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Theorem List for Metamath Proof Explorer - 16601-16700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem2strbas1 16601 The base set of a constructed two-slot structure. Version of 2strbas 16598 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   (Base‘ndx) < 𝑁    &   𝑁 ∈ ℕ       (𝐵𝑉𝐵 = (Base‘𝐺))

Theorem2strop1 16602 The other slot of a constructed two-slot structure. Version of 2strop 16599 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   (Base‘ndx) < 𝑁    &   𝑁 ∈ ℕ    &   𝐸 = Slot 𝑁       ( +𝑉+ = (𝐸𝐺))

Theorembasendxnplusgndx 16603 The slot for the base set is not the slot for the group operation in an extensible structure. (Contributed by AV, 14-Nov-2021.)
(Base‘ndx) ≠ (+g‘ndx)

Theoremgrpstr 16604 A constructed group is a structure on 1...2. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}       𝐺 Struct ⟨1, 2⟩

Theoremgrpbase 16605 The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}       (𝐵𝑉𝐵 = (Base‘𝐺))

Theoremgrpplusg 16606 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 16607 +g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝐻 = (𝐺s 𝐴)    &    + = (+g𝐺)       (𝐴𝑉+ = (+g𝐻))

Theoremgrpbasex 16608 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 16605 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
𝐵 ∈ V    &    + ∈ V    &   𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}       𝐵 = (Base‘𝐺)

Theoremgrpplusgx 16609 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 16606 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
𝐵 ∈ V    &    + ∈ V    &   𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}        + = (+g𝐺)

Theoremmulrndx 16610 Index value of the df-mulr 16574 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(.r‘ndx) = 3

Theoremmulrid 16611 Utility theorem: index-independent form of df-mulr 16574. (Contributed by Mario Carneiro, 8-Jun-2013.)
.r = Slot (.r‘ndx)

Theoremplusgndxnmulrndx 16612 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)

Theorembasendxnmulrndx 16613 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.)
(Base‘ndx) ≠ (.r‘ndx)

Theoremrngstr 16614 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 16615 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 16616 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 16617 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 16618 Index value of the df-starv 16575 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(*𝑟‘ndx) = 4

Theoremstarvid 16619 Utility theorem: index-independent form of df-starv 16575. (Contributed by Mario Carneiro, 6-Oct-2013.)
*𝑟 = Slot (*𝑟‘ndx)

Theoremressmulr 16620 .r is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝑆 = (𝑅s 𝐴)    &    · = (.r𝑅)       (𝐴𝑉· = (.r𝑆))

Theoremressstarv 16621 *𝑟 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝑆 = (𝑅s 𝐴)    &    = (*𝑟𝑅)       (𝐴𝑉 = (*𝑟𝑆))

Theoremsrngstr 16622 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⟩

Theoremsrngbase 16623 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 16624 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 16625 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 16626 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 16627 Index value of the df-sca 16576 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(Scalar‘ndx) = 5

Theoremscaid 16628 Utility theorem: index-independent form of scalar df-sca 16576. (Contributed by Mario Carneiro, 19-Jun-2014.)
Scalar = Slot (Scalar‘ndx)

Theoremvscandx 16629 Index value of the df-vsca 16577 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
( ·𝑠 ‘ndx) = 6

Theoremvscaid 16630 Utility theorem: index-independent form of scalar product df-vsca 16577. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
·𝑠 = Slot ( ·𝑠 ‘ndx)

Theoremlmodstr 16631 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⟩

Theoremlmodbase 16632 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 16633 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 16634 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 16635 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 16636 Index value of the df-ip 16578 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(·𝑖‘ndx) = 8

Theoremipid 16637 Utility theorem: index-independent form of df-ip 16578. (Contributed by Mario Carneiro, 6-Oct-2013.)
·𝑖 = Slot (·𝑖‘ndx)

Theoremipsstr 16638 Lemma to shorten proofs of ipsbase 16639 through ipsvsca 16643. (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 16639 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 16640 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 16641 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 16642 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 16643 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 16644 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 16645 Scalar is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐻 = (𝐺s 𝐴)    &   𝐹 = (Scalar‘𝐺)       (𝐴𝑉𝐹 = (Scalar‘𝐻))

Theoremressvsca 16646 ·𝑠 is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐻 = (𝐺s 𝐴)    &    · = ( ·𝑠𝐺)       (𝐴𝑉· = ( ·𝑠𝐻))

