| Description:
The following gives conventions used in the Metamath Proof Explorer
(MPE, set.mm) regarding labels.
For other conventions, see conventions 30470 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 3053"rgen.1 $e |- ( x e. A -> ph ) $."
or letters corresponding to the (main) class variable used in the
hypothesis, e.g., for mdet0 22571: "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 2663 and stirling 46517.
- 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 1841, 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... 15846. 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 3892, 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 3906. 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 4076. There are many other syntax label fragments, e.g.,
singleton construct {𝐴} has syntax label fragment "sn" (because it
is defined in df-sn 4568), and the pair construct {𝐴, 𝐵} has
fragment "pr" ( from df-pr 4570). 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 4731. An "n" is often used for negation (¬), e.g.,
nan 830.
- 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 11044) and "re" represents real numbers ℝ (Definition df-r 11048).
The empty set ∅ often uses fragment 0, even though it is defined
in df-nul 4274. The syntax construct (𝐴 + 𝐵) usually uses the
fragment "add" (which is consistent with df-add 11049), 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 12311.
- 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 16117 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 16035) we have value cosval 16090 and
closure coscl 16094.
- 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 30473 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 1941 versus 19.21 2215. This
often permits to prove the result using fewer axioms, and/or to
eliminate a nonfreeness hypothesis (such as Ⅎ𝑥𝜑 in 19.21 2215).
If no constraint is put on axiom use, then the v-version can be proved
from the original theorem using nfv 1916. If two (resp. three) such
disjoint variable conditions are added, then the suffix "vv" (resp.
"vvv") is used, e.g., exlimivv 1934.
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 2576 derived from eu6 2574. 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 5401.
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 2413 (cbval 2402 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 3505.
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 585), 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 3130 |
| abl | Abelian group | df-abl 19758 |
Abel | Yes | ablgrp 19760, zringabl 21431 |
| abs | absorption | | | No |
ressabs 17218 |
| abs | absolute value (of a complex number) |
df-abs 15198 | (abs‘𝐴) | Yes |
absval 15200, absneg 15239, abs1 15259 |
| ad | adding | |
| No | adantr 480, ad2antlr 728 |
| add | add (see "p") | df-add 11049 |
(𝐴 + 𝐵) | Yes |
addcl 11120, addcom 11332, addass 11125 |
| al | "for all" | |
∀𝑥𝜑 | No | alim 1812, alex 1828 |
| ALT | alternative/less preferred (suffix) | |
| No | idALT 23 |
| an | and | df-an 396 |
(𝜑 ∧ 𝜓) | Yes |
anor 985, iman 401, imnan 399 |
| ant | antecedent | |
| No | adantr 480 |
| ass | associative | |
