| Description:
The following gives conventions used in the Metamath Proof Explorer
(MPE, set.mm) regarding labels.
For other conventions, see conventions 30329 and links therein.
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
easy-to-remember hints about their contents.
Labels are not a 1-to-1 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 "ax-NAME",
proofs of proven axioms are named "axNAME", and
definitions are named "df-NAME".
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 3046"rgen.1 $e |- ( x e. A -> ph ) $."
or letters corresponding to the (main) class variable used in the
hypothesis, e.g., for mdet0 22493: "mdet0.d $e |- D = ( N maDet R ) $.").
- Common names.
If a theorem has a well-known name, that name (or a short version of it)
is sometimes used directly. Examples include
barbara 2656 and stirling 46087.
- 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 42.
- 19.x series of theorems.
Similar to the conventions for the theorems from Principia Mathematica,
theorems from Section 19 of [Margaris] p. 90 often use a special naming
convention: "19." resp. "r19." (for corresponding restricted quantifier
versions) followed by its identifier.
For example, Theorem 38 from Section 19 of [Margaris] p. 90 is labeled
19.38 1839, and the restricted quantifier version of Theorem 21 from
Section 19 of [Margaris] p. 90 is labeled r19.21 3232.
- Characters to be used for labels.
Although the specification of Metamath allows for dots/periods "." in
any label, it is usually used only in labels for hypotheses (see above).
Exceptions are the labels of theorems from Principia Mathematica and the
19.x series of theorems from Section 19 of [Margaris] p. 90 (see above)
and 0.999... 15847. Furthermore, the underscore "_" should not be used.
Finally, only lower case characters should be used (except the special
suffixes OLD, ALT, and ALTV mentioned in bullet point "Suffixes"), at
least in main set.mm (exceptions are tolerated in mathboxes).
- Syntax label fragments.
Most theorems are named using a concatenation of syntax label fragments
(omitting variables) that represent the important part of the theorem's
main conclusion. Almost every syntactic construct has a definition
labeled "df-NAME", and normally NAME is the syntax label fragment. For
example, the class difference construct (𝐴 ∖ 𝐵) is defined in
df-dif 3917, and thus its syntax label fragment is "dif". Similarly, the
subclass relation 𝐴 ⊆ 𝐵 has syntax label fragment "ss"
because it is defined in df-ss 3931. Most theorem names follow from
these fragments, for example, the theorem proving (𝐴 ∖ 𝐵) ⊆ 𝐴
involves a class difference ("dif") of a subset ("ss"), and thus is
labeled difss 4099. There are many other syntax label fragments, e.g.,
singleton construct {𝐴} has syntax label fragment "sn" (because it
is defined in df-sn 4590), and the pair construct {𝐴, 𝐵} has
fragment "pr" ( from df-pr 4592). 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 is-element-of
(is member of) construct 𝐴 ∈ 𝐵 does not have a df-NAME definition;
in this case its syntax label fragment is "el". Thus, because the
theorem beginning with (𝐴 ∈ (𝐵 ∖ {𝐶}) uses is-element-of
("el") of a class difference ("dif") of a singleton ("sn"), it is
labeled eldifsn 4750. An "n" is often used for negation (¬), e.g.,
nan 829.
- Exceptions.
Sometimes there is a definition df-NAME but the label fragment is not
the NAME part. The definition should note this exception as part of its
definition. In addition, the table below attempts to list all such
cases and marks them in bold. For example, the label fragment "cn"
represents complex numbers ℂ (even though its definition is in
df-c 11074) and "re" represents real numbers ℝ (Definition df-r 11078).
The empty set ∅ often uses fragment 0, even though it is defined
in df-nul 4297. The syntax construct (𝐴 + 𝐵) usually uses the
fragment "add" (which is consistent with df-add 11079), but "p" is used as
the fragment for constant theorems. Equality (𝐴 = 𝐵) often uses
"e" as the fragment. As a result, "two plus two equals four" is labeled
2p2e4 12316.
- 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 16118 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 df-NAME is defined, there is typically a
proof of its value labeled "NAMEval" and of its closure labeled
"NAMEcl". E.g., for cosine (df-cos 16036) we have value cosval 16091 and
closure coscl 16095.
