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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | cvsca 17301 | Extend class notation with scalar product. |
class ·𝑠 | ||
Syntax | cip 17302 | Extend class notation with Hermitian form (inner product). |
class ·𝑖 | ||
Syntax | cts 17303 | Extend class notation with the topology component of a topological space. |
class TopSet | ||
Syntax | cple 17304 | Extend class notation with "less than or equal to" for posets. |
class le | ||
Syntax | coc 17305 | Extend class notation with the class of orthocomplementation extractors. |
class oc | ||
Syntax | cds 17306 | Extend class notation with the metric space distance function. |
class dist | ||
Syntax | cunif 17307 | Extend class notation with the uniform structure. |
class UnifSet | ||
Syntax | chom 17308 | Extend class notation with the hom-set structure. |
class Hom | ||
Syntax | cco 17309 | Extend class notation with the composition operation. |
class comp | ||
Definition | df-plusg 17310 | 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 17324 instead. (New usage is discouraged.) |
⊢ +g = Slot 2 | ||
Definition | df-mulr 17311 | Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form mulrid 11256 instead. (New usage is discouraged.) |
⊢ .r = Slot 3 | ||
Definition | df-starv 17312 | 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 17348 instead. (New usage is discouraged.) |
⊢ *𝑟 = Slot 4 | ||
Definition | df-sca 17313 | 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 17360 instead. (New usage is discouraged.) |
⊢ Scalar = Slot 5 | ||
Definition | df-vsca 17314 | Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form vscaid 17365 instead. (New usage is discouraged.) |
⊢ ·𝑠 = Slot 6 | ||
Definition | df-ip 17315 | Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form ipid 17376 instead. (New usage is discouraged.) |
⊢ ·𝑖 = Slot 8 | ||
Definition | df-tset 17316 | 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 17398 instead. (New usage is discouraged.) |
⊢ TopSet = Slot 9 | ||
Definition | df-ple 17317 | 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 17412 instead. (New usage is discouraged.) |
⊢ le = Slot ;10 | ||
Definition | df-ocomp 17318 | 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 17427 instead. (New usage is discouraged.) |
⊢ oc = Slot ;11 | ||
Definition | df-ds 17319 | 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 17431 instead. (New usage is discouraged.) |
⊢ dist = Slot ;12 | ||
Definition | df-unif 17320 | Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.) Use its index-independent form unifid 17441 instead. (New usage is discouraged.) |
⊢ UnifSet = Slot ;13 | ||
Definition | df-hom 17321 | Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form homid 17457 instead. (New usage is discouraged.) |
⊢ Hom = Slot ;14 | ||
Definition | df-cco 17322 | Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form ccoid 17459 instead. (New usage is discouraged.) |
⊢ comp = Slot ;15 | ||
Theorem | plusgndx 17323 | Index value of the df-plusg 17310 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.) |
⊢ (+g‘ndx) = 2 | ||
Theorem | plusgid 17324 | Utility theorem: index-independent form of df-plusg 17310. (Contributed by NM, 20-Oct-2012.) |
⊢ +g = Slot (+g‘ndx) | ||
Theorem | plusgndxnn 17325 | 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 17326 | 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 17327 | 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 17328 | Obsolete version of basendxnplusgndx 17327 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 17329 | A constructed group is a structure on 1...2. Depending on hard-coded index values. Use grpstrndx 17330 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 17330 | 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 17331 | 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 17332 | Obsolete version of grpbase 17331 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 17333 | 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 17334 | Obsolete version of grpplusg 17333 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 17335 | +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 17336 | 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 17331 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.) |
⊢ 𝐵 ∈ V & ⊢ + ∈ V & ⊢ 𝐺 = {〈1, 𝐵〉, 〈2, + 〉} ⇒ ⊢ 𝐵 = (Base‘𝐺) | ||
Theorem | grpplusgx 17337 | 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 17333 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.) |
⊢ 𝐵 ∈ V & ⊢ + ∈ V & ⊢ 𝐺 = {〈1, 𝐵〉, 〈2, + 〉} ⇒ ⊢ + = (+g‘𝐺) | ||
Theorem | mulrndx 17338 | Index value of the df-mulr 17311 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.) |
⊢ (.r‘ndx) = 3 | ||
Theorem | mulridx 17339 | Utility theorem: index-independent form of df-mulr 17311. (Contributed by Mario Carneiro, 8-Jun-2013.) |
⊢ .r = Slot (.r‘ndx) | ||
Theorem | basendxnmulrndx 17340 | 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 17341 | Obsolete version of basendxnmulrndx 17340 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 17342 | 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 17343 | 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 17344 | 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 17345 | 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 17346 | 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 17347 | Index value of the df-starv 17312 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.) |
⊢ (*𝑟‘ndx) = 4 | ||
Theorem | starvid 17348 | Utility theorem: index-independent form of df-starv 17312. (Contributed by Mario Carneiro, 6-Oct-2013.) |
⊢ *𝑟 = Slot (*𝑟‘ndx) | ||