| No | biass 384, orass 922, mulass 11126 |
| asym | asymmetric, antisymmetric | |
| No | intasym 6078, asymref 6079, posasymb 18285 |
| ax | axiom | |
| No | ax6dgen 2134, ax1cn 11072 |
| bas, base |
base (set of an extensible structure) | df-base 17180 |
(Base‘𝑆) | Yes |
baseval 17181, ressbas 17206, cnfldbas 21356 |
| b, bi | biconditional ("iff", "if and only if")
| df-bi 207 | (𝜑 ↔ 𝜓) | Yes |
impbid 212, sspwb 5401 |
| br | binary relation | df-br 5086 |
𝐴𝑅𝐵 | Yes | brab1 5133, brun 5136 |
| c | commutes, commuted (suffix) | | |
No | biimpac 478 |
| c | contraction (suffix) | | |
No | sylc 65, syl2anc 585 |
| cbv | change bound variable | | |
No | cbvalivw 2009, cbvrex 3325 |
| cdm | codomain | |
| No | ffvelcdm 7033, focdmex 7909 |
| cl | closure | | | No |
ifclda 4502, ovrcl 7408, zaddcl 12567 |
| cn | complex numbers | df-c 11044 |
ℂ | Yes | nnsscn 12179, nncn 12182 |
| cnfld | field of complex numbers | df-cnfld 21353 |
ℂfld | Yes | cnfldbas 21356, cnfldinv 21383 |
| cntz | centralizer | df-cntz 19292 |
(Cntz‘𝑀) | Yes |
cntzfval 19295, dprdfcntz 19992 |
| cnv | converse | df-cnv 5639 |
◡𝐴 | Yes | opelcnvg 5835, f1ocnv 6792 |
| co | composition | df-co 5640 |
(𝐴 ∘ 𝐵) | Yes | cnvco 5840, fmptco 7082 |
| com | commutative | |
| No | orcom 871, bicomi 224, eqcomi 2745 |
| con | contradiction, contraposition | |
| No | condan 818, con2d 134 |
| csb | class substitution | df-csb 3838 |
⦋𝐴 / 𝑥⦌𝐵 | Yes |
csbid 3850, csbie2g 3877 |
| cyg | cyclic group | df-cyg 19853 |
CycGrp | Yes |
iscyg 19854, zringcyg 21449 |
| d | deduction form (suffix) | |
| No | idd 24, impbid 212 |
| df | (alternate) definition (prefix) | |
| No | dfrel2 6153, dffn2 6670 |
| di, distr | distributive | |
| No |
andi 1010, imdi 389, ordi 1008, difindi 4232, ndmovdistr 7556 |
| dif | class difference | df-dif 3892 |
(𝐴 ∖ 𝐵) | Yes |
difss 4076, difindi 4232 |
| div | division | df-div 11808 |
(𝐴 / 𝐵) | Yes |
divcl 11815, divval 11811, divmul 11812 |
| dm | domain | df-dm 5641 |
dom 𝐴 | Yes | dmmpt 6204, iswrddm0 14500 |
| e, eq, equ | equals (equ for setvars, eq for
classes) | df-cleq 2728 |
𝐴 = 𝐵 | Yes |
2p2e4 12311, uneqri 4096, equtr 2023 |
| edg | edge | df-edg 29117 |
(Edg‘𝐺) | Yes |
edgopval 29120, usgredgppr 29265 |
| el | element of | |
𝐴 ∈ 𝐵 | Yes |
eldif 3899, eldifsn 4731, elssuni 4881 |
| en | equinumerous | df-en |
𝐴 ≈ 𝐵 | Yes | domen 8908, enfi 9121 |
| eu | "there exists exactly one" | eu6 2574 |
∃!𝑥𝜑 | Yes | euex 2577, euabsn 4670 |
| ex | exists (i.e. is a set) | |
∈ V | No | brrelex1 5684, 0ex 5242 |
| ex, e | "there exists (at least one)" |
df-ex 1782 |
∃𝑥𝜑 | Yes | exim 1836, alex 1828 |
| exp | export | |
| No | expt 177, expcom 413 |
| f | "not free in" (suffix) | |
| No | equs45f 2463, sbf 2278 |
| f | function | df-f 6502 |
𝐹:𝐴⟶𝐵 | Yes | fssxp 6695, opelf 6701 |
| fal | false | df-fal 1555 |
⊥ | Yes | bifal 1558, falantru 1577 |
| fi | finite intersection | df-fi 9324 |
(fi‘𝐵) | Yes | fival 9325, inelfi 9331 |
| fi, fin | finite | df-fin 8897 |
Fin | Yes |
isfi 8922, snfi 8990, onfin 9149 |
| fld | field (Note: there is an alternative
definition Fld of a field, see df-fld 38313) | df-field 20709 |
Field | Yes | isfld 20717, fldidom 20748 |
| fn | function with domain | df-fn 6501 |
