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Type | Label | Description |
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Statement | ||
Theorem | nninfdclemlt 12501* | Lemma for nninfdc 12503. The function from nninfdclemf 12499 is strictly monotonic. (Contributed by Jim Kingdon, 24-Sep-2024.) |
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Theorem | nninfdclemf1 12502* | Lemma for nninfdc 12503. The function from nninfdclemf 12499 is one-to-one. (Contributed by Jim Kingdon, 23-Sep-2024.) |
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Theorem | nninfdc 12503* | An unbounded decidable set of positive integers is infinite. (Contributed by Jim Kingdon, 23-Sep-2024.) |
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Theorem | unbendc 12504* | An unbounded decidable set of positive integers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Jim Kingdon, 30-Sep-2024.) |
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Theorem | prminf 12505 | There are an infinite number of primes. Theorem 1.7 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 28-Nov-2012.) |
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Theorem | infpn2 12506* |
There exist infinitely many prime numbers: the set of all primes ![]() |
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An "extensible structure" (or "structure" in short, at least in this section) is used to define a specific group, ring, poset, and so on. An extensible structure can contain many components. For example, a group will have at least two components (base set and operation), although it can be further specialized by adding other components such as a multiplicative operation for rings (and still remain a group per our definition). Thus, every ring is also a group. This extensible structure approach allows theorems from more general structures (such as groups) to be reused for more specialized structures (such as rings) without having to reprove anything. Structures are common in mathematics, but in informal (natural language) proofs the details are assumed in ways that we must make explicit.
An extensible structure is implemented as a function (a set of ordered pairs)
on a finite (and not necessarily sequential) subset of
There are many other possible ways to handle structures. We chose this
extensible structure approach because this approach (1) results in simpler
notation than other approaches we are aware of, and (2) is easier to do
proofs with. We cannot use an approach that uses "hidden"
arguments;
Metamath does not support hidden arguments, and in any case we want nothing
hidden. It would be possible to use a categorical approach (e.g., something
vaguely similar to Lean's mathlib). However, instances (the chain of proofs
that an
To create a substructure of a given extensible structure, you can simply use
the multifunction restriction operator for extensible structures
↾s as
defined in df-iress 12519. This 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 Extensible structures only work well when they represent concrete categories, where there is a "base set", morphisms are functions, and subobjects are subsets with induced operations. In short, they primarily work well for "sets with (some) extra structure". Extensible structures may not suffice for more complicated situations. For example, in manifolds, ↾s would not work. That said, extensible structures are sufficient for many of the structures that set.mm currently considers, and offer a good compromise for a goal-oriented formalization. | ||
Syntax | cstr 12507 |
Extend class notation with the class of structures with components
numbered below ![]() |
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Syntax | cnx 12508 | Extend class notation with the structure component index extractor. |
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Syntax | csts 12509 | Set components of a structure. |
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Syntax | cslot 12510 | Extend class notation with the slot function. |
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Syntax | cbs 12511 | Extend class notation with the class of all base set extractors. |
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Syntax | cress 12512 | Extend class notation with the extensible structure builder restriction operator. |
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Definition | df-struct 12513* |
Define a structure with components in ![]() ![]() ![]()
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
Allowing an extensible structure to contain the empty set ensures that
expressions like |
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Definition | df-ndx 12514 |
Define the structure component index extractor. See Theorem ndxarg 12534 to
understand its purpose. The restriction to ![]() ![]() ![]() ![]() ![]() |
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Definition | df-slot 12515* |
Define the slot extractor for extensible structures. The class
Slot ![]()
Note that Slot
The special "structure"
The class Slot cannot be defined as
|
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Theorem | sloteq 12516 |
Equality theorem for the Slot construction. The converse holds if
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Definition | df-base 12517 | 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.) |
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Definition | df-sets 12518* | 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-iress 12519 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.) |
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Definition | df-iress 12519* |
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 ![]() (Credit for this operator, as well as the 2023 modification for iset.mm, goes to Mario Carneiro.) (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 7-Oct-2023.) |
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Theorem | brstruct 12520 | The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.) |
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Theorem | isstruct2im 12521 |
The property of being a structure with components in
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Theorem | isstruct2r 12522 |
The property of being a structure with components in
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Theorem | structex 12523 | A structure is a set. (Contributed by AV, 10-Nov-2021.) |
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Theorem | structn0fun 12524 | A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.) |
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Theorem | isstructim 12525 |
The property of being a structure with components in ![]() ![]() ![]() |
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Theorem | isstructr 12526 |
The property of being a structure with components in ![]() ![]() ![]() |
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Theorem | structcnvcnv 12527 | Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
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Theorem | structfung 12528 | The converse of the converse of a structure is a function. Closed form of structfun 12529. (Contributed by AV, 12-Nov-2021.) |
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Theorem | structfun 12529 | Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Proof shortened by AV, 12-Nov-2021.) |
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Theorem | structfn 12530 | Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
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Theorem | strnfvnd 12531 | Deduction version of strnfvn 12532. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 19-Jan-2023.) |
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Theorem | strnfvn 12532 |
Value of a structure component extractor ![]() ![]() ![]() ![]() ![]() ![]() Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strslfv 12556. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.) |
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Theorem | strfvssn 12533 |
A structure component extractor produces a value which is contained in a
set dependent on ![]() ![]() |
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Theorem | ndxarg 12534 | Get the numeric argument from a defined structure component extractor such as df-base 12517. (Contributed by Mario Carneiro, 6-Oct-2013.) |
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Theorem | ndxid 12535 |
A structure component extractor is defined by its own index. This
theorem, together with strslfv 12556 below, is useful for avoiding direct
reference to the hard-coded numeric index in component extractor
definitions, such as the ![]() (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof shortened by BJ, 27-Dec-2021.) |
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Theorem | ndxslid 12536 | 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 12556. (Contributed by Jim Kingdon, 29-Jan-2023.) |
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Theorem | slotslfn 12537 | A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by Jim Kingdon, 10-Feb-2023.) |
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Theorem | slotex 12538 | Existence of slot value. A corollary of slotslfn 12537. (Contributed by Jim Kingdon, 12-Feb-2023.) |
