| Description:
The following gives conventions used in the Metamath Proof Explorer
(MPE, set.mm) regarding labels.
For other conventions, see conventions 30485 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 3054"rgen.1 $e |- ( x e. A -> ph ) $."
or letters corresponding to the (main) class variable used in the
hypothesis, e.g., for mdet0 22581: "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 2664 and stirling 46535.
- 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 3233.
- 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... 15837. 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 3893, 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 3907. 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 4077. There are many other syntax label fragments, e.g.,
singleton construct {𝐴} has syntax label fragment "sn" (because it
is defined in df-sn 4569), and the pair construct {𝐴, 𝐵} has
fragment "pr" ( from df-pr 4571). 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 4730. 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 11035) and "re" represents real numbers ℝ (Definition df-r 11039).
The empty set ∅ often uses fragment 0, even though it is defined
in df-nul 4275. The syntax construct (𝐴 + 𝐵) usually uses the
fragment "add" (which is consistent with df-add 11040), 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 12302.
- 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 16108 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 16026) we have value cosval 16081 and
closure coscl 16085.
- 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 30488 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 2577 derived from eu6 2575. 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 5396.
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 2414 (cbval 2403 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 3506.
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 3131 |
| abl | Abelian group | df-abl 19749 |
Abel | Yes | ablgrp 19751, zringabl 21441 |
| abs | absorption | | | No |
ressabs 17209 |
| abs | absolute value (of a complex number) |
df-abs 15189 | (abs‘𝐴) | Yes |
absval 15191, absneg 15230, abs1 15250 |
| ad | adding | |
| No | adantr 480, ad2antlr 728 |
| add | add (see "p") | df-add 11040 |
(𝐴 + 𝐵) | Yes |
addcl 11111, addcom 11323, addass 11116 |
| 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 11117 |
| asym | asymmetric, antisymmetric | |
| No | intasym 6072, asymref 6073, posasymb 18276 |
| ax | axiom | |
| No | ax6dgen 2134, ax1cn 11063 |
| bas, base |
base (set of an extensible structure) | df-base 17171 |
(Base‘𝑆) | Yes |
baseval 17172, ressbas 17197, cnfldbas 21348 |
| b, bi | biconditional ("iff", "if and only if")
| df-bi 207 | (𝜑 ↔ 𝜓) | Yes |
impbid 212, sspwb 5396 |
| br | binary relation | df-br 5087 |
𝐴𝑅𝐵 | Yes | brab1 5134, brun 5137 |
| c | commutes, commuted (suffix) | | |
