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
Syntax	cvsca 17201	Extend class notation with scalar product.
class ·𝑠
Syntax	cip 17202	Extend class notation with Hermitian form (inner product).
class ·𝑖
Syntax	cts 17203	Extend class notation with the topology component of a topological space.
class TopSet
Syntax	cple 17204	Extend class notation with "less than or equal to" for posets.
class le
Syntax	coc 17205	Extend class notation with the class of orthocomplementation extractors.
class oc
Syntax	cds 17206	Extend class notation with the metric space distance function.
class dist
Syntax	cunif 17207	Extend class notation with the uniform structure.
class UnifSet
Syntax	chom 17208	Extend class notation with the hom-set structure.
class Hom
Syntax	cco 17209	Extend class notation with the composition operation.
class comp
Definition	df-plusg 17210	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 17224 instead. (New usage is discouraged.)
⊢ +g = Slot 2
Definition	df-mulr 17211	Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form mulrid 11212 instead. (New usage is discouraged.)
⊢ .r = Slot 3
Definition	df-starv 17212	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 17248 instead. (New usage is discouraged.)
⊢ *𝑟 = Slot 4
Definition	df-sca 17213	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 17260 instead. (New usage is discouraged.)
⊢ Scalar = Slot 5
Definition	df-vsca 17214	Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form vscaid 17265 instead. (New usage is discouraged.)
⊢ ·𝑠 = Slot 6
Definition	df-ip 17215	Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form ipid 17276 instead. (New usage is discouraged.)
⊢ ·𝑖 = Slot 8
Definition	df-tset 17216	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 17298 instead. (New usage is discouraged.)
⊢ TopSet = Slot 9
Definition	df-ple 17217	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 17312 instead. (New usage is discouraged.)
⊢ le = Slot ;10
Definition	df-ocomp 17218	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 17327 instead. (New usage is discouraged.)
⊢ oc = Slot ;11
Definition	df-ds 17219	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 17331 instead. (New usage is discouraged.)
⊢ dist = Slot ;12
Definition	df-unif 17220	Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.) Use its index-independent form unifid 17341 instead. (New usage is discouraged.)
⊢ UnifSet = Slot ;13
Definition	df-hom 17221	Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form homid 17357 instead. (New usage is discouraged.)
⊢ Hom = Slot ;14
Definition	df-cco 17222	Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form ccoid 17359 instead. (New usage is discouraged.)
⊢ comp = Slot ;15
Theorem	plusgndx 17223	Index value of the df-plusg 17210 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (+g‘ndx) = 2
Theorem	plusgid 17224	Utility theorem: index-independent form of df-plusg 17210. (Contributed by NM, 20-Oct-2012.)
⊢ +g = Slot (+g‘ndx)
Theorem	plusgndxnn 17225	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 17226	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 17227	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 17228	Obsolete version of basendxnplusgndx 17227 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 17229	A constructed group is a structure on 1...2. Depending on hard-coded index values. Use grpstrndx 17230 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 17230	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 17231	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 17232	Obsolete version of grpbase 17231 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 17233	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 17234	Obsolete version of grpplusg 17233 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 17235	+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 17236	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 17231 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    ⇒   ⊢ 𝐵 = (Base‘𝐺)
Theorem	grpplusgx 17237	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 17233 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    ⇒   ⊢ + = (+g‘𝐺)
Theorem	mulrndx 17238	Index value of the df-mulr 17211 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (.r‘ndx) = 3
Theorem	mulridx 17239	Utility theorem: index-independent form of df-mulr 17211. (Contributed by Mario Carneiro, 8-Jun-2013.)
