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Theorem List for Intuitionistic Logic Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremstrslssd 11601 Deduction version of strslss 11602. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝜑𝑇𝑉)    &   (𝜑 → Fun 𝑇)    &   (𝜑𝑆𝑇)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)       (𝜑 → (𝐸𝑇) = (𝐸𝑆))
 
Theoremstrslss 11602 Propagate component extraction to a structure 𝑇 from a subset structure 𝑆. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Jim Kingdon, 31-Jan-2023.)
𝑇 ∈ V    &   Fun 𝑇    &   𝑆𝑇    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆       (𝐸𝑇) = (𝐸𝑆)
 
Theoremstrsl0 11603 All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 31-Jan-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)       ∅ = (𝐸‘∅)
 
Theorembase0 11604 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
∅ = (Base‘∅)
 
Theoremsetsslid 11605 Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)       ((𝑊𝐴𝐶𝑉) → 𝐶 = (𝐸‘(𝑊 sSet ⟨(𝐸‘ndx), 𝐶⟩)))
 
Theoremsetsslnid 11606 Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝐸‘ndx) ≠ 𝐷    &   𝐷 ∈ ℕ       ((𝑊𝐴𝐶𝑉) → (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩)))
 
Theorembaseval 11607 Value of the base set extractor. (Normally it is preferred to work with (Base‘ndx) rather than the hard-coded 1 in order to make structure theorems portable. This is an example of how to obtain it when needed.) (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
𝐾 ∈ V       (Base‘𝐾) = (𝐾‘1)
 
Theorembaseid 11608 Utility theorem: index-independent form of df-base 11561. (Contributed by NM, 20-Oct-2012.)
Base = Slot (Base‘ndx)
 
Theorembasendx 11609 Index value of the base set extractor. (Normally it is preferred to work with (Base‘ndx) rather than the hard-coded 1 in order to make structure theorems portable. This is an example of how to obtain it when needed.) (New usage is discouraged.) (Contributed by Mario Carneiro, 2-Aug-2013.)
(Base‘ndx) = 1
 
Theorembasendxnn 11610 The index value of the base set extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 23-Sep-2020.)
(Base‘ndx) ∈ ℕ
 
Theorembaseslid 11611 The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.)
(Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ)
 
Theorembasfn 11612 The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
Base Fn V
 
Theoremreldmress 11613 The structure restriction is a proper operator, so it can be used with ovprc1 5699. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Rel dom ↾s
 
Theoremressid2 11614 General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 26-Jan-2023.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = 𝑊)
 
Theoremressval2 11615 Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
 
Theoremressid 11616 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝐵 = (Base‘𝑊)       (𝑊𝑋 → (𝑊s 𝐵) = 𝑊)
 
5.1.2  Slot definitions
 
Syntaxcplusg 11617 Extend class notation with group (addition) operation.
class +g
 
Syntaxcmulr 11618 Extend class notation with ring multiplication.
class .r
 
Syntaxcstv 11619 Extend class notation with involution.
class *𝑟
 
Syntaxcsca 11620 Extend class notation with scalar field.
class Scalar
 
Syntaxcvsca 11621 Extend class notation with scalar product.
class ·𝑠
 
Syntaxcip 11622 Extend class notation with Hermitian form (inner product).
class ·𝑖
 
Syntaxcts 11623 Extend class notation with the topology component of a topological space.
class TopSet
 
Syntaxcple 11624 Extend class notation with "less than or equal to" for posets.
class le
 
Syntaxcoc 11625 Extend class notation with the class of orthocomplementation extractors.
class oc
 
Syntaxcds 11626 Extend class notation with the metric space distance function.
class dist
 
Syntaxcunif 11627 Extend class notation with the uniform structure.
class UnifSet
 
Syntaxchom 11628 Extend class notation with the hom-set structure.
class Hom
 
Syntaxcco 11629 Extend class notation with the composition operation.
class comp
 
Definitiondf-plusg 11630 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
+g = Slot 2
 
Definitiondf-mulr 11631 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
.r = Slot 3
 
Definitiondf-starv 11632 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 11633 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 11634 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑠 = Slot 6
 
Definitiondf-ip 11635 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑖 = Slot 8
 
Definitiondf-tset 11636 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 11637 Define "less than or equal to" 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
 
Definitiondf-ocomp 11638 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 11639 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 11640 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
UnifSet = Slot 13
 
Definitiondf-hom 11641 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hom = Slot 14
 
Definitiondf-cco 11642 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp = Slot 15
 
Theoremstrleund 11643 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
(𝜑𝐹 Struct ⟨𝐴, 𝐵⟩)    &   (𝜑𝐺 Struct ⟨𝐶, 𝐷⟩)    &   (𝜑𝐵 < 𝐶)       (𝜑 → (𝐹𝐺) Struct ⟨𝐴, 𝐷⟩)
 
