Type | Label | Description |
Statement |
|
Theorem | starvslid 12601 |
Slot property of  . (Contributed by Jim
Kingdon, 4-Feb-2023.)
|
  Slot     
       |
|
Theorem | starvndxnbasendx 12602 |
The slot for the involution function is not the slot for the base set in
an extensible structure. (Contributed by AV, 18-Oct-2024.)
|
    
     |
|
Theorem | starvndxnplusgndx 12603 |
The slot for the involution function is not the slot for the base set in
an extensible structure. (Contributed by AV, 18-Oct-2024.)
|
    
    |
|
Theorem | starvndxnmulrndx 12604 |
The slot for the involution function is not the slot for the base set in
an extensible structure. (Contributed by AV, 18-Oct-2024.)
|
    
     |
|
Theorem | ressmulrg 12605 |
is unaffected by
restriction. (Contributed by Stefan O'Rear,
27-Nov-2014.)
|
 ↾s 
             |
|
Theorem | srngstrd 12606 |
A constructed star ring is a structure. (Contributed by Mario Carneiro,
18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
|
                                   
      Struct      |
|
Theorem | srngbased 12607 |
The base set of a constructed star ring. (Contributed by Mario
Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
|
                                   
            |
|
Theorem | srngplusgd 12608 |
The addition operation of a constructed star ring. (Contributed by
Mario Carneiro, 20-Jun-2015.) (Revised by Jim Kingdon, 5-Feb-2023.)
|
                                   
           |
|
Theorem | srngmulrd 12609 |
The multiplication operation of a constructed star ring. (Contributed
by Mario Carneiro, 20-Jun-2015.)
|
                                   
            |
|
Theorem | srnginvld 12610 |
The involution function of a constructed star ring. (Contributed by
Mario Carneiro, 20-Jun-2015.)
|
                                   
             |
|
Theorem | scandx 12611 |
Index value of the df-sca 12554 slot. (Contributed by Mario Carneiro,
14-Aug-2015.)
|
Scalar   |
|
Theorem | scaid 12612 |
Utility theorem: index-independent form of scalar df-sca 12554. (Contributed
by Mario Carneiro, 19-Jun-2014.)
|
Scalar Slot Scalar   |
|
Theorem | scaslid 12613 |
Slot property of Scalar. (Contributed by Jim Kingdon,
5-Feb-2023.)
|
Scalar Slot
Scalar  Scalar 
  |
|
Theorem | scandxnbasendx 12614 |
The slot for the scalar is not the slot for the base set in an extensible
structure. (Contributed by AV, 21-Oct-2024.)
|
Scalar       |
|
Theorem | scandxnplusgndx 12615 |
The slot for the scalar field is not the slot for the group operation in
an extensible structure. (Contributed by AV, 18-Oct-2024.)
|
Scalar      |
|
Theorem | scandxnmulrndx 12616 |
The slot for the scalar field is not the slot for the ring
(multiplication) operation in an extensible structure. (Contributed by
AV, 29-Oct-2024.)
|
Scalar       |
|
Theorem | vscandx 12617 |
Index value of the df-vsca 12555 slot. (Contributed by Mario Carneiro,
14-Aug-2015.)
|
   
 |
|
Theorem | vscaid 12618 |
Utility theorem: index-independent form of scalar product df-vsca 12555.
(Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro,
19-Jun-2014.)
|
Slot
     |
|
Theorem | vscandxnbasendx 12619 |
The slot for the scalar product is not the slot for the base set in an
extensible structure. (Contributed by AV, 18-Oct-2024.)
|
         |
|
Theorem | vscandxnplusgndx 12620 |
The slot for the scalar product is not the slot for the group operation in
an extensible structure. (Contributed by AV, 18-Oct-2024.)
|
        |
|
Theorem | vscandxnmulrndx 12621 |
The slot for the scalar product is not the slot for the ring
(multiplication) operation in an extensible structure. (Contributed by
AV, 29-Oct-2024.)
|
         |
|
Theorem | vscandxnscandx 12622 |
The slot for the scalar product is not the slot for the scalar field in an
extensible structure. (Contributed by AV, 18-Oct-2024.)
|
    Scalar   |
|
Theorem | vscaslid 12623 |
Slot property of .
(Contributed by Jim Kingdon, 5-Feb-2023.)
|
 Slot           |
|
Theorem | lmodstrd 12624 |
A constructed left module or left vector space is a structure.
(Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Jim Kingdon,
5-Feb-2023.)
|
                 Scalar           
        