⊢ .r = Slot (.r‘ndx)
Theorem	basendxnmulrndx 17240	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 17241	Obsolete proof of basendxnmulrndx 17240 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 17242	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 17243	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 17244	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 17245	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 17246	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 17247	Index value of the df-starv 17212 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (*𝑟‘ndx) = 4
Theorem	starvid 17248	Utility theorem: index-independent form of df-starv 17212. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ *𝑟 = Slot (*𝑟‘ndx)
Theorem	starvndxnbasendx 17249	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17253. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (Base‘ndx)
Theorem	starvndxnplusgndx 17250	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17253. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (+g‘ndx)
Theorem	starvndxnmulrndx 17251	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17253. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (.r‘ndx)
Theorem	ressmulr 17252	.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 17253	*𝑟 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ ∗ = (*𝑟‘𝑅)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ∗ = (*𝑟‘𝑆))
Theorem	srngstr 17254	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 17255	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 17256	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 17257	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 17258	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 17259	Index value of the df-sca 17213 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (Scalar‘ndx) = 5
Theorem	scaid 17260	Utility theorem: index-independent form of scalar df-sca 17213. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ Scalar = Slot (Scalar‘ndx)
Theorem	scandxnbasendx 17261	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 17262	The slot for the scalar field is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpsca 19995. (Contributed by AV, 18-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (+g‘ndx)
Theorem	scandxnmulrndx 17263	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 19995. (Contributed by AV, 29-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (.r‘ndx)
Theorem	vscandx 17264	Index value of the df-vsca 17214 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ ( ·𝑠 ‘ndx) = 6
Theorem	vscaid 17265	Utility theorem: index-independent form of scalar product df-vsca 17214. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
Theorem	vscandxnbasendx 17266	The slot for the scalar product is not the slot for the base set in an extensible structure. Formerly part of proof for rmodislmod 20540. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (Base‘ndx)
Theorem	vscandxnplusgndx 17267	The slot for the scalar product is not the slot for the group operation in an extensible structure. Formerly part of proof for rmodislmod 20540. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (+g‘ndx)
Theorem	vscandxnmulrndx 17268	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 20540. (Contributed by AV, 29-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (.r‘ndx)
Theorem	vscandxnscandx 17269	The slot for the scalar product is not the slot for the scalar field in an extensible structure. Formerly part of proof for rmodislmod 20540. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (Scalar‘ndx)
Theorem	lmodstr 17270	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 17271	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 17272	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 17273	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 17274	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 17275	Index value of the df-ip 17215 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (·𝑖‘ndx) = 8
Theorem	ipid 17276	Utility theorem: index-independent form of df-ip 17215. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ ·𝑖 = Slot (·𝑖‘ndx)
Theorem	ipndxnbasendx 17277	The slot for the inner product is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
⊢ (·𝑖‘ndx) ≠ (Base‘ndx)
Theorem	ipndxnplusgndx 17278	The slot for the inner product is not the slot for the group operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
⊢ (·𝑖‘ndx) ≠ (+g‘ndx)
Theorem	ipndxnmulrndx 17279	The slot for the inner product is not the slot for the ring (multiplication) operation in an extensible structure. Formerly part of proof for mgpsca 19995. (Contributed by AV, 29-Oct-2024.)
⊢ (·𝑖‘ndx) ≠ (.r‘ndx)
Theorem	slotsdifipndx 17280	The slot for the scalar is not the index of other slots. Formerly part of proof for srasca 20798 and sravsca 20800. (Contributed by AV, 12-Nov-2024.)
⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx))
Theorem	ipsstr 17281	Lemma to shorten proofs of ipsbase 17282 through ipsvsca 17286. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    ⇒   ⊢ 𝐴 Struct ⟨1, 8⟩
Theorem	ipsbase 17282	The base set of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐴))
Theorem	ipsaddg 17283	The additive operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    ⇒   ⊢ ( + ∈ 𝑉 → + = (+g‘𝐴))
Theorem	ipsmulr 17284	The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    ⇒   ⊢ ( × ∈ 𝑉 → × = (.r‘𝐴))
Theorem	ipssca 17285	The set of scalars of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    ⇒   ⊢ (𝑆 ∈ 𝑉 → 𝑆 = (Scalar‘𝐴))
Theorem	ipsvsca 17286	The scalar product operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    ⇒   ⊢ ( · ∈ 𝑉 → · = ( ·𝑠 ‘𝐴))
Theorem	ipsip 17287	The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐼 = (·𝑖‘𝐴))
Theorem	resssca 17288	Scalar is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝐹 = (Scalar‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐹 = (Scalar‘𝐻))
Theorem	ressvsca 17289	·𝑠 is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ · = ( ·𝑠 ‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘𝐻))
Theorem	ressip 17290	The inner product is unaffected by restriction. (Contributed by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ , = (·𝑖‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → , = (·𝑖‘𝐻))
Theorem	phlstr 17291	A constructed pre-Hilbert space is a structure. Starting from lmodstr 17270 (which has 4 members), we chain strleun 17090 once more, adding an ordered pair to the function, to get all 5 members. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})    ⇒   ⊢ 𝐻 Struct ⟨1, 8⟩
Theorem	phlbase 17292	The base set of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})    ⇒   ⊢ (𝐵 ∈ 𝑋 → 𝐵 = (Base‘𝐻))
Theorem	phlplusg 17293	The additive operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})    ⇒   ⊢ ( + ∈ 𝑋 → + = (+g‘𝐻))
Theorem	phlsca 17294	The ring of scalars of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})    ⇒   ⊢ (𝑇 ∈ 𝑋 → 𝑇 = (Scalar‘𝐻))
Theorem	phlvsca 17295	The scalar product operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})    ⇒   ⊢ ( · ∈ 𝑋 → · = ( ·𝑠 ‘𝐻))
Theorem	phlip 17296	The inner product (Hermitian form) operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})    ⇒   ⊢ ( , ∈ 𝑋 → , = (·𝑖‘𝐻))
Theorem	tsetndx 17297	Index value of the df-tset 17216 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (TopSet‘ndx) = 9
Theorem	tsetid 17298	Utility theorem: index-independent form of df-tset 17216. (Contributed by NM, 20-Oct-2012.)
⊢ TopSet = Slot (TopSet‘ndx)
Theorem	tsetndxnn 17299	The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 31-Oct-2024.)
⊢ (TopSet‘ndx) ∈ ℕ
Theorem	basendxlttsetndx 17300	The index of the slot for the base set is less then the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.)
⊢ (Base‘ndx) < (TopSet‘ndx)