P. 174 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17301-17400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	strfv 17301	Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 (such as a member of Poset, df-poset 18407) with a component extractor 𝐸 (such as the base set extractor df-base 17308). By virtue of ndxid 17295, this can be done without having to refer to the hard-coded numeric index of 𝐸. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑆 Struct 𝑋    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆    ⇒   ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))
Theorem	strfv3 17302	Variant on strfv 17301 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.)
⊢ (𝜑 → 𝑈 = 𝑆)    &   ⊢ 𝑆 Struct 𝑋    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ 𝐴 = (𝐸‘𝑈)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	strssd 17303	Deduction version of strss 17304. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    &   ⊢ (𝜑 → Fun 𝑇)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑇)    &   ⊢ (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐸‘𝑇) = (𝐸‘𝑆))
Theorem	strss 17304	Propagate component extraction to a structure 𝑇 from a subset structure 𝑆. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 15-Jan-2014.)
⊢ 𝑇 ∈ V    &   ⊢ Fun 𝑇    &   ⊢ 𝑆 ⊆ 𝑇    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆    ⇒   ⊢ (𝐸‘𝑇) = (𝐸‘𝑆)
Theorem	setsid 17305	Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐸 = Slot (𝐸‘ndx)    ⇒   ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 = (𝐸‘(𝑊 sSet ⟨(𝐸‘ndx), 𝐶⟩)))
Theorem	setsnid 17306	Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 7-Nov-2024.)
⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ 𝐷    ⇒   ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩))
7.1.1.5  Base sets
Syntax	cbs 17307	Extend class notation with the class of all base set extractors.
class Base
Definition	df-base 17308	Define the base set (also called underlying set, ground set, carrier set, or carrier) extractor for extensible structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form baseid 17310 instead. (New usage is discouraged.)
⊢ Base = Slot 1
Theorem	baseval 17309	Value of the base set extractor. (Normally it is preferred to work with (Base‘ndx) rather than the hard-coded 1 in order to make structure theorems portable. This is an example of how to obtain it when needed.) (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
⊢ 𝐾 ∈ V    ⇒   ⊢ (Base‘𝐾) = (𝐾‘1)
Theorem	baseid 17310	Utility theorem: index-independent form of df-base 17308. (Contributed by NM, 20-Oct-2012.)
⊢ Base = Slot (Base‘ndx)
Theorem	basfn 17311	The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
⊢ Base Fn V
Theorem	base0 17312	The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ ∅ = (Base‘∅)
Theorem	elbasfv 17313	Utility theorem: reverse closure for any structure defined as a function. (Contributed by Stefan O'Rear, 24-Aug-2015.)
⊢ 𝑆 = (𝐹‘𝑍)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ (𝑋 ∈ 𝐵 → 𝑍 ∈ V)
Theorem	elbasov 17314	Utility theorem: reverse closure for any structure defined as a two-argument function. (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ Rel dom 𝑂    &   ⊢ 𝑆 = (𝑋𝑂𝑌)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
Theorem	strov2rcl 17315	Partial reverse closure for any structure defined as a two-argument function. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 2-Dec-2019.)
⊢ 𝑆 = (𝐼𝐹𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ Rel dom 𝐹    ⇒   ⊢ (𝑋 ∈ 𝐵 → 𝐼 ∈ V)
Theorem	basendx 17316	Index value of the base set extractor. (Contributed by Mario Carneiro, 2-Aug-2013.) Use of this theorem is discouraged since the particular value 1 for the index is an implementation detail, see section header comment mmtheorems.html#cnx for more information. (New usage is discouraged.)
⊢ (Base‘ndx) = 1
Theorem	basendxnn 17317	The index value of the base set extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 13-Oct-2024.)
⊢ (Base‘ndx) ∈ ℕ
Theorem	basndxelwund 17318	The index of the base set is an element in a weak universe containing the natural numbers. Formerly part of proof for 1strwun 17324. (Contributed by AV, 27-Mar-2020.) (Revised by AV, 17-Oct-2024.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
Theorem	basprssdmsets 17319	The pair of the base index and another index is a subset of the domain of the structure obtained by replacing/adding a slot at the other index in a structure having a base slot. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
⊢ (𝜑 → 𝑆 Struct 𝑋)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    &   ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)    ⇒   ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
Theorem	opelstrbas 17320	The base set of a structure with a base set. (Contributed by AV, 10-Nov-2021.)
