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Theorem List for Intuitionistic Logic Explorer - 12501-12600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsetscom 12501 Different components can be set in any order. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐴 ∈ V    &   π΅ ∈ V    β‡’   (((𝑆 ∈ 𝑉 ∧ 𝐴 β‰  𝐡) ∧ (𝐢 ∈ π‘Š ∧ 𝐷 ∈ 𝑋)) β†’ ((𝑆 sSet ⟨𝐴, 𝐢⟩) sSet ⟨𝐡, 𝐷⟩) = ((𝑆 sSet ⟨𝐡, 𝐷⟩) sSet ⟨𝐴, 𝐢⟩))
 
Theoremsetscomd 12502 Different components can be set in any order. (Contributed by Jim Kingdon, 20-Feb-2025.)
(πœ‘ β†’ 𝐴 ∈ π‘Œ)    &   (πœ‘ β†’ 𝐡 ∈ 𝑍)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    &   (πœ‘ β†’ 𝐢 ∈ π‘Š)    &   (πœ‘ β†’ 𝐷 ∈ 𝑋)    β‡’   (πœ‘ β†’ ((𝑆 sSet ⟨𝐴, 𝐢⟩) sSet ⟨𝐡, 𝐷⟩) = ((𝑆 sSet ⟨𝐡, 𝐷⟩) sSet ⟨𝐴, 𝐢⟩))
 
Theoremstrslfvd 12503 Deduction version of strslfv 12506. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 30-Jan-2023.)
(𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ Fun 𝑆)    &   (πœ‘ β†’ ⟨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆)    β‡’   (πœ‘ β†’ 𝐢 = (πΈβ€˜π‘†))
 
Theoremstrslfv2d 12504 Deduction version of strslfv 12506. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
(𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ Fun ◑◑𝑆)    &   (πœ‘ β†’ ⟨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆)    &   (πœ‘ β†’ 𝐢 ∈ π‘Š)    β‡’   (πœ‘ β†’ 𝐢 = (πΈβ€˜π‘†))
 
Theoremstrslfv2 12505 A variation on strslfv 12506 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 (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   βŸ¨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆    β‡’   (𝐢 ∈ 𝑉 β†’ 𝐢 = (πΈβ€˜π‘†))
 
Theoremstrslfv 12506 Extract a structure component 𝐢 (such as the base set) from a structure 𝑆 with a component extractor 𝐸 (such as the base set extractor df-base 12467). By virtue of ndxslid 12486, 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 (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   {⟨(πΈβ€˜ndx), 𝐢⟩} βŠ† 𝑆    β‡’   (𝐢 ∈ 𝑉 β†’ 𝐢 = (πΈβ€˜π‘†))
 
Theoremstrslfv3 12507 Variant on strslfv 12506 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.)
(πœ‘ β†’ π‘ˆ = 𝑆)    &   π‘† Struct 𝑋    &   (𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   {⟨(πΈβ€˜ndx), 𝐢⟩} βŠ† 𝑆    &   (πœ‘ β†’ 𝐢 ∈ 𝑉)    &   π΄ = (πΈβ€˜π‘ˆ)    β‡’   (πœ‘ β†’ 𝐴 = 𝐢)
 
Theoremstrslssd 12508 Deduction version of strslss 12509. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.)
(𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (πœ‘ β†’ 𝑇 ∈ 𝑉)    &   (πœ‘ β†’ Fun 𝑇)    &   (πœ‘ β†’ 𝑆 βŠ† 𝑇)    &   (πœ‘ β†’ ⟨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆)    β‡’   (πœ‘ β†’ (πΈβ€˜π‘‡) = (πΈβ€˜π‘†))
 
Theoremstrslss 12509 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    &   π‘† βŠ† 𝑇    &   (𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   βŸ¨(πΈβ€˜ndx), 𝐢⟩ ∈ 𝑆    β‡’   (πΈβ€˜π‘‡) = (πΈβ€˜π‘†)
 
Theoremstrsl0 12510 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 (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    β‡’   βˆ… = (πΈβ€˜βˆ…)
 
Theorembase0 12511 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
βˆ… = (Baseβ€˜βˆ…)
 
