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Theorem List for Metamath Proof Explorer - 16001-16100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremressstarv 16001 *𝑟 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝑆 = (𝑅s 𝐴)    &    = (*𝑟𝑅)       (𝐴𝑉 = (*𝑟𝑆))

Theoremsrngfn 16002 A constructed star ring is a function with domain contained in 1 thru 4. (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⟩

Theoremsrngbase 16003 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), ⟩})       (𝐵𝑋𝐵 = (Base‘𝑅))

Theoremsrngplusg 16004 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+ = (+g𝑅))

Theoremsrngmulr 16005 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· = (.r𝑅))

Theoremsrnginvl 16006 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 = (*𝑟𝑅))

Theoremscandx 16007 Index value of the df-sca 15951 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(Scalar‘ndx) = 5

Theoremscaid 16008 Utility theorem: index-independent form of scalar df-sca 15951. (Contributed by Mario Carneiro, 19-Jun-2014.)
Scalar = Slot (Scalar‘ndx)

Theoremvscandx 16009 Index value of the df-vsca 15952 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
( ·𝑠 ‘ndx) = 6

Theoremvscaid 16010 Utility theorem: index-independent form of scalar product df-vsca 15952. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
·𝑠 = Slot ( ·𝑠 ‘ndx)

Theoremlmodstr 16011 A constructed left module or left vector space is a function. (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⟩

Theoremlmodbase 16012 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), · ⟩})       (𝐵𝑋𝐵 = (Base‘𝑊))

Theoremlmodplusg 16013 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+ = (+g𝑊))

Theoremlmodsca 16014 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), · ⟩})       (𝐹𝑋𝐹 = (Scalar‘𝑊))

Theoremlmodvsca 16015 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· = ( ·𝑠𝑊))

Theoremipndx 16016 Index value of the df-ip 15953 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(·𝑖‘ndx) = 8

Theoremipid 16017 Utility theorem: index-independent form of df-ip 15953. (Contributed by Mario Carneiro, 6-Oct-2013.)
·𝑖 = Slot (·𝑖‘ndx)

Theoremipsstr 16018 Lemma to shorten proofs of ipsbase 16019 through ipsvsca 16023. (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⟩

Theoremipsbase 16019 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⟩})       (𝐵𝑉𝐵 = (Base‘𝐴))

Theoremipsaddg 16020 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+ = (+g𝐴))

Theoremipsmulr 16021 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× = (.r𝐴))

Theoremipssca 16022 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⟩})       (𝑆𝑉𝑆 = (Scalar‘𝐴))

Theoremipsvsca 16023 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· = ( ·𝑠𝐴))

Theoremipsip 16024 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⟩})       (𝐼𝑉𝐼 = (·𝑖𝐴))

Theoremresssca 16025 Scalar is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐻 = (𝐺s 𝐴)    &   𝐹 = (Scalar‘𝐺)       (𝐴𝑉𝐹 = (Scalar‘𝐻))

Theoremressvsca 16026 ·𝑠 is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐻 = (𝐺s 𝐴)    &    · = ( ·𝑠𝐺)       (𝐴𝑉· = ( ·𝑠𝐻))

Theoremressip 16027 The inner product is unaffected by restriction. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (𝐺s 𝐴)    &    , = (·𝑖𝐺)       (𝐴𝑉, = (·𝑖𝐻))

Theoremphlstr 16028 A constructed pre-Hilbert space is a structure. Starting from lmodstr 16011 (which has 4 members), we chain strleun 15966 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⟩

Theoremphlbase 16029 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), , ⟩})       (𝐵𝑋𝐵 = (Base‘𝐻))

Theoremphlplusg 16030 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+ = (+g𝐻))

Theoremphlsca 16031 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), , ⟩})       (𝑇𝑋𝑇 = (Scalar‘𝐻))

Theoremphlvsca 16032 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· = ( ·𝑠𝐻))

Theoremphlip 16033 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, = (·𝑖𝐻))

