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Type | Label | Description |
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Statement | ||
Theorem | setsn0fun 12501 | 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.) |
β’ (π β π Struct π) & β’ (π β πΌ β π) & β’ (π β πΈ β π) β β’ (π β Fun ((π sSet β¨πΌ, πΈβ©) β {β })) | ||
Theorem | setsresg 12502 | The structure replacement function does not affect the value of π away from π΄. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.) |
β’ ((π β π β§ π΄ β π β§ π΅ β π) β ((π sSet β¨π΄, π΅β©) βΎ (V β {π΄})) = (π βΎ (V β {π΄}))) | ||
Theorem | setsabsd 12503 | 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.) |
β’ (π β π β π) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) β β’ (π β ((π sSet β¨π΄, π΅β©) sSet β¨π΄, πΆβ©) = (π sSet β¨π΄, πΆβ©)) | ||
Theorem | setscom 12504 | Different components can be set in any order. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ π΄ β V & β’ π΅ β V β β’ (((π β π β§ π΄ β π΅) β§ (πΆ β π β§ π· β π)) β ((π sSet β¨π΄, πΆβ©) sSet β¨π΅, π·β©) = ((π sSet β¨π΅, π·β©) sSet β¨π΄, πΆβ©)) | ||
Theorem | setscomd 12505 | Different components can be set in any order. (Contributed by Jim Kingdon, 20-Feb-2025.) |
β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β π β π) & β’ (π β π΄ β π΅) & β’ (π β πΆ β π) & β’ (π β π· β π) β β’ (π β ((π sSet β¨π΄, πΆβ©) sSet β¨π΅, π·β©) = ((π sSet β¨π΅, π·β©) sSet β¨π΄, πΆβ©)) | ||
Theorem | strslfvd 12506 | Deduction version of strslfv 12509. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 30-Jan-2023.) |
β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) & β’ (π β π β π) & β’ (π β Fun π) & β’ (π β β¨(πΈβndx), πΆβ© β π) β β’ (π β πΆ = (πΈβπ)) | ||
Theorem | strslfv2d 12507 | Deduction version of strslfv 12509. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.) |
β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) & β’ (π β π β π) & β’ (π β Fun β‘β‘π) & β’ (π β β¨(πΈβndx), πΆβ© β π) & β’ (π β πΆ β π) β β’ (π β πΆ = (πΈβπ)) | ||
Theorem | strslfv2 12508 | A variation on strslfv 12509 to avoid asserting that π itself is a function, which involves sethood of all the ordered pair components of π. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.) |
β’ π β V & β’ Fun β‘β‘π & β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) & β’ β¨(πΈβndx), πΆβ© β π β β’ (πΆ β π β πΆ = (πΈβπ)) | ||
Theorem | strslfv 12509 | Extract a structure component πΆ (such as the base set) from a structure π with a component extractor πΈ (such as the base set extractor df-base 12470). By virtue of ndxslid 12489, this can be done without having to refer to the hard-coded numeric index of πΈ. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 30-Jan-2023.) |
β’ π Struct π & β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) & β’ {β¨(πΈβndx), πΆβ©} β π β β’ (πΆ β π β πΆ = (πΈβπ)) | ||
Theorem | strslfv3 12510 | Variant on strslfv 12509 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.) |
β’ (π β π = π) & β’ π Struct π & β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) & β’ {β¨(πΈβndx), πΆβ©} β π & β’ (π β πΆ β π) & β’ π΄ = (πΈβπ) β β’ (π β π΄ = πΆ) | ||
Theorem | strslssd 12511 | Deduction version of strslss 12512. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.) |
β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) & β’ (π β π β π) & β’ (π β Fun π) & β’ (π β π β π) & β’ (π β β¨(πΈβndx), πΆβ© β π) β β’ (π β (πΈβπ) = (πΈβπ)) | ||
Theorem | strslss 12512 | Propagate component extraction to a structure π from a subset structure π. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Jim Kingdon, 31-Jan-2023.) |
β’ π β V & β’ Fun π & β’ π β π & β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) & β’ β¨(πΈβndx), πΆβ© β π β β’ (πΈβπ) = (πΈβπ) | ||
