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Theorem List for Intuitionistic Logic Explorer - 12401-12500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcsts 12401 Set components of a structure.
class sSet
 
Syntaxcslot 12402 Extend class notation with the slot function.
class Slot 𝐴
 
Syntaxcbs 12403 Extend class notation with the class of all base set extractors.
class Base
 
Syntaxcress 12404 Extend class notation with the extensible structure builder restriction operator.
class s
 
Definitiondf-struct 12405* Define a structure with components in 𝑀...𝑁. This is not a requirement for groups, posets, etc., but it is a useful assumption for component extraction theorems.

As mentioned in the section header, an "extensible structure should be implemented as a function (a set of ordered pairs)". The current definition, however, is less restrictive: it allows for classes which contain the empty set to be extensible structures. Because of 0nelfun 5214, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 12416: 𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅}).

Allowing an extensible structure to contain the empty set ensures that expressions like {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} are structures without asserting or implying that 𝐴, 𝐵, 𝐶 and 𝐷 are sets (if 𝐴 or 𝐵 is a proper class, then 𝐴, 𝐵⟩ = ∅, see opprc 3784). (Contributed by Mario Carneiro, 29-Aug-2015.)

Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
 
Definitiondf-ndx 12406 Define the structure component index extractor. See Theorem ndxarg 12426 to understand its purpose. The restriction to ensures that ndx is a set. The restriction to some set is necessary since I is a proper class. In principle, we could have chosen or (if we revise all structure component definitions such as df-base 12409) another set such as the set of finite ordinals ω (df-iom 4573). (Contributed by NM, 4-Sep-2011.)
ndx = ( I ↾ ℕ)
 
Definitiondf-slot 12407* Define the slot extractor for extensible structures. The class Slot 𝐴 is a function whose argument can be any set, although it is meaningful only if that set is a member of an extensible structure (such as a partially ordered set or a group).

Note that Slot 𝐴 is implemented as "evaluation at 𝐴". That is, (Slot 𝐴𝑆) is defined to be (𝑆𝐴), where 𝐴 will typically be a small nonzero natural number. Each extensible structure 𝑆 is a function defined on specific natural number "slots", and this function extracts the value at a particular slot.

The special "structure" ndx, defined as the identity function restricted to , can be used to extract the number 𝐴 from a slot, since (Slot 𝐴‘ndx) = 𝐴 (see ndxarg 12426). This is typically used to refer to the number of a slot when defining structures without having to expose the detail of what that number is (for instance, we use the expression (Base‘ndx) in theorems and proofs instead of its value 1).

The class Slot cannot be defined as (𝑥 ∈ V ↦ (𝑓 ∈ V ↦ (𝑓𝑥))) because each Slot 𝐴 is a function on the proper class V so is itself a proper class, and the values of functions are sets (fvex 5514). It is necessary to allow proper classes as values of Slot 𝐴 since for instance the class of all (base sets of) groups is proper. (Contributed by Mario Carneiro, 22-Sep-2015.)

Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥𝐴))
 
Theoremsloteq 12408 Equality theorem for the Slot construction. The converse holds if 𝐴 (or 𝐵) is a set. (Contributed by BJ, 27-Dec-2021.)
(𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)
 
Definitiondf-base 12409 Define the base set (also called underlying set, ground set, carrier set, or carrier) extractor for extensible structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Base = Slot 1
 
Definitiondf-sets 12410* Set a component of an extensible structure. This function is useful for taking an existing structure and "overriding" one of its components. For example, df-ress 12411 adjusts the base set to match its second argument, which has the effect of making subgroups, subspaces, subrings etc. from the original structures. (Contributed by Mario Carneiro, 1-Dec-2014.)
sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
 
Definitiondf-ress 12411* Define a multifunction restriction operator for extensible structures, which can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the Base set while leaving operators alone; individual kinds of structures will need to handle this behavior, by ignoring operators' values outside the range, defining a function using the base set and applying that, or explicitly truncating the slot before use.

(Credit for this operator goes to Mario Carneiro.)

