P. 173 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17201-17300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1strbas 17201	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 17202	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 17203	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 17204	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 17205	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 17206	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 17207	Extend class notation with the extensible structure builder restriction operator.
class ↾s
Definition	df-ress 17208*	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 17213 for the altered base set, and resseqnbas 17219 (subrg0 20495, ressplusg 17261, subrg1 20498, ressmulr 17277) 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 17209	The structure restriction is a proper operator, so it can be used with ovprc1 7429. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ Rel dom ↾s
Theorem	ressval 17210	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 17211	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 17212	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 17213	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 17214	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 17215	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 17216	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 17217	Obsolete version of ressbas 17213 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 17218	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 17219	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 17220	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 17221	Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s 𝐵) = 𝑊)
Theorem	ressinbas 17222	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 17223	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 17224	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 17225	Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐵))
Theorem	wunress 17226	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 17227	Extend class notation with group (addition) operation.
class +g
Syntax	cmulr 17228	Extend class notation with ring multiplication.
class .r
Syntax	cstv 17229	Extend class notation with involution.
class *𝑟
Syntax	csca 17230	Extend class notation with scalar field.
class Scalar
Syntax	cvsca 17231	Extend class notation with scalar product.
class ·𝑠
Syntax	cip 17232	Extend class notation with Hermitian form (inner product).
class ·𝑖
Syntax	cts 17233	Extend class notation with the topology component of a topological space.
class TopSet
Syntax	cple 17234	Extend class notation with "less than or equal to" for posets.
class le
Syntax	coc 17235	Extend class notation with the class of orthocomplementation extractors.
class oc
Syntax	cds 17236	Extend class notation with the metric space distance function.
class dist
Syntax	cunif 17237	Extend class notation with the uniform structure.
class UnifSet
Syntax	chom 17238	Extend class notation with the hom-set structure.
class Hom
Syntax	cco 17239	Extend class notation with the composition operation.
class comp
Definition	df-plusg 17240	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 17254 instead. (New usage is discouraged.)
⊢ +g = Slot 2
Definition	df-mulr 17241	Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form mulrid 11179 instead. (New usage is discouraged.)
⊢ .r = Slot 3
Definition	df-starv 17242	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 17273 instead. (New usage is discouraged.)
⊢ *𝑟 = Slot 4
Definition	df-sca 17243	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 17285 instead. (New usage is discouraged.)
⊢ Scalar = Slot 5
Definition	df-vsca 17244	Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form vscaid 17290 instead. (New usage is discouraged.)
⊢ ·𝑠 = Slot 6
Definition	df-ip 17245	Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form ipid 17301 instead. (New usage is discouraged.)
⊢ ·𝑖 = Slot 8
Definition	df-tset 17246	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 17323 instead. (New usage is discouraged.)
⊢ TopSet = Slot 9
Definition	df-ple 17247	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 17337 instead. (New usage is discouraged.)
⊢ le = Slot ;10
Definition	df-ocomp 17248	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 17352 instead. (New usage is discouraged.)
⊢ oc = Slot ;11
Definition	df-ds 17249	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 17356 instead. (New usage is discouraged.)
⊢ dist = Slot ;12
Definition	df-unif 17250	Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.) Use its index-independent form unifid 17366 instead. (New usage is discouraged.)
⊢ UnifSet = Slot ;13
Definition	df-hom 17251	Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form homid 17382 instead. (New usage is discouraged.)
⊢ Hom = Slot ;14
Definition	df-cco 17252	Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form ccoid 17384 instead. (New usage is discouraged.)
⊢ comp = Slot ;15
Theorem	plusgndx 17253	Index value of the df-plusg 17240 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (+g‘ndx) = 2
Theorem	plusgid 17254	Utility theorem: index-independent form of df-plusg 17240. (Contributed by NM, 20-Oct-2012.)
⊢ +g = Slot (+g‘ndx)
Theorem	plusgndxnn 17255	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 17256	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 17257	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 17258	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 17259	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 17260	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 17261	+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 17262	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 17259 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    ⇒   ⊢ 𝐵 = (Base‘𝐺)
Theorem	grpplusgx 17263	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 17260 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    ⇒   ⊢ + = (+g‘𝐺)
Theorem	mulrndx 17264	Index value of the df-mulr 17241 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (.r‘ndx) = 3
Theorem	mulridx 17265	Utility theorem: index-independent form of df-mulr 17241. (Contributed by Mario Carneiro, 8-Jun-2013.)
