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Theorem List for Metamath Proof Explorer - 13501-13600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremstrfv2d 13501 Deduction version of strfv 13503. (Contributed by Mario Carneiro, 30-Apr-2015.)
Slot

Theoremstrfv2 13502 A variation on strfv 13503 to avoid asserting that itself is a function, which involves sethood of all the ordered pair components of . (Contributed by Mario Carneiro, 30-Apr-2015.)
Slot

Theoremstrfv 13503 Extract a structure component (such as the base set) from a structure (such as a member of , df-poset 14405) with a component extractor (such as the base set extractor df-base 13476). By virtue of ndxid 13492, this can be done without having to refer to the hard-coded numeric index of . (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Struct        Slot

Theoremstrfv3 13504 Variant on strfv 13503 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.)
Struct        Slot

Theoremstrssd 13505 Deduction version of strss 13506. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Slot

Theoremstrss 13506 Propagate component extraction to a structure from a subset structure . (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 15-Jan-2014.)
Slot

Theoremstr0 13507 All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
Slot

Theorembase0 13508 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theoremstrfvi 13509 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.)
Slot

Theoremsetsid 13510 Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Slot        sSet

Theoremsetsnid 13511 Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Slot               sSet

Theorembaseval 13512 Value of the base set extractor. (Normally it is preferred to work with rather than the hard-coded 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.)

Theorembaseid 13513 Utility theorem: index-independent form of df-base 13476. (Contributed by NM, 20-Oct-2012.)
Slot

Theoremelbasfv 13514 Utility theorem: reverse closure for any structure defined as a function. (Contributed by Stefan O'Rear, 24-Aug-2015.)

Theoremelbasov 13515 Utility theorem: reverse closure for any structure defined as a two-argument function. (Contributed by Mario Carneiro, 3-Oct-2015.)

Theorembasendx 13516 Index value of the base set extractor. (Normally it is preferred to work with rather than the hard-coded 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.)

Theoremreldmress 13517 The structure restriction is a proper operator, so it can be used with ovprc1 6111. (Contributed by Stefan O'Rear, 29-Nov-2014.)
s

Theoremressval 13518 Value of structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
s               sSet

Theoremressid2 13519 General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
s

Theoremressval2 13520 Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
s               sSet

Theoremressbas 13521 Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.)
s

Theoremressbas2 13522 Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.)
s

Theoremressbasss 13523 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.)
s

Theoremresslem 13524 Other elements of a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
s               Slot

Theoremress0 13525 All restrictions of the null set are trivial. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
s

Theoremressid 13526 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
s

Theoremressinbas 13527 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.)
s s

Theoremressress 13528 Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)
s s s

Theoremressabs 13529 Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
s s s

Theoremwunress 13530 Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
WUni                     s

7.1.2  Slot definitions

Syntaxcplusg 13531 Extend class notation with group (addition) operation.

Syntaxcmulr 13532 Extend class notation with ring multiplication.

Syntaxcstv 13533 Extend class notation with involution.

Syntaxcsca 13534 Extend class notation with scalar field.
Scalar

Syntaxcvsca 13535 Extend class notation with scalar product.

Syntaxcip 13536 Extend class notation with Hermitian form (inner product).

Syntaxcts 13537 Extend class notation with the topology component of a topological space.
TopSet

Syntaxcple 13538 Extend class notation with less-than-or-equal for posets.

Syntaxcoc 13539 Extend class notation with the class of orthocomplementation extractors.

Syntaxcds 13540 Extend class notation with the metric space distance function.

Syntaxcunif 13541 Extend class notation with the uniform structure.

Syntaxchom 13542 Extend class notation with the hom-set structure.

Syntaxcco 13543 Extend class notation with the composition operation.
comp

Definitiondf-plusg 13544 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Slot

Definitiondf-mulr 13545 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Slot

Definitiondf-starv 13546 Define the involution function of a *-ring. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Slot

Definitiondf-sca 13547 Define scalar field component of a vector space . (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Scalar Slot

Definitiondf-vsca 13548 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Slot

Definitiondf-ip 13549 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Slot

Definitiondf-tset 13550 Define the topology component of a topological space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
TopSet Slot

Definitiondf-ple 13551 Define less-than-or-equal ordering extractor for posets and related structures. We use for the index to avoid conflict with through used for other purposes. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Slot

Definitiondf-ocomp 13552 Define the orthocomplementation extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Slot ;

Definitiondf-ds 13553 Define the distance function component of a metric space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Slot ;

Definitiondf-unif 13554 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
Slot ;

Definitiondf-hom 13555 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Slot ;

Definitiondf-cco 13556 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp Slot ;

Theoremstrlemor0 13557 Structure definition utility lemma. To prove that an explicit function is a function using O(n) steps, exploit the order properties of the index set. Zero-pair case. (Contributed by Stefan O'Rear, 3-Jan-2015.)

Theoremstrlemor1 13558 Add one element to the end of a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremstrlemor2 13559 Add two elements to the end of a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremstrlemor3 13560 Add three elements to the end of a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremstrleun 13561 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct        Struct               Struct

Theoremstrle1 13562 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremstrle2 13563 Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremstrle3 13564 Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremplusgndx 13565 Index value of the df-plusg 13544 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremplusgid 13566 Utility theorem: index-independent form of df-plusg 13544. (Contributed by NM, 20-Oct-2012.)
Slot

Theorem2strstr 13567 A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
Slot                      Struct

Theorem2strbas 13568 The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
Slot

Theorem2strop 13569 The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
Slot

Theoremgrpstr 13570 A constructed group is a structure on . (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
Struct

Theoremgrpbase 13571 The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremgrpplusg 13572 The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremressplusg 13573 is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s

Theoremgrpbasex 13574 The base of an explicitly given group. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use grpbase 13571 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)

Theoremgrpplusgx 13575 The operation of an explicitly given group. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use grpplusgx 13575 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)

Theoremmulrndx 13576 Index value of the df-mulr 13545 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremmulrid 13577 Utility theorem: index-independent form of df-mulr 13545. (Contributed by Mario Carneiro, 8-Jun-2013.)
Slot

Theoremrngstr 13578 A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremrngbase 13579 The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremrngplusg 13580 The additive operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremrngmulr 13581 The muliplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremstarvndx 13582 Index value of the df-starv 13546 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremstarvid 13583 Utility theorem: index-independent form of df-starv 13546. (Contributed by Mario Carneiro, 6-Oct-2013.)
Slot

Theoremressmulr 13584 is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s

Theoremressstarv 13585 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
s

Theoremsrngfn 13586 A constructed star ring is a function with domain contained in thru . (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 14-Aug-2015.)
Struct

Theoremsrngbase 13587 The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-May-2015.)

Theoremsrngplusg 13588 The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremsrngmulr 13589 The multiplication operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremsrnginvl 13590 The involution function of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremscandx 13591 Index value of the df-sca 13547 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
Scalar

Theoremscaid 13592 Utility theorem: index-independent form of scalar df-sca 13547. (Contributed by Mario Carneiro, 19-Jun-2014.)
Scalar Slot Scalar

Theoremvscandx 13593 Index value of the df-vsca 13548 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremvscaid 13594 Utility theorem: index-independent form of scalar product df-vsca 13548. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Slot

Theoremlmodstr 13595 A constructed left module or left vector space is a function. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar        Struct

Theoremlmodbase 13596 The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremlmodplusg 13597 The additive operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremlmodsca 13598 The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar        Scalar

Theoremlmodvsca 13599 The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremalgstr 13600 Lemma to shorten proofs of algbase 13601 through algvsca 13605. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar        Struct

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