Theoremstrleun 11644 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
 
Theoremstrle1g 11645 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
𝐼 ∈ ℕ    &   𝐴 = 𝐼       (𝑋𝑉 → {⟨𝐴, 𝑋⟩} Struct ⟨𝐼, 𝐼⟩)
 
Theoremstrle2g 11646 Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
𝐼 ∈ ℕ    &   𝐴 = 𝐼    &   𝐼 < 𝐽    &   𝐽 ∈ ℕ    &   𝐵 = 𝐽       ((𝑋𝑉𝑌𝑊) → {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩} Struct ⟨𝐼, 𝐽⟩)
 
Theoremstrle3g 11647 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       ((𝑋𝑉𝑌𝑊𝑍𝑃) → {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩, ⟨𝐶, 𝑍⟩} Struct ⟨𝐼, 𝐾⟩)
 
Theoremplusgndx 11648 Index value of the df-plusg 11630 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(+g‘ndx) = 2
 
Theoremplusgid 11649 Utility theorem: index-independent form of df-plusg 11630. (Contributed by NM, 20-Oct-2012.)
+g = Slot (+g‘ndx)
 
Theoremplusgslid 11650 Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.)
(+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
 
Theoremopelstrsl 11651 The slot of a structure which contains an ordered pair for that slot. (Contributed by Jim Kingdon, 5-Feb-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝜑𝑆 Struct 𝑋)    &   (𝜑𝑉𝑌)    &   (𝜑 → ⟨(𝐸‘ndx), 𝑉⟩ ∈ 𝑆)       (𝜑𝑉 = (𝐸𝑆))
 
Theoremopelstrbas 11652 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))
 
Theorem1strstrg 11653 A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}       (𝐵𝑉𝐺 Struct ⟨1, 1⟩)
 
Theorem1strbas 11654 The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}       (𝐵𝑉𝐵 = (Base‘𝐺))
 
Theorem2strstrg 11655 A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   𝐸 = Slot 𝑁    &   1 < 𝑁    &   𝑁 ∈ ℕ       ((𝐵𝑉+𝑊) → 𝐺 Struct ⟨1, 𝑁⟩)
 
Theorem2strbasg 11656 The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   𝐸 = Slot 𝑁    &   1 < 𝑁    &   𝑁 ∈ ℕ       ((𝐵𝑉+𝑊) → 𝐵 = (Base‘𝐺))
 
Theorem2stropg 11657 The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   𝐸 = Slot 𝑁    &   1 < 𝑁    &   𝑁 ∈ ℕ       ((𝐵𝑉+𝑊) → + = (𝐸𝐺))
 
Theorem2strstr1g 11658 A constructed two-slot structure. Version of 2strstrg 11655 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   (Base‘ndx) < 𝑁    &   𝑁 ∈ ℕ       ((𝐵𝑉+𝑊) → 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩)
 
Theorem2strbas1g 11659 The base set of a constructed two-slot structure. Version of 2strbasg 11656 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   (Base‘ndx) < 𝑁    &   𝑁 ∈ ℕ       ((𝐵𝑉+𝑊) → 𝐵 = (Base‘𝐺))
 
Theorem2strop1g 11660 The other slot of a constructed two-slot structure. Version of 2stropg 11657 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   (Base‘ndx) < 𝑁    &   𝑁 ∈ ℕ    &   𝐸 = Slot 𝑁       ((𝐵𝑉+𝑊) → + = (𝐸𝐺))
 
Theorembasendxnplusgndx 11661 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)
 
Theoremgrpstrg 11662 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), + ⟩}       ((𝐵𝑉+𝑊) → 𝐺 Struct ⟨1, 2⟩)
 
Theoremgrpbaseg 11663 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), + ⟩}       ((𝐵𝑉+𝑊) → 𝐵 = (Base‘𝐺))
 
Theoremgrpplusgg 11664 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), + ⟩}       ((𝐵𝑉+𝑊) → + = (+g𝐺))
 
Theoremmulrndx 11665 Index value of the df-mulr 11631 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(.r‘ndx) = 3
 
Theoremmulrid 11666 Utility theorem: index-independent form of df-mulr 11631. (Contributed by Mario Carneiro, 8-Jun-2013.)
.r = Slot (.r‘ndx)
 
Theoremmulrslid 11667 Slot property of .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
(.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
 
Theoremplusgndxnmulrndx 11668 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 11669 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)
 
Theoremrngstrg 11670 A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
𝑅 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}       ((𝐵𝑉+𝑊·𝑋) → 𝑅 Struct ⟨1, 3⟩)
 
Theoremrngbaseg 11671 The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
𝑅 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩}       ((𝐵𝑉+𝑊·𝑋) → 𝐵 = (Base‘𝑅))
 
Theoremrngplusgg 11672 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), · ⟩}       ((𝐵𝑉+𝑊·𝑋) → + = (+g𝑅))
 
Theoremrngmulrg 11673 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), · ⟩}       ((𝐵𝑉+𝑊·𝑋) → · = (.r𝑅))
 
Theoremstarvndx 11674 Index value of the df-starv 11632 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(*𝑟‘ndx) = 4
 
Theoremstarvid 11675 Utility theorem: index-independent form of df-starv 11632. (Contributed by Mario Carneiro, 6-Oct-2013.)
*𝑟 = Slot (*𝑟‘ndx)
 
Theoremstarvslid 11676 Slot property of *𝑟. (Contributed by Jim Kingdon, 4-Feb-2023.)
(*𝑟 = Slot (*𝑟‘ndx) ∧ (*𝑟‘ndx) ∈ ℕ)
 
Theoremsrngstrd 11677 A constructed star ring is a structure. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑·𝑋)    &   (𝜑𝑌)       (𝜑𝑅 Struct ⟨1, 4⟩)
 
Theoremsrngbased 11678 The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑·𝑋)    &   (𝜑𝑌)       (𝜑𝐵 = (Base‘𝑅))
 
Theoremsrngplusgd 11679 The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by Jim Kingdon, 5-Feb-2023.)
𝑅 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑·𝑋)    &   (𝜑𝑌)       (𝜑+ = (+g𝑅))
 
Theoremsrngmulrd 11680 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), ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑·𝑋)    &   (𝜑𝑌)       (𝜑· = (.r𝑅))
 
Theoremsrnginvld 11681 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), ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑·𝑋)    &   (𝜑𝑌)       (𝜑 = (*𝑟𝑅))
 
Theoremscandx 11682 Index value of the df-sca 11633 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(Scalar‘ndx) = 5
 
Theoremscaid 11683 Utility theorem: index-independent form of scalar df-sca 11633. (Contributed by Mario Carneiro, 19-Jun-2014.)
Scalar = Slot (Scalar‘ndx)
 
Theoremscaslid 11684 Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
(Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
 
Theoremvscandx 11685 Index value of the df-vsca 11634 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
( ·𝑠 ‘ndx) = 6
 
Theoremvscaid 11686 Utility theorem: index-independent form of scalar product df-vsca 11634. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
·𝑠 = Slot ( ·𝑠 ‘ndx)
 
Theoremvscaslid 11687 Slot property of ·𝑠. (Contributed by Jim Kingdon, 5-Feb-2023.)
( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
 
Theoremlmodstrd 11688 A constructed left module or left vector space is a structure. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑋)    &   (𝜑𝐹𝑌)    &   (𝜑·𝑍)       (𝜑𝑊 Struct ⟨1, 6⟩)
 
Theoremlmodbased 11689 The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑋)    &   (𝜑𝐹𝑌)    &   (𝜑·𝑍)       (𝜑𝐵 = (Base‘𝑊))
 
Theoremlmodplusgd 11690 The additive operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑋)    &   (𝜑𝐹𝑌)    &   (𝜑·𝑍)       (𝜑+ = (+g𝑊))
 
Theoremlmodscad 11691 The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑋)    &   (𝜑𝐹𝑌)    &   (𝜑·𝑍)       (𝜑𝐹 = (Scalar‘𝑊))
 
Theoremlmodvscad 11692 The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝐹⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑋)    &   (𝜑𝐹𝑌)    &   (𝜑·𝑍)       (𝜑· = ( ·𝑠𝑊))
 
Theoremipndx 11693 Index value of the df-ip 11635 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(·𝑖‘ndx) = 8
 
Theoremipid 11694 Utility theorem: index-independent form of df-ip 11635. (Contributed by Mario Carneiro, 6-Oct-2013.)
·𝑖 = Slot (·𝑖‘ndx)
 
Theoremipslid 11695 Slot property of ·𝑖. (Contributed by Jim Kingdon, 7-Feb-2023.)
(·𝑖 = Slot (·𝑖‘ndx) ∧ (·𝑖‘ndx) ∈ ℕ)
 
Theoremipsstrd 11696 A constructed inner product space is a structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑×𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑·𝑄)    &   (𝜑𝐼𝑍)       (𝜑𝐴 Struct ⟨1, 8⟩)
 
Theoremipsbased 11697 The base set of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑×𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑·𝑄)    &   (𝜑𝐼𝑍)       (𝜑𝐵 = (Base‘𝐴))
 
Theoremipsaddgd 11698 The additive operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑×𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑·𝑄)    &   (𝜑𝐼𝑍)       (𝜑+ = (+g𝐴))
 
Theoremipsmulrd 11699 The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑×𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑·𝑄)    &   (𝜑𝐼𝑍)       (𝜑× = (.r𝐴))
 
Theoremipsscad 11700 The set of scalars of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), 𝐼⟩})    &   (𝜑𝐵𝑉)    &   (𝜑+𝑊)    &   (𝜑×𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑·𝑄)    &   (𝜑𝐼𝑍)       (𝜑𝑆 = (Scalar‘𝐴))
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