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	cco 17201	Extend class notation with the composition operation.
class comp
Definition	df-plusg 17202	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 17216 instead. (New usage is discouraged.)
⊢ +g = Slot 2
Definition	df-mulr 17203	Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form mulrid 11142 instead. (New usage is discouraged.)
⊢ .r = Slot 3
Definition	df-starv 17204	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 17235 instead. (New usage is discouraged.)
⊢ *𝑟 = Slot 4
Definition	df-sca 17205	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 17247 instead. (New usage is discouraged.)
⊢ Scalar = Slot 5
Definition	df-vsca 17206	Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form vscaid 17252 instead. (New usage is discouraged.)
⊢ ·𝑠 = Slot 6
Definition	df-ip 17207	Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form ipid 17263 instead. (New usage is discouraged.)
⊢ ·𝑖 = Slot 8
Definition	df-tset 17208	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 17285 instead. (New usage is discouraged.)
⊢ TopSet = Slot 9
Definition	df-ple 17209	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 17299 instead. (New usage is discouraged.)
⊢ le = Slot ;10
Definition	df-ocomp 17210	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 17314 instead. (New usage is discouraged.)
⊢ oc = Slot ;11
Definition	df-ds 17211	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 17318 instead. (New usage is discouraged.)
⊢ dist = Slot ;12
Definition	df-unif 17212	Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.) Use its index-independent form unifid 17328 instead. (New usage is discouraged.)
⊢ UnifSet = Slot ;13
Definition	df-hom 17213	Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form homid 17344 instead. (New usage is discouraged.)
⊢ Hom = Slot ;14
Definition	df-cco 17214	Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form ccoid 17346 instead. (New usage is discouraged.)
⊢ comp = Slot ;15
Theorem	plusgndx 17215	Index value of the df-plusg 17202 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (+g‘ndx) = 2
Theorem	plusgid 17216	Utility theorem: index-independent form of df-plusg 17202. (Contributed by NM, 20-Oct-2012.)
⊢ +g = Slot (+g‘ndx)
Theorem	plusgndxnn 17217	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 17218	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 17219	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 17220	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 17221	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 17222	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 17223	+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 17224	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 17221 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    ⇒   ⊢ 𝐵 = (Base‘𝐺)
Theorem	grpplusgx 17225	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 17222 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    ⇒   ⊢ + = (+g‘𝐺)
Theorem	mulrndx 17226	Index value of the df-mulr 17203 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (.r‘ndx) = 3
Theorem	mulridx 17227	Utility theorem: index-independent form of df-mulr 17203. (Contributed by Mario Carneiro, 8-Jun-2013.)
⊢ .r = Slot (.r‘ndx)
Theorem	basendxnmulrndx 17228	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 17229	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 17230	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 17231	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 17232	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 17233	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 17234	Index value of the df-starv 17204 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (*𝑟‘ndx) = 4
Theorem	starvid 17235	Utility theorem: index-independent form of df-starv 17204. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ *𝑟 = Slot (*𝑟‘ndx)
Theorem	starvndxnbasendx 17236	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17240. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (Base‘ndx)
Theorem	starvndxnplusgndx 17237	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17240. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (+g‘ndx)
Theorem	starvndxnmulrndx 17238	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17240. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (.r‘ndx)
Theorem	ressmulr 17239	.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 17240	*𝑟 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ ∗ = (*𝑟‘𝑅)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ∗ = (*𝑟‘𝑆))
Theorem	srngstr 17241	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 17242	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 17243	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 17244	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 17245	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 17246	Index value of the df-sca 17205 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (Scalar‘ndx) = 5
Theorem	scaid 17247	Utility theorem: index-independent form of scalar df-sca 17205. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ Scalar = Slot (Scalar‘ndx)
Theorem	scandxnbasendx 17248	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 17249	The slot for the scalar field is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpsca 20093. (Contributed by AV, 18-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (+g‘ndx)
Theorem	scandxnmulrndx 17250	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 20093. (Contributed by AV, 29-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (.r‘ndx)
Theorem	vscandx 17251	Index value of the df-vsca 17206 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ ( ·𝑠 ‘ndx) = 6
Theorem	vscaid 17252	Utility theorem: index-independent form of scalar product df-vsca 17206. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
Theorem	vscandxnbasendx 17253	The slot for the scalar product is not the slot for the base set in an extensible structure. Formerly part of proof for rmodislmod 20893. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (Base‘ndx)
Theorem	vscandxnplusgndx 17254	The slot for the scalar product is not the slot for the group operation in an extensible structure. Formerly part of proof for rmodislmod 20893. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (+g‘ndx)
Theorem	vscandxnmulrndx 17255	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 20893. (Contributed by AV, 29-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (.r‘ndx)
Theorem	vscandxnscandx 17256	The slot for the scalar product is not the slot for the scalar field in an extensible structure. Formerly part of proof for rmodislmod 20893. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (Scalar‘ndx)
Theorem	lmodstr 17257	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 17258	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 17259	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 17260	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 17261	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 17262	Index value of the df-ip 17207 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (·𝑖‘ndx) = 8
Theorem	ipid 17263	Utility theorem: index-independent form of df-ip 17207. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ ·𝑖 = Slot (·𝑖‘ndx)
Theorem	ipndxnbasendx 17264	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 17265	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 17266	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 20093. (Contributed by AV, 29-Oct-2024.)