𝐴 Fn 𝐵 | Yes | ffn 6668, fndm 6601 |
| frgp | free group | df-frgp 19685 |
(freeGrp‘𝐼) | Yes |
frgpval 19733, frgpadd 19738 |
| fsupp | finitely supported function |
df-fsupp 9275 | 𝑅 finSupp 𝑍 | Yes |
isfsupp 9278, fdmfisuppfi 9287, fsuppco 9315 |
| fun | function | df-fun 6500 |
Fun 𝐹 | Yes | funrel 6515, ffun 6671 |
| fv | function value | df-fv 6506 |
(𝐹‘𝐴) | Yes | fvres 6859, swrdfv 14611 |
| fz | finite set of sequential integers |
df-fz 13462 |
(𝑀...𝑁) | Yes | fzval 13463, eluzfz 13473 |
| fz0 | finite set of sequential nonnegative integers |
|
(0...𝑁) | Yes | nn0fz0 13579, fz0tp 13582 |
| fzo | half-open integer range | df-fzo 13609 |
(𝑀..^𝑁) | Yes |
elfzo 13615, elfzofz 13630 |
| g | more general (suffix); eliminates "is a set"
hypotheses | |
| No | uniexg 7694 |
| gr | graph | |
| No | uhgrf 29131, isumgr 29164, usgrres1 29384 |
| grp | group | df-grp 18912 |
Grp | Yes | isgrp 18915, tgpgrp 24043 |
| gsum | group sum | df-gsum 17405 |
(𝐺 Σg 𝐹) | Yes |
gsumval 18645, gsumwrev 19341 |
| hash | size (of a set) | df-hash 14293 |
(♯‘𝐴) | Yes |
hashgval 14295, hashfz1 14308, hashcl 14318 |
| hb | hypothesis builder (prefix) | |
| No | hbxfrbi 1827, hbald 2174, hbequid 39355 |
| hm | (monoid, group, ring, ...) homomorphism |
| | No |
ismhm 18753, isghm 19190, isrhm 20458 |
| i | inference (suffix) | |
| No | eleq1i 2827, tcsni 9662 |
| i | implication (suffix) | |
| No | brwdomi 9483, infeq5i 9557 |
| id | identity | |
| No | biid 261 |
| iedg | indexed edge | df-iedg 29068 |
(iEdg‘𝐺) | Yes |
iedgval0 29109, edgiedgb 29123 |
| idm | idempotent | |
| No | anidm 564, tpidm13 4700 |
| im, imp | implication (label often omitted) |
df-im 15063 | (𝐴 → 𝐵) | Yes |
iman 401, imnan 399, impbidd 210 |
| im | (group, ring, ...) isomorphism | |
| No | isgim 19237, rimrcl 20461 |
| ima | image | df-ima 5644 |
(𝐴 “ 𝐵) | Yes | resima 5980, imaundi 6113 |
| imp | import | |
| No | biimpa 476, impcom 407 |
| in | intersection | df-in 3896 |
(𝐴 ∩ 𝐵) | Yes | elin 3905, incom 4149 |
| inf | infimum | df-inf 9356 |
inf(ℝ+, ℝ*, < ) | Yes |
fiinfcl 9416, infiso 9423 |
| is... | is (something a) ...? | |
| No | isring 20218 |
| j | joining, disjoining | |
| No | jc 161, jaoi 858 |
| l | left | |
| No | olcd 875, simpl 482 |
| map | mapping operation or set exponentiation |
df-map 8775 | (𝐴 ↑m 𝐵) | Yes |
mapvalg 8783, elmapex 8795 |
| mat | matrix | df-mat 22373 |
(𝑁 Mat 𝑅) | Yes |
matval 22376, matring 22408 |
| mdet | determinant (of a square matrix) |
df-mdet 22550 | (𝑁 maDet 𝑅) | Yes |
mdetleib 22552, mdetrlin 22567 |
| mgm | magma | df-mgm 18608 |
Magma | Yes |
mgmidmo 18628, mgmlrid 18635, ismgm 18609 |
| mgp | multiplicative group | df-mgp 20122 |
(mulGrp‘𝑅) | Yes |
mgpress 20131, ringmgp 20220 |
| mnd | monoid | df-mnd 18703 |
Mnd | Yes | mndass 18711, mndodcong 19517 |
| mo | "there exists at most one" | df-mo 2539 |
∃*𝑥𝜑 | Yes | eumo 2578, moim 2544 |
| mp | modus ponens | ax-mp 5 |
| No | mpd 15, mpi 20 |
| mpo | maps-to notation for an operation |
df-mpo 7372 | (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | Yes |
mpompt 7481, resmpo 7487 |
| mpt | modus ponendo tollens | |
| No | mptnan 1770, mptxor 1771 |