Theoremsetsslid 12512 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 (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    β‡’   ((π‘Š ∈ 𝐴 ∧ 𝐢 ∈ 𝑉) β†’ 𝐢 = (πΈβ€˜(π‘Š sSet ⟨(πΈβ€˜ndx), 𝐢⟩)))
 
Theoremsetsslnid 12513 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 (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (πΈβ€˜ndx) β‰  𝐷    &   π· ∈ β„•    β‡’   ((π‘Š ∈ 𝐴 ∧ 𝐢 ∈ 𝑉) β†’ (πΈβ€˜π‘Š) = (πΈβ€˜(π‘Š sSet ⟨𝐷, 𝐢⟩)))
 
Theorembaseval 12514 Value of the base set extractor. (Normally it is preferred to work with (Baseβ€˜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    β‡’   (Baseβ€˜πΎ) = (πΎβ€˜1)
 
Theorembaseid 12515 Utility theorem: index-independent form of df-base 12467. (Contributed by NM, 20-Oct-2012.)
Base = Slot (Baseβ€˜ndx)
 
Theorembasendx 12516 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 (Baseβ€˜ndx) and use theorems such as baseid 12515 and basendxnn 12517.

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 12621. Although we have a few theorems such as basendxnplusgndx 12582, 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.)

(Baseβ€˜ndx) = 1
 
Theorembasendxnn 12517 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.)
(Baseβ€˜ndx) ∈ β„•
 
Theorembaseslid 12518 The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.)
(Base = Slot (Baseβ€˜ndx) ∧ (Baseβ€˜ndx) ∈ β„•)
 
Theorembasfn 12519 The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
Base Fn V
 
Theorembasmex 12520 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 18-Nov-2024.)
𝐡 = (Baseβ€˜πΊ)    β‡’   (𝐴 ∈ 𝐡 β†’ 𝐺 ∈ V)
 
Theorembasmexd 12521 A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 28-Nov-2024.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΊ))    &   (πœ‘ β†’ 𝐴 ∈ 𝐡)    β‡’   (πœ‘ β†’ 𝐺 ∈ V)
 
Theoremreldmress 12522 The structure restriction is a proper operator, so it can be used with ovprc1 5910. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Rel dom β†Ύs
 
Theoremressvalsets 12523 Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩))
 
Theoremressex 12524 Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ (π‘Š β†Ύs 𝐴) ∈ V)
 
Theoremressval2 12525 Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝑅 = (π‘Š β†Ύs 𝐴)    &   π΅ = (Baseβ€˜π‘Š)    β‡’   ((Β¬ 𝐡 βŠ† 𝐴 ∧ π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝑅 = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩))
 
Theoremressbasd 12526 Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
(πœ‘ β†’ 𝑅 = (π‘Š β†Ύs 𝐴))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝐴 ∩ 𝐡) = (Baseβ€˜π‘…))
 
Theoremressbas2d 12527 Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.)
(πœ‘ β†’ 𝑅 = (π‘Š β†Ύs 𝐴))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐡)    β‡’   (πœ‘ β†’ 𝐴 = (Baseβ€˜π‘…))
 
Theoremressbasssd 12528 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.)
(πœ‘ β†’ 𝑅 = (π‘Š β†Ύs 𝐴))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    β‡’   (πœ‘ β†’ (Baseβ€˜π‘…) βŠ† 𝐡)
 
Theoremstrressid 12529 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š Struct βŸ¨π‘€, π‘βŸ©)    &   (πœ‘ β†’ Fun π‘Š)    &   (πœ‘ β†’ (Baseβ€˜ndx) ∈ dom π‘Š)    β‡’   (πœ‘ β†’ (π‘Š β†Ύs 𝐡) = π‘Š)
 
Theoremressval3d 12530 Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)
𝑅 = (𝑆 β†Ύs 𝐴)    &   π΅ = (Baseβ€˜π‘†)    &   πΈ = (Baseβ€˜ndx)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ Fun 𝑆)    &   (πœ‘ β†’ 𝐸 ∈ dom 𝑆)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐡)    β‡’   (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
 
