P. 171 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)

Type	Label	Description
Statement
7.1.1.5  Base sets
Syntax	cbs 17001	Extend class notation with the class of all base set extractors.
class Base
Definition	df-base 17002	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 17004 instead. (New usage is discouraged.)
⊢ Base = Slot 1
Theorem	baseval 17003	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 17004	Utility theorem: index-independent form of df-base 17002. (Contributed by NM, 20-Oct-2012.)
⊢ Base = Slot (Base‘ndx)
Theorem	basfn 17005	The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
⊢ Base Fn V
Theorem	base0 17006	The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ ∅ = (Base‘∅)
Theorem	elbasfv 17007	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 17008	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 17009	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 17010	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 17011	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	basendxnnOLD 17012	Obsolete proof of basendxnn 17011 as of 13-Oct-2024. (Contributed by AV, 23-Sep-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (Base‘ndx) ∈ ℕ
Theorem	basndxelwund 17013	The index of the base set is an element in a weak universe containing the natural numbers. Formerly part of proof for 1strwun 17021. (Contributed by AV, 27-Mar-2020.) (Revised by AV, 17-Oct-2024.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
Theorem	basprssdmsets 17014	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 17015	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 17016	A constructed one-slot structure. Depending on hard-coded index. Use 1strstr1 17017 instead. (Contributed by AV, 27-Mar-2020.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ 𝐺 Struct ⟨1, 1⟩
Theorem	1strstr1 17017	A constructed one-slot structure. (Contributed by AV, 15-Nov-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ 𝐺 Struct ⟨(Base‘ndx), (Base‘ndx)⟩
Theorem	1strbas 17018	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	1strbasOLD 17019	Obsolete proof of 1strbas 17018 as of 15-Nov-2024. The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
Theorem	1strwunbndx 17020	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 17021	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	1strwunOLD 17022	Obsolete version of 1strwun 17021 as of 17-Oct-2024. A constructed one-slot structure in a weak universe. (Contributed by AV, 27-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈)
Theorem	2strstr 17023	A constructed two-slot structure. Depending on hard-coded indices. Use 2strstr1 17026 instead. (Contributed by Mario Carneiro, 29-Aug-2015.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 1 < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝐺 Struct ⟨1, 𝑁⟩
Theorem	2strbas 17024	The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) Use the index-independent version 2strbas1 17028 instead. (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 1 < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
Theorem	2strop 17025	The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) Use the index-independent version 2strop1 17029 instead. (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 1 < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ( + ∈ 𝑉 → + = (𝐸‘𝐺))
Theorem	2strstr1 17026	A constructed two-slot structure. Version of 2strstr 17023 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	2strstr1OLD 17027	Obsolete version of 2strstr1 17026 as of 27-Oct-2024. A constructed two-slot structure. Version of 2strstr 17023 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩
Theorem	2strbas1 17028	The base set of a constructed two-slot structure. Version of 2strbas 17024 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	2strop1 17029	The other slot of a constructed two-slot structure. Version of 2strop 17025 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 17030	Extend class notation with the extensible structure builder restriction operator.
class ↾s
Definition	df-ress 17031*	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 17036 for the altered base set, and resseqnbas 17040 (subrg0 20128, ressplusg 17089, subrg1 20131, ressmulr 17106) 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 17032	The structure restriction is a proper operator, so it can be used with ovprc1 7368. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ Rel dom ↾s
Theorem	ressval 17033	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 17034	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 17035	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 17036	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	ressbasOLD 17037	Obsolete proof of ressbas 17036 as of 7-Nov-2024. Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅))
Theorem	ressbas2 17038	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 17039	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.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (Base‘𝑅) ⊆ 𝐵
Theorem	resseqnbas 17040	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	resslemOLD 17041	Obsolete version of resseqnbas 17040 as of 21-Oct-2024. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐶 = (𝐸‘𝑊)    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 1 < 𝑁    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅))
Theorem	ress0 17042	All restrictions of the null set are trivial. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ (∅ ↾s 𝐴) = ∅
Theorem	ressid 17043	Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s 𝐵) = 𝑊)
Theorem	ressinbas 17044	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 17045	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	ressval3dOLD 17046	Obsolete version of ressval3d 17045 as of 17-Oct-2024. Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 3-Jul-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 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 17047	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 17048	Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐵))
Theorem	wunress 17049	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)
Theorem	wunressOLD 17050	Obsolete proof of wunress 17049 as of 28-Oct-2024. Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑊 ↾s 𝐴) ∈ 𝑈)
7.1.2  Slot definitions
Syntax	cplusg 17051	Extend class notation with group (addition) operation.
class +g
Syntax	cmulr 17052	Extend class notation with ring multiplication.
class .r
Syntax	cstv 17053	Extend class notation with involution.