(Contributed by Stefan O'Rear, 29-Nov-2014.)

s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
 
Theorembrstruct 12412 The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
Rel Struct
 
Theoremisstruct2im 12413 The property of being a structure with components in (1st𝑋)...(2nd𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(𝐹 Struct 𝑋 → (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))
 
Theoremisstruct2r 12414 The property of being a structure with components in (1st𝑋)...(2nd𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹𝑉 ∧ dom 𝐹 ⊆ (...‘𝑋))) → 𝐹 Struct 𝑋)
 
Theoremstructex 12415 A structure is a set. (Contributed by AV, 10-Nov-2021.)
(𝐺 Struct 𝑋𝐺 ∈ V)
 
Theoremstructn0fun 12416 A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.)
(𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅}))
 
Theoremisstructim 12417 The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(𝐹 Struct ⟨𝑀, 𝑁⟩ → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁)))
 
Theoremisstructr 12418 The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 Struct ⟨𝑀, 𝑁⟩)
 
Theoremstructcnvcnv 12419 Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
(𝐹 Struct 𝑋𝐹 = (𝐹 ∖ {∅}))
 
Theoremstructfung 12420 The converse of the converse of a structure is a function. Closed form of structfun 12421. (Contributed by AV, 12-Nov-2021.)
(𝐹 Struct 𝑋 → Fun 𝐹)
 
Theoremstructfun 12421 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Proof shortened by AV, 12-Nov-2021.)
𝐹 Struct 𝑋       Fun 𝐹
 
Theoremstructfn 12422 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐹 Struct ⟨𝑀, 𝑁       (Fun 𝐹 ∧ dom 𝐹 ⊆ (1...𝑁))
 
Theoremstrnfvnd 12423 Deduction version of strnfvn 12424. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 19-Jan-2023.)
𝐸 = Slot 𝑁    &   (𝜑𝑆𝑉)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝐸𝑆) = (𝑆𝑁))
 
Theoremstrnfvn 12424 Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 12409) and 𝑁 is a fixed integer such as 1. 𝑆 is a structure, i.e. a specific member of a class of structures.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strslfv 12447. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.)

𝑆 ∈ V    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸𝑆) = (𝑆𝑁)
 
Theoremstrfvssn 12425 A structure component extractor produces a value which is contained in a set dependent on 𝑆, but not 𝐸. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Jim Kingdon, 19-Jan-2023.)
𝐸 = Slot 𝑁    &   (𝜑𝑆𝑉)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝐸𝑆) ⊆ ran 𝑆)
 
Theoremndxarg 12426 Get the numeric argument from a defined structure component extractor such as df-base 12409. (Contributed by Mario Carneiro, 6-Oct-2013.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸‘ndx) = 𝑁
 
Theoremndxid 12427 A structure component extractor is defined by its own index. This theorem, together with strslfv 12447 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 12409, making it easier to change should the need arise.

(Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof shortened by BJ, 27-Dec-2021.)

𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       𝐸 = Slot (𝐸‘ndx)
 
Theoremndxslid 12428 A structure component extractor is defined by its own index. That the index is a natural number will also be needed in quite a few contexts so it is included in the conclusion of this theorem which can be used as a hypothesis of theorems like strslfv 12447. (Contributed by Jim Kingdon, 29-Jan-2023.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
 
Theoremslotslfn 12429 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by Jim Kingdon, 10-Feb-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)       𝐸 Fn V
 
Theoremslotex 12430 Existence of slot value. A corollary of slotslfn 12429. (Contributed by Jim Kingdon, 12-Feb-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)       (𝐴𝑉 → (𝐸𝐴) ∈ V)
 
Theoremstrndxid 12431 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 12432 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Rel dom sSet
 
Theoremsetsvalg 12433 Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝑆𝑉𝐴𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
 
Theoremsetsvala 12434 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 12435 Applying the structure replacement function yields a set. (Contributed by Jim Kingdon, 22-Jan-2023.)
((𝑆𝑉𝐴𝑋𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V)
 