Theoremresseqnbasd 12531 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 = (π‘Š β†Ύs 𝐴)    &   πΆ = (πΈβ€˜π‘Š)    &   (𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (πΈβ€˜ndx) β‰  (Baseβ€˜ndx)    &   (πœ‘ β†’ π‘Š ∈ 𝑋)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝐢 = (πΈβ€˜π‘…))
 
Theoremressinbasd 12532 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.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘Š))    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    &   (πœ‘ β†’ π‘Š ∈ 𝑉)    β‡’   (πœ‘ β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
 
Theoremressressg 12533 Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)
((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ ((π‘Š β†Ύs 𝐴) β†Ύs 𝐡) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
 
Theoremressabsg 12534 Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
((𝐴 ∈ 𝑋 ∧ 𝐡 βŠ† 𝐴 ∧ π‘Š ∈ π‘Œ) β†’ ((π‘Š β†Ύs 𝐴) β†Ύs 𝐡) = (π‘Š β†Ύs 𝐡))
 
6.1.2  Slot definitions
 
Syntaxcplusg 12535 Extend class notation with group (addition) operation.
class +g
 
Syntaxcmulr 12536 Extend class notation with ring multiplication.
class .r
 
Syntaxcstv 12537 Extend class notation with involution.
class *π‘Ÿ
 
Syntaxcsca 12538 Extend class notation with scalar field.
class Scalar
 
Syntaxcvsca 12539 Extend class notation with scalar product.
class ·𝑠
 
Syntaxcip 12540 Extend class notation with Hermitian form (inner product).
class ·𝑖
 
Syntaxcts 12541 Extend class notation with the topology component of a topological space.
class TopSet
 
Syntaxcple 12542 Extend class notation with "less than or equal to" for posets.
class le
 
Syntaxcoc 12543 Extend class notation with the class of orthocomplementation extractors.
class oc
 
Syntaxcds 12544 Extend class notation with the metric space distance function.
class dist
 
Syntaxcunif 12545 Extend class notation with the uniform structure.
class UnifSet
 
Syntaxchom 12546 Extend class notation with the hom-set structure.
class Hom
 
Syntaxcco 12547 Extend class notation with the composition operation.
class comp
 
Definitiondf-plusg 12548 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
+g = Slot 2
 
Definitiondf-mulr 12549 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
.r = Slot 3
 
Definitiondf-starv 12550 Define the involution function of a *-ring. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
*π‘Ÿ = Slot 4
 
Definitiondf-sca 12551 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 12552 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑠 = Slot 6
 
Definitiondf-ip 12553 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑖 = Slot 8
 
Definitiondf-tset 12554 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 12555 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 12556 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 12557 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 12558 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
UnifSet = Slot 13
 
Definitiondf-hom 12559 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hom = Slot 14
 
Definitiondf-cco 12560 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp = Slot 15
 
Theoremstrleund 12561 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
(πœ‘ β†’ 𝐹 Struct ⟨𝐴, 𝐡⟩)    &   (πœ‘ β†’ 𝐺 Struct ⟨𝐢, 𝐷⟩)    &   (πœ‘ β†’ 𝐡 < 𝐢)    β‡’   (πœ‘ β†’ (𝐹 βˆͺ 𝐺) Struct ⟨𝐴, 𝐷⟩)
 
Theoremstrleun 12562 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐹 Struct ⟨𝐴, 𝐡⟩    &   πΊ Struct ⟨𝐢, 𝐷⟩    &   π΅ < 𝐢    β‡’   (𝐹 βˆͺ 𝐺) Struct ⟨𝐴, 𝐷⟩
 
Theoremstrext 12563 Extending the upper range of a structure. This works because when we say that a structure has components in 𝐴...𝐢 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.)
(πœ‘ β†’ 𝐹 Struct ⟨𝐴, 𝐡⟩)    &   (πœ‘ β†’ 𝐢 ∈ (β„€β‰₯β€˜π΅))    β‡’   (πœ‘ β†’ 𝐹 Struct ⟨𝐴, 𝐢⟩)
 
Theoremstrle1g 12564 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
𝐼 ∈ β„•    &   π΄ = 𝐼    β‡’   (𝑋 ∈ 𝑉 β†’ {⟨𝐴, π‘‹βŸ©} Struct ⟨𝐼, 𝐼⟩)
 