  Struct      |
|
Theorem | lmodbased 12625 |
The base set of a constructed left vector space. (Contributed by Mario
Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
|
                 Scalar           
        
        |
|
Theorem | lmodplusgd 12626 |
The additive operation of a constructed left vector space. (Contributed
by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon,
6-Feb-2023.)
|
                 Scalar           
        
       |
|
Theorem | lmodscad 12627 |
The set of scalars of a constructed left vector space. (Contributed by
Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
|
                 Scalar           
        
  Scalar    |
|
Theorem | lmodvscad 12628 |
The scalar product operation of a constructed left vector space.
(Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon,
7-Feb-2023.)
|
                 Scalar           
        
        |
|
Theorem | ipndx 12629 |
Index value of the df-ip 12556 slot. (Contributed by Mario Carneiro,
14-Aug-2015.)
|
   
 |
|
Theorem | ipid 12630 |
Utility theorem: index-independent form of df-ip 12556. (Contributed by
Mario Carneiro, 6-Oct-2013.)
|
Slot
     |
|
Theorem | ipslid 12631 |
Slot property of .
(Contributed by Jim Kingdon, 7-Feb-2023.)
|
 Slot           |
|
Theorem | ipndxnbasendx 12632 |
The slot for the inner product is not the slot for the base set in an
extensible structure. (Contributed by AV, 21-Oct-2024.)
|
         |
|
Theorem | ipndxnplusgndx 12633 |
The slot for the inner product is not the slot for the group operation in
an extensible structure. (Contributed by AV, 29-Oct-2024.)
|
        |
|
Theorem | ipndxnmulrndx 12634 |
The slot for the inner product is not the slot for the ring
(multiplication) operation in an extensible structure. (Contributed by
AV, 29-Oct-2024.)
|
         |
|
Theorem | slotsdifipndx 12635 |
The slot for the scalar is not the index of other slots. (Contributed by
AV, 12-Nov-2024.)
|
    
    Scalar        |
|
Theorem | ipsstrd 12636 |
A constructed inner product space is a structure. (Contributed by
Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
|
                         Scalar                       
     
    Struct      |
|
Theorem | ipsbased 12637 |
The base set of a constructed inner product space. (Contributed by
Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
|
                         Scalar                       
     
          |
|
Theorem | ipsaddgd 12638 |
The additive operation of a constructed inner product space.
(Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon,
7-Feb-2023.)
|
                         Scalar                       
     
         |
|
Theorem | ipsmulrd 12639 |
The multiplicative operation of a constructed inner product space.
(Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon,
7-Feb-2023.)
|
                         Scalar                       
     
          |
|
Theorem | ipsscad 12640 |
The set of scalars of a constructed inner product space. (Contributed
by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon,
8-Feb-2023.)
|
                         Scalar                       
     
    Scalar    |
|
Theorem | ipsvscad 12641 |
The scalar product operation of a constructed inner product space.
(Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon,
8-Feb-2023.)
|
                         Scalar                       
     
          |
|
Theorem | ipsipd 12642 |
The multiplicative operation of a constructed inner product space.
(Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon,
8-Feb-2023.)
|
                         Scalar                       
     
