrnmpt2 6397 
rcl  reverse closure  
 No  ndmfvrcl 5873, nnarcl 7301 
re  real numbers  dfr 9531 
 Yes  recn 9611, 0re 9625 
rel  relation  dfrel 4829  
Yes  brrelex 4861, relmpt2opab 6865 
res  restriction  dfres 4834 
 Yes 
opelres 5098, f1ores 5812 
reu  restricted existential uniqueness 
dfreu 2760   Yes 
nfreud 2979, reurex 3023 
rex  restricted existential quantification 
dfrex 2759   Yes 
rexnal 2851, rexrnmpt2 6398 
rmo  restricted "at most one" 
dfrmo 2761   Yes 
nfrmod 2980, nrexrmo 3026 
rn  range  dfrn 4833  
Yes  elrng 5014, rncnvcnv 5046 
rng  (unital) ring  dfring 17518 
 Yes 
ringidval 17473, isring 17520, ringgrp 17521 
rot  rotation  
 No  3anrot 979, 3orrot 980 
s  eliminates need for syllogism (suffix) 
  No  ancoms 451 
sb  (proper) substitution (of a set) 
dfsb 1764   Yes 
spsbe 1767, sbimi 1769 
sbc  (proper) substitution of a class 
dfsbc 3277   Yes 
sbc2or 3285, sbcth 3291 
sca  scalar  dfsca 14923 
Scalar  Yes 
resssca 14989, mgpsca 17466 
simp  simple, simplification  
 No  simpl 455, simp3r3 1107 
sn  singleton  dfsn 3972 
 Yes  eldifsn 4096 
sp  specialization  
 No  spsbe 1767, spei 2039 
ss  subset  dfss 3427 
 Yes  difss 3569 
struct  structure  dfstruct 14841 
Struct  Yes  brstruct 14847, structfn 14852 
sub  subtract  dfsub 9842 
 Yes 
subval 9846, subaddi 9942 
supp  support (of a function)  dfsupp 6902 
supp  Yes 
ressuppfi 7888, mptsuppd 6925 
swap  swap (two parts within a theorem) 
  No  rabswap 2986, 2reuswap 3251 
syl  syllogism  syl 17 
 No  3syl 20 
sym  symmetric  
 No  dfsymdif 3669, cnvsym 5201 
symg  symmetric group  dfsymg 16725 
 Yes 
symghash 16732, pgrpsubgsymg 16755 
t 
times (see "mul"), for allconstant theorems 
dfmul 9533 
 Yes 
3t2e6 10727 
th  theorem  
 No  nfth 1646, sbcth 3291, weth 8906 
tp  triple  dftp 3976 
 Yes 
eltpi 4015, tpeq1 4059 
tr  transitive  
 No  bitrd 253, biantr 932 
tru  true  dftru 1408 
 Yes  bitru 1417, truanfal 1430 
un  union  dfun 3418 
 Yes 
uneqri 3584, uncom 3586 
unit  unit (in a ring) 
dfunit 17609  Unit  Yes 
isunit 17624, nzrunit 18233 
v  distinct variable conditions used when
a notfree hypothesis (suffix) 
  No  spimv 2036 
vv  2 distinct variables (in a notfree hypothesis)
(suffix)    No  19.23vv 1785 
w  weak (version of a theorem) (suffix)  
 No  ax11w 1850, spnfw 1809 
wrd  word 
dfword 12589  Word  Yes 
iswrdb 12602, wrdfn 12610, ffz0iswrd 12619 
xp  cross product (Cartesian product) 
dfxp 4828   Yes 
elxp 4839, opelxpi 4854, xpundi 4875 
xr  eXtended reals  dfxr 9661 
 Yes  ressxr 9666, rexr 9668, 0xr 9669 
z  integers (from German "Zahlen") 
dfz 10905   Yes 
elz 10906, zcn 10909 
zn  ring of integers  dfzn 18842 
ℤ/nℤ  Yes 
znval 18870, zncrng 18879, znhash 18893 
zring  ring of integers  dfzring 18807 
ℤ_{ring}  Yes  zringbas 18812, zringcrng 18808