No | biimpac 478 |
| c | contraction (suffix) | | |
No | sylc 65, syl2anc 585 |
| cbv | change bound variable | | |
No | cbvalivw 2009, cbvrex 3326 |
| cdm | codomain | |
| No | ffvelcdm 7027, focdmex 7902 |
| cl | closure | | | No |
ifclda 4503, ovrcl 7401, zaddcl 12558 |
| cn | complex numbers | df-c 11035 |
ℂ | Yes | nnsscn 12170, nncn 12173 |
| cnfld | field of complex numbers | df-cnfld 21345 |
ℂfld | Yes | cnfldbas 21348, cnfldinv 21392 |
| cntz | centralizer | df-cntz 19283 |
(Cntz‘𝑀) | Yes |
cntzfval 19286, dprdfcntz 19983 |
| cnv | converse | df-cnv 5632 |
◡𝐴 | Yes | opelcnvg 5829, f1ocnv 6786 |
| co | composition | df-co 5633 |
(𝐴 ∘ 𝐵) | Yes | cnvco 5834, fmptco 7076 |
| com | commutative | |
| No | orcom 871, bicomi 224, eqcomi 2746 |
| con | contradiction, contraposition | |
| No | condan 818, con2d 134 |
| csb | class substitution | df-csb 3839 |
⦋𝐴 / 𝑥⦌𝐵 | Yes |
csbid 3851, csbie2g 3878 |
| cyg | cyclic group | df-cyg 19844 |
CycGrp | Yes |
iscyg 19845, zringcyg 21459 |
| d | deduction form (suffix) | |
| No | idd 24, impbid 212 |
| df | (alternate) definition (prefix) | |
| No | dfrel2 6147, dffn2 6664 |
| di, distr | distributive | |
| No |
andi 1010, imdi 389, ordi 1008, difindi 4233, ndmovdistr 7549 |
| dif | class difference | df-dif 3893 |
(𝐴 ∖ 𝐵) | Yes |
difss 4077, difindi 4233 |
| div | division | df-div 11799 |
(𝐴 / 𝐵) | Yes |
divcl 11806, divval 11802, divmul 11803 |
| dm | domain | df-dm 5634 |
dom 𝐴 | Yes | dmmpt 6198, iswrddm0 14491 |
| e, eq, equ | equals (equ for setvars, eq for
classes) | df-cleq 2729 |
𝐴 = 𝐵 | Yes |
2p2e4 12302, uneqri 4097, equtr 2023 |
| edg | edge | df-edg 29131 |
(Edg‘𝐺) | Yes |
edgopval 29134, usgredgppr 29279 |
| el | element of | |
𝐴 ∈ 𝐵 | Yes |
eldif 3900, eldifsn 4730, elssuni 4882 |
| en | equinumerous | df-en |
𝐴 ≈ 𝐵 | Yes | domen 8901, enfi 9114 |
| eu | "there exists exactly one" | eu6 2575 |
∃!𝑥𝜑 | Yes | euex 2578, euabsn 4671 |
| ex | exists (i.e. is a set) | |
∈ V | No | brrelex1 5677, 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 2464, sbf 2278 |
| f | function | df-f 6496 |
𝐹:𝐴⟶𝐵 | Yes | fssxp 6689, opelf 6695 |
| fal | false | df-fal 1555 |
⊥ | Yes | bifal 1558, falantru 1577 |
| fi | finite intersection | df-fi 9317 |
(fi‘𝐵) | Yes | fival 9318, inelfi 9324 |
| fi, fin | finite | df-fin 8890 |
Fin | Yes |
isfi 8915, snfi 8983, onfin 9142 |
| fld | field (Note: there is an alternative
definition Fld of a field, see df-fld 38327) | df-field 20700 |
Field | Yes | isfld 20708, fldidom 20739 |
| fn | function with domain | df-fn 6495 |
𝐴 Fn 𝐵 | Yes | ffn 6662, fndm 6595 |
| frgp | free group | df-frgp 19676 |
(freeGrp‘𝐼) | Yes |
frgpval 19724, frgpadd 19729 |
| fsupp | finitely supported function |
df-fsupp 9268 | 𝑅 finSupp 𝑍 | Yes |
isfsupp 9271, fdmfisuppfi 9280, fsuppco 9308 |
| fun | function | df-fun 6494 |
Fun 𝐹 | Yes | funrel 6509, ffun 6665 |
| fv | function value | df-fv 6500 |