class *𝑟
Syntax	csca 17054	Extend class notation with scalar field.
class Scalar
Syntax	cvsca 17055	Extend class notation with scalar product.
class ·𝑠
Syntax	cip 17056	Extend class notation with Hermitian form (inner product).
class ·𝑖
Syntax	cts 17057	Extend class notation with the topology component of a topological space.
class TopSet
Syntax	cple 17058	Extend class notation with "less than or equal to" for posets.
class le
Syntax	coc 17059	Extend class notation with the class of orthocomplementation extractors.
class oc
Syntax	cds 17060	Extend class notation with the metric space distance function.
class dist
Syntax	cunif 17061	Extend class notation with the uniform structure.
class UnifSet
Syntax	chom 17062	Extend class notation with the hom-set structure.
class Hom
Syntax	cco 17063	Extend class notation with the composition operation.
class comp
Definition	df-plusg 17064	Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form plusgid 17078 instead. (New usage is discouraged.)
⊢ +g = Slot 2
Definition	df-mulr 17065	Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form mulrid 17093 instead. (New usage is discouraged.)
⊢ .r = Slot 3
Definition	df-starv 17066	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 17102 instead. (New usage is discouraged.)
⊢ *𝑟 = Slot 4
Definition	df-sca 17067	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 17114 instead. (New usage is discouraged.)
⊢ Scalar = Slot 5
Definition	df-vsca 17068	Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form vscaid 17119 instead. (New usage is discouraged.)
⊢ ·𝑠 = Slot 6
Definition	df-ip 17069	Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form ipid 17130 instead. (New usage is discouraged.)
⊢ ·𝑖 = Slot 8
Definition	df-tset 17070	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 17152 instead. (New usage is discouraged.)
⊢ TopSet = Slot 9
Definition	df-ple 17071	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 17166 instead. (New usage is discouraged.)
⊢ le = Slot ;10
Definition	df-ocomp 17072	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 17181 instead. (New usage is discouraged.)
⊢ oc = Slot ;11
Definition	df-ds 17073	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 17185 instead. (New usage is discouraged.)
⊢ dist = Slot ;12
Definition	df-unif 17074	Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.) Use its index-independent form unifid 17195 instead. (New usage is discouraged.)
⊢ UnifSet = Slot ;13
Definition	df-hom 17075	Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form homid 17211 instead. (New usage is discouraged.)
⊢ Hom = Slot ;14
Definition	df-cco 17076	Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form ccoid 17213 instead. (New usage is discouraged.)
⊢ comp = Slot ;15
Theorem	plusgndx 17077	Index value of the df-plusg 17064 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (+g‘ndx) = 2
Theorem	plusgid 17078	Utility theorem: index-independent form of df-plusg 17064. (Contributed by NM, 20-Oct-2012.)
⊢ +g = Slot (+g‘ndx)
Theorem	plusgndxnn 17079	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 17080	The index of the slot for the base set is less then 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 17081	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	basendxnplusgndxOLD 17082	Obsolete version of basendxnplusgndx 17081 as of 17-Oct-2024. 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 modification is discouraged.) (New usage is discouraged.)
⊢ (Base‘ndx) ≠ (+g‘ndx)
Theorem	grpstr 17083	A constructed group is a structure on 1...2. Depending on hard-coded index values. Use grpstrndx 17084 instead. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ 𝐺 Struct ⟨1, 2⟩
Theorem	grpstrndx 17084	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 17085	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	grpbaseOLD 17086	Obsolete version of grpbase 17085 as of 27-Oct-2024. The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
Theorem	grpplusg 17087	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	grpplusgOLD 17088	Obsolete version of grpplusg 17087 as of 27-Oct-2024. The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ ( + ∈ 𝑉 → + = (+g‘𝐺))
Theorem	ressplusg 17089	+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 17090	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 17085 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    ⇒   ⊢ 𝐵 = (Base‘𝐺)
Theorem	grpplusgx 17091	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 17087 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    ⇒   ⊢ + = (+g‘𝐺)
Theorem	mulrndx 17092	Index value of the df-mulr 17065 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (.r‘ndx) = 3
Theorem	mulrid 17093	Utility theorem: index-independent form of df-mulr 17065. (Contributed by Mario Carneiro, 8-Jun-2013.)
⊢ .r = Slot (.r‘ndx)
Theorem	basendxnmulrndx 17094	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	basendxnmulrndxOLD 17095	Obsolete proof of basendxnmulrndx 17094 as of 28-Oct-2024. 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 modification is discouraged.) (New usage is discouraged.)
⊢ (Base‘ndx) ≠ (.r‘ndx)
Theorem	plusgndxnmulrndx 17096	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 17097	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 17098	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 17099	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 17100	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))