0, z 
slashed zero (empty set) (see n0)  dfnul 3738 
 Yes 
n0i 3742, vn0 3745; snnz 4089, prnz 4090 
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 1638 (whose description has some important technical
details). Similarly, is read is not free in (class)
, see dfnfc 2552.
 "$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 $."
Sometimes a theorem is proved using
(dfnf 1638) in place of
"$d $." when a more general result is desired;
ax5 1725 can be used to derive the $d version. For an example of
how to get from the $d version back to the $e version, see the
proof of euf 2248 from dfeu 2242.
 "$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 209 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 2048. 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 185, dfclab 2388, dfcleq 2394, and dfclel 2397);
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 973 or wsb 1763).
 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 976 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 labelled 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 notfreeness; 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 3059.
The definition dfv 3060 is justified
by the justification theorem vjust 3059
.
Another example of a justification theorem is trujust 1407;
the definition dftru 1408
is justified by trujust 1407 .
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 13366 and absmax 13309.
 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.)



17.1.2 Natural deduction


Theorem  natded 25528 
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 530 
jca 530, pm3.2i 453 
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 457 
simpld 457, adantr 463 
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 461 
simpr 459, adantl 464 
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 432  ex 432 
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 666, imp 427 
Definition E in [Pfenning] p. 18,
definition E=>m,n in [Clemente] p. 11, and
definition E in [Indrzejczak] p. 33. 
I_{L}  =>

olcd 391 
olc 382, olci 389, olcd 391 
Definition I in [Pfenning] p. 18,
definition In(1) in [Clemente] p. 12 
I_{R}  =>

orcd 390 
orc 383, orci 388, orcd 390 
Definition I_{R} in [Pfenning] p. 18,
definition In(2) in [Clemente] p. 12. 
E  & &
=> 
mpjaodan 787 
mpjaodan 787, jaodan 786, jaod 378 
Definition E in [Pfenning] p. 18,
definition Em,n,p in [Clemente] p. 12. 
I  => 
inegd 1426  pm2.01d 169 

I  & =>

mtand 657  mtand 657 
definition Im,n,p in [Clemente] p. 13. 
I  & =>

pm2.65da 574  pm2.65da 574 
Contradiction. 
I 
=> 
pm2.01da 440  pm2.01d 169, pm2.65da 574, pm2.65d 175 
For an alternative falsumfree natural deduction ruleset 
E 
& => 
pm2.21fal 1428 
pm2.21dd 174  
E 
=> 

pm2.21dd 174 
definition E in [Indrzejczak] p. 33. 
E 
& => 
pm2.21dd 174  pm2.21dd 174, pm2.21d 106, pm2.21 108 
For an alternative falsumfree natural deduction ruleset.
Definition E in [Pfenning] p. 18. 
I  
a1tru 1421  tru 1409, a1tru 1421, trud 1414 
Definition I in [Pfenning] p. 18. 
E  
falimd 1420  falim 1419 
Definition E in [Pfenning] p. 18. 
I 
=> 
alrimiv 1740  alrimiv 1740, ralrimiva 2817 
Definition I^{a} in [Pfenning] p. 18,
definition In in [Clemente] p. 32. 
E 
=> 
spsbcd 3290  spcv 3149, rspcv 3155 
Definition E in [Pfenning] p. 18,
definition En,t in [Clemente] p. 32. 
I 
=> 
spesbcd 3359  spcev 3150, rspcev 3159 
Definition I in [Pfenning] p. 18,
definition In,t in [Clemente] p. 32. 
E 
& =>