Theoremstrle2g 12565 Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
𝐼 ∈ β„•    &   π΄ = 𝐼    &   πΌ < 𝐽    &   π½ ∈ β„•    &   π΅ = 𝐽    β‡’   ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ π‘Š) β†’ {⟨𝐴, π‘‹βŸ©, ⟨𝐡, π‘ŒβŸ©} Struct ⟨𝐼, 𝐽⟩)
 
Theoremstrle3g 12566 Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐼 ∈ β„•    &   π΄ = 𝐼    &   πΌ < 𝐽    &   π½ ∈ β„•    &   π΅ = 𝐽    &   π½ < 𝐾    &   πΎ ∈ β„•    &   πΆ = 𝐾    β‡’   ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ π‘Š ∧ 𝑍 ∈ 𝑃) β†’ {⟨𝐴, π‘‹βŸ©, ⟨𝐡, π‘ŒβŸ©, ⟨𝐢, π‘βŸ©} Struct ⟨𝐼, 𝐾⟩)
 
Theoremplusgndx 12567 Index value of the df-plusg 12548 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(+gβ€˜ndx) = 2
 
Theoremplusgid 12568 Utility theorem: index-independent form of df-plusg 12548. (Contributed by NM, 20-Oct-2012.)
+g = Slot (+gβ€˜ndx)
 
Theoremplusgndxnn 12569 The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 17-Oct-2024.)
(+gβ€˜ndx) ∈ β„•
 
Theoremplusgslid 12570 Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.)
(+g = Slot (+gβ€˜ndx) ∧ (+gβ€˜ndx) ∈ β„•)
 
Theorembasendxltplusgndx 12571 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.)
(Baseβ€˜ndx) < (+gβ€˜ndx)
 
Theoremopelstrsl 12572 The slot of a structure which contains an ordered pair for that slot. (Contributed by Jim Kingdon, 5-Feb-2023.)
(𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (πœ‘ β†’ 𝑆 Struct 𝑋)    &   (πœ‘ β†’ 𝑉 ∈ π‘Œ)    &   (πœ‘ β†’ ⟨(πΈβ€˜ndx), π‘‰βŸ© ∈ 𝑆)    β‡’   (πœ‘ β†’ 𝑉 = (πΈβ€˜π‘†))
 
Theoremopelstrbas 12573 The base set of a structure with a base set. (Contributed by AV, 10-Nov-2021.)
(πœ‘ β†’ 𝑆 Struct 𝑋)    &   (πœ‘ β†’ 𝑉 ∈ π‘Œ)    &   (πœ‘ β†’ ⟨(Baseβ€˜ndx), π‘‰βŸ© ∈ 𝑆)    β‡’   (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘†))
 
Theorem1strstrg 12574 A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩}    β‡’   (𝐡 ∈ 𝑉 β†’ 𝐺 Struct ⟨1, 1⟩)
 
Theorem1strbas 12575 The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩}    β‡’   (𝐡 ∈ 𝑉 β†’ 𝐡 = (Baseβ€˜πΊ))
 
Theorem2strstrg 12576 A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩}    &   πΈ = Slot 𝑁    &   1 < 𝑁    &   π‘ ∈ β„•    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ 𝐺 Struct ⟨1, π‘βŸ©)
 
Theorem2strbasg 12577 The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩}    &   πΈ = Slot 𝑁    &   1 < 𝑁    &   π‘ ∈ β„•    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ 𝐡 = (Baseβ€˜πΊ))
 
Theorem2stropg 12578 The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(πΈβ€˜ndx), + ⟩}    &   πΈ = Slot 𝑁    &   1 < 𝑁    &   π‘ ∈ β„•    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ + = (πΈβ€˜πΊ))
 
Theorem2strstr1g 12579 A constructed two-slot structure. Version of 2strstrg 12576 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 = {⟨(Baseβ€˜ndx), 𝐡⟩, βŸ¨π‘, + ⟩}    &   (Baseβ€˜ndx) < 𝑁    &   π‘ ∈ β„•    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ 𝐺 Struct ⟨(Baseβ€˜ndx), π‘βŸ©)
 
