Theorem List for Intuitionistic Logic Explorer - 12401-12500 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Syntax | coc 12401 |
Extend class notation with the class of orthocomplementation
extractors.
|
class oc |
|
Syntax | cds 12402 |
Extend class notation with the metric space distance function.
|
class dist |
|
Syntax | cunif 12403 |
Extend class notation with the uniform structure.
|
class UnifSet |
|
Syntax | chom 12404 |
Extend class notation with the hom-set structure.
|
class Hom |
|
Syntax | cco 12405 |
Extend class notation with the composition operation.
|
class comp |
|
Definition | df-plusg 12406 |
Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by
Mario Carneiro, 14-Aug-2015.)
|
⊢ +g = Slot 2 |
|
Definition | df-mulr 12407 |
Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by
Mario Carneiro, 14-Aug-2015.)
|
⊢ .r = Slot 3 |
|
Definition | df-starv 12408 |
Define the involution function of a *-ring. (Contributed by NM,
4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
|
⊢ *𝑟 = Slot
4 |
|
Definition | df-sca 12409 |
Define scalar field component of a vector space 𝑣. (Contributed by
NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
|
⊢ Scalar = Slot 5 |
|
Definition | df-vsca 12410 |
Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by
Mario Carneiro, 14-Aug-2015.)
|
⊢ ·𝑠 = Slot
6 |
|
Definition | df-ip 12411 |
Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.)
(Revised by Mario Carneiro, 14-Aug-2015.)
|
⊢ ·𝑖 = Slot
8 |
|
Definition | df-tset 12412 |
Define the topology component of a topological space (structure).
(Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro,
14-Aug-2015.)
|
⊢ TopSet = Slot 9 |
|
Definition | df-ple 12413 |
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.)
|
⊢ le = Slot ;10 |
|
Definition | df-ocomp 12414 |
Define the orthocomplementation extractor for posets and related
structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro,
14-Aug-2015.)
|
⊢ oc = Slot ;11 |
|
Definition | df-ds 12415 |
Define the distance function component of a metric space (structure).
(Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro,
14-Aug-2015.)
|
⊢ dist = Slot ;12 |
|
Definition | df-unif 12416 |
Define the uniform structure component of a uniform space. (Contributed
by Mario Carneiro, 14-Aug-2015.)
|
⊢ UnifSet = Slot ;13 |
|
Definition | df-hom 12417 |
Define the hom-set component of a category. (Contributed by Mario
Carneiro, 2-Jan-2017.)
|
⊢ Hom = Slot ;14 |
|
Definition | df-cco 12418 |
Define the composition operation of a category. (Contributed by Mario
Carneiro, 2-Jan-2017.)
|
⊢ comp = Slot ;15 |
|
Theorem | strleund 12419 |
Combine two structures into one. (Contributed by Mario Carneiro,
29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
|
⊢ (𝜑 → 𝐹 Struct 〈𝐴, 𝐵〉) & ⊢ (𝜑 → 𝐺 Struct 〈𝐶, 𝐷〉) & ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∪ 𝐺) Struct 〈𝐴, 𝐷〉) |
|
Theorem | strleun 12420 |
Combine two structures into one. (Contributed by Mario Carneiro,
29-Aug-2015.)
|
⊢ 𝐹 Struct 〈𝐴, 𝐵〉 & ⊢ 𝐺 Struct 〈𝐶, 𝐷〉 & ⊢ 𝐵 < 𝐶 ⇒ ⊢ (𝐹 ∪ 𝐺) Struct 〈𝐴, 𝐷〉 |
|
Theorem | strle1g 12421 |
Make a structure from a singleton. (Contributed by Mario Carneiro,
29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
|
⊢ 𝐼 ∈ ℕ & ⊢ 𝐴 = 𝐼 ⇒ ⊢ (𝑋 ∈ 𝑉 → {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉) |
|
Theorem | strle2g 12422 |
Make a structure from a pair. (Contributed by Mario Carneiro,
29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
|
⊢ 𝐼 ∈ ℕ & ⊢ 𝐴 = 𝐼
& ⊢ 𝐼 < 𝐽
& ⊢ 𝐽 ∈ ℕ & ⊢ 𝐵 = 𝐽 ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} Struct 〈𝐼, 𝐽〉) |
|
Theorem | strle3g 12423 |
Make a structure from a triple. (Contributed by Mario Carneiro,
29-Aug-2015.)
