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

Definitiondf-vsca 13501 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Slot

Definitiondf-ip 13502 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Slot

Definitiondf-tset 13503 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 13504 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 13505 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 13506 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 13507 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
Slot ;

Definitiondf-hom 13508 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Slot ;

Definitiondf-cco 13509 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp Slot ;

Theoremstrlemor0 13510 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 13511 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 13512 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 13513 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 13514 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct        Struct               Struct

Theoremstrle1 13515 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremstrle2 13516 Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremstrle3 13517 Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremplusgndx 13518 Index value of the df-plusg 13497 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremplusgid 13519 Utility theorem: index-independent form of df-plusg 13497. (Contributed by NM, 20-Oct-2012.)
Slot

Theorem2strstr 13520 A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
Slot                      Struct

Theorem2strbas 13521 The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
Slot

Theorem2strop 13522 The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
Slot

Theoremgrpstr 13523 A constructed group is a structure on . (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
Struct

Theoremgrpbase 13524 The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremgrpplusg 13525 The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremressplusg 13526 is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s

Theoremgrpbasex 13527 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 13524 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)

Theoremgrpplusgx 13528 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 13528 instead. (New usage is discouraged.) (Contributed by NM, 17-Oct-2012.)

Theoremmulrndx 13529 Index value of the df-mulr 13498 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremmulrid 13530 Utility theorem: index-independent form of df-mulr 13498. (Contributed by Mario Carneiro, 8-Jun-2013.)
Slot

Theoremrngstr 13531 A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremrngbase 13532 The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremrngplusg 13533 The additive operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremrngmulr 13534 The muliplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremstarvndx 13535 Index value of the df-starv 13499 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremstarvid 13536 Utility theorem: index-independent form of df-starv 13499. (Contributed by Mario Carneiro, 6-Oct-2013.)
Slot

Theoremressmulr 13537 is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
s

Theoremressstarv 13538 is unaffected by restriction. (Contributed by Mario Carneiro, 9-Oct-2015.)
s

Theoremsrngfn 13539 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 13540 The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-May-2015.)

Theoremsrngplusg 13541 The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremsrngmulr 13542 The multiplication operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremsrnginvl 13543 The involution function of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremscandx 13544 Index value of the df-sca 13500 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
Scalar

Theoremscaid 13545 Utility theorem: index-independent form of scalar df-sca 13500. (Contributed by Mario Carneiro, 19-Jun-2014.)
Scalar Slot Scalar

Theoremvscandx 13546 Index value of the df-vsca 13501 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremvscaid 13547 Utility theorem: index-independent form of scalar product df-vsca 13501. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Slot

Theoremlmodstr 13548 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 13549 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 13550 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 13551 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 13552 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 13553 Lemma to shorten proofs of algbase 13554 through algvsca 13558. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar        Struct

Theoremalgbase 13554 The base set of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremalgaddg 13555 The additive operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremalgmulr 13556 The multiplicative operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremalgsca 13557 The set of scalars of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar        Scalar

Theoremalgvsca 13558 The scalar product operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremresssca 13559 Scalar is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
s        Scalar       Scalar

Theoremressvsca 13560 is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
s

Theoremipndx 13561 Index value of the df-ip 13502 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremipid 13562 Utility theorem: index-independent form of df-ip 13502. (Contributed by Mario Carneiro, 6-Oct-2013.)
Slot

Theoremphlstr 13563 A constructed pre-Hilbert space is a structure. Starting from lmodstr 13548 (which has 4 members), we chain strleun 13514 once more, adding an ordered pair to the function, to get all 5 members. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar        Struct

Theoremphlbase 13564 The base set of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremphlplusg 13565 The additive operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremphlsca 13566 The ring of scalars of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar        Scalar

Theoremphlvsca 13567 The scalar product operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremphlip 13568 The inner product (Hermitian form) operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
Scalar

Theoremtsetndx 13569 Index value of the df-tset 13503 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
TopSet

Theoremtsetid 13570 Utility theorem: index-independent form of df-tset 13503. (Contributed by NM, 20-Oct-2012.)
TopSet Slot TopSet

Theoremtopgrpstr 13571 A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
TopSet        Struct

Theoremtopgrpbas 13572 The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
TopSet

Theoremtopgrpplusg 13573 The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
TopSet

Theoremtopgrptset 13574 The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
TopSet        TopSet

Theoremresstset 13575 TopSet is unaffected by restriction. (Contributed by Mario Carneiro, 13-Aug-2015.)
s        TopSet       TopSet

Theoremplendx 13576 Index value of the df-ple 13504 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theorempleid 13577 Utility theorem: self-referencing, index-independent form of df-ple 13504. (Contributed by NM, 9-Nov-2012.)
Slot

Theoremotpsstr 13578 Functionality of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
TopSet        Struct

Theoremotpsbas 13579 The base set of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
TopSet

Theoremotpstset 13580 The open sets of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
TopSet        TopSet

Theoremotpsle 13581 The order of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
TopSet

Theoremressle 13582 is unaffected by restriction. (Contributed by Mario Carneiro, 3-Nov-2015.)
s

Theoremocndx 13583 Index value of the df-ocomp 13505 slot. (Contributed by Mario Carneiro, 25-Oct-2015.)
;

Theoremocid 13584 Utility theorem: index-independent form of df-ocomp 13505. (Contributed by Mario Carneiro, 25-Oct-2015.)
Slot

Theoremdsndx 13585 Index value of the df-ds 13506 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
;

Theoremdsid 13586 Utility theorem: index-independent form of df-ds 13506. (Contributed by Mario Carneiro, 23-Dec-2013.)
Slot

Theoremunifndx 13587 Index value of the df-unif 13507 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.)
;

Theoremunifid 13588 Utility theorem: index-independent form of df-unif 13507. (Contributed by Thierry Arnoux, 17-Dec-2017.)
Slot

Theoremodrngstr 13589 Functionality of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet        Struct ;

Theoremodrngbas 13590 The base set of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet

Theoremodrngplusg 13591 The addition operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet

Theoremodrngmulr 13592 The multiplication operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet

Theoremodrngtset 13593 The open sets of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet        TopSet

Theoremodrngle 13594 The order of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet

Theoremodrngds 13595 The metric of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
TopSet

Theoremressds 13596 is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015.)
s

Theoremhomndx 13597 Index value of the df-hom 13508 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
;

Theoremhomid 13598 Utility theorem: index-independent form of df-hom 13508. (Contributed by Mario Carneiro, 7-Jan-2017.)
Slot

Theoremccondx 13599 Index value of the df-cco 13509 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
comp ;

Theoremccoid 13600 Utility theorem: index-independent form of df-cco 13509. (Contributed by Mario Carneiro, 7-Jan-2017.)
comp Slot comp

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