Theorem conventions-labels 30198

Description:

The following gives conventions used in the Metamath Proof Explorer (MPE, set.mm) regarding labels. For other conventions, see conventions 30197 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 3058"rgen.1 $e |- ( x e. A -> ph ) $." or letters corresponding to the (main) class variable used in the hypothesis, e.g., for mdet0 22495: "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 2653 and stirling 45400.
• 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 1834, and the restricted quantifier version of Theorem 21 from Section 19 of [Margaris] p. 90 is labeled r19.21 3246.
• 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... 15851. 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 3947, 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 3961. 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 4127. There are many other syntax label fragments, e.g., singleton construct {𝐴} has syntax label fragment "sn" (because it is defined in df-sn 4625), and the pair construct {𝐴, 𝐵} has fragment "pr" ( from df-pr 4627). 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 4786. 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 11136) and "re" represents real numbers ℝ (Definition df-r 11140). The empty set ∅ often uses fragment 0, even though it is defined in df-nul 4319. The syntax construct (𝐴 + 𝐵) usually uses the fragment "add" (which is consistent with df-add 11141), 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 12369.
• 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 16038) 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 30200 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 1935 versus 19.21 2193. This often permits to prove the result using fewer axioms, and/or to eliminate a nonfreeness hypothesis (such as Ⅎ𝑥𝜑 in 19.21 2193). If no constraint is put on axiom use, then the v-version can be proved from the original theorem using nfv 1910. If two (resp. three) such disjoint variable conditions are added, then the suffix "vv" (resp. "vvv") is used, e.g., exlimivv 1928. 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 2565 derived from eu6 2563. 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 5445. 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 2403 (cbval 2392 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 3542. 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 583), 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 3142
abl	Abelian group	df-abl 19729	Abel	Yes	ablgrp 19731, zringabl 21364
abs	absorption			No	ressabs 17221
abs	absolute value (of a complex number)	df-abs 15207	(abs‘𝐴)	Yes	absval 15209, absneg 15248, abs1 15268
ad	adding			No	adantr 480, ad2antlr 726
add	add (see "p")	df-add 11141	(𝐴 + 𝐵)	Yes	addcl 11212, addcom 11422, addass 11217