| mpt | maps-to notation for a function |
df-mpt 5167 | (𝑥 ∈ 𝐴 ↦ 𝐵) | Yes |
fconstmpt 5693, resmpt 6002 |
| mul | multiplication (see "t") | df-mul 11050 |
(𝐴 · 𝐵) | Yes |
mulcl 11122, divmul 11812, mulcom 11124, mulass 11126 |
| n, not | not | |
¬ 𝜑 | Yes |
nan 830, notnotr 130 |
| ne | not equal | df-ne | 𝐴 ≠ 𝐵 |
Yes | exmidne 2942, neeqtrd 3001 |
| nel | not element of | df-nel | 𝐴 ∉ 𝐵
|
Yes | neli 3038, nnel 3046 |
| ne0 | not equal to zero (see n0) | |
≠ 0 | No |
negne0d 11503, ine0 11585, gt0ne0 11615 |
| nf | "not free in" (prefix) | df-nf 1786 |
Ⅎ𝑥𝜑 | Yes | nfnd 1860 |
| ngp | normed group | df-ngp 24548 |
NrmGrp | Yes | isngp 24561, ngptps 24567 |
| nm | norm (on a group or ring) | df-nm 24547 |
(norm‘𝑊) | Yes |
nmval 24554, subgnm 24598 |
| nn | positive integers | df-nn 12175 |
ℕ | Yes | nnsscn 12179, nncn 12182 |
| nn0 | nonnegative integers | df-n0 12438 |
ℕ0 | Yes | nnnn0 12444, nn0cn 12447 |
| n0 | not the empty set (see ne0) | |
≠ ∅ | No | n0i 4280, vn0 4285, ssn0 4344 |
| OLD | old, obsolete (to be removed soon) | |
| No | 19.43OLD 1885 |
| on | ordinal number | df-on 6327 |
𝐴 ∈ On | Yes |
elon 6332, 1on 8417 onelon 6348 |
| op | ordered pair | df-op 4574 |
⟨𝐴, 𝐵⟩ | Yes | dfopif 4813, opth 5429 |
| or | or | df-or 849 |
(𝜑 ∨ 𝜓) | Yes |
orcom 871, anor 985 |
| ot | ordered triple | df-ot 4576 |
⟨𝐴, 𝐵, 𝐶⟩ | Yes |
euotd 5467, fnotovb 7419 |
| ov | operation value | df-ov 7370 |
(𝐴𝐹𝐵) | Yes
| fnotovb 7419, fnovrn 7542 |
| p | plus (see "add"), for all-constant
theorems | df-add 11049 |
(3 + 2) = 5 | Yes |
3p2e5 12327 |
| pfx | prefix | df-pfx 14634 |
(𝑊 prefix 𝐿) | Yes |
pfxlen 14646, ccatpfx 14663 |
| pm | Principia Mathematica | |
| No | pm2.27 42 |
| pm | partial mapping (operation) | df-pm 8776 |
(𝐴 ↑pm 𝐵) | Yes | elpmi 8793, pmsspw 8825 |
| pr | pair | df-pr 4570 |
{𝐴, 𝐵} | Yes |
elpr 4592, prcom 4676, prid1g 4704, prnz 4721 |
| prm, prime | prime (number) | df-prm 16641 |
ℙ | Yes | 1nprm 16648, dvdsprime 16656 |
| pss | proper subset | df-pss 3909 |
𝐴 ⊊ 𝐵 | Yes | pssss 4038, sspsstri 4045 |
| q | rational numbers ("quotients") | df-q 12899 |
ℚ | Yes | elq 12900 |
| r | reversed (suffix) | |
| No | pm4.71r 558, caovdir 7601 |
| r | right | |
| No | orcd 874, simprl 771 |
| rab | restricted class abstraction |
df-rab 3390 | {𝑥 ∈ 𝐴 ∣ 𝜑} | Yes |
rabswap 3398, df-oprab 7371 |
| ral | restricted universal quantification |
df-ral 3052 | ∀𝑥 ∈ 𝐴𝜑 | Yes |
ralnex 3063, ralrnmpo 7506 |
| rcl | reverse closure | |
| No | ndmfvrcl 6873, nnarcl 8552 |
| re | real numbers | df-r 11048 |
ℝ | Yes | recn 11128, 0re 11146 |
| rel | relation | df-rel 5638 | Rel 𝐴 |
Yes | brrelex1 5684, relmpoopab 8044 |
| res | restriction | df-res 5643 |
(𝐴 ↾ 𝐵) | Yes |
opelres 5950, f1ores 6794 |
| reu | restricted existential uniqueness |
df-reu 3343 | ∃!𝑥 ∈ 𝐴𝜑 | Yes |
nfreud 3386, reurex 3346 |
| rex | restricted existential quantification |
df-rex 3062 | ∃𝑥 ∈ 𝐴𝜑 | Yes |
rexnal 3089, rexrnmpo 7507 |
| rmo | restricted "at most one" |
df-rmo 3342 | ∃*𝑥 ∈ 𝐴𝜑 | Yes |
nfrmod 3385, nrexrmo 3361 |
| rn | range | df-rn 5642 | ran 𝐴 |
Yes | elrng 5846, rncnvcnv 5889 |
| ring | (unital) ring | df-ring 20216 |
Ring | Yes |
ringidval 20164, isring 20218, ringgrp 20219 |
