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Theorem List for Intuitionistic Logic Explorer - 12001-12100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremstrndxid 12001 The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.)
(𝜑𝑆𝑉)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸𝑆))

Theoremreldmsets 12002 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Rel dom sSet

Theoremsetsvalg 12003 Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝑆𝑉𝐴𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))

Theoremsetsvala 12004 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 20-Jan-2023.)
((𝑆𝑉𝐴𝑋𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))

Theoremsetsex 12005 Applying the structure replacement function yields a set. (Contributed by Jim Kingdon, 22-Jan-2023.)
((𝑆𝑉𝐴𝑋𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V)

Theoremstrsetsid 12006 Value of the structure replacement function. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 30-Jan-2023.)
𝐸 = Slot (𝐸‘ndx)    &   (𝜑𝑆 Struct ⟨𝑀, 𝑁⟩)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → (𝐸‘ndx) ∈ dom 𝑆)       (𝜑𝑆 = (𝑆 sSet ⟨(𝐸‘ndx), (𝐸𝑆)⟩))

Theoremfvsetsid 12007 The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.)
((𝐹𝑉𝑋𝑊𝑌𝑈) → ((𝐹 sSet ⟨𝑋, 𝑌⟩)‘𝑋) = 𝑌)

Theoremsetsfun 12008 A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.)
(((𝐺𝑉 ∧ Fun 𝐺) ∧ (𝐼𝑈𝐸𝑊)) → Fun (𝐺 sSet ⟨𝐼, 𝐸⟩))

Theoremsetsfun0 12009 A structure with replacement without the empty set is a function if the original structure without the empty set is a function. This variant of setsfun 12008 is useful for proofs based on isstruct2r 11984 which requires Fun (𝐹 ∖ {∅}) for 𝐹 to be an extensible structure. (Contributed by AV, 7-Jun-2021.)
(((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼𝑈𝐸𝑊)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))

Theoremsetsn0fun 12010 The value of the structure replacement function (without the empty set) is a function if the structure (without the empty set) is a function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
(𝜑𝑆 Struct 𝑋)    &   (𝜑𝐼𝑈)    &   (𝜑𝐸𝑊)       (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))

Theoremsetsresg 12011 The structure replacement function does not affect the value of 𝑆 away from 𝐴. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
((𝑆𝑉𝐴𝑊𝐵𝑋) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴})))

Theoremsetsabsd 12012 Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
(𝜑𝑆𝑉)    &   (𝜑𝐴𝑊)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑈)       (𝜑 → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))

Theoremsetscom 12013 Component-setting is commutative when the x-values are different. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (((𝑆𝑉𝐴𝐵) ∧ (𝐶𝑊𝐷𝑋)) → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))

Theoremstrslfvd 12014 Deduction version of strslfv 12017. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 30-Jan-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)       (𝜑𝐶 = (𝐸𝑆))

Theoremstrslfv2d 12015 Deduction version of strslfv 12017. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)    &   (𝜑𝐶𝑊)       (𝜑𝐶 = (𝐸𝑆))

Theoremstrslfv2 12016 A variation on strslfv 12017 to avoid asserting that 𝑆 itself is a function, which involves sethood of all the ordered pair components of 𝑆. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
𝑆 ∈ V    &   Fun 𝑆    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆       (𝐶𝑉𝐶 = (𝐸𝑆))

Theoremstrslfv 12017 Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 with a component extractor 𝐸 (such as the base set extractor df-base 11979). By virtue of ndxslid 11998, this can be done without having to refer to the hard-coded numeric index of 𝐸. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 30-Jan-2023.)
𝑆 Struct 𝑋    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆       (𝐶𝑉𝐶 = (𝐸𝑆))

Theoremstrslfv3 12018 Variant on strslfv 12017 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.)
(𝜑𝑈 = 𝑆)    &   𝑆 Struct 𝑋    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆    &   (𝜑𝐶𝑉)    &   𝐴 = (𝐸𝑈)       (𝜑𝐴 = 𝐶)

Theoremstrslssd 12019 Deduction version of strslss 12020. (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 12020 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 12021 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 12022 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
∅ = (Base‘∅)

