P. 174 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17301-17400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	grpbase 17301	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 17302	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 17303	+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 17304	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 17301 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    ⇒   ⊢ 𝐵 = (Base‘𝐺)
Theorem	grpplusgx 17305	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 17302 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
⊢ 𝐵 ∈ V    &   ⊢ + ∈ V    &   ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}    ⇒   ⊢ + = (+g‘𝐺)
Theorem	mulrndx 17306	Index value of the df-mulr 17283 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (.r‘ndx) = 3
Theorem	mulridx 17307	Utility theorem: index-independent form of df-mulr 17283. (Contributed by Mario Carneiro, 8-Jun-2013.)
⊢ .r = Slot (.r‘ndx)
Theorem	basendxnmulrndx 17308	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 17309	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 17310	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 17311	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 17312	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 17313	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 17314	Index value of the df-starv 17284 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (*𝑟‘ndx) = 4
Theorem	starvid 17315	Utility theorem: index-independent form of df-starv 17284. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ *𝑟 = Slot (*𝑟‘ndx)
Theorem	starvndxnbasendx 17316	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17320. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (Base‘ndx)
Theorem	starvndxnplusgndx 17317	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17320. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (+g‘ndx)
Theorem	starvndxnmulrndx 17318	The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17320. (Contributed by AV, 18-Oct-2024.)
⊢ (*𝑟‘ndx) ≠ (.r‘ndx)
Theorem	ressmulr 17319	.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 17320	*𝑟 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ ∗ = (*𝑟‘𝑅)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ∗ = (*𝑟‘𝑆))
Theorem	srngstr 17321	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 17322	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 17323	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 17324	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 17325	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 17326	Index value of the df-sca 17285 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (Scalar‘ndx) = 5
Theorem	scaid 17327	Utility theorem: index-independent form of scalar df-sca 17285. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ Scalar = Slot (Scalar‘ndx)
Theorem	scandxnbasendx 17328	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 17329	The slot for the scalar field is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpsca 20104. (Contributed by AV, 18-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (+g‘ndx)
Theorem	scandxnmulrndx 17330	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 20104. (Contributed by AV, 29-Oct-2024.)
⊢ (Scalar‘ndx) ≠ (.r‘ndx)
Theorem	vscandx 17331	Index value of the df-vsca 17286 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ ( ·𝑠 ‘ndx) = 6
Theorem	vscaid 17332	Utility theorem: index-independent form of scalar product df-vsca 17286. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
Theorem	vscandxnbasendx 17333	The slot for the scalar product is not the slot for the base set in an extensible structure. Formerly part of proof for rmodislmod 20885. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (Base‘ndx)
Theorem	vscandxnplusgndx 17334	The slot for the scalar product is not the slot for the group operation in an extensible structure. Formerly part of proof for rmodislmod 20885. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (+g‘ndx)
Theorem	vscandxnmulrndx 17335	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 20885. (Contributed by AV, 29-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (.r‘ndx)
Theorem	vscandxnscandx 17336	The slot for the scalar product is not the slot for the scalar field in an extensible structure. Formerly part of proof for rmodislmod 20885. (Contributed by AV, 18-Oct-2024.)
⊢ ( ·𝑠 ‘ndx) ≠ (Scalar‘ndx)
Theorem	lmodstr 17337	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 17338	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 17339	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 17340	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 17341	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 17342	Index value of the df-ip 17287 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (·𝑖‘ndx) = 8
Theorem	ipid 17343	Utility theorem: index-independent form of df-ip 17287. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ ·𝑖 = Slot (·𝑖‘ndx)
Theorem	ipndxnbasendx 17344	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 17345	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 17346	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 20104. (Contributed by AV, 29-Oct-2024.)
⊢ (·𝑖‘ndx) ≠ (.r‘ndx)
Theorem	slotsdifipndx 17347	The slot for the scalar is not the index of other slots. Formerly part of proof for srasca 21136 and sravsca 21137. (Contributed by AV, 12-Nov-2024.)
⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx))
Theorem	ipsstr 17348	Lemma to shorten proofs of ipsbase 17349 through ipsvsca 17353. (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 17349	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 17350	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 17351	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 17352	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 17353	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 17354	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 17355	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 17356	·𝑠 is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ · = ( ·𝑠 ‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘𝐻))
Theorem	ressip 17357	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 17358	A constructed pre-Hilbert space is a structure. Starting from lmodstr 17337 (which has 4 members), we chain strleun 17174 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 17359	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 17360	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 17361	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 17362	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 17363	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 17364	Index value of the df-tset 17288 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (TopSet‘ndx) = 9
Theorem	tsetid 17365	Utility theorem: index-independent form of df-tset 17288. (Contributed by NM, 20-Oct-2012.)
⊢ TopSet = Slot (TopSet‘ndx)
Theorem	tsetndxnn 17366	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 17367	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)
Theorem	tsetndxnbasendx 17368	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 17369	The slot for the topology is not the slot for the group operation in an extensible structure. Formerly part of proof for oppgtset 19333. (Contributed by AV, 18-Oct-2024.)
⊢ (TopSet‘ndx) ≠ (+g‘ndx)
Theorem	tsetndxnmulrndx 17370	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 17371	The slot for the topology is not the slot for the involution in an extensible structure. Formerly part of proof for cnfldfunALT 21328. (Contributed by AV, 11-Nov-2024.)