- Special cases.
Sometimes, syntax and related markings are insufficient to distinguish
different theorems. For example, there are over a hundred different
implication-only theorems. They are grouped in a more ad-hoc way that
attempts to make their distinctions clearer. These often use
abbreviations such as "mp" for "modus ponens", "syl" for syllogism, and
"id" for "identity". It is especially hard to give good names in the
propositional calculus section because there are so few primitives.
However, in most cases this is not a serious problem. There are a few
very common theorems like ax-mp 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 30332 for a list), and that is
about all you need for most things. As for the rest, you can just
assume that if it involves at most three connectives, then it is
probably already proved in set.mm, and searching for it will give you
the label.
- Suffixes.
Suffixes are used to indicate the form of a theorem (inference,
deduction, or closed form, see above).
Additionally, we sometimes suffix with "v" the label of a theorem adding
a disjoint variable condition, as in 19.21v 1939 versus 19.21 2208. This
often permits to prove the result using fewer axioms, and/or to
eliminate a nonfreeness hypothesis (such as Ⅎ𝑥𝜑 in 19.21 2208).
If no constraint is put on axiom use, then the v-version can be proved
from the original theorem using nfv 1914. If two (resp. three) such
disjoint variable conditions are added, then the suffix "vv" (resp.
"vvv") is used, e.g., exlimivv 1932.
Conversely, we sometimes suffix with "f" the label of a theorem
introducing such a hypothesis to eliminate the need for the disjoint
variable condition; e.g., euf 2569 derived from eu6 2567. 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 5409.
We sometimes suffix with "s" the label of an inference that manipulates
an antecedent, leaving the consequent unchanged. The "s" means that the
inference eliminates the need for a syllogism (syl 17) -type inference
in a proof. A theorem label is suffixed with "ALT" if it provides an
alternate less-preferred proof of a theorem (e.g., the proof is
clearer but uses more axioms than the preferred version).
The "ALT" may be further suffixed with a number if there is more
than one alternate theorem.
Furthermore, a theorem label is suffixed with "OLD" if there is a new
version of it and the OLD version is obsolete (and will be removed
within one year).
Finally, it should be mentioned that suffixes can be combined, for
example in cbvaldva 2407 (cbval 2396 in deduction form "d" with a not free
variable replaced by a disjoint variable condition "v" with a
conjunction as antecedent "a"). As a general rule, the suffixes for
the theorem forms ("i", "d" or "g") should be the first of multiple
suffixes, as for example in vtocldf 3526.
Here is a non-exhaustive list of common suffixes:
- a : theorem having a conjunction as antecedent
- b : theorem expressing a logical equivalence
- c : contraction (e.g., sylc 65, syl2anc 584), commutes
(e.g., biimpac 478)
- d : theorem in deduction form
- f : theorem with a hypothesis such as Ⅎ𝑥𝜑
- g : theorem in closed form having an "is a set" antecedent
- i : theorem in inference form
- l : theorem concerning something at the left
- r : theorem concerning something at the right
- r : theorem with something reversed (e.g., a biconditional)
- s : inference that manipulates an antecedent ("s" refers to an
application of syl 17 that is eliminated)
- t : theorem in closed form (not having an "is a set" antecedent)
- v : theorem with one (main) disjoint variable condition
- vv : theorem with two (main) disjoint variable conditions
- w : weak(er) form of a theorem
- ALT : alternate proof of a theorem
- ALTV : alternate version of a theorem or definition (mathbox
only)
- OLD : old/obsolete version of a theorem (or proof) or definition
- Reuse.
When creating a new theorem or axiom, try to reuse abbreviations used
elsewhere. A comment should explain the first use of an abbreviation.