Theorem | strsl0 12513 | All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 31-Jan-2023.) |
β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) β β’ β = (πΈββ ) | ||
Theorem | base0 12514 | The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.) |
β’ β = (Baseββ ) | ||
Theorem | setsslid 12515 | Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.) |
β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) β β’ ((π β π΄ β§ πΆ β π) β πΆ = (πΈβ(π sSet β¨(πΈβndx), πΆβ©))) | ||
Theorem | setsslnid 12516 | Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.) |
β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) & β’ (πΈβndx) β π· & β’ π· β β β β’ ((π β π΄ β§ πΆ β π) β (πΈβπ) = (πΈβ(π sSet β¨π·, πΆβ©))) | ||
Theorem | baseval 12517 | Value of the base set extractor. (Normally it is preferred to work with (Baseβndx) 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.) |
β’ πΎ β V β β’ (BaseβπΎ) = (πΎβ1) | ||
Theorem | baseid 12518 | Utility theorem: index-independent form of df-base 12470. (Contributed by NM, 20-Oct-2012.) |
β’ Base = Slot (Baseβndx) | ||
Theorem | basendx 12519 |
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 (Baseβndx) and use theorems such as baseid 12518 and basendxnn 12520. 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 12624. Although we have a few theorems such as basendxnplusgndx 12585, 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.) |
β’ (Baseβndx) = 1 | ||
Theorem | basendxnn 12520 | 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.) |
β’ (Baseβndx) β β | ||
Theorem | baseslid 12521 | The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.) |
β’ (Base = Slot (Baseβndx) β§ (Baseβndx) β β) | ||
Theorem | basfn 12522 | The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.) |
β’ Base Fn V | ||
Theorem | basmex 12523 | A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 18-Nov-2024.) |
β’ π΅ = (BaseβπΊ) β β’ (π΄ β π΅ β πΊ β V) | ||
Theorem | basmexd 12524 | A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 28-Nov-2024.) |
β’ (π β π΅ = (BaseβπΊ)) & β’ (π β π΄ β π΅) β β’ (π β πΊ β V) | ||
Theorem | reldmress 12525 | The structure restriction is a proper operator, so it can be used with ovprc1 5913. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
β’ Rel dom βΎs | ||
Theorem | ressvalsets 12526 | Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.) |
β’ ((π β π β§ π΄ β π) β (π βΎs π΄) = (π sSet β¨(Baseβndx), (π΄ β© (Baseβπ))β©)) | ||
Theorem | ressex 12527 | Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.) |
β’ ((π β π β§ π΄ β π) β (π βΎs π΄) β V) | ||
Theorem | ressval2 12528 | Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
β’ π = (π βΎs π΄) & β’ π΅ = (Baseβπ) β β’ ((Β¬ π΅ β π΄ β§ π β π β§ π΄ β π) β π = (π sSet β¨(Baseβndx), (π΄ β© π΅)β©)) | ||
Theorem | ressbasd 12529 | Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.) |
β’ (π β π = (π βΎs π΄)) & β’ (π β π΅ = (Baseβπ)) & β’ (π β π β π) & β’ (π β π΄ β π) β β’ (π β (π΄ β© π΅) = (Baseβπ )) | ||
Theorem | ressbas2d 12530 | Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.) |
β’ (π β π = (π βΎs π΄)) & β’ (π β π΅ = (Baseβπ)) & β’ (π β π β π) & β’ (π β π΄ β π΅) β β’ (π β π΄ = (Baseβπ )) | ||
Theorem | ressbasssd 12531 | 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.) |
β’ (π β π = (π βΎs π΄)) & β’ (π β π΅ = (Baseβπ)) & β’ (π β π β π) & β’ (π β π΄ β π) β β’ (π β (Baseβπ ) β π΅) | ||
Theorem | strressid 12532 | Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.) |
β’ (π β π΅ = (Baseβπ)) & β’ (π β π Struct β¨π, πβ©) & β’ (π β Fun π) & β’ (π β (Baseβndx) β dom π) β β’ (π β (π βΎs π΅) = π) | ||