⊢ .r = Slot (.r‘ndx)
Theorem	basendxnmulrndx 17266	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 17267	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 17268	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 17269	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 17270	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 17271	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 17272	Index value of the df-starv 17242 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (*𝑟‘ndx) = 4
Theorem	starvid 17273	Utility theorem: index-independent form of df-starv 17242. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ *𝑟 = Slot (*𝑟‘ndx)
Theorem	starvndxnbasendx 17274	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17278. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (Base‘ndx)
Theorem	starvndxnplusgndx 17275	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17278. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (+g‘ndx)
Theorem	starvndxnmulrndx 17276	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17278. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (.r‘ndx)
Theorem	ressmulr 17277	.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 17278	*𝑟 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ ∗ = (*𝑟‘𝑅)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ∗ = (*𝑟‘𝑆))
Theorem	srngstr 17279	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⟩
Theorem	srngbase 17280	The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ 𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    ⇒   ⊢ (𝐵 ∈ 𝑋 → 𝐵 = (Base‘𝑅))
Theorem	srngplusg 17281	The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ 𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗ ⟩})    ⇒   ⊢ ( + ∈ 𝑋 → + = (+g‘𝑅))
Theorem	srngmulr 17282	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), ∗ ⟩})    ⇒   ⊢ ( · ∈ 𝑋 → · = (.r‘𝑅))
Theorem	srnginvl 17283	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), ∗ ⟩})    ⇒   ⊢ ( ∗ ∈ 𝑋 → ∗ = (*𝑟‘𝑅))
Theorem	scandx 17284	Index value of the df-sca 17243 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (Scalar‘ndx) = 5
Theorem	scaid 17285	Utility theorem: index-independent form of scalar df-sca 17243. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ Scalar = Slot (Scalar‘ndx)
Theorem	scandxnbasendx 17286	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 17287	The slot for the scalar field is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpsca 20062. (Contributed by AV, 18-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (+g‘ndx)
Theorem	scandxnmulrndx 17288	The slot for the scalar field is not the slot for the ring (multiplication) operation in an extensible structure. Formerly part of proof for mgpsca 20062. (Contributed by AV, 29-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (.r‘ndx)
Theorem	vscandx 17289	Index value of the df-vsca 17244 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ ( ·𝑠 ‘ndx) = 6
Theorem	vscaid 17290	Utility theorem: index-independent form of scalar product df-vsca 17244. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
Theorem	vscandxnbasendx 17291	The slot for the scalar product is not the slot for the base set in an extensible structure. Formerly part of proof for rmodislmod 20843. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (Base‘ndx)
Theorem	vscandxnplusgndx 17292	The slot for the scalar product is not the slot for the group operation in an extensible structure. Formerly part of proof for rmodislmod 20843. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (+g‘ndx)
Theorem	vscandxnmulrndx 17293	The slot for the scalar product is not the slot for the ring (multiplication) operation in an extensible structure. Formerly part of proof for rmodislmod 20843. (Contributed by AV, 29-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (.r‘ndx)
Theorem	vscandxnscandx 17294	The slot for the scalar product is not the slot for the scalar field in an extensible structure. Formerly part of proof for rmodislmod 20843. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (Scalar‘ndx)
Theorem	lmodstr 17295	A constructed left module or left vector space is a structure. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ 𝑊 Struct ⟨1, 6⟩
Theorem	lmodbase 17296	The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ (𝐵 ∈ 𝑋 → 𝐵 = (Base‘𝑊))
Theorem	lmodplusg 17297	The additive operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ ( + ∈ 𝑋 → + = (+g‘𝑊))
Theorem	lmodsca 17298	The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ (𝐹 ∈ 𝑋 → 𝐹 = (Scalar‘𝑊))
Theorem	lmodvsca 17299	The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ ( · ∈ 𝑋 → · = ( ·𝑠 ‘𝑊))
Theorem	ipndx 17300	Index value of the df-ip 17245 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (·𝑖‘ndx) = 8