The following table shows some commonly used abbreviations in labels, in
alphabetical order. For each abbreviation we provide a mnenomic, the
source theorem or the assumption defining it, an expression showing what
it looks like, whether or not it is a "syntax fragment" (an abbreviation
that indicates a particular kind of syntax), and hyperlinks to label
examples that use the abbreviation. The abbreviation is bolded if there
is a df-NAME 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 476, rexlimiva 3126 |
| abl | Abelian group | df-abl 19713 |
Abel | Yes | ablgrp 19715, zringabl 21361 |
| abs | absorption | | | No |
ressabs 17218 |
| abs | absolute value (of a complex number) |
df-abs 15202 | (abs‘𝐴) | Yes |
absval 15204, absneg 15243, abs1 15263 |
| ad | adding | |
| No | adantr 480, ad2antlr 727 |
| add | add (see "p") | df-add 11079 |
(𝐴 + 𝐵) | Yes |
addcl 11150, addcom 11360, addass 11155 |
| al | "for all" | |
∀𝑥𝜑 | No | alim 1810, alex 1826 |
| ALT | alternative/less preferred (suffix) | |
| No | idALT 23 |
| an | and | df-an 396 |
(𝜑 ∧ 𝜓) | Yes |
anor 984, iman 401, imnan 399 |
| ant | antecedent | |
| No | adantr 480 |
| ass | associative | |
| No | biass 384, orass 921, mulass 11156 |
| asym | asymmetric, antisymmetric | |
| No | intasym 6088, asymref 6089, posasymb 18280 |
| ax | axiom | |
| No | ax6dgen 2129, ax1cn 11102 |
| bas, base |
base (set of an extensible structure) | df-base 17180 |
(Base‘𝑆) | Yes |
baseval 17181, ressbas 17206, cnfldbas 21268 |
| b, bi | biconditional ("iff", "if and only if")
| df-bi 207 | (𝜑 ↔ 𝜓) | Yes |
impbid 212, sspwb 5409 |
| br | binary relation | df-br 5108 |
𝐴𝑅𝐵 | Yes | brab1 5155, brun 5158 |
| c | commutes, commuted (suffix) | | |
No | biimpac 478 |
| c | contraction (suffix) | | |
No | sylc 65, syl2anc 584 |
| cbv | change bound variable | | |
No | cbvalivw 2007, cbvrex 3337 |
| cdm | codomain | |
| No | ffvelcdm 7053, focdmex 7934 |
| cl | closure | | | No |
ifclda 4524, ovrcl 7428, zaddcl 12573 |
| cn | complex numbers | df-c 11074 |
ℂ | Yes | nnsscn 12191, nncn 12194 |
| cnfld | field of complex numbers | df-cnfld 21265 |
ℂfld | Yes | cnfldbas 21268, cnfldinv 21314 |
| cntz | centralizer | df-cntz 19249 |
(Cntz‘𝑀) | Yes |
cntzfval 19252, dprdfcntz 19947 |
| cnv | converse | df-cnv 5646 |
◡𝐴 | Yes | opelcnvg 5844, f1ocnv 6812 |
| co | composition | df-co 5647 |
(𝐴 ∘ 𝐵) | Yes | cnvco 5849, fmptco 7101 |
| com | commutative | |
| No | orcom 870, bicomi 224, eqcomi 2738 |
| con | contradiction, contraposition | |
| No | condan 817, con2d 134 |
| csb | class substitution | df-csb 3863 |
⦋𝐴 / 𝑥⦌𝐵 | Yes |
csbid 3875, csbie2g 3902 |
| cyg | cyclic group | df-cyg 19808 |
CycGrp | Yes |
iscyg 19809, zringcyg 21379 |
| d | deduction form (suffix) | |
| No | idd 24, impbid 212 |
| df | (alternate) definition (prefix) | |
