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Theorem List for Metamath Proof Explorer - 16501-16600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreldmsets 16501 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Rel dom sSet
 
Theoremsetsvalg 16502 Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝑆𝑉𝐴𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
 
Theoremsetsval 16503 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
((𝑆𝑉𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
 
Theoremsetsidvald 16504 Value of the structure replacement function, deduction version. (Contributed by AV, 14-Mar-2020.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → (𝐸‘ndx) ∈ dom 𝑆)       (𝜑𝑆 = (𝑆 sSet ⟨(𝐸‘ndx), (𝐸𝑆)⟩))
 
Theoremfvsetsid 16505 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)
 
Theoremfsets 16506 The structure replacement function is a function. (Contributed by SO, 12-Jul-2018.)
(((𝐹𝑉𝐹:𝐴𝐵) ∧ 𝑋𝐴𝑌𝐵) → (𝐹 sSet ⟨𝑋, 𝑌⟩):𝐴𝐵)
 
Theoremsetsdm 16507 The domain of a structure with replacement is the domain of the original structure extended by the index of the replacement. (Contributed by AV, 7-Jun-2021.)
((𝐺𝑉𝐸𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
 
Theoremsetsfun 16508 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 16509 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 16508 is useful for proofs based on isstruct2 16483 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 16510 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⟩) ∖ {∅}))
 
Theoremsetsstruct2 16511 An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)
(((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) ∧ 𝑌 = ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑌)
 
Theoremsetsexstruct2 16512* An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)
((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → ∃𝑦(𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑦)
 
Theoremsetsstruct 16513 An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 9-Jun-2021.) (Revised by AV, 14-Nov-2021.)
((𝐸𝑉𝐼 ∈ (ℤ𝑀) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩)
 
Theoremwunsets 16514 Closure of structure replacement in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝑆𝑈)    &   (𝜑𝐴𝑈)       (𝜑 → (𝑆 sSet 𝐴) ∈ 𝑈)
 
Theoremsetsres 16515 The structure replacement function does not affect the value of 𝑆 away from 𝐴. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝑆𝑉 → ((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴})))
 
Theoremsetsabs 16516 Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.)
((𝑆𝑉𝐶𝑊) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))
 
Theoremsetscom 16517 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⟩))
 
Theoremstrfvd 16518 Deduction version of strfv 16521. (Contributed by Mario Carneiro, 15-Nov-2014.)
𝐸 = Slot (𝐸‘ndx)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)       (𝜑𝐶 = (𝐸𝑆))
 
Theoremstrfv2d 16519 Deduction version of strfv2 16520. (Contributed by Mario Carneiro, 30-Apr-2015.)
𝐸 = Slot (𝐸‘ndx)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)    &   (𝜑𝐶𝑊)       (𝜑𝐶 = (𝐸𝑆))
 
Theoremstrfv2 16520 A variation on strfv 16521 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.)
𝑆 ∈ V    &   Fun 𝑆    &   𝐸 = Slot (𝐸‘ndx)    &   ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆       (𝐶𝑉𝐶 = (𝐸𝑆))
 
Theoremstrfv 16521 Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 (such as a member of Poset, df-poset 17546) with a component extractor 𝐸 (such as the base set extractor df-base 16479). By virtue of ndxid 16499, 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 Mario Carneiro, 29-Aug-2015.)
𝑆 Struct 𝑋    &   𝐸 = Slot (𝐸‘ndx)    &   {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆       (𝐶𝑉𝐶 = (𝐸𝑆))
 
Theoremstrfv3 16522 Variant on strfv 16521 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.)
(𝜑𝑈 = 𝑆)    &   𝑆 Struct 𝑋    &   𝐸 = Slot (𝐸‘ndx)    &   {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆    &   (𝜑𝐶𝑉)    &   𝐴 = (𝐸𝑈)       (𝜑𝐴 = 𝐶)
 
Theoremstrssd 16523 Deduction version of strss 16524. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐸 = Slot (𝐸‘ndx)    &   (𝜑𝑇𝑉)    &   (𝜑 → Fun 𝑇)    &   (𝜑𝑆𝑇)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)       (𝜑 → (𝐸𝑇) = (𝐸𝑆))
 
