P. 135 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13401-13500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ressvalsets 13401	Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
Theorem	ressex 13402	Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) ∈ V)
Theorem	ressval2 13403	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	ressbasd 13404	Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
⊢ (𝜑 → 𝑅 = (𝑊 ↾s 𝐴))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = (Base‘𝑅))
Theorem	ressbas2d 13405	Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ (𝜑 → 𝑅 = (𝑊 ↾s 𝐴))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 = (Base‘𝑅))
Theorem	ressbasssd 13406	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.)
⊢ (𝜑 → 𝑅 = (𝑊 ↾s 𝐴))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (Base‘𝑅) ⊆ 𝐵)
Theorem	ressbasid 13407	The trivial structure restriction leaves the base set unchanged. (Contributed by Jim Kingdon, 29-Apr-2025.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑉 → (Base‘(𝑊 ↾s 𝐵)) = 𝐵)
Theorem	strressid 13408	Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝑊))    &   ⊢ (𝜑 → 𝑊 Struct ⟨𝑀, 𝑁⟩)    &   ⊢ (𝜑 → Fun 𝑊)    &   ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑊)    ⇒   ⊢ (𝜑 → (𝑊 ↾s 𝐵) = 𝑊)
Theorem	ressval3d 13409	Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)
⊢ 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	resseqnbasd 13410	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) ∈ ℕ)    &   ⊢ (𝐸‘ndx) ≠ (Base‘ndx)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐶 = (𝐸‘𝑅))
Theorem	ressinbasd 13411	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.)
⊢ (𝜑 → 𝐵 = (Base‘𝑊))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
Theorem	ressressg 13412	Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
Theorem	ressabsg 13413	Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑌) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐵))
6.1.2  Slot definitions
Syntax	cplusg 13414	Extend class notation with group (addition) operation.
class +g
Syntax	cmulr 13415	Extend class notation with ring multiplication.
class .r
Syntax	cstv 13416	Extend class notation with involution.
class *𝑟
Syntax	csca 13417	Extend class notation with scalar field.
class Scalar
Syntax	cvsca 13418	Extend class notation with scalar product.
class ·𝑠
Syntax	cip 13419	Extend class notation with Hermitian form (inner product).
class ·𝑖
Syntax	cts 13420	Extend class notation with the topology component of a topological space.
class TopSet
Syntax	cple 13421	Extend class notation with "less than or equal to" for posets.
class le
Syntax	coc 13422	Extend class notation with the class of orthocomplementation extractors.
class oc
Syntax	cds 13423	Extend class notation with the metric space distance function.
class dist
Syntax	cunif 13424	Extend class notation with the uniform structure.
class UnifSet
Syntax	chom 13425	Extend class notation with the hom-set structure.
class Hom
Syntax	cco 13426	Extend class notation with the composition operation.
class comp
Definition	df-plusg 13427	Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ +g = Slot 2
Definition	df-mulr 13428	Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ .r = Slot 3
Definition	df-starv 13429	Define the involution function of a *-ring. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ *𝑟 = Slot 4
Definition	df-sca 13430	Define scalar field component of a vector space 𝑣. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ Scalar = Slot 5
Definition	df-vsca 13431	Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ ·𝑠 = Slot 6
Definition	df-ip 13432	Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ ·𝑖 = Slot 8
Definition	df-tset 13433	Define the topology component of a topological space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ TopSet = Slot 9
Definition	df-ple 13434	Define "less than or equal to" ordering extractor for posets and related structures. We use ;10 for the index to avoid conflict with 1 through 9 used for other purposes. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
⊢ le = Slot ;10
Definition	df-ocomp 13435	Define the orthocomplementation extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ oc = Slot ;11
Definition	df-ds 13436	Define the distance function component of a metric space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ dist = Slot ;12
Definition	df-unif 13437	Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ UnifSet = Slot ;13
Definition	df-hom 13438	Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ Hom = Slot ;14
Definition	df-cco 13439	Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ comp = Slot ;15
Theorem	strleund 13440	Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
⊢ (𝜑 → 𝐹 Struct ⟨𝐴, 𝐵⟩)    &   ⊢ (𝜑 → 𝐺 Struct ⟨𝐶, 𝐷⟩)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∪ 𝐺) Struct ⟨𝐴, 𝐷⟩)
Theorem	strleun 13441	Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐹 Struct ⟨𝐴, 𝐵⟩    &   ⊢ 𝐺 Struct ⟨𝐶, 𝐷⟩    &   ⊢ 𝐵 < 𝐶    ⇒   ⊢ (𝐹 ∪ 𝐺) Struct ⟨𝐴, 𝐷⟩
Theorem	strext 13442	Extending the upper range of a structure. This works because when we say that a structure has components in 𝐴...𝐶 we are not saying that every slot in that range is present, just that all the slots that are present are within that range. (Contributed by Jim Kingdon, 26-Feb-2025.)