exlimddv 1747  exlimddv 1747, exlimdd 2008,
exlimdv 1745, rexlimdva 2895 
Definition E^{a,u} in [Pfenning] p. 18,
definition Em,n,p,a in [Clemente] p. 32. 
C 
=> 
efald 1427  efald 1427 
Proof by contradiction (classical logic),
definition C in [Pfenning] p. 17. 
C 
=> 
pm2.18da 441  pm2.18da 441, pm2.18d 111, pm2.18 110 
For an alternative falsumfree natural deduction ruleset 
C 
=> 
notnotrd 113  notnotrd 113, notnot2 112 
Double negation rule (classical logic),
definition NNC in [Pfenning] p. 17,
definition En in [Clemente] p. 14. 
EM  
exmidd 414  exmid 413 
Excluded middle (classical logic),
definition XM in [Pfenning] p. 17,
proof 5.11 in [Clemente] p. 14. 
I  
eqidd 2403  eqid 2402, eqidd 2403 
Introduce equality,
definition =I in [Pfenning] p. 127. 
E  & =>

sbceq1dd 3282  sbceq1d 3281, 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 25529, exnatded5.3 25532,
exnatded5.5 25535, exnatded5.7 25536, exnatded5.8 25538, exnatded5.13 25540,
exnatded9.20 25542, and exnatded9.26 25544.
(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 25528 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 25529 
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 530, 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 25530.
A proof without context is shown in exnatded5.2i 25531.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)



Theorem  exnatded5.22 25530 
A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with
exnatded5.2 25529 and exnatded5.2i 25531. (New usage is discouraged.)
(Proof modification is discouraged.) (Contributed by Mario Carneiro,
9Feb2017.)



Theorem  exnatded5.2i 25531 
The same as exnatded5.2 25529 and exnatded5.22 25530 but with no context.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)



Theorem  exnatded5.3 25532 
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 25533.
A proof without context is shown in exnatded5.3i 25534.
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 463 to move it into the ND hypothesis 
2  5;6  

Given 
$e; adantr 463 to move it into the ND hypothesis 
3  1  ... 

ND hypothesis assumption 
simpr 459, to access the new assumption 
4  4  ... 

E 1,3 
mpd 15, the MPE equivalent of E, 1.3.
adantr 463 was used to transform its dependency
(we could also use imp 427 to get this directly from 1)

5  7  ... 

E 2,4 
mpd 15, the MPE equivalent of E, 4,6.
adantr 463 was used to transform its dependency 
6  8  ... 

I 4,5 
jca 530, the MPE equivalent of I, 4,7 
7  9  

I 3,6 
ex 432, 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 25533 
A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with
exnatded5.3 25532 and exnatded5.3i 25534. (New usage is discouraged.)
(Proof modification is discouraged.) (Contributed by Mario Carneiro,
9Feb2017.)



Theorem  exnatded5.3i 25534 
The same as exnatded5.3 25532 and exnatded5.32 25533 but with no context.
Identical to jccir 537, which should be used instead.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)



Theorem  exnatded5.5 25535 
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 463 to move it into the ND hypothesis 
2  5  
 Given 
$e; we'll use adantr 463 to move it into the ND hypothesis 
3  1 
...  
ND hypothesis assumption 
simpr 459 
4  4  ... 

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

IT 2 
adantr 463 5 
6  7  

I 3,4,5 
pm2.65da 574 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 463; simpr 459 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 177;
a proof without context is shown in mto 176.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by David A. Wheeler, 19Feb2017.)



Theorem  exnatded5.7 25536 
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 25537.
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 463 because we do not move this
into an ND hypothesis 
2  1  ... 

ND hypothesis assumption (new scope) 
simpr 459 
3  2  ... 

I_{L} 2 
orcd 390, the MPE equivalent of I_{L}, 1 
4  3  ... 

ND hypothesis assumption (new scope) 
simpr 459 
5  4  ... 

E_{L} 4 
simpld 457, the MPE equivalent of E_{L}, 3 
6  6  ... 

I_{R} 5 
olcd 391, the MPE equivalent of I_{R}, 4 
7  7  

E 1,3,6 
mpjaodan 787, 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 25537 
A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with
exnatded5.7 25536. (Proof modification is discouraged.)
(New usage is discouraged.) (Contributed by Mario Carneiro,
9Feb2017.)