⊢ (𝜑 → 𝑆 Struct 𝑋)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝑆)    ⇒   ⊢ (𝜑 → 𝑉 = (Base‘𝑆))
Theorem	1strstr 17321	A constructed one-slot structure. (Contributed by AV, 15-Nov-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ 𝐺 Struct ⟨(Base‘ndx), (Base‘ndx)⟩
Theorem	1strbas 17322	The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Proof shortened by AV, 15-Nov-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
Theorem	1strwunbndx 17323	A constructed one-slot structure in a weak universe containing the index of the base set extractor. (Contributed by AV, 27-Mar-2020.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈)
Theorem	1strwun 17324	A constructed one-slot structure in a weak universe. (Contributed by AV, 27-Mar-2020.) (Proof shortened by AV, 17-Oct-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈)
Theorem	2strstr 17325	A constructed two-slot structure not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Proof shortened by AV, 17-Oct-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩
Theorem	2strbas 17326	The base set of a constructed two-slot structure not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
Theorem	2strop 17327	The other slot of a constructed two-slot structure not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐸 = Slot 𝑁    ⇒   ⊢ ( + ∈ 𝑉 → + = (𝐸‘𝐺))
7.1.1.6  Base set restrictions
Syntax	cress 17328	Extend class notation with the extensible structure builder restriction operator.
class ↾s
Definition	df-ress 17329*	Define a multifunction restriction operator for extensible structures, which can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the Base set while leaving operators alone; individual kinds of structures will need to handle this behavior, by ignoring operators' values outside the range (like Ring), defining a function using the base set and applying that (like TopGrp), or explicitly truncating the slot before use (like MetSp). (Credit for this operator goes to Mario Carneiro.) See ressbas 17334 for the altered base set, and resseqnbas 17340 (subrg0 20747, ressplusg 17382, subrg1 20750, ressmulr 17398) for the (un)altered other operations. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
Theorem	reldmress 17330	The structure restriction is a proper operator, so it can be used with ovprc1 7456. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ Rel dom ↾s
Theorem	ressval 17331	Value of structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))
Theorem	ressid2 17332	General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊)
Theorem	ressval2 17333	Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
Theorem	ressbas 17334	Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅))
Theorem	ressbasssg 17335	The base set of a restriction to 𝐴 is a subset of 𝐴 and the base set 𝐵 of the original structure. (Contributed by SN, 10-Jan-2025.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (Base‘𝑅) ⊆ (𝐴 ∩ 𝐵)
Theorem	ressbas2 17336	Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑅))
Theorem	ressbasss 17337	The base set of a restriction is a subset of the base set of the original structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof shortened by SN, 25-Feb-2025.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (Base‘𝑅) ⊆ 𝐵
Theorem	ressbasssOLD 17338	Obsolete version of ressbas 17334 as of 25-Feb-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (Base‘𝑅) ⊆ 𝐵
Theorem	ressbasss2 17339	The base set of a restriction to 𝐴 is a subset of 𝐴. (Contributed by SN, 10-Jan-2025.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    ⇒   ⊢ (Base‘𝑅) ⊆ 𝐴
Theorem	resseqnbas 17340	The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐶 = (𝐸‘𝑊)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (Base‘ndx)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅))
Theorem	ress0 17341	All restrictions of the empty set are trivial. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ (∅ ↾s 𝐴) = ∅
Theorem	ressid 17342	Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s 𝐵) = 𝑊)
Theorem	ressinbas 17343	Restriction only cares about the part of the second set which intersects the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝐴 ∈ 𝑋 → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
Theorem	ressval3d 17344	Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 3-Jul-2022.) (Proof shortened by AV, 17-Oct-2024.)
⊢ 𝑅 = (𝑆 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐸 = (Base‘ndx)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → Fun 𝑆)    &   ⊢ (𝜑 → 𝐸 ∈ dom 𝑆)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
Theorem	ressress 17345	Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
Theorem	ressabs 17346	Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐵))
Theorem	wunress 17347	Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 28-Oct-2024.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑊 ↾s 𝐴) ∈ 𝑈)
7.1.2  Slot definitions
Syntax	cplusg 17348	Extend class notation with group (addition) operation.
class +g
Syntax	cmulr 17349	Extend class notation with ring multiplication.
class .r
Syntax	cstv 17350	Extend class notation with involution.
class *𝑟
Syntax	csca 17351	Extend class notation with scalar field.
class Scalar
Syntax	cvsca 17352	Extend class notation with scalar product.
class ·𝑠
Syntax	cip 17353	Extend class notation with Hermitian form (inner product).
class ·𝑖
Syntax	cts 17354	Extend class notation with the topology component of a topological space.
class TopSet
Syntax	cple 17355	Extend class notation with "less than or equal to" for posets.
class le
Syntax	coc 17356	Extend class notation with the class of orthocomplementation extractors.
class oc
Syntax	cds 17357	Extend class notation with the metric space distance function.
class dist
Syntax	cunif 17358	Extend class notation with the uniform structure.
class UnifSet
Syntax	chom 17359	Extend class notation with the hom-set structure.
class Hom
Syntax	cco 17360	Extend class notation with the composition operation.
class comp
Definition	df-plusg 17361	Define group operation. In the context of less restrictive structures, this operation is also called magma, semigroup or monoid operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form plusgid 17375 instead. (New usage is discouraged.)
⊢ +g = Slot 2
Definition	df-mulr 17362	Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form mulrid 11234 instead. (New usage is discouraged.)
⊢ .r = Slot 3
Definition	df-starv 17363	Define the involution function of a *-ring. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form starvid 17394 instead. (New usage is discouraged.)
⊢ *𝑟 = Slot 4
Definition	df-sca 17364	Define scalar field component of a vector space 𝑣. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form scaid 17406 instead. (New usage is discouraged.)
⊢ Scalar = Slot 5
Definition	df-vsca 17365	Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form vscaid 17411 instead. (New usage is discouraged.)
