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Theorem List for Metamath Proof Explorer - 15701-15800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgrpplusg 15701 The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}       ( +𝑉+ = (+g𝐺))
 
Theoremressplusg 15702 +g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝐻 = (𝐺s 𝐴)    &    + = (+g𝐺)       (𝐴𝑉+ = (+g𝐻))
 
Theoremgrpbasex 15703 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 15700 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
𝐵 ∈ V    &    + ∈ V    &   𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}       𝐵 = (Base‘𝐺)
 
Theoremgrpplusgx 15704 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 15701 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
𝐵 ∈ V    &    + ∈ V    &   𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}        + = (+g𝐺)
 
Theoremmulrndx 15705 Index value of the df-mulr 15666 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(.r‘ndx) = 3
 
Theoremmulrid 15706 Utility theorem: index-independent form of df-mulr 15666. (Contributed by Mario Carneiro, 8-Jun-2013.)
.r = Slot (.r‘ndx)
 
Theoremrngstr 15707 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⟩
 
Theoremrngbase 15708 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), · ⟩}       (𝐵𝑉𝐵 = (Base‘𝑅))
 
Theoremrngplusg 15709 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+ = (+g𝑅))
 
Theoremrngmulr 15710 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· = (.r𝑅))
 
Theoremstarvndx 15711 Index value of the df-starv 15667 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(*𝑟‘ndx) = 4
 
Theoremstarvid 15712 Utility theorem: index-independent form of df-starv 15667. (Contributed by Mario Carneiro, 6-Oct-2013.)
*𝑟 = Slot (*𝑟‘ndx)
 
Theoremressmulr 15713 .r is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝑆 = (𝑅s 𝐴)    &    · = (.r𝑅)       (𝐴𝑉· = (.r𝑆))
 
Theoremressstarv 15714 *𝑟 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝑆 = (𝑅s 𝐴)    &    = (*𝑟𝑅)       (𝐴𝑉 = (*𝑟𝑆))
 
Theoremsrngfn 15715 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 15716 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 15717 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 15718 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 15719 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 15720 Index value of the df-sca 15668 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(Scalar‘ndx) = 5
 
Theoremscaid 15721 Utility theorem: index-independent form of scalar df-sca 15668. (Contributed by Mario Carneiro, 19-Jun-2014.)
Scalar = Slot (Scalar‘ndx)
 
Theoremvscandx 15722 Index value of the df-vsca 15669 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
( ·𝑠 ‘ndx) = 6
 
Theoremvscaid 15723 Utility theorem: index-independent form of scalar product df-vsca 15669. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
·𝑠 = Slot ( ·𝑠 ‘ndx)
 
Theoremlmodstr 15724 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 15725 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 15726 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 15727 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 15728 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 15729 Index value of the df-ip 15670 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(·𝑖‘ndx) = 8
 
Theoremipid 15730 Utility theorem: index-independent form of df-ip 15670. (Contributed by Mario Carneiro, 6-Oct-2013.)
·𝑖 = Slot (·𝑖‘ndx)
 
Theoremipsstr 15731 Lemma to shorten proofs of ipsbase 15732 through ipsvsca 15736. (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 15732 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 15733 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 15734 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 15735 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 15736 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 15737 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 15738 Scalar is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐻 = (𝐺s 𝐴)    &   𝐹 = (Scalar‘𝐺)       (𝐴𝑉𝐹 = (Scalar‘𝐻))
 
Theoremressvsca 15739 ·𝑠 is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐻 = (𝐺s 𝐴)    &    · = ( ·𝑠𝐺)       (𝐴𝑉· = ( ·𝑠𝐻))
 
Theoremressip 15740 The inner product is unaffected by restriction. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (𝐺s 𝐴)    &    , = (·𝑖𝐺)       (𝐴𝑉, = (·𝑖𝐻))
 
Theoremphlstr 15741 A constructed pre-Hilbert space is a structure. Starting from lmodstr 15724 (which has 4 members), we chain strleun 15683 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 15742 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 15743 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 15744 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 15745 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 15746 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 15747 Index value of the df-tset 15671 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(TopSet‘ndx) = 9
 
Theoremtsetid 15748 Utility theorem: index-independent form of df-tset 15671. (Contributed by NM, 20-Oct-2012.)
TopSet = Slot (TopSet‘ndx)
 
