![]() |
Metamath
Proof Explorer Theorem List (p. 173 of 480) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30209) |
![]() (30210-31732) |
![]() (31733-47936) |
Type | Label | Description |
---|---|---|
Statement | ||
Syntax | cstv 17201 | Extend class notation with involution. |
class *𝑟 | ||
Syntax | csca 17202 | Extend class notation with scalar field. |
class Scalar | ||
Syntax | cvsca 17203 | Extend class notation with scalar product. |
class ·𝑠 | ||
Syntax | cip 17204 | Extend class notation with Hermitian form (inner product). |
class ·𝑖 | ||
Syntax | cts 17205 | Extend class notation with the topology component of a topological space. |
class TopSet | ||
Syntax | cple 17206 | Extend class notation with "less than or equal to" for posets. |
class le | ||
Syntax | coc 17207 | Extend class notation with the class of orthocomplementation extractors. |
class oc | ||
Syntax | cds 17208 | Extend class notation with the metric space distance function. |
class dist | ||
Syntax | cunif 17209 | Extend class notation with the uniform structure. |
class UnifSet | ||
Syntax | chom 17210 | Extend class notation with the hom-set structure. |
class Hom | ||
Syntax | cco 17211 | Extend class notation with the composition operation. |
class comp | ||
Definition | df-plusg 17212 | 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 17226 instead. (New usage is discouraged.) |
⊢ +g = Slot 2 | ||
Definition | df-mulr 17213 | Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form mulrid 11214 instead. (New usage is discouraged.) |
⊢ .r = Slot 3 | ||
Definition | df-starv 17214 | 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 17250 instead. (New usage is discouraged.) |
⊢ *𝑟 = Slot 4 | ||
Definition | df-sca 17215 | 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 17262 instead. (New usage is discouraged.) |
⊢ Scalar = Slot 5 | ||
Definition | df-vsca 17216 | Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form vscaid 17267 instead. (New usage is discouraged.) |
⊢ ·𝑠 = Slot 6 | ||
Definition | df-ip 17217 | Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form ipid 17278 instead. (New usage is discouraged.) |
⊢ ·𝑖 = Slot 8 | ||
Definition | df-tset 17218 | 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 17300 instead. (New usage is discouraged.) |
⊢ TopSet = Slot 9 | ||
Definition | df-ple 17219 | 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 17314 instead. (New usage is discouraged.) |
⊢ le = Slot ;10 | ||
Definition | df-ocomp 17220 | 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 17329 instead. (New usage is discouraged.) |
⊢ oc = Slot ;11 | ||
Definition | df-ds 17221 | 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 17333 instead. (New usage is discouraged.) |
⊢ dist = Slot ;12 | ||
Definition | df-unif 17222 | Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.) Use its index-independent form unifid 17343 instead. (New usage is discouraged.) |
⊢ UnifSet = Slot ;13 | ||
Definition | df-hom 17223 | Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form homid 17359 instead. (New usage is discouraged.) |
⊢ Hom = Slot ;14 | ||
Definition | df-cco 17224 | Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) Use its index-independent form ccoid 17361 instead. (New usage is discouraged.) |
⊢ comp = Slot ;15 | ||
Theorem | plusgndx 17225 | Index value of the df-plusg 17212 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.) |
⊢ (+g‘ndx) = 2 | ||
Theorem | plusgid 17226 | Utility theorem: index-independent form of df-plusg 17212. (Contributed by NM, 20-Oct-2012.) |
⊢ +g = Slot (+g‘ndx) | ||
Theorem | plusgndxnn 17227 | 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 17228 | 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 17229 | 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 17230 | Obsolete version of basendxnplusgndx 17229 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 17231 | A constructed group is a structure on 1...2. Depending on hard-coded index values. Use grpstrndx 17232 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 17232 | 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 17233 | 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 17234 | Obsolete version of grpbase 17233 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 17235 | 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 17236 | Obsolete version of grpplusg 17235 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 17237 | +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 17238 | 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 17233 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.) |
⊢ 𝐵 ∈ V & ⊢ + ∈ V & ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩} ⇒ ⊢ 𝐵 = (Base‘𝐺) | ||
Theorem | grpplusgx 17239 | 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 17235 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.) |
⊢ 𝐵 ∈ V & ⊢ + ∈ V & ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩} ⇒ ⊢ + = (+g‘𝐺) | ||
Theorem | mulrndx 17240 | Index value of the df-mulr 17213 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.) |
⊢ (.r‘ndx) = 3 | ||
Theorem | mulridx 17241 | Utility theorem: index-independent form of df-mulr 17213. (Contributed by Mario Carneiro, 8-Jun-2013.) |
⊢ .r = Slot (.r‘ndx) | ||
Theorem | basendxnmulrndx 17242 | 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 17243 | Obsolete proof of basendxnmulrndx 17242 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 17244 | 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 17245 | 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 17246 | 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 17247 | 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 17248 | 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 17249 | Index value of the df-starv 17214 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.) |