Theoremressip 16647 The inner product is unaffected by restriction. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (𝐺s 𝐴)    &    , = (·𝑖𝐺)       (𝐴𝑉, = (·𝑖𝐻))

Theoremphlstr 16648 A constructed pre-Hilbert space is a structure. Starting from lmodstr 16631 (which has 4 members), we chain strleun 16586 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 16649 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 16650 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 16651 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 16652 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 16653 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 16654 Index value of the df-tset 16579 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(TopSet‘ndx) = 9

Theoremtsetid 16655 Utility theorem: index-independent form of df-tset 16579. (Contributed by NM, 20-Oct-2012.)
TopSet = Slot (TopSet‘ndx)

Theoremtopgrpstr 16656 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 16657 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 16658 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 16659 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 16660 TopSet is unaffected by restriction. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐻 = (𝐺s 𝐴)    &   𝐽 = (TopSet‘𝐺)       (𝐴𝑉𝐽 = (TopSet‘𝐻))

Theoremplendx 16661 Index value of the df-ple 16580 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
(le‘ndx) = 10

Theorempleid 16662 Utility theorem: self-referencing, index-independent form of df-ple 16580. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
le = Slot (le‘ndx)

Theoremotpsstr 16663 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 16664 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 16665 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 16666 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))

Theoremressle 16667 le is unaffected by restriction. (Contributed by Mario Carneiro, 3-Nov-2015.)
𝑊 = (𝐾s 𝐴)    &    = (le‘𝐾)       (𝐴𝑉 = (le‘𝑊))

Theoremocndx 16668 Index value of the df-ocomp 16581 slot. (Contributed by Mario Carneiro, 25-Oct-2015.)
(oc‘ndx) = 11

Theoremocid 16669 Utility theorem: index-independent form of df-ocomp 16581. (Contributed by Mario Carneiro, 25-Oct-2015.)
oc = Slot (oc‘ndx)

Theoremdsndx 16670 Index value of the df-ds 16582 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(dist‘ndx) = 12

Theoremdsid 16671 Utility theorem: index-independent form of df-ds 16582. (Contributed by Mario Carneiro, 23-Dec-2013.)
dist = Slot (dist‘ndx)

Theoremunifndx 16672 Index value of the df-unif 16583 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.)
(UnifSet‘ndx) = 13

Theoremunifid 16673 Utility theorem: index-independent form of df-unif 16583. (Contributed by Thierry Arnoux, 17-Dec-2017.)
UnifSet = Slot (UnifSet‘ndx)

Theoremodrngstr 16674 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 16675 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 16676 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 16677 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 16678 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 16679 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 16680 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 16681 dist is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐻 = (𝐺s 𝐴)    &   𝐷 = (dist‘𝐺)       (𝐴𝑉𝐷 = (dist‘𝐻))

Theoremhomndx 16682 Index value of the df-hom 16584 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
(Hom ‘ndx) = 14

Theoremhomid 16683 Utility theorem: index-independent form of df-hom 16584. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hom = Slot (Hom ‘ndx)

Theoremccondx 16684 Index value of the df-cco 16585 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
(comp‘ndx) = 15

Theoremccoid 16685 Utility theorem: index-independent form of df-cco 16585. (Contributed by Mario Carneiro, 7-Jan-2017.)
comp = Slot (comp‘ndx)

Theoremresshom 16686 Hom is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
𝐷 = (𝐶s 𝐴)    &   𝐻 = (Hom ‘𝐶)       (𝐴𝑉𝐻 = (Hom ‘𝐷))

Theoremressco 16687 comp is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
𝐷 = (𝐶s 𝐴)    &    · = (comp‘𝐶)       (𝐴𝑉· = (comp‘𝐷))

Theoremslotsbhcdif 16688 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 16689 Extend class notation with the function returning a subspace topology.
class t

Syntaxctopn 16690 Extend class notation with the topology extractor function.
class TopOpen

Definitiondf-rest 16691* 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 16692 Define the topology extractor function. This differs from df-tset 16579 when a structure has been restricted using df-ress 16486; 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 16693 The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
t Fn (V × V)

Theoremtopnfn 16694 The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOpen Fn V

Theoremrestval 16695* 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 16696* 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 16697 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 16698 Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.)
(∅ ↾t 𝐴) = ∅

Theoremrestid2 16699 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 16700 The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽t 𝐴) ⊆ 𝒫 𝐴

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