          |
|
Theorem | ressscag 12643 |
Scalar is unaffected by restriction. (Contributed by Mario
Carneiro, 7-Dec-2014.)
|
 ↾s  Scalar     Scalar    |
|
Theorem | ressvscag 12644 |
is unaffected by
restriction. (Contributed by Mario Carneiro,
7-Dec-2014.)
|
 ↾s 
             |
|
Theorem | ressipg 12645 |
The inner product is unaffected by restriction. (Contributed by
Thierry Arnoux, 16-Jun-2019.)
|
 ↾s 
             |
|
Theorem | tsetndx 12646 |
Index value of the df-tset 12557 slot. (Contributed by Mario Carneiro,
14-Aug-2015.)
|
TopSet   |
|
Theorem | tsetid 12647 |
Utility theorem: index-independent form of df-tset 12557. (Contributed by
NM, 20-Oct-2012.)
|
TopSet Slot TopSet   |
|
Theorem | tsetslid 12648 |
Slot property of TopSet. (Contributed by Jim Kingdon,
9-Feb-2023.)
|
TopSet Slot
TopSet  TopSet 
  |
|
Theorem | tsetndxnn 12649 |
The index of the slot for the group operation in an extensible structure
is a positive integer. (Contributed by AV, 31-Oct-2024.)
|
TopSet   |
|
Theorem | basendxlttsetndx 12650 |
The index of the slot for the base set is less then the index of the slot
for the topology in an extensible structure. (Contributed by AV,
31-Oct-2024.)
|
    TopSet   |
|
Theorem | tsetndxnbasendx 12651 |
The slot for the topology is not the slot for the base set in an
extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened
by AV, 31-Oct-2024.)
|
TopSet       |
|
Theorem | tsetndxnplusgndx 12652 |
The slot for the topology is not the slot for the group operation in an
extensible structure. (Contributed by AV, 18-Oct-2024.)
|
TopSet      |
|
Theorem | tsetndxnmulrndx 12653 |
The slot for the topology is not the slot for the ring multiplication
operation in an extensible structure. (Contributed by AV,
31-Oct-2024.)
|
TopSet       |
|
Theorem | tsetndxnstarvndx 12654 |
The slot for the topology is not the slot for the involution in an
extensible structure. (Contributed by AV, 11-Nov-2024.)
|
TopSet        |
|
Theorem | slotstnscsi 12655 |
The slots Scalar,
and are different
from the slot
TopSet. (Contributed by AV, 29-Oct-2024.)
|
 TopSet  Scalar  TopSet     
TopSet        |
|
Theorem | topgrpstrd 12656 |
A constructed topological group is a structure. (Contributed by Mario
Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|
                TopSet       
    Struct      |
|
Theorem | topgrpbasd 12657 |
The base set of a constructed topological group. (Contributed by Mario
Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|
                TopSet       
          |
|
Theorem | topgrpplusgd 12658 |
The additive operation of a constructed topological group. (Contributed
by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon,
9-Feb-2023.)
|
                TopSet       
         |
|
Theorem | topgrptsetd 12659 |
The topology of a constructed topological group. (Contributed by Mario
Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|
                TopSet       
    TopSet    |
|
Theorem | plendx 12660 |
Index value of the df-ple 12558 slot. (Contributed by Mario Carneiro,
14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
|
   
;  |
|
Theorem | pleid 12661 |
Utility theorem: self-referencing, index-independent form of df-ple 12558.
(Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
|
Slot
     |
|
Theorem | pleslid 12662 |
Slot property of .
(Contributed by Jim Kingdon, 9-Feb-2023.)
|
 Slot           |
|
Theorem | plendxnn 12663 |
The index value of the order slot 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, 30-Oct-2024.)
|
   
 |
|
Theorem | basendxltplendx 12664 |
The index value of the slot is less than the index value of the
slot.
(Contributed by AV, 30-Oct-2024.)
|
         |
|
Theorem | plendxnbasendx 12665 |
The slot for the order is not the slot for the base set in an extensible
structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV,
30-Oct-2024.)
|
         |
|
Theorem | plendxnplusgndx 12666 |
The slot for the "less than or equal to" ordering is not the slot for
the
group operation in an extensible structure. (Contributed by AV,
18-Oct-2024.)
|
        |
|
Theorem | plendxnmulrndx 12667 |
The slot for the "less than or equal to" ordering is not the slot for
the
ring multiplication operation in an extensible structure. (Contributed by
AV, 1-Nov-2024.)
|
         |
|
Theorem | plendxnscandx 12668 |
The slot for the "less than or equal to" ordering is not the slot for
the
scalar in an extensible structure. (Contributed by AV, 1-Nov-2024.)
|
    Scalar   |
|
Theorem | plendxnvscandx 12669 |
The slot for the "less than or equal to" ordering is not the slot for
the
scalar product in an extensible structure. (Contributed by AV,
1-Nov-2024.)
|
         |
|
Theorem | slotsdifplendx 12670 |
The index of the slot for the distance is not the index of other slots.
(Contributed by AV, 11-Nov-2024.)
|
          TopSet        |
|
Theorem | dsndx 12671 |
Index value of the df-ds 12560 slot. (Contributed by Mario Carneiro,
14-Aug-2015.)
|
    ;  |
|
Theorem | dsid 12672 |
Utility theorem: index-independent form of df-ds 12560. (Contributed by
Mario Carneiro, 23-Dec-2013.)
|
Slot      |
|
Theorem | dsslid 12673 |
Slot property of . (Contributed by Jim Kingdon, 6-May-2023.)
|