⊢ ·𝑠 = Slot 6
Definition	df-ip 17366	Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form ipid 17422 instead. (New usage is discouraged.)
⊢ ·𝑖 = Slot 8
Definition	df-tset 17367	Define the topology component of a topological space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form tsetid 17444 instead. (New usage is discouraged.)
⊢ TopSet = Slot 9
Definition	df-ple 17368	Define "less than or equal to" ordering extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.) Use its index-independent form pleid 17458 instead. (New usage is discouraged.)
⊢ le = Slot ;10
Definition	df-ocomp 17369	Define the orthocomplementation extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form ocid 17473 instead. (New usage is discouraged.)
⊢ oc = Slot ;11
Definition	df-ds 17370	Define the distance function component of a metric space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form dsid 17477 instead. (New usage is discouraged.)
⊢ dist = Slot ;12
Definition	df-unif 17371	Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.) Use its index-independent form unifid 17487 instead. (New usage is discouraged.)
⊢ UnifSet = Slot ;13
Definition	df-hom 17372	Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form homid 17503 instead. (New usage is discouraged.)
⊢ Hom = Slot ;14
Definition	df-cco 17373	Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form ccoid 17505 instead. (New usage is discouraged.)
⊢ comp = Slot ;15
Theorem	plusgndx 17374	Index value of the df-plusg 17361 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (+g‘ndx) = 2
Theorem	plusgid 17375	Utility theorem: index-independent form of df-plusg 17361. (Contributed by NM, 20-Oct-2012.)
⊢ +g = Slot (+g‘ndx)
Theorem	plusgndxnn 17376	The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 17-Oct-2024.)
⊢ (+g‘ndx) ∈ ℕ
Theorem	basendxltplusgndx 17377	The index of the slot for the base set is less than the index of the slot for the group operation in an extensible structure. (Contributed by AV, 17-Oct-2024.)
⊢ (Base‘ndx) < (+g‘ndx)
Theorem	basendxnplusgndx 17378	The slot for the base set is not the slot for the group operation in an extensible structure. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 17-Oct-2024.)
⊢ (Base‘ndx) ≠ (+g‘ndx)
Theorem	grpstr 17379	A constructed group is a structure. Version not depending on the implementation of the indices. (Contributed by AV, 27-Oct-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ 𝐺 Struct ⟨(Base‘ndx), (+g‘ndx)⟩
Theorem	grpbase 17380	The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by AV, 27-Oct-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
Theorem	grpplusg 17381	The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by AV, 27-Oct-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ ( + ∈ 𝑉 → + = (+g‘𝐺))
Theorem	ressplusg 17382	+g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → + = (+g‘𝐻))
Theorem	grpbasex 17383	The base of an explicitly given group. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use grpbase 17380 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    ⇒   ⊢ 𝐵 = (Base‘𝐺)
Theorem	grpplusgx 17384	The operation of an explicitly given group. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use grpplusg 17381 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    ⇒   ⊢ + = (+g‘𝐺)
Theorem	mulrndx 17385	Index value of the df-mulr 17362 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (.r‘ndx) = 3
Theorem	mulridx 17386	Utility theorem: index-independent form of df-mulr 17362. (Contributed by Mario Carneiro, 8-Jun-2013.)
⊢ .r = Slot (.r‘ndx)
Theorem	basendxnmulrndx 17387	The slot for the base set is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.) (Proof shortened by AV, 28-Oct-2024.)
⊢ (Base‘ndx) ≠ (.r‘ndx)
Theorem	plusgndxnmulrndx 17388	The slot for the group (addition) operation is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.)
⊢ (+g‘ndx) ≠ (.r‘ndx)
Theorem	rngstr 17389	A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑅 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}    ⇒   ⊢ 𝑅 Struct ⟨1, 3⟩
Theorem	rngbase 17390	The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝑅 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝑅))
Theorem	rngplusg 17391	The additive operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝑅 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}    ⇒   ⊢ ( + ∈ 𝑉 → + = (+g‘𝑅))
Theorem	rngmulr 17392	The multiplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝑅 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}    ⇒   ⊢ ( · ∈ 𝑉 → · = (.r‘𝑅))
Theorem	starvndx 17393	Index value of the df-starv 17363 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (*𝑟‘ndx) = 4
Theorem	starvid 17394	Utility theorem: index-independent form of df-starv 17363. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ *𝑟 = Slot (*𝑟‘ndx)
Theorem	starvndxnbasendx 17395	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17399. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (Base‘ndx)
Theorem	starvndxnplusgndx 17396	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17399. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (+g‘ndx)
Theorem	starvndxnmulrndx 17397	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17399. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (.r‘ndx)
Theorem	ressmulr 17398	.r is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝐴 ∈ 𝑉 → · = (.r‘𝑆))
Theorem	ressstarv 17399	*𝑟 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ ∗ = (*𝑟‘𝑅)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ∗ = (*𝑟‘𝑆))
Theorem	srngstr 17400	A constructed star ring is a structure. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ 𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    ⇒   ⊢ 𝑅 Struct ⟨1, 4⟩