| No | dfrel2 6162, dffn2 6690 |
| di, distr | distributive | |
| No |
andi 1009, imdi 389, ordi 1007, difindi 4255, ndmovdistr 7578 |
| dif | class difference | df-dif 3917 |
(𝐴 ∖ 𝐵) | Yes |
difss 4099, difindi 4255 |
| div | division | df-div 11836 |
(𝐴 / 𝐵) | Yes |
divcl 11843, divval 11839, divmul 11840 |
| dm | domain | df-dm 5648 |
dom 𝐴 | Yes | dmmpt 6213, iswrddm0 14503 |
| e, eq, equ | equals (equ for setvars, eq for
classes) | df-cleq 2721 |
𝐴 = 𝐵 | Yes |
2p2e4 12316, uneqri 4119, equtr 2021 |
| edg | edge | df-edg 28975 |
(Edg‘𝐺) | Yes |
edgopval 28978, usgredgppr 29123 |
| el | element of | |
𝐴 ∈ 𝐵 | Yes |
eldif 3924, eldifsn 4750, elssuni 4901 |
| en | equinumerous | df-en |
𝐴 ≈ 𝐵 | Yes | domen 8933, enfi 9151 |
| eu | "there exists exactly one" | eu6 2567 |
∃!𝑥𝜑 | Yes | euex 2570, euabsn 4690 |
| ex | exists (i.e. is a set) | |
∈ V | No | brrelex1 5691, 0ex 5262 |
| ex, e | "there exists (at least one)" |
df-ex 1780 |
∃𝑥𝜑 | Yes | exim 1834, alex 1826 |
| exp | export | |
| No | expt 177, expcom 413 |
| f | "not free in" (suffix) | |
| No | equs45f 2457, sbf 2271 |
| f | function | df-f 6515 |
𝐹:𝐴⟶𝐵 | Yes | fssxp 6715, opelf 6721 |
| fal | false | df-fal 1553 |
⊥ | Yes | bifal 1556, falantru 1575 |
| fi | finite intersection | df-fi 9362 |
(fi‘𝐵) | Yes | fival 9363, inelfi 9369 |
| fi, fin | finite | df-fin 8922 |
Fin | Yes |
isfi 8947, snfi 9014, onfin 9179 |
| fld | field (Note: there is an alternative
definition Fld of a field, see df-fld 37986) | df-field 20641 |
Field | Yes | isfld 20649, fldidom 20680 |
| fn | function with domain | df-fn 6514 |
𝐴 Fn 𝐵 | Yes | ffn 6688, fndm 6621 |
| frgp | free group | df-frgp 19640 |
(freeGrp‘𝐼) | Yes |
frgpval 19688, frgpadd 19693 |
| fsupp | finitely supported function |
df-fsupp 9313 | 𝑅 finSupp 𝑍 | Yes |
isfsupp 9316, fdmfisuppfi 9325, fsuppco 9353 |
| fun | function | df-fun 6513 |
Fun 𝐹 | Yes | funrel 6533, ffun 6691 |
| fv | function value | df-fv 6519 |
(𝐹‘𝐴) | Yes | fvres 6877, swrdfv 14613 |
| fz | finite set of sequential integers |
df-fz 13469 |
(𝑀...𝑁) | Yes | fzval 13470, eluzfz 13480 |
| fz0 | finite set of sequential nonnegative integers |
|
(0...𝑁) | Yes | nn0fz0 13586, fz0tp 13589 |
| fzo | half-open integer range | df-fzo 13616 |
(𝑀..^𝑁) | Yes |
elfzo 13622, elfzofz 13636 |
| g | more general (suffix); eliminates "is a set"
hypotheses | |
| No | uniexg 7716 |
| gr | graph | |
| No | uhgrf 28989, isumgr 29022, usgrres1 29242 |
| grp | group | df-grp 18868 |
Grp | Yes | isgrp 18871, tgpgrp 23965 |
| gsum | group sum | df-gsum 17405 |
(𝐺 Σg 𝐹) | Yes |
gsumval 18604, gsumwrev 19298 |
| hash | size (of a set) | df-hash 14296 |
(♯‘𝐴) | Yes |
hashgval 14298, hashfz1 14311, hashcl 14321 |
| hb | hypothesis builder (prefix) | |
| No | hbxfrbi 1825, hbald 2169, hbequid 38902 |
| hm | (monoid, group, ring, ...) homomorphism |