Theorem  exnatded5.8 25538 
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 463 to move it into the ND hypothesis 
2  3;4  
 Given 
$e; adantr 463 to move it into the ND hypothesis 
3  7;8 
 
Given 
$e; adantr 463 to move it into the ND hypothesis 
4  1;2   
Given 
$e. adantr 463 to move it into the ND hypothesis 
5  6  ... 

ND Hypothesis/Assumption 
simpr 459. New ND hypothesis scope, each reference outside
the scope must change antecedent to . 
6  9  ... 

I 5,3 
jca 530 (I), 6,8 (adantr 463 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 574 (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 463; simpr 459 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 25539.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)



Theorem  exnatded5.82 25539 
A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer
linebyline translation, see exnatded5.8 25538.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)



Theorem  exnatded5.13 25540 
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 25541.
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 463 to move it into the ND hypothesis 
3  9  

Given 
$e. ad2antrr 724 to move it into the ND subhypothesis 
4  1  ... 

ND hypothesis assumption 
simpr 459 
5  4  ... 

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

I 5 
orcd 390 4 
7  6  ... 

ND hypothesis assumption 
simpr 459 
8  8  ... ... 

(sub) ND hypothesis assumption 
simpr 459 
9  11  ... ... 

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

IT 7 
adantr 463 6 
11  12  ... 

I 8,9,10 
pm2.65da 574 7,11 
12  13  ... 

E 11 
notnotrd 113 12 
13  14  ... 

I 12 
olcd 391 13 
14  16  

E 1,6,13 
mpjaodan 787 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 463; simpr 459 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 25541 
A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare
with exnatded5.13 25540. (Proof modification is discouraged.)
(New usage is discouraged.) (Contributed by Mario Carneiro,
9Feb2017.)



Theorem  exnatded9.20 25542 
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 457 1 
3  11 


E_{R} 1 
simprd 461 1 
4  4 
... 

ND hypothesis assumption 
simpr 459 
5  5 
... 

I 2,4 
jca 530 3,4 
6  6 
... 

I_{R} 5 
orcd 390 5 
7  8 
... 

ND hypothesis assumption 
simpr 459 
8  9 
... 

I 2,7 
jca 530 7,8 
9  10 
... 

I_{L} 8 
olcd 391 9 
10  12 


E 3,6,9 
mpjaodan 787 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 463; simpr 459 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 25543.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by David A. Wheeler, 19Feb2017.)



Theorem  exnatded9.202 25543 
A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare
with exnatded9.20 25542. (Proof modification is discouraged.)
(New usage is discouraged.) (Contributed by David A. Wheeler,
19Feb2017.)



Theorem  exnatded9.26 25544* 
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 459. Later statements will have this scope. 
3  7;5,4  ... 

E 2,y 
spsbcd 3290 (E), 5,6. To use it we need a1i 11 and vex 3061.
This could be immediately done with 19.21bi 1893, but we want to show
the general approach for substitution.

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

I 3,a 
spesbcd 3359 (I), 11.
To use it we need sylibr 212, which in turn requires sylib 196 and
two uses of sbcid 3293.
This could be more immediately done using 19.8a 1881, but we want to show
the general approach for substitution.

5  13;1,2  
 E 1,2,4,a 
exlimdd 2008 (E), 1,2,3,12.
We'll need supporting
assertions that the variable is free (not bound),
as provided in nfv 1728 and nfe1 1864 (MPE# 1,2) 
6  14  

I 5 
alrimiv 1740 (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 25545.
(Proof modification is discouraged.) (New usage is discouraged.)
(Contributed by Mario Carneiro, 9Feb2017.)
(Revised by David A. Wheeler, 18Feb2017.)



Theorem  exnatded9.262 25545* 
A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare
with exnatded9.26 25544. (Proof modification is discouraged.)
(New usage is discouraged.) (Contributed by Mario Carneiro,
9Feb2017.)