al	"for all"		∀𝑥𝜑	No	alim 1805, alex 1821
ALT	alternative/less preferred (suffix)			No	idALT 23
an	and	df-an 396	(𝜑 ∧ 𝜓)	Yes	anor 981, iman 401, imnan 399
ant	antecedent			No	adantr 480
ass	associative			No	biass 384, orass 920, mulass 11218
asym	asymmetric, antisymmetric			No	intasym 6115, asymref 6116, posasymb 18302
ax	axiom			No	ax6dgen 2117, ax1cn 11164
bas, base	base (set of an extensible structure)	df-base 17172	(Base‘𝑆)	Yes	baseval 17173, ressbas 17206, cnfldbas 21270
b, bi	biconditional ("iff", "if and only if")	df-bi 206	(𝜑 ↔ 𝜓)	Yes	impbid 211, sspwb 5445
br	binary relation	df-br 5143	𝐴𝑅𝐵	Yes	brab1 5190, brun 5193
c	commutes, commuted (suffix)			No	biimpac 478
c	contraction (suffix)			No	sylc 65, syl2anc 583
cbv	change bound variable			No	cbvalivw 2003, cbvrex 3354
cdm	codomain			No	ffvelcdm 7085, focdmex 7953
cl	closure			No	ifclda 4559, ovrcl 7455, zaddcl 12624
cn	complex numbers	df-c 11136	ℂ	Yes	nnsscn 12239, nncn 12242
cnfld	field of complex numbers	df-cnfld 21267	ℂfld	Yes	cnfldbas 21270, cnfldinv 21317
cntz	centralizer	df-cntz 19259	(Cntz‘𝑀)	Yes	cntzfval 19262, dprdfcntz 19963
cnv	converse	df-cnv 5680	◡𝐴	Yes	opelcnvg 5877, f1ocnv 6845
co	composition	df-co 5681	(𝐴 ∘ 𝐵)	Yes	cnvco 5882, fmptco 7132
com	commutative			No	orcom 869, bicomi 223, eqcomi 2736
con	contradiction, contraposition			No	condan 817, con2d 134
csb	class substitution	df-csb 3890	⦋𝐴 / 𝑥⦌𝐵	Yes	csbid 3902, csbie2g 3932
cyg	cyclic group	df-cyg 19824	CycGrp	Yes	iscyg 19825, zringcyg 21382
d	deduction form (suffix)			No	idd 24, impbid 211
df	(alternate) definition (prefix)			No	dfrel2 6187, dffn2 6718
di, distr	distributive			No	andi 1006, imdi 389, ordi 1004, difindi 4277, ndmovdistr 7604
dif	class difference	df-dif 3947	(𝐴 ∖ 𝐵)	Yes	difss 4127, difindi 4277
div	division	df-div 11894	(𝐴 / 𝐵)	Yes	divcl 11900, divval 11896, divmul 11897
dm	domain	df-dm 5682	dom 𝐴	Yes	dmmpt 6238, iswrddm0 14512
e, eq, equ	equals (equ for setvars, eq for classes)	df-cleq 2719	𝐴 = 𝐵	Yes	2p2e4 12369, uneqri 4147, equtr 2017
edg	edge	df-edg 28848	(Edg‘𝐺)	Yes	edgopval 28851, usgredgppr 28996
el	element of		𝐴 ∈ 𝐵	Yes	eldif 3954, eldifsn 4786, elssuni 4935
en	equinumerous	df-en	𝐴 ≈ 𝐵	Yes	domen 8973, enfi 9206
eu	"there exists exactly one"	eu6 2563	∃!𝑥𝜑	Yes	euex 2566, euabsn 4726
ex	exists (i.e. is a set)		∈ V	No	brrelex1 5725, 0ex 5301
ex, e	"there exists (at least one)"	df-ex 1775	∃𝑥𝜑	Yes	exim 1829, alex 1821
exp	export			No	expt 177, expcom 413
f	"not free in" (suffix)			No	equs45f 2453, sbf 2255
f	function	df-f 6546	𝐹:𝐴⟶𝐵	Yes	fssxp 6745, opelf 6752
fal	false	df-fal 1547	⊥	Yes	bifal 1550, falantru 1569
fi	finite intersection	df-fi 9426	(fi‘𝐵)	Yes	fival 9427, inelfi 9433
fi, fin	finite	df-fin 8959	Fin	Yes	isfi 8988, snfi 9060, onfin 9246
fld	field (Note: there is an alternative definition Fld of a field, see df-fld 37400)	df-field 20616	Field	Yes	isfld 20624, fldidom 21245