⊢ (*𝑟‘ndx) = 4 | ||
Theorem | starvid 17250 | Utility theorem: index-independent form of df-starv 17214. (Contributed by Mario Carneiro, 6-Oct-2013.) |
⊢ *𝑟 = Slot (*𝑟‘ndx) | ||
Theorem | starvndxnbasendx 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 17255. (Contributed by AV, 18-Oct-2024.) |
⊢ (*𝑟‘ndx) ≠ (Base‘ndx) | ||
Theorem | starvndxnplusgndx 17252 | The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17255. (Contributed by AV, 18-Oct-2024.) |
⊢ (*𝑟‘ndx) ≠ (+g‘ndx) | ||
Theorem | starvndxnmulrndx 17253 | The slot for the involution function is not the slot for the base set in an extensible structure. Formerly part of proof for ressstarv 17255. (Contributed by AV, 18-Oct-2024.) |
⊢ (*𝑟‘ndx) ≠ (.r‘ndx) | ||
Theorem | ressmulr 17254 | .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 17255 | *𝑟 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ ∗ = (*𝑟‘𝑅) ⇒ ⊢ (𝐴 ∈ 𝑉 → ∗ = (*𝑟‘𝑆)) | ||
Theorem | srngstr 17256 | 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 17257 | 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 17258 | 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 17259 | 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 17260 | 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 17261 | Index value of the df-sca 17215 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.) |
⊢ (Scalar‘ndx) = 5 | ||
Theorem | scaid 17262 | Utility theorem: index-independent form of scalar df-sca 17215. (Contributed by Mario Carneiro, 19-Jun-2014.) |
⊢ Scalar = Slot (Scalar‘ndx) | ||
Theorem | scandxnbasendx 17263 | 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 17264 | The slot for the scalar field is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpsca 19997. (Contributed by AV, 18-Oct-2024.) |
⊢ (Scalar‘ndx) ≠ (+g‘ndx) | ||
Theorem | scandxnmulrndx 17265 | 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 19997. (Contributed by AV, 29-Oct-2024.) |
⊢ (Scalar‘ndx) ≠ (.r‘ndx) | ||
Theorem | vscandx 17266 | Index value of the df-vsca 17216 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.) |
⊢ ( ·𝑠 ‘ndx) = 6 | ||
Theorem | vscaid 17267 | Utility theorem: index-independent form of scalar product df-vsca 17216. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx) | ||
Theorem | vscandxnbasendx 17268 | The slot for the scalar product is not the slot for the base set in an extensible structure. Formerly part of proof for rmodislmod 20545. (Contributed by AV, 18-Oct-2024.) |
⊢ ( ·𝑠 ‘ndx) ≠ (Base‘ndx) | ||
Theorem | vscandxnplusgndx 17269 | The slot for the scalar product is not the slot for the group operation in an extensible structure. Formerly part of proof for rmodislmod 20545. (Contributed by AV, 18-Oct-2024.) |
⊢ ( ·𝑠 ‘ndx) ≠ (+g‘ndx) | ||
Theorem | vscandxnmulrndx 17270 | 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 20545. (Contributed by AV, 29-Oct-2024.) |
⊢ ( ·𝑠 ‘ndx) ≠ (.r‘ndx) | ||
Theorem | vscandxnscandx 17271 | The slot for the scalar product is not the slot for the scalar field in an extensible structure. Formerly part of proof for rmodislmod 20545. (Contributed by AV, 18-Oct-2024.) |
⊢ ( ·𝑠 ‘ndx) ≠ (Scalar‘ndx) | ||
Theorem | lmodstr 17272 | 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 17273 | 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 17274 | 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 17275 | 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 17276 | 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 17277 | Index value of the df-ip 17217 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.) |
⊢ (·𝑖‘ndx) = 8 | ||
Theorem | ipid 17278 | Utility theorem: index-independent form of df-ip 17217. (Contributed by Mario Carneiro, 6-Oct-2013.) |
⊢ ·𝑖 = Slot (·𝑖‘ndx) | ||
Theorem | ipndxnbasendx 17279 | 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 17280 | 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 17281 | 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 19997. (Contributed by AV, 29-Oct-2024.) |
⊢ (·𝑖‘ndx) ≠ (.r‘ndx) | ||
Theorem | slotsdifipndx 17282 | The slot for the scalar is not the index of other slots. Formerly part of proof for srasca 20804 and sravsca 20806. (Contributed by AV, 12-Nov-2024.) |
⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx)) | ||
Theorem | ipsstr 17283 | Lemma to shorten proofs of ipsbase 17284 through ipsvsca 17288. (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 17284 | 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 17285 | 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 17286 | 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 17287 | 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 17288 | 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 17289 | 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 17290 | 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 17291 | ·𝑠 is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ · = ( ·𝑠 ‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘𝐻)) | ||
Theorem | ressip 17292 | 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 17293 | A constructed pre-Hilbert space is a structure. Starting from lmodstr 17272 (which has 4 members), we chain strleun 17092 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 17294 | 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 17295 | 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 17296 | 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 17297 | 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 17298 | 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 17299 | Index value of the df-tset 17218 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.) |
⊢ (TopSet‘ndx) = 9 | ||
Theorem | tsetid 17300 | Utility theorem: index-independent form of df-tset 17218. (Contributed by NM, 20-Oct-2012.) |
⊢ TopSet = Slot (TopSet‘ndx) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |