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Theorem List for Intuitionistic Logic Explorer - 12501-12600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremndxslid 12501 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 12521. (Contributed by Jim Kingdon, 29-Jan-2023.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
 
Theoremslotslfn 12502 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 12503 Existence of slot value. A corollary of slotslfn 12502. (Contributed by Jim Kingdon, 12-Feb-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)       (𝐴𝑉 → (𝐸𝐴) ∈ V)
 
Theoremstrndxid 12504 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 12505 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Rel dom sSet
 
Theoremsetsvalg 12506 Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝑆𝑉𝐴𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
 
Theoremsetsvala 12507 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 12508 Applying the structure replacement function yields a set. (Contributed by Jim Kingdon, 22-Jan-2023.)
((𝑆𝑉𝐴𝑋𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V)
 
Theoremstrsetsid 12509 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 12510 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 12511 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 12512 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 12511 is useful for proofs based on isstruct2r 12487 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 12513 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 12514 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 12515 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 12516 Different components can be set in any order. (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⟩))
 
Theoremsetscomd 12517 Different components can be set in any order. (Contributed by Jim Kingdon, 20-Feb-2025.)
(𝜑𝐴𝑌)    &   (𝜑𝐵𝑍)    &   (𝜑𝑆𝑉)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝑊)    &   (𝜑𝐷𝑋)       (𝜑 → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
 
Theoremstrslfvd 12518 Deduction version of strslfv 12521. (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 12519 Deduction version of strslfv 12521. (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 12520 A variation on strslfv 12521 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 12521 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 12482). By virtue of ndxslid 12501, 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 12522 Variant on strslfv 12521 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 12523 Deduction version of strslss 12524. (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 12524 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 12525 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 12526 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
∅ = (Base‘∅)
 
Theoremsetsslid 12527 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 12528 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 12529 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 12530 Utility theorem: index-independent form of df-base 12482. (Contributed by NM, 20-Oct-2012.)
Base = Slot (Base‘ndx)
 
Theorembasendx 12531 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 12530 and basendxnn 12532.

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 12637. Although we have a few theorems such as basendxnplusgndx 12598, 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 12532 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 12533 The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.)
(Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ)
 
Theorembasfn 12534 The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
Base Fn V
 
Theorembasmex 12535 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 18-Nov-2024.)
𝐵 = (Base‘𝐺)       (𝐴𝐵𝐺 ∈ V)
 
Theorembasmexd 12536 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 28-Nov-2024.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑𝐴𝐵)       (𝜑𝐺 ∈ V)
 
Theoremreldmress 12537 The structure restriction is a proper operator, so it can be used with ovprc1 5924. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Rel dom ↾s
 
Theoremressvalsets 12538 Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
((𝑊𝑋𝐴𝑌) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
 
Theoremressex 12539 Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
((𝑊𝑋𝐴𝑌) → (𝑊s 𝐴) ∈ V)
 
Theoremressval2 12540 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)⟩))
 
Theoremressbasd 12541 Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
(𝜑𝑅 = (𝑊s 𝐴))    &   (𝜑𝐵 = (Base‘𝑊))    &   (𝜑𝑊𝑋)    &   (𝜑𝐴𝑉)       (𝜑 → (𝐴𝐵) = (Base‘𝑅))
 
Theoremressbas2d 12542 Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.)
(𝜑𝑅 = (𝑊s 𝐴))    &   (𝜑𝐵 = (Base‘𝑊))    &   (𝜑𝑊𝑋)    &   (𝜑𝐴𝐵)       (𝜑𝐴 = (Base‘𝑅))
 
Theoremressbasssd 12543 The base set of a restriction is a subset of the base set of the original structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝜑𝑅 = (𝑊s 𝐴))    &   (𝜑𝐵 = (Base‘𝑊))    &   (𝜑𝑊𝑋)    &   (𝜑𝐴𝑉)       (𝜑 → (Base‘𝑅) ⊆ 𝐵)
 
Theoremressbasid 12544 The trivial structure restriction leaves the base set unchanged. (Contributed by Jim Kingdon, 29-Apr-2025.)
𝐵 = (Base‘𝑊)       (𝑊𝑉 → (Base‘(𝑊s 𝐵)) = 𝐵)
 
Theoremstrressid 12545 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.)
(𝜑𝐵 = (Base‘𝑊))    &   (𝜑𝑊 Struct ⟨𝑀, 𝑁⟩)    &   (𝜑 → Fun 𝑊)    &   (𝜑 → (Base‘ndx) ∈ dom 𝑊)       (𝜑 → (𝑊s 𝐵) = 𝑊)
 