|
⊢ 𝐼 ∈ ℕ & ⊢ 𝐴 = 𝐼
& ⊢ 𝐼 < 𝐽
& ⊢ 𝐽 ∈ ℕ & ⊢ 𝐵 = 𝐽
& ⊢ 𝐽 < 𝐾
& ⊢ 𝐾 ∈ ℕ & ⊢ 𝐶 = 𝐾 ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑃) → {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉, 〈𝐶, 𝑍〉} Struct 〈𝐼, 𝐾〉) |
|
Theorem | plusgndx 12424 |
Index value of the df-plusg 12406 slot. (Contributed by Mario Carneiro,
14-Aug-2015.)
|
⊢ (+g‘ndx) =
2 |
|
Theorem | plusgid 12425 |
Utility theorem: index-independent form of df-plusg 12406. (Contributed by
NM, 20-Oct-2012.)
|
⊢ +g = Slot
(+g‘ndx) |
|
Theorem | plusgslid 12426 |
Slot property of +g. (Contributed by Jim
Kingdon, 3-Feb-2023.)
|
⊢ (+g = Slot
(+g‘ndx) ∧ (+g‘ndx) ∈
ℕ) |
|
Theorem | opelstrsl 12427 |
The slot of a structure which contains an ordered pair for that slot.
(Contributed by Jim Kingdon, 5-Feb-2023.)
|
⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) & ⊢ (𝜑 → 𝑆 Struct 𝑋)
& ⊢ (𝜑 → 𝑉 ∈ 𝑌)
& ⊢ (𝜑 → 〈(𝐸‘ndx), 𝑉〉 ∈ 𝑆) ⇒ ⊢ (𝜑 → 𝑉 = (𝐸‘𝑆)) |
|
Theorem | opelstrbas 12428 |
The base set of a structure with a base set. (Contributed by AV,
10-Nov-2021.)
|
⊢ (𝜑 → 𝑆 Struct 𝑋)
& ⊢ (𝜑 → 𝑉 ∈ 𝑌)
& ⊢ (𝜑 → 〈(Base‘ndx), 𝑉〉 ∈ 𝑆) ⇒ ⊢ (𝜑 → 𝑉 = (Base‘𝑆)) |
|
Theorem | 1strstrg 12429 |
A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
(Revised by Jim Kingdon, 28-Jan-2023.)
|
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉} ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐺 Struct 〈1, 1〉) |
|
Theorem | 1strbas 12430 |
The base set of a constructed one-slot structure. (Contributed by AV,
27-Mar-2020.)
|
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉} ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺)) |
|
Theorem | 2strstrg 12431 |
A constructed two-slot structure. (Contributed by Mario Carneiro,
29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
|
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(𝐸‘ndx), + 〉} & ⊢ 𝐸 = Slot 𝑁
& ⊢ 1 < 𝑁
& ⊢ 𝑁 ∈ ℕ
⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct 〈1, 𝑁〉) |
|
Theorem | 2strbasg 12432 |
The base set of a constructed two-slot structure. (Contributed by Mario
Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
|
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(𝐸‘ndx), + 〉} & ⊢ 𝐸 = Slot 𝑁
& ⊢ 1 < 𝑁
& ⊢ 𝑁 ∈ ℕ
⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐵 = (Base‘𝐺)) |
|
Theorem | 2stropg 12433 |
The other slot of a constructed two-slot structure. (Contributed by
Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
|
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(𝐸‘ndx), + 〉} & ⊢ 𝐸 = Slot 𝑁
& ⊢ 1 < 𝑁
& ⊢ 𝑁 ∈ ℕ
⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + = (𝐸‘𝐺)) |
|
Theorem | 2strstr1g 12434 |
A constructed two-slot structure. Version of 2strstrg 12431 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.)