Theoremsetsslid 12023 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 12024 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 12025 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 12026 Utility theorem: index-independent form of df-base 11979. (Contributed by NM, 20-Oct-2012.)
Base = Slot (Base‘ndx)

Theorembasendx 12027 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 12028 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 12029 The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.)
(Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ)

Theorembasfn 12030 The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
Base Fn V

Theoremreldmress 12031 The structure restriction is a proper operator, so it can be used with ovprc1 5807. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Rel dom ↾s

Theoremressid2 12032 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 12033 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 12034 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝐵 = (Base‘𝑊)       (𝑊𝑋 → (𝑊s 𝐵) = 𝑊)

6.1.2  Slot definitions

Syntaxcplusg 12035 Extend class notation with group (addition) operation.
class +g

Syntaxcmulr 12036 Extend class notation with ring multiplication.
class .r

Syntaxcstv 12037 Extend class notation with involution.
class *𝑟

Syntaxcsca 12038 Extend class notation with scalar field.
class Scalar

Syntaxcvsca 12039 Extend class notation with scalar product.
class ·𝑠

Syntaxcip 12040 Extend class notation with Hermitian form (inner product).
class ·𝑖

Syntaxcts 12041 Extend class notation with the topology component of a topological space.
class TopSet

Syntaxcple 12042 Extend class notation with "less than or equal to" for posets.
class le

Syntaxcoc 12043 Extend class notation with the class of orthocomplementation extractors.
class oc

Syntaxcds 12044 Extend class notation with the metric space distance function.
class dist

Syntaxcunif 12045 Extend class notation with the uniform structure.
class UnifSet

Syntaxchom 12046 Extend class notation with the hom-set structure.
class Hom

Syntaxcco 12047 Extend class notation with the composition operation.
class comp

Definitiondf-plusg 12048 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
+g = Slot 2

Definitiondf-mulr 12049 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
.r = Slot 3

Definitiondf-starv 12050 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 12051 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 12052 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑠 = Slot 6

Definitiondf-ip 12053 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑖 = Slot 8

Definitiondf-tset 12054 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 12055 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 12056 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 12057 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 12058 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
UnifSet = Slot 13

Definitiondf-hom 12059 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hom = Slot 14

Definitiondf-cco 12060 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp = Slot 15

Theoremstrleund 12061 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 12062 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 12063 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 12064 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 12065 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 12066 Index value of the df-plusg 12048 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(+g‘ndx) = 2

Theoremplusgid 12067 Utility theorem: index-independent form of df-plusg 12048. (Contributed by NM, 20-Oct-2012.)
+g = Slot (+g‘ndx)

Theoremplusgslid 12068 Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.)
(+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)

Theoremopelstrsl 12069 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 12070 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 12071 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 12072 The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}       (𝐵𝑉𝐵 = (Base‘𝐺))

Theorem2strstrg 12073 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 12074 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 12075 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 12076 A constructed two-slot structure. Version of 2strstrg 12073 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 12077 The base set of a constructed two-slot structure. Version of 2strbasg 12074 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 12078 The other slot of a constructed two-slot structure. Version of 2stropg 12075 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 12079 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 12080 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 12081 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 12082 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 12083 Index value of the df-mulr 12049 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(.r‘ndx) = 3

Theoremmulrid 12084 Utility theorem: index-independent form of df-mulr 12049. (Contributed by Mario Carneiro, 8-Jun-2013.)
.r = Slot (.r‘ndx)

Theoremmulrslid 12085 Slot property of .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
(.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)

Theoremplusgndxnmulrndx 12086 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 12087 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 12088 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 12089 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 12090 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 12091 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 12092 Index value of the df-starv 12050 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(*𝑟‘ndx) = 4

Theoremstarvid 12093 Utility theorem: index-independent form of df-starv 12050. (Contributed by Mario Carneiro, 6-Oct-2013.)
*𝑟 = Slot (*𝑟‘ndx)

Theoremstarvslid 12094 Slot property of *𝑟. (Contributed by Jim Kingdon, 4-Feb-2023.)
(*𝑟 = Slot (*𝑟‘ndx) ∧ (*𝑟‘ndx) ∈ ℕ)

Theoremsrngstrd 12095 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 12096 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 12097 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 12098 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 12099 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 12100 Index value of the df-sca 12051 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(Scalar‘ndx) = 5

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