Theorem | starvndxnbasendx 17349 | The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17353. (Contributed by AV, 18-Oct-2024.) |
⊢ (*𝑟‘ndx) ≠ (Base‘ndx) | ||
Theorem | starvndxnplusgndx 17350 | The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17353. (Contributed by AV, 18-Oct-2024.) |
⊢ (*𝑟‘ndx) ≠ (+g‘ndx) | ||
Theorem | starvndxnmulrndx 17351 | The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17353. (Contributed by AV, 18-Oct-2024.) |
⊢ (*𝑟‘ndx) ≠ (.r‘ndx) | ||
Theorem | ressmulr 17352 | .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 17353 | *𝑟 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ ∗ = (*𝑟‘𝑅) ⇒ ⊢ (𝐴 ∈ 𝑉 → ∗ = (*𝑟‘𝑆)) | ||
Theorem | srngstr 17354 | 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 17355 | 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 17356 | 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 17357 | 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 17358 | 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 17359 | Index value of the df-sca 17313 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.) |
⊢ (Scalar‘ndx) = 5 | ||
Theorem | scaid 17360 | Utility theorem: index-independent form of scalar df-sca 17313. (Contributed by Mario Carneiro, 19-Jun-2014.) |
⊢ Scalar = Slot (Scalar‘ndx) | ||
Theorem | scandxnbasendx 17361 | 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 17362 | The slot for the scalar field is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpsca 20159. (Contributed by AV, 18-Oct-2024.) |
⊢ (Scalar‘ndx) ≠ (+g‘ndx) | ||
Theorem | scandxnmulrndx 17363 | 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 20159. (Contributed by AV, 29-Oct-2024.) |
⊢ (Scalar‘ndx) ≠ (.r‘ndx) | ||
Theorem | vscandx 17364 | Index value of the df-vsca 17314 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.) |
⊢ ( ·𝑠 ‘ndx) = 6 | ||
Theorem | vscaid 17365 | Utility theorem: index-independent form of scalar product df-vsca 17314. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx) | ||
Theorem | vscandxnbasendx 17366 | The slot for the scalar product is not the slot for the base set in an extensible structure. Formerly part of proof for rmodislmod 20944. (Contributed by AV, 18-Oct-2024.) |
⊢ ( ·𝑠 ‘ndx) ≠ (Base‘ndx) | ||
Theorem | vscandxnplusgndx 17367 | The slot for the scalar product is not the slot for the group operation in an extensible structure. Formerly part of proof for rmodislmod 20944. (Contributed by AV, 18-Oct-2024.) |
⊢ ( ·𝑠 ‘ndx) ≠ (+g‘ndx) | ||
Theorem | vscandxnmulrndx 17368 | 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 20944. (Contributed by AV, 29-Oct-2024.) |
⊢ ( ·𝑠 ‘ndx) ≠ (.r‘ndx) | ||
Theorem | vscandxnscandx 17369 | The slot for the scalar product is not the slot for the scalar field in an extensible structure. Formerly part of proof for rmodislmod 20944. (Contributed by AV, 18-Oct-2024.) |
⊢ ( ·𝑠 ‘ndx) ≠ (Scalar‘ndx) | ||
Theorem | lmodstr 17370 | 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 17371 | 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 17372 | 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 17373 | 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 17374 | 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 17375 | Index value of the df-ip 17315 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.) |
⊢ (·𝑖‘ndx) = 8 | ||
Theorem | ipid 17376 | Utility theorem: index-independent form of df-ip 17315. (Contributed by Mario Carneiro, 6-Oct-2013.) |
⊢ ·𝑖 = Slot (·𝑖‘ndx) | ||
Theorem | ipndxnbasendx 17377 | 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 17378 | 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 17379 | 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 20159. (Contributed by AV, 29-Oct-2024.) |
⊢ (·𝑖‘ndx) ≠ (.r‘ndx) | ||
Theorem | slotsdifipndx 17380 | The slot for the scalar is not the index of other slots. Formerly part of proof for srasca 21200 and sravsca 21202. (Contributed by AV, 12-Nov-2024.) |
⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx)) | ||
Theorem | ipsstr 17381 | Lemma to shorten proofs of ipsbase 17382 through ipsvsca 17386. (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 17382 | 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 17383 | 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 17384 | 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 17385 | 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 17386 | 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 17387 | 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 17388 | 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 17389 | ·𝑠 is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ · = ( ·𝑠 ‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘𝐻)) | ||
Theorem | ressip 17390 | 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 17391 | A constructed pre-Hilbert space is a structure. Starting from lmodstr 17370 (which has 4 members), we chain strleun 17190 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 17392 | 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 17393 | 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 17394 | 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 17395 | 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 17396 | 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 17397 | Index value of the df-tset 17316 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.) |
⊢ (TopSet‘ndx) = 9 | ||
Theorem | tsetid 17398 | Utility theorem: index-independent form of df-tset 17316. (Contributed by NM, 20-Oct-2012.) |
⊢ TopSet = Slot (TopSet‘ndx) | ||
Theorem | tsetndxnn 17399 | 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 17400 | 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) |
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