|
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈𝑁, + 〉} & ⊢
(Base‘ndx) < 𝑁
& ⊢ 𝑁 ∈ ℕ
⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct 〈(Base‘ndx), 𝑁〉) |
|
Theorem | 2strbas1g 12435 |
The base set of a constructed two-slot structure. Version of 2strbasg 12432
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.)
|
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈𝑁, + 〉} & ⊢
(Base‘ndx) < 𝑁
& ⊢ 𝑁 ∈ ℕ
⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐵 = (Base‘𝐺)) |
|
Theorem | 2strop1g 12436 |
The other slot of a constructed two-slot structure. Version of
2stropg 12433 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.)
|
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈𝑁, + 〉} & ⊢
(Base‘ndx) < 𝑁
& ⊢ 𝑁 ∈ ℕ & ⊢ 𝐸 = Slot 𝑁 ⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + = (𝐸‘𝐺)) |
|
Theorem | basendxnplusgndx 12437 |
The slot for the base set is not the slot for the group operation in an
extensible structure. (Contributed by AV, 14-Nov-2021.)
|
⊢ (Base‘ndx) ≠
(+g‘ndx) |
|
Theorem | grpstrg 12438 |
A constructed group is a structure on 1...2.
(Contributed by
Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro,
30-Apr-2015.)
|
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), +
〉} ⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐺 Struct 〈1, 2〉) |
|
Theorem | grpbaseg 12439 |
The base set of a constructed group. (Contributed by Mario Carneiro,
2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
|
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), +
〉} ⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → 𝐵 = (Base‘𝐺)) |
|
Theorem | grpplusgg 12440 |
The operation of a constructed group. (Contributed by Mario Carneiro,
2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
|
⊢ 𝐺 = {〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), +
〉} ⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊) → + =
(+g‘𝐺)) |
|
Theorem | mulrndx 12441 |
Index value of the df-mulr 12407 slot. (Contributed by Mario Carneiro,
14-Aug-2015.)
|
⊢ (.r‘ndx) =
3 |
|
Theorem | mulrid 12442 |
Utility theorem: index-independent form of df-mulr 12407. (Contributed by
Mario Carneiro, 8-Jun-2013.)
|
⊢ .r = Slot
(.r‘ndx) |
|
Theorem | mulrslid 12443 |
Slot property of .r. (Contributed by Jim
Kingdon, 3-Feb-2023.)
|
⊢ (.r = Slot
(.r‘ndx) ∧ (.r‘ndx) ∈
ℕ) |
|
Theorem | plusgndxnmulrndx 12444 |
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.)
|
⊢ (+g‘ndx) ≠
(.r‘ndx) |
|
Theorem | basendxnmulrndx 12445 |
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.)
|
⊢ (Base‘ndx) ≠
(.r‘ndx) |
|
Theorem | rngstrg 12446 |
A constructed ring is a structure. (Contributed by Mario Carneiro,
28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
|
⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), ·
〉} ⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝑅 Struct 〈1, 3〉) |
|
Theorem | rngbaseg 12447 |
The base set of a constructed ring. (Contributed by Mario Carneiro,
2-Oct-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
|
⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), ·
〉} ⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → 𝐵 = (Base‘𝑅)) |
|
Theorem | rngplusgg 12448 |
The additive operation of a constructed ring. (Contributed by Mario
Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
|
⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), ·
〉} ⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → + =
(+g‘𝑅)) |
|
Theorem | rngmulrg 12449 |
The multiplicative operation of a constructed ring. (Contributed by
Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro,
30-Apr-2015.)
|
⊢ 𝑅 = {〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), ·
〉} ⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑊 ∧ · ∈ 𝑋) → · =
(.r‘𝑅)) |
|
Theorem | starvndx 12450 |
Index value of the df-starv 12408 slot. (Contributed by Mario Carneiro,
14-Aug-2015.)
|
⊢ (*𝑟‘ndx) =
4 |
|
Theorem | starvid 12451 |
Utility theorem: index-independent form of df-starv 12408. (Contributed by
Mario Carneiro, 6-Oct-2013.)