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Theorem | strndxid 12539 | The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.) |
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Theorem | reldmsets 12540 | The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
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Theorem | setsvalg 12541 | Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.) |
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Theorem | setsvala 12542 | Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 20-Jan-2023.) |
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Theorem | setsex 12543 | Applying the structure replacement function yields a set. (Contributed by Jim Kingdon, 22-Jan-2023.) |
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Theorem | strsetsid 12544 | Value of the structure replacement function. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 30-Jan-2023.) |
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Theorem | fvsetsid 12545 | The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.) |
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Theorem | setsfun 12546 | A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.) |
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Theorem | setsfun0 12547 |
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 12546 is useful for proofs based on isstruct2r 12522 which requires
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Theorem | setsn0fun 12548 | 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.) |
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Theorem | setsresg 12549 |
The structure replacement function does not affect the value of ![]() ![]() |
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Theorem | setsabsd 12550 | 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.) |
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Theorem | setscom 12551 | Different components can be set in any order. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
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Theorem | setscomd 12552 | Different components can be set in any order. (Contributed by Jim Kingdon, 20-Feb-2025.) |
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Theorem | strslfvd 12553 | Deduction version of strslfv 12556. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 30-Jan-2023.) |
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Theorem | strslfv2d 12554 | Deduction version of strslfv 12556. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.) |
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Theorem | strslfv2 12555 |
A variation on strslfv 12556 to avoid asserting that ![]() ![]() |
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Theorem | strslfv 12556 |
Extract a structure component ![]() ![]() ![]() ![]() |
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Theorem | strslfv3 12557 | Variant on strslfv 12556 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.) |
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Theorem | strslssd 12558 | Deduction version of strslss 12559. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.) |
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Theorem | strslss 12559 |
Propagate component extraction to a structure ![]() ![]() |
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Theorem | strsl0 12560 | All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 31-Jan-2023.) |
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Theorem | base0 12561 | The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.) |
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Theorem | setsslid 12562 | Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.) |
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Theorem | setsslnid 12563 | Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.) |
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Theorem | baseval 12564 |
Value of the base set extractor. (Normally it is preferred to work with
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Theorem | baseid 12565 | Utility theorem: index-independent form of df-base 12517. (Contributed by NM, 20-Oct-2012.) |
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Theorem | basendx 12566 |
Index value of the base set extractor.
Use of this theorem is discouraged since the particular value 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 12672. Although we have a few theorems such as basendxnplusgndx 12633, 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.) |
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Theorem | basendxnn 12567 | 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.) |
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Theorem | baseslid 12568 | The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.) |
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Theorem | basfn 12569 |
The base set extractor is a function on ![]() |
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Theorem | basmex 12570 | A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 18-Nov-2024.) |
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Theorem | basmexd 12571 | A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 28-Nov-2024.) |
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Theorem | reldmress 12572 | The structure restriction is a proper operator, so it can be used with ovprc1 5931. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
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Theorem | ressvalsets 12573 | Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.) |
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Theorem | ressex 12574 | Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.) |
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Theorem | ressval2 12575 | Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
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Theorem | ressbasd 12576 | Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.) |
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Theorem | ressbas2d 12577 | Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.) |
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Theorem | ressbasssd 12578 | 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.) |
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Theorem | ressbasid 12579 | The trivial structure restriction leaves the base set unchanged. (Contributed by Jim Kingdon, 29-Apr-2025.) |
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Theorem | strressid 12580 | Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.) |
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Theorem | ressval3d 12581 | Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.) |
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Theorem | resseqnbasd 12582 | 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.) |
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Theorem | ressinbasd 12583 | 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.) |
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Theorem | ressressg 12584 | Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.) |
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Theorem | ressabsg 12585 | Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.) |
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Syntax | cplusg 12586 | Extend class notation with group (addition) operation. |
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Syntax | cmulr 12587 | Extend class notation with ring multiplication. |
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Syntax | cstv 12588 | Extend class notation with involution. |
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Syntax | csca 12589 | Extend class notation with scalar field. |
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Syntax | cvsca 12590 | Extend class notation with scalar product. |
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Syntax | cip 12591 | Extend class notation with Hermitian form (inner product). |
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Syntax | cts 12592 | Extend class notation with the topology component of a topological space. |
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Syntax | cple 12593 | Extend class notation with "less than or equal to" for posets. |
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Syntax | coc 12594 | Extend class notation with the class of orthocomplementation extractors. |
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Syntax | cds 12595 | Extend class notation with the metric space distance function. |
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Syntax | cunif 12596 | Extend class notation with the uniform structure. |
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Syntax | chom 12597 | Extend class notation with the hom-set structure. |
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Syntax | cco 12598 | Extend class notation with the composition operation. |
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Definition | df-plusg 12599 | Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
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Definition | df-mulr 12600 | Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
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