Slot           |
|
Theorem | dsndxnn 12674 |
The index of the slot for the distance in an extensible structure is a
positive integer. (Contributed by AV, 28-Oct-2024.)
|
     |
|
Theorem | basendxltdsndx 12675 |
The index of the slot for the base set is less then the index of the slot
for the distance in an extensible structure. (Contributed by AV,
28-Oct-2024.)
|
         |
|
Theorem | dsndxnbasendx 12676 |
The slot for the distance is not the slot for the base set in an
extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened
by AV, 28-Oct-2024.)
|
         |
|
Theorem | dsndxnplusgndx 12677 |
The slot for the distance function is not the slot for the group operation
in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|
        |
|
Theorem | dsndxnmulrndx 12678 |
The slot for the distance function is not the slot for the ring
multiplication operation in an extensible structure. (Contributed by AV,
31-Oct-2024.)
|
         |
|
Theorem | slotsdnscsi 12679 |
The slots Scalar,
and are different
from the slot
.
(Contributed by AV, 29-Oct-2024.)
|
     Scalar         
          |
|
Theorem | dsndxntsetndx 12680 |
The slot for the distance function is not the slot for the topology in an
extensible structure. (Contributed by AV, 29-Oct-2024.)
|
    TopSet   |
|
Theorem | slotsdifdsndx 12681 |
The index of the slot for the distance is not the index of other slots.
(Contributed by AV, 11-Nov-2024.)
|
                    |
|
Theorem | unifndx 12682 |
Index value of the df-unif 12561 slot. (Contributed by Thierry Arnoux,
17-Dec-2017.) (New usage is discouraged.)
|
    ;  |
|
Theorem | unifid 12683 |
Utility theorem: index-independent form of df-unif 12561. (Contributed by
Thierry Arnoux, 17-Dec-2017.)
|
Slot      |
|
Theorem | unifndxnn 12684 |
The index of the slot for the uniform set in an extensible structure is a
positive integer. (Contributed by AV, 28-Oct-2024.)
|
     |
|
Theorem | basendxltunifndx 12685 |
The index of the slot for the base set is less then the index of the slot
for the uniform set in an extensible structure. (Contributed by AV,
28-Oct-2024.)
|
         |
|
Theorem | unifndxnbasendx 12686 |
The slot for the uniform set is not the slot for the base set in an
extensible structure. (Contributed by AV, 21-Oct-2024.)
|
         |
|
Theorem | unifndxntsetndx 12687 |
The slot for the uniform set is not the slot for the topology in an
extensible structure. (Contributed by AV, 28-Oct-2024.)
|
    TopSet   |
|
Theorem | slotsdifunifndx 12688 |
The index of the slot for the uniform set is not the index of other slots.
(Contributed by AV, 10-Nov-2024.)
|
                          
                    |
|
Theorem | homid 12689 |
Utility theorem: index-independent form of df-hom 12562. (Contributed by
Mario Carneiro, 7-Jan-2017.)
|
Slot     |
|
Theorem | homslid 12690 |
Slot property of . (Contributed by Jim Kingdon, 20-Mar-2025.)
|
 Slot         |
|
Theorem | ccoid 12691 |
Utility theorem: index-independent form of df-cco 12563. (Contributed by
Mario Carneiro, 7-Jan-2017.)
|
comp Slot comp   |
|
Theorem | ccoslid 12692 |
Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.)
|
comp Slot
comp  comp 
  |
|
6.1.3 Definition of the structure
product
|
|
Syntax | crest 12693 |
Extend class notation with the function returning a subspace topology.
|
↾t |
|
Syntax | ctopn 12694 |
Extend class notation with the topology extractor function.
|
 |
|
Definition | df-rest 12695* |
Function returning the subspace topology induced by the topology
and the set .
(Contributed by FL, 20-Sep-2010.) (Revised by
Mario Carneiro, 1-May-2015.)
|
↾t  
      |
|
Definition | df-topn 12696 |
Define the topology extractor function. This differs from df-tset 12557
when a structure has been restricted using df-iress 12472; in this case the
TopSet component will still have a topology over the larger set, and
this function fixes this by restricting the topology as well.
(Contributed by Mario Carneiro, 13-Aug-2015.)
|
  TopSet  ↾t        |
|
Theorem | restfn 12697 |
The subspace topology operator is a function on pairs. (Contributed by
Mario Carneiro, 1-May-2015.)
|
↾t    |
|
Theorem | topnfn 12698 |
The topology extractor function is a function on the universe.
(Contributed by Mario Carneiro, 13-Aug-2015.)
|
 |
|
Theorem | restval 12699* |
The subspace topology induced by the topology on the set .
(Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro,
1-May-2015.)
|
    ↾t        |
|
Theorem | elrest 12700* |
The predicate "is an open set of a subspace topology". (Contributed
by
FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
|
    
↾t  
     |