⊢ (𝜑 → 𝐹 Struct ⟨𝐴, 𝐵⟩)    &   ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐵))    ⇒   ⊢ (𝜑 → 𝐹 Struct ⟨𝐴, 𝐶⟩)
Theorem	strle1g 13443	Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
⊢ 𝐼 ∈ ℕ    &   ⊢ 𝐴 = 𝐼    ⇒   ⊢ (𝑋 ∈ 𝑉 → {⟨𝐴, 𝑋⟩} Struct ⟨𝐼, 𝐼⟩)
Theorem	strle2g 13444	Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
⊢ 𝐼 ∈ ℕ    &   ⊢ 𝐴 = 𝐼    &   ⊢ 𝐼 < 𝐽    &   ⊢ 𝐽 ∈ ℕ    &   ⊢ 𝐵 = 𝐽    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩} Struct ⟨𝐼, 𝐽⟩)
Theorem	strle3g 13445	Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐼 ∈ ℕ    &   ⊢ 𝐴 = 𝐼    &   ⊢ 𝐼 < 𝐽    &   ⊢ 𝐽 ∈ ℕ    &   ⊢ 𝐵 = 𝐽    &   ⊢ 𝐽 < 𝐾    &   ⊢ 𝐾 ∈ ℕ    &   ⊢ 𝐶 = 𝐾    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑃) → {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩, ⟨𝐶, 𝑍⟩} Struct ⟨𝐼, 𝐾⟩)
Theorem	plusgndx 13446	Index value of the df-plusg 13427 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (+g‘ndx) = 2
Theorem	plusgid 13447	Utility theorem: index-independent form of df-plusg 13427. (Contributed by NM, 20-Oct-2012.)
⊢ +g = Slot (+g‘ndx)
Theorem	plusgndxnn 13448	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	plusgslid 13449	Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.)
⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
Theorem	basendxltplusgndx 13450	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	opelstrsl 13451	The slot of a structure which contains an ordered pair for that slot. (Contributed by Jim Kingdon, 5-Feb-2023.)
⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   ⊢ (𝜑 → 𝑆 Struct 𝑋)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → ⟨(𝐸‘ndx), 𝑉⟩ ∈ 𝑆)    ⇒   ⊢ (𝜑 → 𝑉 = (𝐸‘𝑆))
Theorem	opelstrbas 13452	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	1strstrg 13453	A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Revised by Jim Kingdon, 28-Jan-2023.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐺 Struct ⟨1, 1⟩)
Theorem	1strbas 13454	The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
Theorem	2strstrndx 13455	A constructed two-slot structure not depending on the hard-coded index value of the base set. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 14-Dec-2025.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩)
Theorem	2strstrg 13456	A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.) Use 2strstrndx 13455 instead. (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 1 < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct ⟨1, 𝑁⟩)
Theorem	2strbasg 13457	The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 1 < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐵 = (Base‘𝐺))
Theorem	2stropg 13458	The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 1 < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + = (𝐸‘𝐺))
Theorem	2strstr1g 13459	A constructed two-slot structure. Version of 2strstrg 13456 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩)
Theorem	2strbas1g 13460	The base set of a constructed two-slot structure. Version of 2strbasg 13457 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐵 = (Base‘𝐺))
Theorem	2strop1g 13461	The other slot of a constructed two-slot structure. Version of 2stropg 13458 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐸 = Slot 𝑁    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + = (𝐸‘𝐺))
Theorem	basendxnplusgndx 13462	The slot for the base set is not the slot for the group operation in an extensible structure. (Contributed by AV, 14-Nov-2021.)
⊢ (Base‘ndx) ≠ (+g‘ndx)
Theorem	grpstrg 13463	A constructed group is a structure on 1...2. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct ⟨1, 2⟩)
Theorem	grpbaseg 13464	The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐵 = (Base‘𝐺))
Theorem	grpplusgg 13465	The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + = (+g‘𝐺))
Theorem	ressplusgd 13466	+g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ (𝜑 → 𝐻 = (𝐺 ↾s 𝐴))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    ⇒   ⊢ (𝜑 → + = (+g‘𝐻))
Theorem	mulrndx 13467	Index value of the df-mulr 13428 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (.r‘ndx) = 3
Theorem	mulridx 13468	Utility theorem: index-independent form of df-mulr 13428. (Contributed by Mario Carneiro, 8-Jun-2013.)