Theoremstrsetsid 12436 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 12437 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 12438 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 12439 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 12438 is useful for proofs based on isstruct2r 12414 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 12440 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 12441 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 12442 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 12443 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 12444 Deduction version of strslfv 12447. (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 12445 Deduction version of strslfv 12447. (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 12446 A variation on strslfv 12447 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 12447 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 12409). By virtue of ndxslid 12428, 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 12448 Variant on strslfv 12447 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 12449 Deduction version of strslss 12450. (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 12450 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 12451 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 12452 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
∅ = (Base‘∅)
 
Theoremsetsslid 12453 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 12454 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 12455 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 12456 Utility theorem: index-independent form of df-base 12409. (Contributed by NM, 20-Oct-2012.)
Base = Slot (Base‘ndx)
 
Theorembasendx 12457 Index value of the base set extractor.

Use of this theorem is discouraged since the particular value 1 for the index is an implementation detail. It is generally sufficient to work with (Base‘ndx) and use theorems such as baseid 12456 and basendxnn 12458.

The main circumstance in which it is necessary to look at indices directly is when showing that a set of indices are disjoint, in proofs such as lmodstrd 12538. Although we have a few theorems such as basendxnplusgndx 12511, we do not intend to add such theorems for every pair of indices (which would be quadradically many in the number of indices).

(New usage is discouraged.) (Contributed by Mario Carneiro, 2-Aug-2013.)

(Base‘ndx) = 1
 
Theorembasendxnn 12458 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 12459 The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.)
(Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ)
 
Theorembasfn 12460 The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
Base Fn V
 
Theorembasmex 12461 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 18-Nov-2024.)
𝐵 = (Base‘𝐺)       (𝐴𝐵𝐺 ∈ V)
 
Theorembasmexd 12462 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 28-Nov-2024.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑𝐴𝐵)       (𝜑𝐺 ∈ V)
 
Theoremreldmress 12463 The structure restriction is a proper operator, so it can be used with ovprc1 5886. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Rel dom ↾s
 
Theoremressid2 12464 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 12465 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 12466 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 12467 Extend class notation with group (addition) operation.
class +g
 
Syntaxcmulr 12468 Extend class notation with ring multiplication.
class .r
 
Syntaxcstv 12469 Extend class notation with involution.
class *𝑟
 
Syntaxcsca 12470 Extend class notation with scalar field.
class Scalar
 
Syntaxcvsca 12471 Extend class notation with scalar product.
class ·𝑠
 
Syntaxcip 12472 Extend class notation with Hermitian form (inner product).
class ·𝑖
 
Syntaxcts 12473 Extend class notation with the topology component of a topological space.
class TopSet
 
Syntaxcple 12474 Extend class notation with "less than or equal to" for posets.
class le
 
Syntaxcoc 12475 Extend class notation with the class of orthocomplementation extractors.
class oc
 
Syntaxcds 12476 Extend class notation with the metric space distance function.
class dist
 
Syntaxcunif 12477 Extend class notation with the uniform structure.
class UnifSet
 
Syntaxchom 12478 Extend class notation with the hom-set structure.
class Hom
 
Syntaxcco 12479 Extend class notation with the composition operation.
class comp
 
Definitiondf-plusg 12480 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
+g = Slot 2
 
Definitiondf-mulr 12481 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
.r = Slot 3
 
Definitiondf-starv 12482 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 12483 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 12484 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑠 = Slot 6
 
Definitiondf-ip 12485 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑖 = Slot 8
 
Definitiondf-tset 12486 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 12487 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 12488 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 12489 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 12490 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
UnifSet = Slot 13
 
Definitiondf-hom 12491 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hom = Slot 14
 
Definitiondf-cco 12492 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp = Slot 15
 
Theoremstrleund 12493 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 12494 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 12495 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 12496 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 12497 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 12498 Index value of the df-plusg 12480 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(+g‘ndx) = 2
 
Theoremplusgid 12499 Utility theorem: index-independent form of df-plusg 12480. (Contributed by NM, 20-Oct-2012.)
+g = Slot (+g‘ndx)
 
Theoremplusgslid 12500 Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.)
(+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
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