Theorem2strbas1g 12580 The base set of a constructed two-slot structure. Version of 2strbasg 12577 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 = {⟨(Baseβ€˜ndx), 𝐡⟩, βŸ¨π‘, + ⟩}    &   (Baseβ€˜ndx) < 𝑁    &   π‘ ∈ β„•    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ 𝐡 = (Baseβ€˜πΊ))
 
Theorem2strop1g 12581 The other slot of a constructed two-slot structure. Version of 2stropg 12578 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 = {⟨(Baseβ€˜ndx), 𝐡⟩, βŸ¨π‘, + ⟩}    &   (Baseβ€˜ndx) < 𝑁    &   π‘ ∈ β„•    &   πΈ = Slot 𝑁    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ + = (πΈβ€˜πΊ))
 
Theorembasendxnplusgndx 12582 The slot for the base set is not the slot for the group operation in an extensible structure. (Contributed by AV, 14-Nov-2021.)
(Baseβ€˜ndx) β‰  (+gβ€˜ndx)
 
Theoremgrpstrg 12583 A constructed group is a structure on 1...2. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩}    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ 𝐺 Struct ⟨1, 2⟩)
 
Theoremgrpbaseg 12584 The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩}    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ 𝐡 = (Baseβ€˜πΊ))
 
Theoremgrpplusgg 12585 The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩}    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š) β†’ + = (+gβ€˜πΊ))
 
Theoremressplusgd 12586 +g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
(πœ‘ β†’ 𝐻 = (𝐺 β†Ύs 𝐴))    &   (πœ‘ β†’ + = (+gβ€˜πΊ))    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐺 ∈ π‘Š)    β‡’   (πœ‘ β†’ + = (+gβ€˜π»))
 
Theoremmulrndx 12587 Index value of the df-mulr 12549 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(.rβ€˜ndx) = 3
 
Theoremmulridx 12588 Utility theorem: index-independent form of df-mulr 12549. (Contributed by Mario Carneiro, 8-Jun-2013.)
.r = Slot (.rβ€˜ndx)
 
Theoremmulrslid 12589 Slot property of .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
(.r = Slot (.rβ€˜ndx) ∧ (.rβ€˜ndx) ∈ β„•)
 
Theoremplusgndxnmulrndx 12590 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.)
(+gβ€˜ndx) β‰  (.rβ€˜ndx)
 
Theorembasendxnmulrndx 12591 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.)
(Baseβ€˜ndx) β‰  (.rβ€˜ndx)
 
Theoremrngstrg 12592 A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
𝑅 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Β· ⟩}    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š ∧ Β· ∈ 𝑋) β†’ 𝑅 Struct ⟨1, 3⟩)
 
Theoremrngbaseg 12593 The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
𝑅 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Β· ⟩}    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š ∧ Β· ∈ 𝑋) β†’ 𝐡 = (Baseβ€˜π‘…))
 
Theoremrngplusgg 12594 The additive operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝑅 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Β· ⟩}    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š ∧ Β· ∈ 𝑋) β†’ + = (+gβ€˜π‘…))
 
Theoremrngmulrg 12595 The multiplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝑅 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(.rβ€˜ndx), Β· ⟩}    β‡’   ((𝐡 ∈ 𝑉 ∧ + ∈ π‘Š ∧ Β· ∈ 𝑋) β†’ Β· = (.rβ€˜π‘…))
 
Theoremstarvndx 12596 Index value of the df-starv 12550 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(*π‘Ÿβ€˜ndx) = 4
 
Theoremstarvid 12597 Utility theorem: index-independent form of df-starv 12550. (Contributed by Mario Carneiro, 6-Oct-2013.)
*π‘Ÿ = Slot (*π‘Ÿβ€˜ndx)
 
Theoremstarvslid 12598 Slot property of *π‘Ÿ. (Contributed by Jim Kingdon, 4-Feb-2023.)
(*π‘Ÿ = Slot (*π‘Ÿβ€˜ndx) ∧ (*π‘Ÿβ€˜ndx) ∈ β„•)
 
Theoremstarvndxnbasendx 12599 The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(*π‘Ÿβ€˜ndx) β‰  (Baseβ€˜ndx)
 
Theoremstarvndxnplusgndx 12600 The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
(*π‘Ÿβ€˜ndx) β‰  (+gβ€˜ndx)
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