Theoremstrss 16524 Propagate component extraction to a structure 𝑇 from a subset structure 𝑆. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 15-Jan-2014.)
𝑇 ∈ V    &   Fun 𝑇    &   𝑆𝑇    &   𝐸 = Slot (𝐸‘ndx)    &   ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆       (𝐸𝑇) = (𝐸𝑆)
 
Theoremstr0 16525 All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
𝐹 = Slot 𝐼       ∅ = (𝐹‘∅)
 
Theorembase0 16526 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
∅ = (Base‘∅)
 
Theoremstrfvi 16527 Structure slot extractors cannot distinguish between proper classes and , so they can be protected using the identity function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
𝐸 = Slot 𝑁    &   𝑋 = (𝐸𝑆)       𝑋 = (𝐸‘( I ‘𝑆))
 
Theoremsetsid 16528 Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐸 = Slot (𝐸‘ndx)       ((𝑊𝐴𝐶𝑉) → 𝐶 = (𝐸‘(𝑊 sSet ⟨(𝐸‘ndx), 𝐶⟩)))
 
Theoremsetsnid 16529 Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ 𝐷       (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩))
 
Theoremsbcie2s 16530* A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝐴 = (𝐸𝑊)    &   𝐵 = (𝐹𝑊)    &   ((𝑎 = 𝐴𝑏 = 𝐵) → (𝜑𝜓))       (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏]𝜓𝜑))
 
Theoremsbcie3s 16531* A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 15-Mar-2019.)
𝐴 = (𝐸𝑊)    &   𝐵 = (𝐹𝑊)    &   𝐶 = (𝐺𝑊)    &   ((𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶) → (𝜑𝜓))       (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏][(𝐺𝑤) / 𝑐]𝜓𝜑))
 
Theorembaseval 16532 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 16533 Utility theorem: index-independent form of df-base 16479. (Contributed by NM, 20-Oct-2012.)
Base = Slot (Base‘ndx)
 
Theoremelbasfv 16534 Utility theorem: reverse closure for any structure defined as a function. (Contributed by Stefan O'Rear, 24-Aug-2015.)
𝑆 = (𝐹𝑍)    &   𝐵 = (Base‘𝑆)       (𝑋𝐵𝑍 ∈ V)
 
Theoremelbasov 16535 Utility theorem: reverse closure for any structure defined as a two-argument function. (Contributed by Mario Carneiro, 3-Oct-2015.)
Rel dom 𝑂    &   𝑆 = (𝑋𝑂𝑌)    &   𝐵 = (Base‘𝑆)       (𝐴𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
 
Theoremstrov2rcl 16536 Partial reverse closure for any structure defined as a two-argument function. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 2-Dec-2019.)
𝑆 = (𝐼𝐹𝑅)    &   𝐵 = (Base‘𝑆)    &   Rel dom 𝐹       (𝑋𝐵𝐼 ∈ V)
 
Theorembasendx 16537 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 16538 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
 
Theorembasprssdmsets 16539 The pair of the base index and another index is a subset of the domain of the structure obtained by replacing/adding a slot at the other index in a structure having a base slot. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
(𝜑𝑆 Struct 𝑋)    &   (𝜑𝐼𝑈)    &   (𝜑𝐸𝑊)    &   (𝜑 → (Base‘ndx) ∈ dom 𝑆)       (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
 
Theoremreldmress 16540 The structure restriction is a proper operator, so it can be used with ovprc1 7184. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Rel dom ↾s
 
Theoremressval 16541 Value of structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       ((𝑊𝑋𝐴𝑌) → 𝑅 = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))
 
Theoremressid2 16542 General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = 𝑊)
 
Theoremressval2 16543 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)⟩))
 
Theoremressbas 16544 Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       (𝐴𝑉 → (𝐴𝐵) = (Base‘𝑅))
 
Theoremressbas2 16545 Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       (𝐴𝐵𝐴 = (Base‘𝑅))
 