Theoremtsetndx 16034 Index value of the df-tset 15954 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(TopSet‘ndx) = 9

Theoremtsetid 16035 Utility theorem: index-independent form of df-tset 15954. (Contributed by NM, 20-Oct-2012.)
TopSet = Slot (TopSet‘ndx)

Theoremtopgrpstr 16036 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⟩

Theoremtopgrpbas 16037 The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}       (𝐵𝑋𝐵 = (Base‘𝑊))

Theoremtopgrpplusg 16038 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+ = (+g𝑊))

Theoremtopgrptset 16039 The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}       (𝐽𝑋𝐽 = (TopSet‘𝑊))

Theoremresstset 16040 TopSet is unaffected by restriction. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐻 = (𝐺s 𝐴)    &   𝐽 = (TopSet‘𝐺)       (𝐴𝑉𝐽 = (TopSet‘𝐻))

Theoremplendx 16041 Index value of the df-ple 15955 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
(le‘ndx) = 10

TheoremplendxOLD 16042 Obsolete version of df-ple 15955 as of 9-Sep-2021. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(le‘ndx) = 10

Theorempleid 16043 Utility theorem: self-referencing, index-independent form of df-ple 15955. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
le = Slot (le‘ndx)

TheorempleidOLD 16044 Obsolete version of otpsstr 16045 as of 9-Sep-2021. (Contributed by Mario Carneiro, 9-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
le = Slot (le‘ndx)

Theoremotpsstr 16045 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⟩

Theoremotpsbas 16046 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), ⟩}       (𝐵𝑉𝐵 = (Base‘𝐾))

Theoremotpstset 16047 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), ⟩}       (𝐽𝑉𝐽 = (TopSet‘𝐾))

Theoremotpsle 16048 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))

TheoremotpsstrOLD 16049 Obsolete version of otpsstr 16045 as of 9-Sep-2021. (Contributed by Mario Carneiro, 12-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩}       𝐾 Struct ⟨1, 10⟩

TheoremotpsbasOLD 16050 Obsolete version of otpsbas 16046 as of 9-Sep-2021. (Contributed by Mario Carneiro, 12-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩}       (𝐵𝑉𝐵 = (Base‘𝐾))

TheoremotpstsetOLD 16051 Obsolete version of otpstset 16047 as of 9-Sep-2021. (Contributed by Mario Carneiro, 12-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩}       (𝐽𝑉𝐽 = (TopSet‘𝐾))

TheoremotpsleOLD 16052 Obsolete version of otpsle 16048 as of 9-Sep-2021. (Contributed by Mario Carneiro, 12-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩}       ( 𝑉 = (le‘𝐾))

Theoremressle 16053 le is unaffected by restriction. (Contributed by Mario Carneiro, 3-Nov-2015.)
𝑊 = (𝐾s 𝐴)    &    = (le‘𝐾)       (𝐴𝑉 = (le‘𝑊))

Theoremocndx 16054 Index value of the df-ocomp 15957 slot. (Contributed by Mario Carneiro, 25-Oct-2015.)
(oc‘ndx) = 11

Theoremocid 16055 Utility theorem: index-independent form of df-ocomp 15957. (Contributed by Mario Carneiro, 25-Oct-2015.)
oc = Slot (oc‘ndx)

Theoremdsndx 16056 Index value of the df-ds 15958 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(dist‘ndx) = 12

Theoremdsid 16057 Utility theorem: index-independent form of df-ds 15958. (Contributed by Mario Carneiro, 23-Dec-2013.)
dist = Slot (dist‘ndx)

Theoremunifndx 16058 Index value of the df-unif 15959 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.)
(UnifSet‘ndx) = 13

Theoremunifid 16059 Utility theorem: index-independent form of df-unif 15959. (Contributed by Thierry Arnoux, 17-Dec-2017.)
UnifSet = Slot (UnifSet‘ndx)

Theoremodrngstr 16060 Functionality of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.) (Proof shortened by AV, 15-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩})       𝑊 Struct ⟨1, 12⟩