|
⊢ *𝑟 = Slot
(*𝑟‘ndx) |
|
Theorem | starvslid 12452 |
Slot property of *𝑟. (Contributed
by Jim Kingdon, 4-Feb-2023.)
|
⊢ (*𝑟 = Slot
(*𝑟‘ndx) ∧ (*𝑟‘ndx)
∈ ℕ) |
|
Theorem | srngstrd 12453 |
A constructed star ring is a structure. (Contributed by Mario Carneiro,
18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
|
⊢ 𝑅 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), · 〉} ∪
{〈(*𝑟‘ndx), ∗
〉})
& ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → · ∈ 𝑋) & ⊢ (𝜑 → ∗ ∈ 𝑌)
⇒ ⊢ (𝜑 → 𝑅 Struct 〈1, 4〉) |
|
Theorem | srngbased 12454 |
The base set of a constructed star ring. (Contributed by Mario
Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
|
⊢ 𝑅 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), · 〉} ∪
{〈(*𝑟‘ndx), ∗
〉})
& ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → · ∈ 𝑋) & ⊢ (𝜑 → ∗ ∈ 𝑌)
⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) |
|
Theorem | srngplusgd 12455 |
The addition operation of a constructed star ring. (Contributed by
Mario Carneiro, 20-Jun-2015.) (Revised by Jim Kingdon, 5-Feb-2023.)
|
⊢ 𝑅 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), · 〉} ∪
{〈(*𝑟‘ndx), ∗
〉})
& ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → · ∈ 𝑋) & ⊢ (𝜑 → ∗ ∈ 𝑌)
⇒ ⊢ (𝜑 → + =
(+g‘𝑅)) |
|
Theorem | srngmulrd 12456 |
The multiplication operation of a constructed star ring. (Contributed
by Mario Carneiro, 20-Jun-2015.)
|
⊢ 𝑅 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), · 〉} ∪
{〈(*𝑟‘ndx), ∗
〉})
& ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → · ∈ 𝑋) & ⊢ (𝜑 → ∗ ∈ 𝑌)
⇒ ⊢ (𝜑 → · =
(.r‘𝑅)) |
|
Theorem | srnginvld 12457 |
The involution function of a constructed star ring. (Contributed by
Mario Carneiro, 20-Jun-2015.)
|
⊢ 𝑅 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), · 〉} ∪
{〈(*𝑟‘ndx), ∗
〉})
& ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → · ∈ 𝑋) & ⊢ (𝜑 → ∗ ∈ 𝑌)
⇒ ⊢ (𝜑 → ∗ =
(*𝑟‘𝑅)) |
|
Theorem | scandx 12458 |
Index value of the df-sca 12409 slot. (Contributed by Mario Carneiro,
14-Aug-2015.)
|
⊢ (Scalar‘ndx) = 5 |
|
Theorem | scaid 12459 |
Utility theorem: index-independent form of scalar df-sca 12409. (Contributed
by Mario Carneiro, 19-Jun-2014.)
|
⊢ Scalar = Slot
(Scalar‘ndx) |
|
Theorem | scaslid 12460 |
Slot property of Scalar. (Contributed by Jim Kingdon,
5-Feb-2023.)
|
⊢ (Scalar = Slot (Scalar‘ndx) ∧
(Scalar‘ndx) ∈ ℕ) |
|
Theorem | vscandx 12461 |
Index value of the df-vsca 12410 slot. (Contributed by Mario Carneiro,
14-Aug-2015.)
|
⊢ ( ·𝑠
‘ndx) = 6 |
|
Theorem | vscaid 12462 |
Utility theorem: index-independent form of scalar product df-vsca 12410.
(Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro,
19-Jun-2014.)
|
⊢ ·𝑠 = Slot
( ·𝑠 ‘ndx) |
|
Theorem | vscaslid 12463 |
Slot property of ·𝑠.