⊢ (TopSet‘ndx) ≠ (*𝑟‘ndx)
Theorem	slotstnscsi 17372	The slots Scalar, ·𝑠 and ·𝑖 are different from the slot TopSet. Formerly part of sralem 21132 and proofs using it. (Contributed by AV, 29-Oct-2024.)
⊢ ((TopSet‘ndx) ≠ (Scalar‘ndx) ∧ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (TopSet‘ndx) ≠ (·𝑖‘ndx))
Theorem	topgrpstr 17373	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 17374	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 17375	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 17376	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 17377	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 17378	Index value of the df-ple 17289 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.) (New usage is discouraged.)
⊢ (le‘ndx) = ;10
Theorem	pleid 17379	Utility theorem: self-referencing, index-independent form of df-ple 17289. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
⊢ le = Slot (le‘ndx)
Theorem	plendxnn 17380	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
Theorem	basendxltplendx 17381	The index value of the Base slot is less than the index value of the le slot. (Contributed by AV, 30-Oct-2024.)
⊢ (Base‘ndx) < (le‘ndx)
Theorem	plendxnbasendx 17382	The slot for the order is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 30-Oct-2024.)
⊢ (le‘ndx) ≠ (Base‘ndx)
Theorem	plendxnplusgndx 17383	The slot for the "less than or equal to" ordering is not the slot for the group operation in an extensible structure. Formerly part of proof for oppgle 32888. (Contributed by AV, 18-Oct-2024.)
⊢ (le‘ndx) ≠ (+g‘ndx)
Theorem	plendxnmulrndx 17384	The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. Formerly part of proof for opsrmulr 22008. (Contributed by AV, 1-Nov-2024.)
⊢ (le‘ndx) ≠ (.r‘ndx)
Theorem	plendxnscandx 17385	The slot for the "less than or equal to" ordering is not the slot for the scalar in an extensible structure. Formerly part of proof for opsrsca 22010. (Contributed by AV, 1-Nov-2024.)
⊢ (le‘ndx) ≠ (Scalar‘ndx)
Theorem	plendxnvscandx 17386	The slot for the "less than or equal to" ordering is not the slot for the scalar product in an extensible structure. Formerly part of proof for opsrvsca 22009. (Contributed by AV, 1-Nov-2024.)
⊢ (le‘ndx) ≠ ( ·𝑠 ‘ndx)
Theorem	slotsdifplendx 17387	The index of the slot for the distance is not the index of other slots. Formerly part of proof for cnfldfunALT 21328. (Contributed by AV, 11-Nov-2024.)
⊢ ((*𝑟‘ndx) ≠ (le‘ndx) ∧ (TopSet‘ndx) ≠ (le‘ndx))
Theorem	otpsstr 17388	Functionality of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩}    ⇒   ⊢ 𝐾 Struct ⟨1, ;10⟩
Theorem	otpsbas 17389	The base set of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐾))
Theorem	otpstset 17390	The open sets of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩}    ⇒   ⊢ (𝐽 ∈ 𝑉 → 𝐽 = (TopSet‘𝐾))
Theorem	otpsle 17391	The order of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩}    ⇒   ⊢ ( ≤ ∈ 𝑉 → ≤ = (le‘𝐾))
Theorem	ressle 17392	le is unaffected by restriction. (Contributed by Mario Carneiro, 3-Nov-2015.)
⊢ 𝑊 = (𝐾 ↾s 𝐴)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ≤ = (le‘𝑊))
Theorem	ocndx 17393	Index value of the df-ocomp 17290 slot. (Contributed by Mario Carneiro, 25-Oct-2015.) (New usage is discouraged.)
⊢ (oc‘ndx) = ;11
Theorem	ocid 17394	Utility theorem: index-independent form of df-ocomp 17290. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ oc = Slot (oc‘ndx)
Theorem	basendxnocndx 17395	The slot for the orthocomplementation is not the slot for the base set in an extensible structure. Formerly part of proof for thlbas 21654. (Contributed by AV, 11-Nov-2024.)
⊢ (Base‘ndx) ≠ (oc‘ndx)
Theorem	plendxnocndx 17396	The slot for the orthocomplementation is not the slot for the order in an extensible structure. Formerly part of proof for thlle 21655. (Contributed by AV, 11-Nov-2024.)
⊢ (le‘ndx) ≠ (oc‘ndx)
Theorem	dsndx 17397	Index value of the df-ds 17291 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (dist‘ndx) = ;12
Theorem	dsid 17398	Utility theorem: index-independent form of df-ds 17291. (Contributed by Mario Carneiro, 23-Dec-2013.)
⊢ dist = Slot (dist‘ndx)
Theorem	dsndxnn 17399	The index of the slot for the distance in an extensible structure is a positive integer. Formerly part of proof for tmslem 24419. (Contributed by AV, 28-Oct-2024.)
⊢ (dist‘ndx) ∈ ℕ
Theorem	basendxltdsndx 17400	The index of the slot for the base set is less then the index of the slot for the distance in an extensible structure. Formerly part of proof for tmslem 24419. (Contributed by AV, 28-Oct-2024.)
⊢ (Base‘ndx) < (dist‘ndx)