⊢ .r = Slot (.r‘ndx)
Theorem	mulrslid 13469	Slot property of .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
Theorem	plusgndxnmulrndx 13470	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	basendxnmulrndx 13471	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.)
⊢ (Base‘ndx) ≠ (.r‘ndx)
Theorem	rngstrg 13472	A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
⊢ 𝑅 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝑅 Struct ⟨1, 3⟩)
Theorem	rngbaseg 13473	The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
⊢ 𝑅 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝐵 = (Base‘𝑅))
Theorem	rngplusgg 13474	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), · ⟩}    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → + = (+g‘𝑅))
Theorem	rngmulrg 13475	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), · ⟩}    ⇒   ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → · = (.r‘𝑅))
Theorem	starvndx 13476	Index value of the df-starv 13429 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (*𝑟‘ndx) = 4
Theorem	starvid 13477	Utility theorem: index-independent form of df-starv 13429. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ *𝑟 = Slot (*𝑟‘ndx)
Theorem	starvslid 13478	Slot property of *𝑟. (Contributed by Jim Kingdon, 4-Feb-2023.)
⊢ (*𝑟 = Slot (*𝑟‘ndx) ∧ (*𝑟‘ndx) ∈ ℕ)
Theorem	starvndxnbasendx 13479	The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (Base‘ndx)
Theorem	starvndxnplusgndx 13480	The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (+g‘ndx)
Theorem	starvndxnmulrndx 13481	The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (.r‘ndx)
Theorem	ressmulrg 13482	.r is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → · = (.r‘𝑆))
Theorem	srngstrd 13483	A constructed star ring is a structure. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
⊢ 𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → · ∈ 𝑋)    &   ⊢ (𝜑 → ∗ ∈ 𝑌)    ⇒   ⊢ (𝜑 → 𝑅 Struct ⟨1, 4⟩)
Theorem	srngbased 13484	The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
⊢ 𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → · ∈ 𝑋)    &   ⊢ (𝜑 → ∗ ∈ 𝑌)    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
Theorem	srngplusgd 13485	The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by Jim Kingdon, 5-Feb-2023.)
⊢ 𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → · ∈ 𝑋)    &   ⊢ (𝜑 → ∗ ∈ 𝑌)    ⇒   ⊢ (𝜑 → + = (+g‘𝑅))
Theorem	srngmulrd 13486	The multiplication operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ 𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → · ∈ 𝑋)    &   ⊢ (𝜑 → ∗ ∈ 𝑌)    ⇒   ⊢ (𝜑 → · = (.r‘𝑅))
Theorem	srnginvld 13487	The involution function of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ 𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → + ∈ 𝑊)    &   ⊢ (𝜑 → · ∈ 𝑋)    &   ⊢ (𝜑 → ∗ ∈ 𝑌)    ⇒   ⊢ (𝜑 → ∗ = (*𝑟‘𝑅))
Theorem	scandx 13488	Index value of the df-sca 13430 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ (Scalar‘ndx) = 5
Theorem	scaid 13489	Utility theorem: index-independent form of scalar df-sca 13430. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ Scalar = Slot (Scalar‘ndx)
Theorem	scaslid 13490	Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
Theorem	scandxnbasendx 13491	The slot for the scalar is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (Base‘ndx)
Theorem	scandxnplusgndx 13492	The slot for the scalar field is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (+g‘ndx)
Theorem	scandxnmulrndx 13493	The slot for the scalar field is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (.r‘ndx)
Theorem	vscandx 13494	Index value of the df-vsca 13431 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ ( ·𝑠 ‘ndx) = 6
Theorem	vscaid 13495	Utility theorem: index-independent form of scalar product df-vsca 13431. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
Theorem	vscandxnbasendx 13496	The slot for the scalar product is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (Base‘ndx)
Theorem	vscandxnplusgndx 13497	The slot for the scalar product is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (+g‘ndx)
Theorem	vscandxnmulrndx 13498	The slot for the scalar product is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (.r‘ndx)
Theorem	vscandxnscandx 13499	The slot for the scalar product is not the slot for the scalar field in an extensible structure. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (Scalar‘ndx)
Theorem	vscaslid 13500	Slot property of ·𝑠. (Contributed by Jim Kingdon, 5-Feb-2023.)
⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)