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Theorem List for Metamath Proof Explorer - 16001-16100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-plusg 16001 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
+g = Slot 2
 
Definitiondf-mulr 16002 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
.r = Slot 3
 
Definitiondf-starv 16003 Define the involution function of a *-ring. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
*𝑟 = Slot 4
 
Definitiondf-sca 16004 Define scalar field component of a vector space 𝑣. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Scalar = Slot 5
 
Definitiondf-vsca 16005 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑠 = Slot 6
 
Definitiondf-ip 16006 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑖 = Slot 8
 
Definitiondf-tset 16007 Define the topology component of a topological space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
TopSet = Slot 9
 
Definitiondf-ple 16008 Define less-than-or-equal ordering extractor for posets and related structures. We use 10 for the index to avoid conflict with 1 through 9 used for other purposes. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
le = Slot 10
 
TheoremdfpleOLD 16009 Obsolete version of df-ple 16008 as of 9-Sep-2021. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
le = Slot 10
 
Definitiondf-ocomp 16010 Define the orthocomplementation extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
oc = Slot 11
 
Definitiondf-ds 16011 Define the distance function component of a metric space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
dist = Slot 12
 
Definitiondf-unif 16012 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
UnifSet = Slot 13
 
Definitiondf-hom 16013 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hom = Slot 14
 
Definitiondf-cco 16014 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp = Slot 15
 
Theoremstrlemor0OLD 16015 Structure definition utility lemma. To prove that an explicit function is a function using O(n) steps, exploit the order properties of the index set. Zero-pair case. Obsolete as of 26-Nov-2021. Theorems strlemor0OLD 16015, strlemor1OLD 16016, strlemor2OLD 16017, strlemor3OLD 16018 were replaced by strleun 16019, strle1 16020, strle2 16021, strle3 16022 following the introduction df-struct 15906. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(Fun ∅ ∧ dom ∅ ⊆ (1...0))
 
Theoremstrlemor1OLD 16016 Add one element to the end of a structure. Obsolete as of 26-Nov-2021. See comment of strlemor0OLD 16015. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(Fun 𝐹 ∧ dom 𝐹 ⊆ (1...𝐼))    &   𝐼 ∈ ℕ0    &   𝐼 < 𝐽    &   𝐽 ∈ ℕ    &   𝐴 = 𝐽    &   𝐺 = (𝐹 ∪ {⟨𝐴, 𝑋⟩})       (Fun 𝐺 ∧ dom 𝐺 ⊆ (1...𝐽))
 
Theoremstrlemor2OLD 16017 Add two elements to the end of a structure. Obsolete as of 26-Nov-2021. See comment of strlemor0OLD 16015. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(Fun 𝐹 ∧ dom 𝐹 ⊆ (1...𝐼))    &   𝐼 ∈ ℕ0    &   𝐼 < 𝐽    &   𝐽 ∈ ℕ    &   𝐴 = 𝐽    &   𝐽 < 𝐾    &   𝐾 ∈ ℕ    &   𝐵 = 𝐾    &   𝐺 = (𝐹 ∪ {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩})       (Fun 𝐺 ∧ dom 𝐺 ⊆ (1...𝐾))
 
Theoremstrlemor3OLD 16018 Add three elements to the end of a structure. Obsolete as of 26-Nov-2021. See comment of strlemor0OLD 16015. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(Fun 𝐹 ∧ dom 𝐹 ⊆ (1...𝐼))    &   𝐼 ∈ ℕ0    &   𝐼 < 𝐽    &   𝐽 ∈ ℕ    &   𝐴 = 𝐽    &   𝐽 < 𝐾    &   𝐾 ∈ ℕ    &   𝐵 = 𝐾    &   𝐾 < 𝐿    &   𝐿 ∈ ℕ    &   𝐶 = 𝐿    &   𝐺 = (𝐹 ∪ {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩, ⟨𝐶, 𝑍⟩})       (Fun 𝐺 ∧ dom 𝐺 ⊆ (1...𝐿))
 