(𝐹‘𝐴) | Yes | fvres 6853, swrdfv 14602 |
| fz | finite set of sequential integers |
df-fz 13453 |
(𝑀...𝑁) | Yes | fzval 13454, eluzfz 13464 |
| fz0 | finite set of sequential nonnegative integers |
|
(0...𝑁) | Yes | nn0fz0 13570, fz0tp 13573 |
| fzo | half-open integer range | df-fzo 13600 |
(𝑀..^𝑁) | Yes |
elfzo 13606, elfzofz 13621 |
| g | more general (suffix); eliminates "is a set"
hypotheses | |
| No | uniexg 7687 |
| gr | graph | |
| No | uhgrf 29145, isumgr 29178, usgrres1 29398 |
| grp | group | df-grp 18903 |
Grp | Yes | isgrp 18906, tgpgrp 24053 |
| gsum | group sum | df-gsum 17396 |
(𝐺 Σg 𝐹) | Yes |
gsumval 18636, gsumwrev 19332 |
| hash | size (of a set) | df-hash 14284 |
(♯‘𝐴) | Yes |
hashgval 14286, hashfz1 14299, hashcl 14309 |
| hb | hypothesis builder (prefix) | |
| No | hbxfrbi 1827, hbald 2174, hbequid 39369 |
| hm | (monoid, group, ring, ...) homomorphism |
| | No |
ismhm 18744, isghm 19181, isrhm 20449 |
| i | inference (suffix) | |
| No | eleq1i 2828, tcsni 9653 |
| i | implication (suffix) | |
| No | brwdomi 9476, infeq5i 9548 |
| id | identity | |
| No | biid 261 |
| iedg | indexed edge | df-iedg 29082 |
(iEdg‘𝐺) | Yes |
iedgval0 29123, edgiedgb 29137 |
| idm | idempotent | |
| No | anidm 564, tpidm13 4701 |
| im, imp | implication (label often omitted) |
df-im 15054 | (𝐴 → 𝐵) | Yes |
iman 401, imnan 399, impbidd 210 |
| im | (group, ring, ...) isomorphism | |
| No | isgim 19228, rimrcl 20452 |
| ima | image | df-ima 5637 |
(𝐴 “ 𝐵) | Yes | resima 5974, imaundi 6107 |
| imp | import | |
| No | biimpa 476, impcom 407 |
| in | intersection | df-in 3897 |
(𝐴 ∩ 𝐵) | Yes | elin 3906, incom 4150 |
| inf | infimum | df-inf 9349 |
inf(ℝ+, ℝ*, < ) | Yes |
fiinfcl 9409, infiso 9416 |
| is... | is (something a) ...? | |
| No | isring 20209 |
| j | joining, disjoining | |
| No | jc 161, jaoi 858 |
| l | left | |
| No | olcd 875, simpl 482 |
| map | mapping operation or set exponentiation |
df-map 8768 | (𝐴 ↑m 𝐵) | Yes |
mapvalg 8776, elmapex 8788 |
| mat | matrix | df-mat 22383 |
(𝑁 Mat 𝑅) | Yes |
matval 22386, matring 22418 |
| mdet | determinant (of a square matrix) |
df-mdet 22560 | (𝑁 maDet 𝑅) | Yes |
mdetleib 22562, mdetrlin 22577 |
| mgm | magma | df-mgm 18599 |
Magma | Yes |
mgmidmo 18619, mgmlrid 18626, ismgm 18600 |
| mgp | multiplicative group | df-mgp 20113 |
(mulGrp‘𝑅) | Yes |
mgpress 20122, ringmgp 20211 |
| mnd | monoid | df-mnd 18694 |
Mnd | Yes | mndass 18702, mndodcong 19508 |
| mo | "there exists at most one" | df-mo 2540 |
∃*𝑥𝜑 | Yes | eumo 2579, moim 2545 |
| mp | modus ponens | ax-mp 5 |
| No | mpd 15, mpi 20 |
| mpo | maps-to notation for an operation |
df-mpo 7365 | (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | Yes |
mpompt 7474, resmpo 7480 |
| mpt | modus ponendo tollens | |
| No | mptnan 1770, mptxor 1771 |
| mpt | maps-to notation for a function |
df-mpt 5168 | (𝑥 ∈ 𝐴 ↦ 𝐵) | Yes |
fconstmpt 5686, resmpt 5996 |