Theoremodrngbas 16061 The base set of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩})       (𝐵𝑉𝐵 = (Base‘𝑊))

Theoremodrngplusg 16062 The addition operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩})       ( +𝑉+ = (+g𝑊))

Theoremodrngmulr 16063 The multiplication operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩})       ( ·𝑉· = (.r𝑊))

Theoremodrngtset 16064 The open sets of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩})       (𝐽𝑉𝐽 = (TopSet‘𝑊))

Theoremodrngle 16065 The order of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩})       ( 𝑉 = (le‘𝑊))

Theoremodrngds 16066 The metric of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩, ⟨(dist‘ndx), 𝐷⟩})       (𝐷𝑉𝐷 = (dist‘𝑊))

Theoremressds 16067 dist is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐻 = (𝐺s 𝐴)    &   𝐷 = (dist‘𝐺)       (𝐴𝑉𝐷 = (dist‘𝐻))

Theoremhomndx 16068 Index value of the df-hom 15960 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
(Hom ‘ndx) = 14

Theoremhomid 16069 Utility theorem: index-independent form of df-hom 15960. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hom = Slot (Hom ‘ndx)

Theoremccondx 16070 Index value of the df-cco 15961 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
(comp‘ndx) = 15

Theoremccoid 16071 Utility theorem: index-independent form of df-cco 15961. (Contributed by Mario Carneiro, 7-Jan-2017.)
comp = Slot (comp‘ndx)

Theoremresshom 16072 Hom is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
𝐷 = (𝐶s 𝐴)    &   𝐻 = (Hom ‘𝐶)       (𝐴𝑉𝐻 = (Hom ‘𝐷))

Theoremressco 16073 comp is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
𝐷 = (𝐶s 𝐴)    &    · = (comp‘𝐶)       (𝐴𝑉· = (comp‘𝐷))

Theoremslotsbhcdif 16074 The slots Base, Hom and comp are different. (Contributed by AV, 5-Mar-2020.)
((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx))

7.1.3  Definition of the structure product

Syntaxcrest 16075 Extend class notation with the function returning a subspace topology.
class t

Syntaxctopn 16076 Extend class notation with the topology extractor function.
class TopOpen

Definitiondf-rest 16077* Function returning the subspace topology induced by the topology 𝑦 and the set 𝑥. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦𝑗 ↦ (𝑦𝑥)))

Definitiondf-topn 16078 Define the topology extractor function. This differs from df-tset 15954 when a structure has been restricted using df-ress 15859; in this case the TopSet component will still have a topology over the larger set, and this function fixes this by restricting the topology as well. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))

Theoremrestfn 16079 The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
t Fn (V × V)

Theoremtopnfn 16080 The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOpen Fn V

Theoremrestval 16081* The subspace topology induced by the topology 𝐽 on the set 𝐴. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))

Theoremelrest 16082* The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
((𝐽𝑉𝐵𝑊) → (𝐴 ∈ (𝐽t 𝐵) ↔ ∃𝑥𝐽 𝐴 = (𝑥𝐵)))

Theoremelrestr 16083 Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
((𝐽𝑉𝑆𝑊𝐴𝐽) → (𝐴𝑆) ∈ (𝐽t 𝑆))

Theorem0rest 16084 Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.)
(∅ ↾t 𝐴) = ∅

Theoremrestid2 16085 The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → (𝐽t 𝐴) = 𝐽)

Theoremrestsspw 16086 The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽t 𝐴) ⊆ 𝒫 𝐴

Theoremfirest 16087 The finite intersections operator commutes with restriction. (Contributed by Mario Carneiro, 30-Aug-2015.)
(fi‘(𝐽t 𝐴)) = ((fi‘𝐽) ↾t 𝐴)

Theoremrestid 16088 The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
𝑋 = 𝐽       (𝐽𝑉 → (𝐽t 𝑋) = 𝐽)