Theorem | ressval3d 12533 | Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.) |
β’ π = (π βΎs π΄) & β’ π΅ = (Baseβπ) & β’ πΈ = (Baseβndx) & β’ (π β π β π) & β’ (π β Fun π) & β’ (π β πΈ β dom π) & β’ (π β π΄ β π΅) β β’ (π β π = (π sSet β¨πΈ, π΄β©)) | ||
Theorem | resseqnbasd 12534 | 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.) |
β’ π = (π βΎs π΄) & β’ πΆ = (πΈβπ) & β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) & β’ (πΈβndx) β (Baseβndx) & β’ (π β π β π) & β’ (π β π΄ β π) β β’ (π β πΆ = (πΈβπ )) | ||
Theorem | ressinbasd 12535 | 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.) |
β’ (π β π΅ = (Baseβπ)) & β’ (π β π΄ β π) & β’ (π β π β π) β β’ (π β (π βΎs π΄) = (π βΎs (π΄ β© π΅))) | ||
Theorem | ressressg 12536 | Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.) |
β’ ((π΄ β π β§ π΅ β π β§ π β π) β ((π βΎs π΄) βΎs π΅) = (π βΎs (π΄ β© π΅))) | ||
Theorem | ressabsg 12537 | Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.) |
β’ ((π΄ β π β§ π΅ β π΄ β§ π β π) β ((π βΎs π΄) βΎs π΅) = (π βΎs π΅)) | ||
Syntax | cplusg 12538 | Extend class notation with group (addition) operation. |
class +g | ||
Syntax | cmulr 12539 | Extend class notation with ring multiplication. |
class .r | ||
Syntax | cstv 12540 | Extend class notation with involution. |
class *π | ||
Syntax | csca 12541 | Extend class notation with scalar field. |
class Scalar | ||
Syntax | cvsca 12542 | Extend class notation with scalar product. |
class Β·π | ||
Syntax | cip 12543 | Extend class notation with Hermitian form (inner product). |
class Β·π | ||
Syntax | cts 12544 | Extend class notation with the topology component of a topological space. |
class TopSet | ||
Syntax | cple 12545 | Extend class notation with "less than or equal to" for posets. |
class le | ||
Syntax | coc 12546 | Extend class notation with the class of orthocomplementation extractors. |
class oc | ||
Syntax | cds 12547 | Extend class notation with the metric space distance function. |
class dist | ||
Syntax | cunif 12548 | Extend class notation with the uniform structure. |
class UnifSet | ||
Syntax | chom 12549 | Extend class notation with the hom-set structure. |
class Hom | ||
Syntax | cco 12550 | Extend class notation with the composition operation. |
class comp | ||
Definition | df-plusg 12551 | Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
β’ +g = Slot 2 | ||
Definition | df-mulr 12552 | Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
β’ .r = Slot 3 | ||
Definition | df-starv 12553 | Define the involution function of a *-ring. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
β’ *π = Slot 4 | ||
Definition | df-sca 12554 | Define scalar field component of a vector space π£. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
β’ Scalar = Slot 5 | ||
Definition | df-vsca 12555 | Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
β’ Β·π = Slot 6 | ||
Definition | df-ip 12556 | Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
β’ Β·π = Slot 8 | ||
Definition | df-tset 12557 | Define the topology component of a topological space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
β’ TopSet = Slot 9 | ||
Definition | df-ple 12558 | 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.) |
β’ le = Slot ;10 | ||
Definition | df-ocomp 12559 | Define the orthocomplementation extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
β’ oc = Slot ;11 | ||
Definition | df-ds 12560 | Define the distance function component of a metric space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
β’ dist = Slot ;12 | ||
Definition | df-unif 12561 | Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ UnifSet = Slot ;13 | ||
Definition | df-hom 12562 | Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ Hom = Slot ;14 | ||