| | No |
ismhm 18712, isghm 19147, isrhm 20387 |
| i | inference (suffix) | |
| No | eleq1i 2819, tcsni 9696 |
| i | implication (suffix) | |
| No | brwdomi 9521, infeq5i 9589 |
| id | identity | |
| No | biid 261 |
| iedg | indexed edge | df-iedg 28926 |
(iEdg‘𝐺) | Yes |
iedgval0 28967, edgiedgb 28981 |
| idm | idempotent | |
| No | anidm 564, tpidm13 4720 |
| im, imp | implication (label often omitted) |
df-im 15067 | (𝐴 → 𝐵) | Yes |
iman 401, imnan 399, impbidd 210 |
| im | (group, ring, ...) isomorphism | |
| No | isgim 19194, rimrcl 20391 |
| ima | image | df-ima 5651 |
(𝐴 “ 𝐵) | Yes | resima 5986, imaundi 6122 |
| imp | import | |
| No | biimpa 476, impcom 407 |
| in | intersection | df-in 3921 |
(𝐴 ∩ 𝐵) | Yes | elin 3930, incom 4172 |
| inf | infimum | df-inf 9394 |
inf(ℝ+, ℝ*, < ) | Yes |
fiinfcl 9454, infiso 9461 |
| is... | is (something a) ...? | |
| No | isring 20146 |
| j | joining, disjoining | |
| No | jc 161, jaoi 857 |
| l | left | |
| No | olcd 874, simpl 482 |
| map | mapping operation or set exponentiation |
df-map 8801 | (𝐴 ↑m 𝐵) | Yes |
mapvalg 8809, elmapex 8821 |
| mat | matrix | df-mat 22295 |
(𝑁 Mat 𝑅) | Yes |
matval 22298, matring 22330 |
| mdet | determinant (of a square matrix) |
df-mdet 22472 | (𝑁 maDet 𝑅) | Yes |
mdetleib 22474, mdetrlin 22489 |
| mgm | magma | df-mgm 18567 |
Magma | Yes |
mgmidmo 18587, mgmlrid 18594, ismgm 18568 |
| mgp | multiplicative group | df-mgp 20050 |
(mulGrp‘𝑅) | Yes |
mgpress 20059, ringmgp 20148 |
| mnd | monoid | df-mnd 18662 |
Mnd | Yes | mndass 18670, mndodcong 19472 |
| mo | "there exists at most one" | df-mo 2533 |
∃*𝑥𝜑 | Yes | eumo 2571, moim 2537 |
| mp | modus ponens | ax-mp 5 |
| No | mpd 15, mpi 20 |
| mpo | maps-to notation for an operation |
df-mpo 7392 | (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | Yes |
mpompt 7503, resmpo 7509 |
| mpt | modus ponendo tollens | |
| No | mptnan 1768, mptxor 1769 |
| mpt | maps-to notation for a function |
df-mpt 5189 | (𝑥 ∈ 𝐴 ↦ 𝐵) | Yes |
fconstmpt 5700, resmpt 6008 |
| mul | multiplication (see "t") | df-mul 11080 |
(𝐴 · 𝐵) | Yes |
mulcl 11152, divmul 11840, mulcom 11154, mulass 11156 |
| n, not | not | |
¬ 𝜑 | Yes |
nan 829, notnotr 130 |
| ne | not equal | df-ne | 𝐴 ≠ 𝐵 |
Yes | exmidne 2935, neeqtrd 2994 |
| nel | not element of | df-nel | 𝐴 ∉ 𝐵
|
Yes | neli 3031, nnel 3039 |
| ne0 | not equal to zero (see n0) | |
≠ 0 | No |
negne0d 11531, ine0 11613, gt0ne0 11643 |
| nf | "not free in" (prefix) | df-nf 1784 |
Ⅎ𝑥𝜑 | Yes | nfnd 1858 |
| ngp | normed group | df-ngp 24471 |
NrmGrp | Yes | isngp 24484, ngptps 24490 |
| nm | norm (on a group or ring) | df-nm 24470 |
(norm‘𝑊) | Yes |
nmval 24477, subgnm 24521 |
| nn | positive integers | df-nn 12187 |
ℕ | Yes | nnsscn 12191, nncn 12194 |
| nn0 | nonnegative integers | df-n0 12443 |