Theoremstrleun 16019 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐹 Struct ⟨𝐴, 𝐵    &   𝐺 Struct ⟨𝐶, 𝐷    &   𝐵 < 𝐶       (𝐹𝐺) Struct ⟨𝐴, 𝐷
 
Theoremstrle1 16020 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐼 ∈ ℕ    &   𝐴 = 𝐼       {⟨𝐴, 𝑋⟩} Struct ⟨𝐼, 𝐼
 
Theoremstrle2 16021 Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐼 ∈ ℕ    &   𝐴 = 𝐼    &   𝐼 < 𝐽    &   𝐽 ∈ ℕ    &   𝐵 = 𝐽       {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩} Struct ⟨𝐼, 𝐽
 
Theoremstrle3 16022 Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐼 ∈ ℕ    &   𝐴 = 𝐼    &   𝐼 < 𝐽    &   𝐽 ∈ ℕ    &   𝐵 = 𝐽    &   𝐽 < 𝐾    &   𝐾 ∈ ℕ    &   𝐶 = 𝐾       {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩, ⟨𝐶, 𝑍⟩} Struct ⟨𝐼, 𝐾
 
Theoremplusgndx 16023 Index value of the df-plusg 16001 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(+g‘ndx) = 2
 
Theoremplusgid 16024 Utility theorem: index-independent form of df-plusg 16001. (Contributed by NM, 20-Oct-2012.)
+g = Slot (+g‘ndx)
 
Theoremopelstrbas 16025 The base set of a structure with a base set. (Contributed by AV, 10-Nov-2021.)
(𝜑𝑆 Struct 𝑋)    &   (𝜑𝑉𝑌)    &   (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝑆)       (𝜑𝑉 = (Base‘𝑆))
 
Theorem1strstr 16026 A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}       𝐺 Struct ⟨1, 1⟩
 
Theorem1strbas 16027 The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}       (𝐵𝑉𝐵 = (Base‘𝐺))
 
Theorem1strwunbndx 16028 A constructed one-slot structure in a weak universe containing the index of the base set extractor. (Contributed by AV, 27-Mar-2020.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → (Base‘ndx) ∈ 𝑈)       ((𝜑𝐵𝑈) → 𝐺𝑈)
 
Theorem1strwun 16029 A constructed one-slot structure in a weak universe. (Contributed by AV, 27-Mar-2020.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)       ((𝜑𝐵𝑈) → 𝐺𝑈)
 
Theorem2strstr 16030 A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   𝐸 = Slot 𝑁    &   1 < 𝑁    &   𝑁 ∈ ℕ       𝐺 Struct ⟨1, 𝑁
 
Theorem2strbas 16031 The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   𝐸 = Slot 𝑁    &   1 < 𝑁    &   𝑁 ∈ ℕ       (𝐵𝑉𝐵 = (Base‘𝐺))
 
Theorem2strop 16032 The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   𝐸 = Slot 𝑁    &   1 < 𝑁    &   𝑁 ∈ ℕ       ( +𝑉+ = (𝐸𝐺))
 
Theorem2strstr1 16033 A constructed two-slot structure. Version of 2strstr 16030 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       𝐺 Struct ⟨(Base‘ndx), 𝑁
 
Theorem2strbas1 16034 The base set of a constructed two-slot structure. Version of 2strbas 16031 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 16035 The other slot of a constructed two-slot structure. Version of 2strop 16032 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 16036 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 16037 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 16038 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 16039 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 16040 +g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝐻 = (𝐺s 𝐴)    &    + = (+g𝐺)       (𝐴𝑉+ = (+g𝐻))
 
Theoremgrpbasex 16041 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 16038 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
𝐵 ∈ V    &    + ∈ V    &   𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}       𝐵 = (Base‘𝐺)
 
Theoremgrpplusgx 16042 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 16039 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)
𝐵 ∈ V    &    + ∈ V    &   𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩}        + = (+g𝐺)
 
Theoremmulrndx 16043 Index value of the df-mulr 16002 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(.r‘ndx) = 3
 