Definition | df-cco 12563 | Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ comp = Slot ;15 | ||
Theorem | strleund 12564 | Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.) |
β’ (π β πΉ Struct β¨π΄, π΅β©) & β’ (π β πΊ Struct β¨πΆ, π·β©) & β’ (π β π΅ < πΆ) β β’ (π β (πΉ βͺ πΊ) Struct β¨π΄, π·β©) | ||
Theorem | strleun 12565 | Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) |
β’ πΉ Struct β¨π΄, π΅β© & β’ πΊ Struct β¨πΆ, π·β© & β’ π΅ < πΆ β β’ (πΉ βͺ πΊ) Struct β¨π΄, π·β© | ||
Theorem | strext 12566 | Extending the upper range of a structure. This works because when we say that a structure has components in π΄...πΆ 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.) |
β’ (π β πΉ Struct β¨π΄, π΅β©) & β’ (π β πΆ β (β€β₯βπ΅)) β β’ (π β πΉ Struct β¨π΄, πΆβ©) | ||
Theorem | strle1g 12567 | Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.) |
β’ πΌ β β & β’ π΄ = πΌ β β’ (π β π β {β¨π΄, πβ©} Struct β¨πΌ, πΌβ©) | ||
Theorem | strle2g 12568 | Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.) |
β’ πΌ β β & β’ π΄ = πΌ & β’ πΌ < π½ & β’ π½ β β & β’ π΅ = π½ β β’ ((π β π β§ π β π) β {β¨π΄, πβ©, β¨π΅, πβ©} Struct β¨πΌ, π½β©) | ||
Theorem | strle3g 12569 | Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.) |
β’ πΌ β β & β’ π΄ = πΌ & β’ πΌ < π½ & β’ π½ β β & β’ π΅ = π½ & β’ π½ < πΎ & β’ πΎ β β & β’ πΆ = πΎ β β’ ((π β π β§ π β π β§ π β π) β {β¨π΄, πβ©, β¨π΅, πβ©, β¨πΆ, πβ©} Struct β¨πΌ, πΎβ©) | ||
Theorem | plusgndx 12570 | Index value of the df-plusg 12551 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ (+gβndx) = 2 | ||
Theorem | plusgid 12571 | Utility theorem: index-independent form of df-plusg 12551. (Contributed by NM, 20-Oct-2012.) |
β’ +g = Slot (+gβndx) | ||
Theorem | plusgndxnn 12572 | The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 17-Oct-2024.) |
β’ (+gβndx) β β | ||
Theorem | plusgslid 12573 | Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.) |
β’ (+g = Slot (+gβndx) β§ (+gβndx) β β) | ||
Theorem | basendxltplusgndx 12574 | 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.) |
β’ (Baseβndx) < (+gβndx) | ||
Theorem | opelstrsl 12575 | The slot of a structure which contains an ordered pair for that slot. (Contributed by Jim Kingdon, 5-Feb-2023.) |
β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) & β’ (π β π Struct π) & β’ (π β π β π) & β’ (π β β¨(πΈβndx), πβ© β π) β β’ (π β π = (πΈβπ)) | ||
Theorem | opelstrbas 12576 | The base set of a structure with a base set. (Contributed by AV, 10-Nov-2021.) |
β’ (π β π Struct π) & β’ (π β π β π) & β’ (π β β¨(Baseβndx), πβ© β π) β β’ (π β π = (Baseβπ)) | ||
Theorem | 1strstrg 12577 | A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Revised by Jim Kingdon, 28-Jan-2023.) |
β’ πΊ = {β¨(Baseβndx), π΅β©} β β’ (π΅ β π β πΊ Struct β¨1, 1β©) | ||
Theorem | 1strbas 12578 | The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) |
β’ πΊ = {β¨(Baseβndx), π΅β©} β β’ (π΅ β π β π΅ = (BaseβπΊ)) | ||
Theorem | 2strstrg 12579 | A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.) |
β’ πΊ = {β¨(Baseβndx), π΅β©, β¨(πΈβndx), + β©} & β’ πΈ = Slot π & β’ 1 < π & β’ π β β β β’ ((π΅ β π β§ + β π) β πΊ Struct β¨1, πβ©) | ||
Theorem | 2strbasg 12580 | The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.) |
β’ πΊ = {β¨(Baseβndx), π΅β©, β¨(πΈβndx), + β©} & β’ πΈ = Slot π & β’ 1 < π & β’ π β β β β’ ((π΅ β π β§ + β π) β π΅ = (BaseβπΊ)) | ||
Theorem | 2stropg 12581 | The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.) |
β’ πΊ = {β¨(Baseβndx), π΅β©, β¨(πΈβndx), + β©} & β’ πΈ = Slot π & β’ 1 < π & β’ π β β β β’ ((π΅ β π β§ + β π) β + = (πΈβπΊ)) | ||