Theoremtopgrpstr 15749 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 15750 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 15751 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 15752 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 15753 TopSet is unaffected by restriction. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐻 = (𝐺s 𝐴)    &   𝐽 = (TopSet‘𝐺)       (𝐴𝑉𝐽 = (TopSet‘𝐻))
 
Theoremplendx 15754 Index value of the df-ple 15672 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
(le‘ndx) = 10
 
TheoremplendxOLD 15755 Obsolete version of df-ple 15672 as of 9-Sep-2021. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(le‘ndx) = 10
 
Theorempleid 15756 Utility theorem: self-referencing, index-independent form of df-ple 15672. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
le = Slot (le‘ndx)
 
TheorempleidOLD 15757 Obsolete version of otpsstr 15758 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 15758 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 15759 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 15760 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 15761 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 15762 Obsolete version of otpsstr 15758 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 15763 Obsolete version of otpsbas 15759 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 15764 Obsolete version of otpstset 15760 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 15765 Obsolete version of otpsle 15761 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 15766 le is unaffected by restriction. (Contributed by Mario Carneiro, 3-Nov-2015.)
𝑊 = (𝐾s 𝐴)    &    = (le‘𝐾)       (𝐴𝑉 = (le‘𝑊))
 
Theoremocndx 15767 Index value of the df-ocomp 15674 slot. (Contributed by Mario Carneiro, 25-Oct-2015.)
(oc‘ndx) = 11
 
Theoremocid 15768 Utility theorem: index-independent form of df-ocomp 15674. (Contributed by Mario Carneiro, 25-Oct-2015.)
oc = Slot (oc‘ndx)
 
Theoremdsndx 15769 Index value of the df-ds 15675 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(dist‘ndx) = 12
 
Theoremdsid 15770 Utility theorem: index-independent form of df-ds 15675. (Contributed by Mario Carneiro, 23-Dec-2013.)
dist = Slot (dist‘ndx)
 
Theoremunifndx 15771 Index value of the df-unif 15676 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.)
(UnifSet‘ndx) = 13
 
Theoremunifid 15772 Utility theorem: index-independent form of df-unif 15676. (Contributed by Thierry Arnoux, 17-Dec-2017.)
UnifSet = Slot (UnifSet‘ndx)
 
Theoremodrngstr 15773 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 15774 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 15775 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 15776 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 15777 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 15778 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 15779 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 15780 dist is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐻 = (𝐺s 𝐴)    &   𝐷 = (dist‘𝐺)       (𝐴𝑉𝐷 = (dist‘𝐻))
 
Theoremhomndx 15781 Index value of the df-hom 15677 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
(Hom ‘ndx) = 14
 
Theoremhomid 15782 Utility theorem: index-independent form of df-hom 15677. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hom = Slot (Hom ‘ndx)
 
Theoremccondx 15783 Index value of the df-cco 15678 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
(comp‘ndx) = 15
 
Theoremccoid 15784 Utility theorem: index-independent form of df-cco 15678. (Contributed by Mario Carneiro, 7-Jan-2017.)
comp = Slot (comp‘ndx)
 
Theoremresshom 15785 Hom is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
𝐷 = (𝐶s 𝐴)    &   𝐻 = (Hom ‘𝐶)       (𝐴𝑉𝐻 = (Hom ‘𝐷))
 
Theoremressco 15786 comp is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
𝐷 = (𝐶s 𝐴)    &    · = (comp‘𝐶)       (𝐴𝑉· = (comp‘𝐷))
 
Theoremslotsbhcdif 15787 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 15788 Extend class notation with the function returning a subspace topology.
class t
 
Syntaxctopn 15789 Extend class notation with the topology extractor function.
class TopOpen
 
Definitiondf-rest 15790* 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 15791 Define the topology extractor function. This differs from df-tset 15671 when a structure has been restricted using df-ress 15586; 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 15792 The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
t Fn (V × V)
 
Theoremtopnfn 15793 The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOpen Fn V
 
Theoremrestval 15794* 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 15795* 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 15796 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 15797 Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.)
(∅ ↾t 𝐴) = ∅
 
Theoremrestid2 15798 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 15799 The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽t 𝐴) ⊆ 𝒫 𝐴
 
Theoremfirest 15800 The finite intersections operator commutes with restriction. (Contributed by Mario Carneiro, 30-Aug-2015.)
(fi‘(𝐽t 𝐴)) = ((fi‘𝐽) ↾t 𝐴)
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