17.1.4 Definitional examples


Theorem  exor 25546 
Example for dfor 368. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 9May2015.)



Theorem  exan 25547 
Example for dfan 369. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 9May2015.)



Theorem  exdif 25548 
Example for dfdif 3416. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)



Theorem  exun 25549 
Example for dfun 3418. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)



Theorem  exin 25550 
Example for dfin 3420. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)



Theorem  exuni 25551 
Example for dfuni 4191. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 2Jul2016.)



Theorem  exss 25552 
Example for dfss 3427. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)



Theorem  expss 25553 
Example for dfpss 3429. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)



Theorem  expw 25554 
Example for dfpw 3956. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 2Jul2016.)



Theorem  expr 25555 
Example for dfpr 3974. (Contributed by Mario Carneiro,
7May2015.)



Theorem  exbr 25556 
Example for dfbr 4395. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)



Theorem  exopab 25557* 
Example for dfopab 4453. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)



Theorem  exeprel 25558 
Example for dfeprel 4733. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)



Theorem  exid 25559 
Example for dfid 4737. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)



Theorem  expo 25560 
Example for dfpo 4743. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)



Theorem  exxp 25561 
Example for dfxp 4828. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)



Theorem  excnv 25562 
Example for dfcnv 4830. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)



Theorem  exco 25563 
Example for dfco 4831. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)



Theorem  exdm 25564 
Example for dfdm 4832. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)



Theorem  exrn 25565 
Example for dfrn 4833. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)



Theorem  exres 25566 
Example for dfres 4834. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 7May2015.)



Theorem  exima 25567 
Example for dfima 4835. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 7May2015.)



Theorem  exfv 25568 
Example for dffv 5576. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)



Theorem  ex1st 25569 
Example for df1st 6783. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)



Theorem  ex2nd 25570 
Example for df2nd 6784. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)



Theorem  1kp2ke3k 25571 
Example for dfdec 11019, 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 11019 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 25572 
Example for dffl 11964. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)



Theorem  exdvds 25573 
3 divides into 6. A demonstration of dfdvds 14194. (Contributed by David
A. Wheeler, 19May2015.)



17.1.5 Other examples


Theorem  exinddvds 25574 
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 25575 
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 25576 
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 25577 
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 25578 
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 25579 
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 25580 
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.)



17.3 (Future  to be reviewed and
classified)


17.3.1 Planar incidence geometry


Syntax  cplig 25581 
Extend class notation with the class of all planar incidence
geometries.



Definition  dfplig 25582* 
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 25583* 
The predicate "is a planar incidence geometry". (Contributed by FL,
2Aug2009.)



Theorem  tncp 25584* 
There exist three non colinear points. (Contributed by FL,
3Aug2009.)



Theorem  lpni 25585* 
For any line, there exists a point not on the line. (Contributed by
Jeff Hankins, 15Aug2009.)



17.3.2 Algebra preliminaries


Syntax  crpm 25586 
Ring primes.

RPrime 

Definition  dfrprm 25587* 
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 14456. (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 25636 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 25588 
Extend class notation with the class of all group operations.



Syntax  cgi 25589 
Extend class notation with a function mapping a group operation to the
group's identity element.

GId 

Syntax  cgn 25590 
Extend class notation with a function mapping a group operation to the
inverse function for the group.



Syntax  cgs 25591 
Extend class notation with a function mapping a group operation to the
division (or subtraction) operation for the group.



Syntax  cgx 25592 
Extend class notation with a function mapping a group operation to the
power operation for the group.



Definition  dfgrpo 25593* 
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 25594* 
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 25595* 
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 25596* 
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 25597* 
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 25598* 
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 25599* 
The predicate "is a group operation." (Contributed by NM,
23Oct2012.) (New usage is discouraged.)



Theorem  isgrpoi 25600* 
Properties that determine a group operation. Read as
. (Contributed by NM, 4Nov2006.)
(New usage is discouraged.)