ℕ0 | Yes | nnnn0 12449, nn0cn 12452 |
| n0 | not the empty set (see ne0) | |
≠ ∅ | No | n0i 4303, vn0 4308, ssn0 4367 |
| OLD | old, obsolete (to be removed soon) | |
| No | 19.43OLD 1883 |
| on | ordinal number | df-on 6336 |
𝐴 ∈ On | Yes |
elon 6341, 1on 8446 onelon 6357 |
| op | ordered pair | df-op 4596 |
⟨𝐴, 𝐵⟩ | Yes | dfopif 4834, opth 5436 |
| or | or | df-or 848 |
(𝜑 ∨ 𝜓) | Yes |
orcom 870, anor 984 |
| ot | ordered triple | df-ot 4598 |
⟨𝐴, 𝐵, 𝐶⟩ | Yes |
euotd 5473, fnotovb 7439 |
| ov | operation value | df-ov 7390 |
(𝐴𝐹𝐵) | Yes
| fnotovb 7439, fnovrn 7564 |
| p | plus (see "add"), for all-constant
theorems | df-add 11079 |
(3 + 2) = 5 | Yes |
3p2e5 12332 |
| pfx | prefix | df-pfx 14636 |
(𝑊 prefix 𝐿) | Yes |
pfxlen 14648, ccatpfx 14666 |
| pm | Principia Mathematica | |
| No | pm2.27 42 |
| pm | partial mapping (operation) | df-pm 8802 |
(𝐴 ↑pm 𝐵) | Yes | elpmi 8819, pmsspw 8850 |
| pr | pair | df-pr 4592 |
{𝐴, 𝐵} | Yes |
elpr 4614, prcom 4696, prid1g 4724, prnz 4741 |
| prm, prime | prime (number) | df-prm 16642 |
ℙ | Yes | 1nprm 16649, dvdsprime 16657 |
| pss | proper subset | df-pss 3934 |
𝐴 ⊊ 𝐵 | Yes | pssss 4061, sspsstri 4068 |
| q | rational numbers ("quotients") | df-q 12908 |
ℚ | Yes | elq 12909 |
| r | reversed (suffix) | |
| No | pm4.71r 558, caovdir 7623 |
| r | right | |
| No | orcd 873, simprl 770 |
| rab | restricted class abstraction |
df-rab 3406 | {𝑥 ∈ 𝐴 ∣ 𝜑} | Yes |
rabswap 3415, df-oprab 7391 |
| ral | restricted universal quantification |
df-ral 3045 | ∀𝑥 ∈ 𝐴𝜑 | Yes |
ralnex 3055, ralrnmpo 7528 |
| rcl | reverse closure | |
| No | ndmfvrcl 6894, nnarcl 8580 |
| re | real numbers | df-r 11078 |
ℝ | Yes | recn 11158, 0re 11176 |
| rel | relation | df-rel 5645 | Rel 𝐴 |
Yes | brrelex1 5691, relmpoopab 8073 |
| res | restriction | df-res 5650 |
(𝐴 ↾ 𝐵) | Yes |
opelres 5956, f1ores 6814 |
| reu | restricted existential uniqueness |
df-reu 3355 | ∃!𝑥 ∈ 𝐴𝜑 | Yes |
nfreud 3402, reurex 3358 |
| rex | restricted existential quantification |
df-rex 3054 | ∃𝑥 ∈ 𝐴𝜑 | Yes |
rexnal 3082, rexrnmpo 7529 |
| rmo | restricted "at most one" |
df-rmo 3354 | ∃*𝑥 ∈ 𝐴𝜑 | Yes |
nfrmod 3401, nrexrmo 3375 |
| rn | range | df-rn 5649 | ran 𝐴 |
Yes | elrng 5855, rncnvcnv 5898 |
| ring | (unital) ring | df-ring 20144 |
Ring | Yes |
ringidval 20092, isring 20146, ringgrp 20147 |
| rng | non-unital ring | df-rng 20062 |
Rng | Yes |
isrng 20063, rngabl 20064, rnglz 20074 |
| rot | rotation | |
| No | 3anrot 1099, 3orrot 1091 |
| s | eliminates need for syllogism (suffix) |
| | No | ancoms 458 |
| sb | (proper) substitution (of a set) |
df-sb 2066 | [𝑦 / 𝑥]𝜑 | Yes |
spsbe 2083, sbimi 2075 |
| sbc | (proper) substitution of a class |
df-sbc 3754 | [𝐴 / 𝑥]𝜑 | Yes |
sbc2or 3762, sbcth 3768 |
| sca | scalar | df-sca 17236 |
(Scalar‘𝐻) | Yes |
resssca 17306, mgpsca 20055 |