Theoremressbasss 16546 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.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       (Base‘𝑅) ⊆ 𝐵
 
Theoremresslem 16547 Other elements of a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐶 = (𝐸𝑊)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   1 < 𝑁       (𝐴𝑉𝐶 = (𝐸𝑅))
 
Theoremress0 16548 All restrictions of the null set are trivial. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
(∅ ↾s 𝐴) = ∅
 
Theoremressid 16549 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝐵 = (Base‘𝑊)       (𝑊𝑋 → (𝑊s 𝐵) = 𝑊)
 
Theoremressinbas 16550 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.)
𝐵 = (Base‘𝑊)       (𝐴𝑋 → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
 
Theoremressval3d 16551 Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 3-Jul-2022.)
𝑅 = (𝑆s 𝐴)    &   𝐵 = (Base‘𝑆)    &   𝐸 = (Base‘ndx)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑𝐸 ∈ dom 𝑆)    &   (𝜑𝐴𝐵)       (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
 
Theoremressress 16552 Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)
((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
 
Theoremressabs 16553 Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
((𝐴𝑋𝐵𝐴) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐵))
 
Theoremwunress 16554 Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝑊𝑈)       (𝜑 → (𝑊s 𝐴) ∈ 𝑈)
 
7.1.2  Slot definitions
 
Syntaxcplusg 16555 Extend class notation with group (addition) operation.
class +g
 
Syntaxcmulr 16556 Extend class notation with ring multiplication.
class .r
 
Syntaxcstv 16557 Extend class notation with involution.
class *𝑟
 
Syntaxcsca 16558 Extend class notation with scalar field.
class Scalar
 
Syntaxcvsca 16559 Extend class notation with scalar product.
class ·𝑠
 
Syntaxcip 16560 Extend class notation with Hermitian form (inner product).
class ·𝑖
 
Syntaxcts 16561 Extend class notation with the topology component of a topological space.
class TopSet
 
Syntaxcple 16562 Extend class notation with "less than or equal to" for posets.
class le
 
Syntaxcoc 16563 Extend class notation with the class of orthocomplementation extractors.
class oc
 
Syntaxcds 16564 Extend class notation with the metric space distance function.
class dist
 
Syntaxcunif 16565 Extend class notation with the uniform structure.
class UnifSet
 
Syntaxchom 16566 Extend class notation with the hom-set structure.
class Hom
 
Syntaxcco 16567 Extend class notation with the composition operation.
class comp
 
Definitiondf-plusg 16568 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
+g = Slot 2
 
Definitiondf-mulr 16569 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
.r = Slot 3
 
Definitiondf-starv 16570 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 16571 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 16572 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑠 = Slot 6
 
Definitiondf-ip 16573 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑖 = Slot 8
 
Definitiondf-tset 16574 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 16575 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 16576 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 16577 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 16578 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
UnifSet = Slot 13
 
Definitiondf-hom 16579 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hom = Slot 14
 
Definitiondf-cco 16580 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp = Slot 15
 
Theoremstrleun 16581 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 16582 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐼 ∈ ℕ    &   𝐴 = 𝐼       {⟨𝐴, 𝑋⟩} Struct ⟨𝐼, 𝐼
 
Theoremstrle2 16583 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 16584 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 16585 Index value of the df-plusg 16568 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(+g‘ndx) = 2
 
Theoremplusgid 16586 Utility theorem: index-independent form of df-plusg 16568. (Contributed by NM, 20-Oct-2012.)
+g = Slot (+g‘ndx)
 
Theoremopelstrbas 16587 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 16588 A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}       𝐺 Struct ⟨1, 1⟩
 
Theorem1strbas 16589 The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}       (𝐵𝑉𝐵 = (Base‘𝐺))
 
Theorem1strwunbndx 16590 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 16591 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 16592 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 16593 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 16594 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 16595 A constructed two-slot structure. Version of 2strstr 16592 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 16596 The base set of a constructed two-slot structure. Version of 2strbas 16593 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 16597 The other slot of a constructed two-slot structure. Version of 2strop 16594 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 16598 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 16599 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 16600 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))
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