| mul | multiplication (see "t") | df-mul 11041 |
(𝐴 · 𝐵) | Yes |
mulcl 11113, divmul 11803, mulcom 11115, mulass 11117 |
| n, not | not | |
¬ 𝜑 | Yes |
nan 830, notnotr 130 |
| ne | not equal | df-ne | 𝐴 ≠ 𝐵 |
Yes | exmidne 2943, neeqtrd 3002 |
| nel | not element of | df-nel | 𝐴 ∉ 𝐵
|
Yes | neli 3039, nnel 3047 |
| ne0 | not equal to zero (see n0) | |
≠ 0 | No |
negne0d 11494, ine0 11576, gt0ne0 11606 |
| nf | "not free in" (prefix) | df-nf 1786 |
Ⅎ𝑥𝜑 | Yes | nfnd 1860 |
| ngp | normed group | df-ngp 24558 |
NrmGrp | Yes | isngp 24571, ngptps 24577 |
| nm | norm (on a group or ring) | df-nm 24557 |
(norm‘𝑊) | Yes |
nmval 24564, subgnm 24608 |
| nn | positive integers | df-nn 12166 |
ℕ | Yes | nnsscn 12170, nncn 12173 |
| nn0 | nonnegative integers | df-n0 12429 |
ℕ0 | Yes | nnnn0 12435, nn0cn 12438 |
| n0 | not the empty set (see ne0) | |
≠ ∅ | No | n0i 4281, vn0 4286, ssn0 4345 |
| OLD | old, obsolete (to be removed soon) | |
| No | 19.43OLD 1885 |
| on | ordinal number | df-on 6321 |
𝐴 ∈ On | Yes |
elon 6326, 1on 8410 onelon 6342 |
| op | ordered pair | df-op 4575 |
⟨𝐴, 𝐵⟩ | Yes | dfopif 4814, opth 5424 |
| or | or | df-or 849 |
(𝜑 ∨ 𝜓) | Yes |
orcom 871, anor 985 |
| ot | ordered triple | df-ot 4577 |
⟨𝐴, 𝐵, 𝐶⟩ | Yes |
euotd 5461, fnotovb 7412 |
| ov | operation value | df-ov 7363 |
(𝐴𝐹𝐵) | Yes
| fnotovb 7412, fnovrn 7535 |
| p | plus (see "add"), for all-constant
theorems | df-add 11040 |
(3 + 2) = 5 | Yes |
3p2e5 12318 |
| pfx | prefix | df-pfx 14625 |
(𝑊 prefix 𝐿) | Yes |
pfxlen 14637, ccatpfx 14654 |
| pm | Principia Mathematica | |
| No | pm2.27 42 |
| pm | partial mapping (operation) | df-pm 8769 |
(𝐴 ↑pm 𝐵) | Yes | elpmi 8786, pmsspw 8818 |
| pr | pair | df-pr 4571 |
{𝐴, 𝐵} | Yes |
elpr 4593, prcom 4677, prid1g 4705, prnz 4722 |
| prm, prime | prime (number) | df-prm 16632 |
ℙ | Yes | 1nprm 16639, dvdsprime 16647 |
| pss | proper subset | df-pss 3910 |
𝐴 ⊊ 𝐵 | Yes | pssss 4039, sspsstri 4046 |
| q | rational numbers ("quotients") | df-q 12890 |
ℚ | Yes | elq 12891 |
| r | reversed (suffix) | |
| No | pm4.71r 558, caovdir 7594 |
| r | right | |
| No | orcd 874, simprl 771 |
| rab | restricted class abstraction |
df-rab 3391 | {𝑥 ∈ 𝐴 ∣ 𝜑} | Yes |
rabswap 3399, df-oprab 7364 |
| ral | restricted universal quantification |
df-ral 3053 | ∀𝑥 ∈ 𝐴𝜑 | Yes |
ralnex 3064, ralrnmpo 7499 |
| rcl | reverse closure | |
| No | ndmfvrcl 6867, nnarcl 8545 |
| re | real numbers | df-r 11039 |
ℝ | Yes | recn 11119, 0re 11137 |
| rel | relation | df-rel 5631 | Rel 𝐴 |
Yes | brrelex1 5677, relmpoopab 8037 |
| res | restriction | df-res 5636 |
(𝐴 ↾ 𝐵) | Yes |
opelres 5944, f1ores 6788 |
| reu | restricted existential uniqueness |
df-reu 3344 | ∃!𝑥 ∈ 𝐴𝜑 | Yes |
nfreud 3387, reurex 3347 |
| rex | restricted existential quantification |
df-rex 3063 | ∃𝑥 ∈ 𝐴𝜑 | Yes |
rexnal 3090, rexrnmpo 7500 |
| rmo | restricted "at most one" |
df-rmo 3343 | ∃*𝑥 ∈ 𝐴𝜑 | Yes |
nfrmod 3386, nrexrmo 3362 |