Theoremmulrid 16044 Utility theorem: index-independent form of df-mulr 16002. (Contributed by Mario Carneiro, 8-Jun-2013.)
.r = Slot (.r‘ndx)
 
Theoremplusgndxnmulrndx 16045 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 16046 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 16047 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 16048 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 16049 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 16050 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 16051 Index value of the df-starv 16003 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(*𝑟‘ndx) = 4
 
Theoremstarvid 16052 Utility theorem: index-independent form of df-starv 16003. (Contributed by Mario Carneiro, 6-Oct-2013.)
*𝑟 = Slot (*𝑟‘ndx)
 
Theoremressmulr 16053 .r is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝑆 = (𝑅s 𝐴)    &    · = (.r𝑅)       (𝐴𝑉· = (.r𝑆))
 
Theoremressstarv 16054 *𝑟 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝑆 = (𝑅s 𝐴)    &    = (*𝑟𝑅)       (𝐴𝑉 = (*𝑟𝑆))
 
Theoremsrngfn 16055 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 16056 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 16057 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 16058 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 16059 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 16060 Index value of the df-sca 16004 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(Scalar‘ndx) = 5
 
Theoremscaid 16061 Utility theorem: index-independent form of scalar df-sca 16004. (Contributed by Mario Carneiro, 19-Jun-2014.)
Scalar = Slot (Scalar‘ndx)
 
Theoremvscandx 16062 Index value of the df-vsca 16005 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
( ·𝑠 ‘ndx) = 6
 
Theoremvscaid 16063 Utility theorem: index-independent form of scalar product df-vsca 16005. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
·𝑠 = Slot ( ·𝑠 ‘ndx)
 
Theoremlmodstr 16064 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 16065 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 16066 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 16067 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 16068 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 16069 Index value of the df-ip 16006 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(·𝑖‘ndx) = 8
 
Theoremipid 16070 Utility theorem: index-independent form of df-ip 16006. (Contributed by Mario Carneiro, 6-Oct-2013.)
·𝑖 = Slot (·𝑖‘ndx)
 
Theoremipsstr 16071 Lemma to shorten proofs of ipsbase 16072 through ipsvsca 16076. (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 16072 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 16073 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 16074 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 16075 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 16076 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 16077 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 16078 Scalar is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐻 = (𝐺s 𝐴)    &   𝐹 = (Scalar‘𝐺)       (𝐴𝑉𝐹 = (Scalar‘𝐻))
 
Theoremressvsca 16079 ·𝑠 is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐻 = (𝐺s 𝐴)    &    · = ( ·𝑠𝐺)       (𝐴𝑉· = ( ·𝑠𝐻))
 
Theoremressip 16080 The inner product is unaffected by restriction. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (𝐺s 𝐴)    &    , = (·𝑖𝐺)       (𝐴𝑉, = (·𝑖𝐻))
 
Theoremphlstr 16081 A constructed pre-Hilbert space is a structure. Starting from lmodstr 16064 (which has 4 members), we chain strleun 16019 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 16082 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 16083 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 16084 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 16085 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 16086 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 16087 Index value of the df-tset 16007 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(TopSet‘ndx) = 9
 
Theoremtsetid 16088 Utility theorem: index-independent form of df-tset 16007. (Contributed by NM, 20-Oct-2012.)
TopSet = Slot (TopSet‘ndx)
 
Theoremtopgrpstr 16089 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 16090 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 16091 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 16092 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 16093 TopSet is unaffected by restriction. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐻 = (𝐺s 𝐴)    &   𝐽 = (TopSet‘𝐺)       (𝐴𝑉𝐽 = (TopSet‘𝐻))
 
Theoremplendx 16094 Index value of the df-ple 16008 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
(le‘ndx) = 10
 
TheoremplendxOLD 16095 Obsolete version of df-ple 16008 as of 9-Sep-2021. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(le‘ndx) = 10
 
Theorempleid 16096 Utility theorem: self-referencing, index-independent form of df-ple 16008. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
le = Slot (le‘ndx)
 
TheorempleidOLD 16097 Obsolete version of otpsstr 16098 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 16098 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 16099 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 16100 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))
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