fn	function with domain	df-fn 6545	𝐴 Fn 𝐵	Yes	ffn 6716, fndm 6651
frgp	free group	df-frgp 19656	(freeGrp‘𝐼)	Yes	frgpval 19704, frgpadd 19709
fsupp	finitely supported function	df-fsupp 9378	𝑅 finSupp 𝑍	Yes	isfsupp 9381, fdmfisuppfi 9389, fsuppco 9417
fun	function	df-fun 6544	Fun 𝐹	Yes	funrel 6564, ffun 6719
fv	function value	df-fv 6550	(𝐹‘𝐴)	Yes	fvres 6910, swrdfv 14622
fz	finite set of sequential integers	df-fz 13509	(𝑀...𝑁)	Yes	fzval 13510, eluzfz 13520
fz0	finite set of sequential nonnegative integers		(0...𝑁)	Yes	nn0fz0 13623, fz0tp 13626
fzo	half-open integer range	df-fzo 13652	(𝑀..^𝑁)	Yes	elfzo 13658, elfzofz 13672
g	more general (suffix); eliminates "is a set" hypotheses			No	uniexg 7739
gr	graph			No	uhgrf 28862, isumgr 28895, usgrres1 29115
grp	group	df-grp 18884	Grp	Yes	isgrp 18887, tgpgrp 23969
gsum	group sum	df-gsum 17415	(𝐺 Σg 𝐹)	Yes	gsumval 18628, gsumwrev 19311
hash	size (of a set)	df-hash 14314	(♯‘𝐴)	Yes	hashgval 14316, hashfz1 14329, hashcl 14339
hb	hypothesis builder (prefix)			No	hbxfrbi 1820, hbald 2161, hbequid 38318
hm	(monoid, group, ring, ...) homomorphism			No	ismhm 18733, isghm 19161, isrhm 20406
i	inference (suffix)			No	eleq1i 2819, tcsni 9758
i	implication (suffix)			No	brwdomi 9583, infeq5i 9651
id	identity			No	biid 261
iedg	indexed edge	df-iedg 28799	(iEdg‘𝐺)	Yes	iedgval0 28840, edgiedgb 28854
idm	idempotent			No	anidm 564, tpidm13 4756
im, imp	implication (label often omitted)	df-im 15072	(𝐴 → 𝐵)	Yes	iman 401, imnan 399, impbidd 209
im	(group, ring, ...) isomorphism			No	isgim 19207, rimrcl 20410
ima	image	df-ima 5685	(𝐴 “ 𝐵)	Yes	resima 6013, imaundi 6148
imp	import			No	biimpa 476, impcom 407
in	intersection	df-in 3951	(𝐴 ∩ 𝐵)	Yes	elin 3960, incom 4197
inf	infimum	df-inf 9458	inf(ℝ+, ℝ*, < )	Yes	fiinfcl 9516, infiso 9523
is...	is (something a) ...?			No	isring 20168
j	joining, disjoining			No	jc 161, jaoi 856
l	left			No	olcd 873, simpl 482
map	mapping operation or set exponentiation	df-map 8838	(𝐴 ↑m 𝐵)	Yes	mapvalg 8846, elmapex 8858
mat	matrix	df-mat 22295	(𝑁 Mat 𝑅)	Yes	matval 22298, matring 22332
mdet	determinant (of a square matrix)	df-mdet 22474	(𝑁 maDet 𝑅)	Yes	mdetleib 22476, mdetrlin 22491
mgm	magma	df-mgm 18591	Magma	Yes	mgmidmo 18611, mgmlrid 18618, ismgm 18592
mgp	multiplicative group	df-mgp 20066	(mulGrp‘𝑅)	Yes	mgpress 20080, ringmgp 20170
mnd	monoid	df-mnd 18686	Mnd	Yes	mndass 18694, mndodcong 19488
mo	"there exists at most one"	df-mo 2529	∃*𝑥𝜑	Yes	eumo 2567, moim 2533
mp	modus ponens	ax-mp 5		No	mpd 15, mpi 20
mpo	maps-to notation for an operation	df-mpo 7419	(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)	Yes	mpompt 7528, resmpo 7534
mpt	modus ponendo tollens			No	mptnan 1763, mptxor 1764
mpt	maps-to notation for a function	df-mpt 5226	(𝑥 ∈ 𝐴 ↦ 𝐵)	Yes	fconstmpt 5734, resmpt 6035