⊢ (·𝑖‘ndx) ≠ (.r‘ndx)
Theorem	slotsdifipndx 17267	The slot for the scalar is not the index of other slots. Formerly part of proof for srasca 21144 and sravsca 21145. (Contributed by AV, 12-Nov-2024.)
⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx))
Theorem	ipsstr 17268	Lemma to shorten proofs of ipsbase 17269 through ipsvsca 17273. (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 17269	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 17270	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 17271	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 17272	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 17273	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 17274	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 17275	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 17276	·𝑠 is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ · = ( ·𝑠 ‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘𝐻))
Theorem	ressip 17277	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 17278	A constructed pre-Hilbert space is a structure. Starting from lmodstr 17257 (which has 4 members), we chain strleun 17096 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 17279	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 17280	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 17281	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 17282	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 17283	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 17284	Index value of the df-tset 17208 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (TopSet‘ndx) = 9
Theorem	tsetid 17285	Utility theorem: index-independent form of df-tset 17208. (Contributed by NM, 20-Oct-2012.)
⊢ TopSet = Slot (TopSet‘ndx)
Theorem	tsetndxnn 17286	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 17287	The index of the slot for the base set is less than the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.)
⊢ (Base‘ndx) < (TopSet‘ndx)
Theorem	tsetndxnbasendx 17288	The slot for the topology is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 31-Oct-2024.)
⊢ (TopSet‘ndx) ≠ (Base‘ndx)
Theorem	tsetndxnplusgndx 17289	The slot for the topology is not the slot for the group operation in an extensible structure. Formerly part of proof for oppgtset 19293. (Contributed by AV, 18-Oct-2024.)
⊢ (TopSet‘ndx) ≠ (+g‘ndx)
Theorem	tsetndxnmulrndx 17290	The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
⊢ (TopSet‘ndx) ≠ (.r‘ndx)
Theorem	tsetndxnstarvndx 17291	The slot for the topology is not the slot for the involution in an extensible structure. Formerly part of proof for cnfldfunALT 21336. (Contributed by AV, 11-Nov-2024.)
⊢ (TopSet‘ndx) ≠ (*𝑟‘ndx)
Theorem	slotstnscsi 17292	The slots Scalar, ·𝑠 and ·𝑖 are different from the slot TopSet. Formerly part of sralem 21140 and proofs using it. (Contributed by AV, 29-Oct-2024.)
⊢ ((TopSet‘ndx) ≠ (Scalar‘ndx) ∧ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (TopSet‘ndx) ≠ (·𝑖‘ndx))
Theorem	topgrpstr 17293	A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    ⇒   ⊢ 𝑊 Struct ⟨1, 9⟩
Theorem	topgrpbas 17294	The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    ⇒   ⊢ (𝐵 ∈ 𝑋 → 𝐵 = (Base‘𝑊))
Theorem	topgrpplusg 17295	The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    ⇒   ⊢ ( + ∈ 𝑋 → + = (+g‘𝑊))
Theorem	topgrptset 17296	The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}    ⇒   ⊢ (𝐽 ∈ 𝑋 → 𝐽 = (TopSet‘𝑊))
Theorem	resstset 17297	TopSet is unaffected by restriction. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝐽 = (TopSet‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐽 = (TopSet‘𝐻))
Theorem	plendx 17298	Index value of the df-ple 17209 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.) (New usage is discouraged.)
⊢ (le‘ndx) = ;10
Theorem	pleid 17299	Utility theorem: self-referencing, index-independent form of df-ple 17209. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
⊢ le = Slot (le‘ndx)
Theorem	plendxnn 17300	The index value of the order slot 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, 30-Oct-2024.)
⊢ (le‘ndx) ∈ ℕ