Theoremressval3d 12546 Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)
𝑅 = (𝑆s 𝐴)    &   𝐵 = (Base‘𝑆)    &   𝐸 = (Base‘ndx)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑𝐸 ∈ dom 𝑆)    &   (𝜑𝐴𝐵)       (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
 
Theoremresseqnbasd 12547 The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)
𝑅 = (𝑊s 𝐴)    &   𝐶 = (𝐸𝑊)    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝐸‘ndx) ≠ (Base‘ndx)    &   (𝜑𝑊𝑋)    &   (𝜑𝐴𝑉)       (𝜑𝐶 = (𝐸𝑅))
 
Theoremressinbasd 12548 Restriction only cares about the part of the second set which intersects the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.)
(𝜑𝐵 = (Base‘𝑊))    &   (𝜑𝐴𝑋)    &   (𝜑𝑊𝑉)       (𝜑 → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
 
Theoremressressg 12549 Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)
((𝐴𝑋𝐵𝑌𝑊𝑍) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
 
Theoremressabsg 12550 Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
((𝐴𝑋𝐵𝐴𝑊𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐵))
 
6.1.2  Slot definitions
 
Syntaxcplusg 12551 Extend class notation with group (addition) operation.
class +g
 
Syntaxcmulr 12552 Extend class notation with ring multiplication.
class .r
 
Syntaxcstv 12553 Extend class notation with involution.
class *𝑟
 
Syntaxcsca 12554 Extend class notation with scalar field.
class Scalar
 
Syntaxcvsca 12555 Extend class notation with scalar product.
class ·𝑠
 
Syntaxcip 12556 Extend class notation with Hermitian form (inner product).
class ·𝑖
 
Syntaxcts 12557 Extend class notation with the topology component of a topological space.
class TopSet
 
Syntaxcple 12558 Extend class notation with "less than or equal to" for posets.
class le
 
Syntaxcoc 12559 Extend class notation with the class of orthocomplementation extractors.
class oc
 
Syntaxcds 12560 Extend class notation with the metric space distance function.
class dist
 
Syntaxcunif 12561 Extend class notation with the uniform structure.
class UnifSet
 
Syntaxchom 12562 Extend class notation with the hom-set structure.
class Hom
 
Syntaxcco 12563 Extend class notation with the composition operation.
class comp
 
Definitiondf-plusg 12564 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
+g = Slot 2
 
Definitiondf-mulr 12565 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
.r = Slot 3
 
Definitiondf-starv 12566 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 12567 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 12568 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑠 = Slot 6
 
Definitiondf-ip 12569 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑖 = Slot 8
 
Definitiondf-tset 12570 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 12571 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 12572 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 12573 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 12574 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
UnifSet = Slot 13
 
Definitiondf-hom 12575 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hom = Slot 14
 
Definitiondf-cco 12576 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp = Slot 15
 
Theoremstrleund 12577 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 12578 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
 
Theoremstrext 12579 Extending the upper range of a structure. This works because when we say that a structure has components in 𝐴...𝐶 we are not saying that every slot in that range is present, just that all the slots that are present are within that range. (Contributed by Jim Kingdon, 26-Feb-2025.)
(𝜑𝐹 Struct ⟨𝐴, 𝐵⟩)    &   (𝜑𝐶 ∈ (ℤ𝐵))       (𝜑𝐹 Struct ⟨𝐴, 𝐶⟩)
 
Theoremstrle1g 12580 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 12581 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 12582 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 12583 Index value of the df-plusg 12564 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(+g‘ndx) = 2
 
Theoremplusgid 12584 Utility theorem: index-independent form of df-plusg 12564. (Contributed by NM, 20-Oct-2012.)
+g = Slot (+g‘ndx)
 
Theoremplusgndxnn 12585 The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 17-Oct-2024.)
(+g‘ndx) ∈ ℕ
 
Theoremplusgslid 12586 Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.)
(+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
 
Theorembasendxltplusgndx 12587 The index of the slot for the base set is less then the index of the slot for the group operation in an extensible structure. (Contributed by AV, 17-Oct-2024.)
(Base‘ndx) < (+g‘ndx)
 
Theoremopelstrsl 12588 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 12589 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 12590 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 12591 The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}       (𝐵𝑉𝐵 = (Base‘𝐺))
 
Theorem2strstrg 12592 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 12593 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 12594 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 12595 A constructed two-slot structure. Version of 2strstrg 12592 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 12596 The base set of a constructed two-slot structure. Version of 2strbasg 12593 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 12597 The other slot of a constructed two-slot structure. Version of 2stropg 12594 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 12598 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 12599 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 12600 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))
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