mpt2	maps-to notation for an operation (deprecated). We are in the process of replacing mpt2 with mpo in labels.	df-mpo 7419	(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)	Yes	mpompt 7528, resmpo 7534
mul	multiplication (see "t")	df-mul 11142	(𝐴 · 𝐵)	Yes	mulcl 11214, divmul 11897, mulcom 11216, mulass 11218
n, not	not		¬ 𝜑	Yes	nan 829, notnotr 130
ne	not equal	df-ne	𝐴 ≠ 𝐵	Yes	exmidne 2945, neeqtrd 3005
nel	not element of	df-nel	𝐴 ∉ 𝐵	Yes	neli 3043, nnel 3051
ne0	not equal to zero (see n0)		≠ 0	No	negne0d 11591, ine0 11671, gt0ne0 11701
nf	"not free in" (prefix)	df-nf 1779	Ⅎ𝑥𝜑	Yes	nfnd 1854
ngp	normed group	df-ngp 24479	NrmGrp	Yes	isngp 24492, ngptps 24498
nm	norm (on a group or ring)	df-nm 24478	(norm‘𝑊)	Yes	nmval 24485, subgnm 24529
nn	positive integers	df-nn 12235	ℕ	Yes	nnsscn 12239, nncn 12242
nn0	nonnegative integers	df-n0 12495	ℕ0	Yes	nnnn0 12501, nn0cn 12504
n0	not the empty set (see ne0)		≠ ∅	No	n0i 4329, vn0 4334, ssn0 4396
OLD	old, obsolete (to be removed soon)			No	19.43OLD 1879
on	ordinal number	df-on 6367	𝐴 ∈ On	Yes	elon 6372, 1on 8492 onelon 6388
op	ordered pair	df-op 4631	⟨𝐴, 𝐵⟩	Yes	dfopif 4866, opth 5472
or	or	df-or 847	(𝜑 ∨ 𝜓)	Yes	orcom 869, anor 981
ot	ordered triple	df-ot 4633	⟨𝐴, 𝐵, 𝐶⟩	Yes	euotd 5509, fnotovb 7464
ov	operation value	df-ov 7417	(𝐴𝐹𝐵)	Yes	fnotovb 7464, fnovrn 7590
p	plus (see "add"), for all-constant theorems	df-add 11141	(3 + 2) = 5	Yes	3p2e5 12385
pfx	prefix	df-pfx 14645	(𝑊 prefix 𝐿)	Yes	pfxlen 14657, ccatpfx 14675
pm	Principia Mathematica			No	pm2.27 42
pm	partial mapping (operation)	df-pm 8839	(𝐴 ↑pm 𝐵)	Yes	elpmi 8856, pmsspw 8887
pr	pair	df-pr 4627	{𝐴, 𝐵}	Yes	elpr 4647, prcom 4732, prid1g 4760, prnz 4777
prm, prime	prime (number)	df-prm 16634	ℙ	Yes	1nprm 16641, dvdsprime 16649
pss	proper subset	df-pss 3963	𝐴 ⊊ 𝐵	Yes	pssss 4091, sspsstri 4098
q	rational numbers ("quotients")	df-q 12955	ℚ	Yes	elq 12956
r	reversed (suffix)			No	pm4.71r 558, caovdir 7649
r	right			No	orcd 872, simprl 770
rab	restricted class abstraction	df-rab 3428	{𝑥 ∈ 𝐴 ∣ 𝜑}	Yes	rabswap 3436, df-oprab 7418
ral	restricted universal quantification	df-ral 3057	∀𝑥 ∈ 𝐴𝜑	Yes	ralnex 3067, ralrnmpo 7554
rcl	reverse closure			No	ndmfvrcl 6927, nnarcl 8630
re	real numbers	df-r 11140	ℝ	Yes	recn 11220, 0re 11238
rel	relation	df-rel 5679	Rel 𝐴	Yes	brrelex1 5725, relmpoopab 8093
res	restriction	df-res 5684	(𝐴 ↾ 𝐵)	Yes	opelres 5985, f1ores 6847
reu	restricted existential uniqueness	df-reu 3372	∃!𝑥 ∈ 𝐴𝜑	Yes	nfreud 3424, reurex 3375
rex	restricted existential quantification	df-rex 3066	∃𝑥 ∈ 𝐴𝜑	Yes	rexnal 3095, rexrnmpo 7555
rmo	restricted "at most one"	df-rmo 3371	∃*𝑥 ∈ 𝐴𝜑	Yes	nfrmod 3423, nrexrmo 3392
rn	range	df-rn 5683	ran 𝐴	Yes	elrng 5888, rncnvcnv 5930
ring	(unital) ring	df-ring 20166	Ring	Yes	ringidval 20114, isring 20168, ringgrp 20169
rng	non-unital ring	df-rng 20084	Rng	Yes	isrng 20085, rngabl 20086, rnglz 20096
rot	rotation			No	3anrot 1098, 3orrot 1090