| rn | range | df-rn 5635 | ran 𝐴 |
Yes | elrng 5840, rncnvcnv 5883 |
| ring | (unital) ring | df-ring 20207 |
Ring | Yes |
ringidval 20155, isring 20209, ringgrp 20210 |
| rng | non-unital ring | df-rng 20125 |
Rng | Yes |
isrng 20126, rngabl 20127, rnglz 20137 |
| 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 3730 | [𝐴 / 𝑥]𝜑 | Yes |
sbc2or 3738, sbcth 3744 |
| sca | scalar | df-sca 17227 |
(Scalar‘𝐻) | Yes |
resssca 17297, mgpsca 20118 |
| simp | simple, simplification | |
| No | simpl 482, simp3r3 1285 |
| sn | singleton | df-sn 4569 |
{𝐴} | Yes | eldifsn 4730 |
| sp | specialization | |
| No | spsbe 2088, spei 2399 |
| ss | subset | df-ss 3907 |
𝐴 ⊆ 𝐵 | Yes | difss 4077 |
| struct | structure | df-struct 17108 |
Struct | Yes | brstruct 17109, structfn 17117 |
| sub | subtract | df-sub 11370 |
(𝐴 − 𝐵) | Yes |
subval 11375, subaddi 11472 |
| sup | supremum | df-sup 9348 |
sup(𝐴, 𝐵, < ) | Yes |
fisupcl 9376, supmo 9358 |
| supp | support (of a function) | df-supp 8104 |
(𝐹 supp 𝑍) | Yes |
ressuppfi 9301, mptsuppd 8130 |
| swap | swap (two parts within a theorem) |
| | No | rabswap 3399, 2reuswap 3693 |
| syl | syllogism | syl 17 |
| No | 3syl 18 |
| sym | symmetric | |
| No | df-symdif 4194, cnvsym 6071 |
| symg | symmetric group | df-symg 19336 |
(SymGrp‘𝐴) | Yes |
symghash 19344, pgrpsubgsymg 19375 |
| t |
times (see "mul"), for all-constant theorems |
df-mul 11041 |
(3 · 2) = 6 | Yes |
3t2e6 12333 |
| th, t |
theorem |
|
|
No |
nfth 1803, sbcth 3744, weth 10408, ancomst 464 |
| tp | triple | df-tp 4573 |
{𝐴, 𝐵, 𝐶} | Yes |
eltpi 4633, tpeq1 4687 |
| 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 3895 |
(𝐴 ∪ 𝐵) | Yes |
uneqri 4097, uncom 4099 |
| unit | unit (in a ring) |
df-unit 20329 | (Unit‘𝑅) | Yes |
isunit 20344, nzrunit 20492 |
| v |
setvar (especially for specializations of
theorems when a class is replaced by a setvar variable) |
|
x |
Yes |
cv 1541, vex 3434, velpw 4547, vtoclf 3510 |
| v |
disjoint variable condition used in place of nonfreeness
hypothesis (suffix) |
|
|
No |
spimv 2395 |
| vtx |
vertex |
df-vtx 29081 |
(Vtx‘𝐺) |
Yes |
vtxval0 29122, opvtxov 29088 |
| 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 14467 | Word 𝑆 | Yes |
iswrdb 14473, wrdfn 14481, ffz0iswrd 14494 |
| xp | cross product (Cartesian product) |
df-xp 5630 | (𝐴 × 𝐵) | Yes |
elxp 5647, opelxpi 5661, xpundi 5693 |
| xr | eXtended reals | df-xr 11174 |
ℝ* | Yes | ressxr 11180, rexr 11182, 0xr 11183 |
| z | integers (from German "Zahlen") |
df-z 12516 | ℤ | Yes |
elz 12517, zcn 12520 |
| zn | ring of integers mod 𝑁 | df-zn 21496 |
(ℤ/nℤ‘𝑁) | Yes |
znval 21525, zncrng 21534, znhash 21548 |
| zring | ring of integers | df-zring 21437 |
ℤring | Yes | zringbas 21443, zringcrng 21438
|
| 0, z |
slashed zero (empty set) | df-nul 4275 |
∅ | Yes |
n0i 4281, vn0 4286; snnz 4721, prnz 4722 |
(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.) |