| simp | simple, simplification | |
| No | simpl 482, simp3r3 1284 |
| sn | singleton | df-sn 4590 |
{𝐴} | Yes | eldifsn 4750 |
| sp | specialization | |
| No | spsbe 2083, spei 2392 |
| ss | subset | df-ss 3931 |
𝐴 ⊆ 𝐵 | Yes | difss 4099 |
| struct | structure | df-struct 17117 |
Struct | Yes | brstruct 17118, structfn 17126 |
| sub | subtract | df-sub 11407 |
(𝐴 − 𝐵) | Yes |
subval 11412, subaddi 11509 |
| sup | supremum | df-sup 9393 |
sup(𝐴, 𝐵, < ) | Yes |
fisupcl 9421, supmo 9403 |
| supp | support (of a function) | df-supp 8140 |
(𝐹 supp 𝑍) | Yes |
ressuppfi 9346, mptsuppd 8166 |
| swap | swap (two parts within a theorem) |
| | No | rabswap 3415, 2reuswap 3717 |
| syl | syllogism | syl 17 |
| No | 3syl 18 |
| sym | symmetric | |
| No | df-symdif 4216, cnvsym 6085 |
| symg | symmetric group | df-symg 19300 |
(SymGrp‘𝐴) | Yes |
symghash 19308, pgrpsubgsymg 19339 |
| t |
times (see "mul"), for all-constant theorems |
df-mul 11080 |
(3 · 2) = 6 | Yes |
3t2e6 12347 |
| th, t |
theorem |
|
|
No |
nfth 1801, sbcth 3768, weth 10448, ancomst 464 |
| tp | triple | df-tp 4594 |
{𝐴, 𝐵, 𝐶} | Yes |
eltpi 4652, tpeq1 4706 |
| tr | transitive | |
| No | bitrd 279, biantr 805 |
| tru, t |
true, truth |
df-tru 1543 |
⊤ |
Yes |
bitru 1549, truanfal 1574, biimt 360 |
| un | union | df-un 3919 |
(𝐴 ∪ 𝐵) | Yes |
uneqri 4119, uncom 4121 |
| unit | unit (in a ring) |
df-unit 20267 | (Unit‘𝑅) | Yes |
isunit 20282, nzrunit 20433 |
| v |
setvar (especially for specializations of
theorems when a class is replaced by a setvar variable) |
|
x |
Yes |
cv 1539, vex 3451, velpw 4568, vtoclf 3530 |
| v |
disjoint variable condition used in place of nonfreeness
hypothesis (suffix) |
|
|
No |
spimv 2388 |
| vtx |
vertex |
df-vtx 28925 |
(Vtx‘𝐺) |
Yes |
vtxval0 28966, opvtxov 28932 |
| vv |
two disjoint variable conditions used in place of nonfreeness
hypotheses (suffix) |
|
|
No |
19.23vv 1943 |
| w | weak (version of a theorem) (suffix) | |
| No | ax11w 2131, spnfw 1979 |
| wrd | word |
df-word 14479 | Word 𝑆 | Yes |
iswrdb 14485, wrdfn 14493, ffz0iswrd 14506 |
| xp | cross product (Cartesian product) |
df-xp 5644 | (𝐴 × 𝐵) | Yes |
elxp 5661, opelxpi 5675, xpundi 5707 |
| xr | eXtended reals | df-xr 11212 |
ℝ* | Yes | ressxr 11218, rexr 11220, 0xr 11221 |
| z | integers (from German "Zahlen") |
df-z 12530 | ℤ | Yes |
elz 12531, zcn 12534 |
| zn | ring of integers mod 𝑁 | df-zn 21416 |
(ℤ/nℤ‘𝑁) | Yes |
znval 21445, zncrng 21454, znhash 21468 |
| zring | ring of integers | df-zring 21357 |
ℤring | Yes | zringbas 21363, zringcrng 21358
|
| 0, z |
slashed zero (empty set) | df-nul 4297 |
∅ | Yes |
n0i 4303, vn0 4308; snnz 4740, prnz 4741 |
(Contributed by the Metamath team, 27-Dec-2016.) Date of last revision.
(Revised by the Metamath team, 22-Sep-2022.)
(Proof modification is discouraged.) (New usage is
discouraged.) |