Theoremtopnval 16089 Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopSet‘𝑊)       (𝐽t 𝐵) = (TopOpen‘𝑊)

Theoremtopnid 16090 Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopSet‘𝑊)       (𝐽 ⊆ 𝒫 𝐵𝐽 = (TopOpen‘𝑊))

Theoremtopnpropd 16091 The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))       (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

Syntaxctg 16092 Extend class notation with a function that converts a basis to its corresponding topology.
class topGen

Syntaxcpt 16093 Extend class notation with a function whose value is a product topology.
class t

Syntaxc0g 16094 Extend class notation with group identity element.
class 0g

Syntaxcgsu 16095 Extend class notation to include finitely supported group sums.
class Σg

Definitiondf-0g 16096* Define group identity element. Remark: this definition is required here because the symbol 0g is already used in df-gsum 16097. The related theorems are provided later, see grpidval 17254. (Contributed by NM, 20-Aug-2011.)
0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))))

Definitiondf-gsum 16097* Define the group sum for the structure 𝐺 of a finite sequence of elements whose values are defined by the expression 𝐵 and whose set of indices is 𝐴. It may be viewed as a product (if 𝐺 is a multiplication), a sum (if 𝐺 is an addition) or whatever. The variable 𝑘 is normally a free variable in 𝐵 ( i.e. 𝐵 can be thought of as 𝐵(𝑘)). The definition is meaningful in different contexts, depending on the size of the index set 𝐴 and each demanding different properties of 𝐺.

1. If 𝐴 = ∅ and 𝐺 has an identity element, then the sum equals this identity.

2. If 𝐴 = (𝑀...𝑁) and 𝐺 is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e. (𝐵(1) + 𝐵(2)) + 𝐵(3) etc.

3. If 𝐴 is a finite set (or is nonzero for finitely many indices) and 𝐺 is a commutative monoid, then the sum adds up these elements in some order, which is then uniquely defined.

4. If 𝐴 is an infinite set and 𝐺 is a Hausdorff topological group, then there is a meaningful sum, but Σg cannot handle this case. See df-tsms 21924. Remark: this definition is required here because the symbol Σg is already used in df-prds 16102 and df-imas 16162. The related theorems are provided later, see gsumvalx 17264. (Contributed by FL, 5-Sep-2010.) (Revised by FL, 17-Oct-2011.) (Revised by Mario Carneiro, 7-Dec-2014.)

Σg = (𝑤 ∈ V, 𝑓 ∈ V ↦ {𝑥 ∈ (Base‘𝑤) ∣ ∀𝑦 ∈ (Base‘𝑤)((𝑥(+g𝑤)𝑦) = 𝑦 ∧ (𝑦(+g𝑤)𝑥) = 𝑦)} / 𝑜if(ran 𝑓𝑜, (0g𝑤), if(dom 𝑓 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝑤), 𝑓)‘𝑛))), (℩𝑥𝑔[(𝑓 “ (V ∖ 𝑜)) / 𝑦](𝑔:(1...(#‘𝑦))–1-1-onto𝑦𝑥 = (seq1((+g𝑤), (𝑓𝑔))‘(#‘𝑦)))))))

Definitiondf-topgen 16098* Define a function that converts a basis to its corresponding topology. Equivalent to the definition of a topology generated by a basis in [Munkres] p. 78 (see tgval2 20754). The first use of this definition is tgval 20753 but the token is used in df-pt 16099. See tgval3 20761 for an alternate expression for the value. (Contributed by NM, 16-Jul-2006.)
topGen = (𝑥 ∈ V ↦ {𝑦𝑦 (𝑥 ∩ 𝒫 𝑦)})

Definitiondf-pt 16099* Define the product topology on a collection of topologies. For convenience, it is defined on arbitrary collections of sets, expressed as a function from some index set to the subbases of each factor space. (Contributed by Mario Carneiro, 3-Feb-2015.)
t = (𝑓 ∈ V ↦ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦))}))

Syntaxcprds 16100 The function constructing structure products.
class Xs

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