s	eliminates need for syllogism (suffix)			No	ancoms 458
sb	(proper) substitution (of a set)	df-sb 2061	[𝑦 / 𝑥]𝜑	Yes	spsbe 2078, sbimi 2070
sbc	(proper) substitution of a class	df-sbc 3775	[𝐴 / 𝑥]𝜑	Yes	sbc2or 3783, sbcth 3789
sca	scalar	df-sca 17240	(Scalar‘𝐻)	Yes	resssca 17315, mgpsca 20073
simp	simple, simplification			No	simpl 482, simp3r3 1281
sn	singleton	df-sn 4625	{𝐴}	Yes	eldifsn 4786
sp	specialization			No	spsbe 2078, spei 2388
ss	subset	df-ss 3961	𝐴 ⊆ 𝐵	Yes	difss 4127
struct	structure	df-struct 17107	Struct	Yes	brstruct 17108, structfn 17116
sub	subtract	df-sub 11468	(𝐴 − 𝐵)	Yes	subval 11473, subaddi 11569
sup	supremum	df-sup 9457	sup(𝐴, 𝐵, < )	Yes	fisupcl 9484, supmo 9467
supp	support (of a function)	df-supp 8160	(𝐹 supp 𝑍)	Yes	ressuppfi 9410, mptsuppd 8185
swap	swap (two parts within a theorem)			No	rabswap 3436, 2reuswap 3739
syl	syllogism	syl 17		No	3syl 18
sym	symmetric			No	df-symdif 4238, cnvsym 6112
symg	symmetric group	df-symg 19313	(SymGrp‘𝐴)	Yes	symghash 19323, pgrpsubgsymg 19355
t	times (see "mul"), for all-constant theorems	df-mul 11142	(3 · 2) = 6	Yes	3t2e6 12400
th, t	theorem			No	nfth 1796, sbcth 3789, weth 10510, ancomst 464
tp	triple	df-tp 4629	{𝐴, 𝐵, 𝐶}	Yes	eltpi 4687, tpeq1 4742
tr	transitive			No	bitrd 279, biantr 805
tru, t	true, truth	df-tru 1537	⊤	Yes	bitru 1543, truanfal 1568, biimt 360
un	union	df-un 3949	(𝐴 ∪ 𝐵)	Yes	uneqri 4147, uncom 4149
unit	unit (in a ring)	df-unit 20286	(Unit‘𝑅)	Yes	isunit 20301, nzrunit 20450
v	setvar (especially for specializations of theorems when a class is replaced by a setvar variable)		x	Yes	cv 1533, vex 3473, velpw 4603, vtoclf 3547
v	disjoint variable condition used in place of nonfreeness hypothesis (suffix)			No	spimv 2384
vtx	vertex	df-vtx 28798	(Vtx‘𝐺)	Yes	vtxval0 28839, opvtxov 28805
vv	two disjoint variable conditions used in place of nonfreeness hypotheses (suffix)			No	19.23vv 1939
w	weak (version of a theorem) (suffix)			No	ax11w 2119, spnfw 1976
wrd	word	df-word 14489	Word 𝑆	Yes	iswrdb 14494, wrdfn 14502, ffz0iswrd 14515
xp	cross product (Cartesian product)	df-xp 5678	(𝐴 × 𝐵)	Yes	elxp 5695, opelxpi 5709, xpundi 5740
xr	eXtended reals	df-xr 11274	ℝ*	Yes	ressxr 11280, rexr 11282, 0xr 11283
z	integers (from German "Zahlen")	df-z 12581	ℤ	Yes	elz 12582, zcn 12585
zn	ring of integers mod 𝑁	df-zn 21419	(ℤ/nℤ‘𝑁)	Yes	znval 21452, zncrng 21465, znhash 21479
zring	ring of integers	df-zring 21360	ℤring	Yes	zringbas 21366, zringcrng 21361
0, z	slashed zero (empty set)	df-nul 4319	∅	Yes	n0i 4329, vn0 4334; snnz 4776, prnz 4777

(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.)

Hypothesis

Ref	Expression
conventions-labels.1	⊢ 𝜑

Assertion

Ref	Expression
conventions-labels	⊢ 𝜑

Proof of Theorem conventions-labels

Step	Hyp	Ref	Expression
1		conventions-labels.1	1 ⊢ 𝜑

This theorem is referenced by: (None)