Theorem | 2strstr1g 12582 | A constructed two-slot structure. Version of 2strstrg 12579 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.) |
β’ πΊ = {β¨(Baseβndx), π΅β©, β¨π, + β©} & β’ (Baseβndx) < π & β’ π β β β β’ ((π΅ β π β§ + β π) β πΊ Struct β¨(Baseβndx), πβ©) | ||
Theorem | 2strbas1g 12583 | The base set of a constructed two-slot structure. Version of 2strbasg 12580 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.) |
β’ πΊ = {β¨(Baseβndx), π΅β©, β¨π, + β©} & β’ (Baseβndx) < π & β’ π β β β β’ ((π΅ β π β§ + β π) β π΅ = (BaseβπΊ)) | ||
Theorem | 2strop1g 12584 | The other slot of a constructed two-slot structure. Version of 2stropg 12581 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.) |
β’ πΊ = {β¨(Baseβndx), π΅β©, β¨π, + β©} & β’ (Baseβndx) < π & β’ π β β & β’ πΈ = Slot π β β’ ((π΅ β π β§ + β π) β + = (πΈβπΊ)) | ||
Theorem | basendxnplusgndx 12585 | The slot for the base set is not the slot for the group operation in an extensible structure. (Contributed by AV, 14-Nov-2021.) |
β’ (Baseβndx) β (+gβndx) | ||
Theorem | grpstrg 12586 | A constructed group is a structure on 1...2. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ πΊ = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©} β β’ ((π΅ β π β§ + β π) β πΊ Struct β¨1, 2β©) | ||
Theorem | grpbaseg 12587 | The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ πΊ = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©} β β’ ((π΅ β π β§ + β π) β π΅ = (BaseβπΊ)) | ||
Theorem | grpplusgg 12588 | The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ πΊ = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©} β β’ ((π΅ β π β§ + β π) β + = (+gβπΊ)) | ||
Theorem | ressplusgd 12589 | +g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
β’ (π β π» = (πΊ βΎs π΄)) & β’ (π β + = (+gβπΊ)) & β’ (π β π΄ β π) & β’ (π β πΊ β π) β β’ (π β + = (+gβπ»)) | ||
Theorem | mulrndx 12590 | Index value of the df-mulr 12552 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ (.rβndx) = 3 | ||
Theorem | mulridx 12591 | Utility theorem: index-independent form of df-mulr 12552. (Contributed by Mario Carneiro, 8-Jun-2013.) |
β’ .r = Slot (.rβndx) | ||
Theorem | mulrslid 12592 | Slot property of .r. (Contributed by Jim Kingdon, 3-Feb-2023.) |
β’ (.r = Slot (.rβndx) β§ (.rβndx) β β) | ||
Theorem | plusgndxnmulrndx 12593 | 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.) |
β’ (+gβndx) β (.rβndx) | ||
Theorem | basendxnmulrndx 12594 | 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.) |
β’ (Baseβndx) β (.rβndx) | ||
Theorem | rngstrg 12595 | A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.) |
β’ π = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(.rβndx), Β· β©} β β’ ((π΅ β π β§ + β π β§ Β· β π) β π Struct β¨1, 3β©) | ||
Theorem | rngbaseg 12596 | The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 3-Feb-2023.) |
β’ π = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(.rβndx), Β· β©} β β’ ((π΅ β π β§ + β π β§ Β· β π) β π΅ = (Baseβπ )) | ||
Theorem | rngplusgg 12597 | The additive operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ π = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(.rβndx), Β· β©} β β’ ((π΅ β π β§ + β π β§ Β· β π) β + = (+gβπ )) | ||
Theorem | rngmulrg 12598 | The multiplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ π = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(.rβndx), Β· β©} β β’ ((π΅ β π β§ + β π β§ Β· β π) β Β· = (.rβπ )) | ||
Theorem | starvndx 12599 | Index value of the df-starv 12553 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ (*πβndx) = 4 | ||
Theorem | starvid 12600 | Utility theorem: index-independent form of df-starv 12553. (Contributed by Mario Carneiro, 6-Oct-2013.) |
β’ *π = Slot (*πβndx) |
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