| rng | non-unital ring | df-rng 20134 |
Rng | Yes |
isrng 20135, rngabl 20136, rnglz 20146 |
| rot | rotation | |
| No | 3anrot 1100, 3orrot 1092 |
| s | eliminates need for syllogism (suffix) |
| | No | ancoms 458 |
| sb | (proper) substitution (of a set) |
df-sb 2069 | [𝑦 / 𝑥]𝜑 | Yes |
spsbe 2088, sbimi 2080 |
| sbc | (proper) substitution of a class |
df-sbc 3729 | [𝐴 / 𝑥]𝜑 | Yes |
sbc2or 3737, sbcth 3743 |
| sca | scalar | df-sca 17236 |
(Scalar‘𝐻) | Yes |
resssca 17306, mgpsca 20127 |
| simp | simple, simplification | |
| No | simpl 482, simp3r3 1285 |
| sn | singleton | df-sn 4568 |
{𝐴} | Yes | eldifsn 4731 |
| sp | specialization | |
| No | spsbe 2088, spei 2398 |
| ss | subset | df-ss 3906 |
𝐴 ⊆ 𝐵 | Yes | difss 4076 |
| struct | structure | df-struct 17117 |
Struct | Yes | brstruct 17118, structfn 17126 |
| sub | subtract | df-sub 11379 |
(𝐴 − 𝐵) | Yes |
subval 11384, subaddi 11481 |
| sup | supremum | df-sup 9355 |
sup(𝐴, 𝐵, < ) | Yes |
fisupcl 9383, supmo 9365 |
| supp | support (of a function) | df-supp 8111 |
(𝐹 supp 𝑍) | Yes |
ressuppfi 9308, mptsuppd 8137 |
| swap | swap (two parts within a theorem) |
| | No | rabswap 3398, 2reuswap 3692 |
| syl | syllogism | syl 17 |
| No | 3syl 18 |
| sym | symmetric | |
| No | df-symdif 4193, cnvsym 6077 |
| symg | symmetric group | df-symg 19345 |
(SymGrp‘𝐴) | Yes |
symghash 19353, pgrpsubgsymg 19384 |
| t |
times (see "mul"), for all-constant theorems |
df-mul 11050 |
(3 · 2) = 6 | Yes |
3t2e6 12342 |
| th, t |
theorem |
|
|
No |
nfth 1803, sbcth 3743, weth 10417, ancomst 464 |
| tp | triple | df-tp 4572 |
{𝐴, 𝐵, 𝐶} | Yes |
eltpi 4632, tpeq1 4686 |
| tr | transitive | |
| No | bitrd 279, biantr 806 |
| tru, t |
true, truth |
df-tru 1545 |
⊤ |
Yes |
bitru 1551, truanfal 1576, biimt 360 |
| un | union | df-un 3894 |
(𝐴 ∪ 𝐵) | Yes |
uneqri 4096, uncom 4098 |
| unit | unit (in a ring) |
df-unit 20338 | (Unit‘𝑅) | Yes |
isunit 20353, nzrunit 20501 |
| v |
setvar (especially for specializations of
theorems when a class is replaced by a setvar variable) |
|
x |
Yes |
cv 1541, vex 3433, velpw 4546, vtoclf 3509 |
| v |
disjoint variable condition used in place of nonfreeness
hypothesis (suffix) |
|
|
No |
spimv 2394 |
| vtx |
vertex |
df-vtx 29067 |
(Vtx‘𝐺) |
Yes |
vtxval0 29108, opvtxov 29074 |
| vv |
two disjoint variable conditions used in place of nonfreeness
hypotheses (suffix) |
|
|
No |
19.23vv 1945 |
| w | weak (version of a theorem) (suffix) | |
| No | ax11w 2136, spnfw 1981 |
| wrd | word |
df-word 14476 | Word 𝑆 | Yes |
iswrdb 14482, wrdfn 14490, ffz0iswrd 14503 |
| xp | cross product (Cartesian product) |
df-xp 5637 | (𝐴 × 𝐵) | Yes |
elxp 5654, opelxpi 5668, xpundi 5700 |
| xr | eXtended reals | df-xr 11183 |
ℝ* | Yes | ressxr 11189, rexr 11191, 0xr 11192 |
| z | integers (from German "Zahlen") |
df-z 12525 | ℤ | Yes |
elz 12526, zcn 12529 |
| zn | ring of integers mod 𝑁 | df-zn 21486 |
(ℤ/nℤ‘𝑁) | Yes |
znval 21515, zncrng 21524, znhash 21538 |
| zring | ring of integers | df-zring 21427 |
ℤring | Yes | zringbas 21433, zringcrng 21428
|
| 0, z |
slashed zero (empty set) | df-nul 4274 |
∅ | Yes |
n0i 4280, vn0 4285; snnz 4720, prnz 4721 |
(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.) |