(Contributed by Jim Kingdon, 5-Feb-2023.)
|
⊢ ( ·𝑠 = Slot
( ·𝑠 ‘ndx) ∧ (
·𝑠 ‘ndx) ∈
ℕ) |
|
Theorem | lmodstrd 12464 |
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.)
|
⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(Scalar‘ndx), 𝐹〉} ∪ {〈(
·𝑠 ‘ndx), ·
〉})
& ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑋)
& ⊢ (𝜑 → 𝐹 ∈ 𝑌)
& ⊢ (𝜑 → · ∈ 𝑍)
⇒ ⊢ (𝜑 → 𝑊 Struct 〈1, 6〉) |
|
Theorem | lmodbased 12465 |
The base set of a constructed left vector space. (Contributed by Mario
Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
|
⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(Scalar‘ndx), 𝐹〉} ∪ {〈(
·𝑠 ‘ndx), ·
〉})
& ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑋)
& ⊢ (𝜑 → 𝐹 ∈ 𝑌)
& ⊢ (𝜑 → · ∈ 𝑍)
⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝑊)) |
|
Theorem | lmodplusgd 12466 |
The additive operation of a constructed left vector space. (Contributed
by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon,
6-Feb-2023.)
|
⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(Scalar‘ndx), 𝐹〉} ∪ {〈(
·𝑠 ‘ndx), ·
〉})
& ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑋)
& ⊢ (𝜑 → 𝐹 ∈ 𝑌)
& ⊢ (𝜑 → · ∈ 𝑍)
⇒ ⊢ (𝜑 → + =
(+g‘𝑊)) |
|
Theorem | lmodscad 12467 |
The set of scalars of a constructed left vector space. (Contributed by
Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
|
⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(Scalar‘ndx), 𝐹〉} ∪ {〈(
·𝑠 ‘ndx), ·
〉})
& ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑋)
& ⊢ (𝜑 → 𝐹 ∈ 𝑌)
& ⊢ (𝜑 → · ∈ 𝑍)
⇒ ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊)) |
|
Theorem | lmodvscad 12468 |
The scalar product operation of a constructed left vector space.
(Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon,
7-Feb-2023.)
|
⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(Scalar‘ndx), 𝐹〉} ∪ {〈(
·𝑠 ‘ndx), ·
〉})
& ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑋)
& ⊢ (𝜑 → 𝐹 ∈ 𝑌)
& ⊢ (𝜑 → · ∈ 𝑍)
⇒ ⊢ (𝜑 → · = (
·𝑠 ‘𝑊)) |
|
Theorem | ipndx 12469 |
Index value of the df-ip 12411 slot. (Contributed by Mario Carneiro,
14-Aug-2015.)
|
⊢
(·𝑖‘ndx) = 8 |
|
Theorem | ipid 12470 |
Utility theorem: index-independent form of df-ip 12411. (Contributed by
Mario Carneiro, 6-Oct-2013.)
|
⊢ ·𝑖 = Slot
(·𝑖‘ndx) |
|
Theorem | ipslid 12471 |
Slot property of ·𝑖.
(Contributed by Jim Kingdon, 7-Feb-2023.)
|
⊢ (·𝑖 = Slot
(·𝑖‘ndx) ∧
(·𝑖‘ndx) ∈
ℕ) |
|
Theorem | ipsstrd 12472 |
A constructed inner product space is a structure. (Contributed by
Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
|
⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), 𝐼〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → × ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌)
& ⊢ (𝜑 → · ∈ 𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝐴 Struct 〈1, 8〉) |
|
Theorem | ipsbased 12473 |
The base set of a constructed inner product space. (Contributed by
Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
|
⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), 𝐼〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → × ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌)
& ⊢ (𝜑 → · ∈ 𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝐴)) |
|
Theorem | ipsaddgd 12474 |
The additive operation of a constructed inner product space.
(Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon,
7-Feb-2023.)
|
⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), 𝐼〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → × ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌)
& ⊢ (𝜑 → · ∈ 𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) ⇒ ⊢ (𝜑 → + =
(+g‘𝐴)) |
|
Theorem | ipsmulrd 12475 |
The multiplicative operation of a constructed inner product space.
(Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon,
7-Feb-2023.)
|
⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), 𝐼〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → × ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌)
& ⊢ (𝜑 → · ∈ 𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) ⇒ ⊢ (𝜑 → × =
(.r‘𝐴)) |
|
Theorem | ipsscad 12476 |
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.)
|
⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), 𝐼〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → × ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌)
& ⊢ (𝜑 → · ∈ 𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝑆 = (Scalar‘𝐴)) |
|
Theorem | ipsvscad 12477 |
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.)
|
⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), 𝐼〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → × ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌)
& ⊢ (𝜑 → · ∈ 𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) ⇒ ⊢ (𝜑 → · = (
·𝑠 ‘𝐴)) |
|
Theorem | ipsipd 12478 |
The multiplicative operation of a constructed inner product space.
(Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon,
8-Feb-2023.)
|
⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), 𝐼〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → × ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌)
& ⊢ (𝜑 → · ∈ 𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝐼 =
(·𝑖‘𝐴)) |
|
Theorem | tsetndx 12479 |
Index value of the df-tset 12412 slot. (Contributed by Mario Carneiro,
14-Aug-2015.)
|
⊢ (TopSet‘ndx) = 9 |
|
Theorem | tsetid 12480 |
Utility theorem: index-independent form of df-tset 12412. (Contributed by
NM, 20-Oct-2012.)
|
⊢ TopSet = Slot
(TopSet‘ndx) |
|
Theorem | tsetslid 12481 |
Slot property of TopSet. (Contributed by Jim Kingdon,
9-Feb-2023.)
|
⊢ (TopSet = Slot (TopSet‘ndx) ∧
(TopSet‘ndx) ∈ ℕ) |
|
Theorem | topgrpstrd 12482 |
A constructed topological group is a structure. (Contributed by Mario
Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|
⊢ 𝑊 = {〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(TopSet‘ndx), 𝐽〉} & ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → 𝐽 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝑊 Struct 〈1, 9〉) |
|
Theorem | topgrpbasd 12483 |
The base set of a constructed topological group. (Contributed by Mario
Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|
⊢ 𝑊 = {〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(TopSet‘ndx), 𝐽〉} & ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → 𝐽 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝑊)) |
|
Theorem | topgrpplusgd 12484 |
The additive operation of a constructed topological group. (Contributed
by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon,
9-Feb-2023.)
|
⊢ 𝑊 = {〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(TopSet‘ndx), 𝐽〉} & ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → 𝐽 ∈ 𝑋) ⇒ ⊢ (𝜑 → + =
(+g‘𝑊)) |
|
Theorem | topgrptsetd 12485 |
The topology of a constructed topological group. (Contributed by Mario
Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|
⊢ 𝑊 = {〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(TopSet‘ndx), 𝐽〉} & ⊢ (𝜑 → 𝐵 ∈ 𝑉)
& ⊢ (𝜑 → + ∈ 𝑊)
& ⊢ (𝜑 → 𝐽 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝐽 = (TopSet‘𝑊)) |
|
Theorem | plendx 12486 |
Index value of the df-ple 12413 slot. (Contributed by Mario Carneiro,
14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
|
⊢ (le‘ndx) = ;10 |
|
Theorem | pleid 12487 |
Utility theorem: self-referencing, index-independent form of df-ple 12413.
(Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
|
⊢ le = Slot (le‘ndx) |
|
Theorem | pleslid 12488 |
Slot property of le. (Contributed by Jim Kingdon,
9-Feb-2023.)
|
⊢ (le = Slot (le‘ndx) ∧
(le‘ndx) ∈ ℕ) |
|
Theorem | dsndx 12489 |
Index value of the df-ds 12415 slot. (Contributed by Mario Carneiro,
14-Aug-2015.)
|
⊢ (dist‘ndx) = ;12 |
|
Theorem | dsid 12490 |
Utility theorem: index-independent form of df-ds 12415. (Contributed by
Mario Carneiro, 23-Dec-2013.)
|
⊢ dist = Slot
(dist‘ndx) |
|
Theorem | dsslid 12491 |
Slot property of dist. (Contributed by Jim Kingdon,
6-May-2023.)
|
⊢ (dist = Slot (dist‘ndx) ∧
(dist‘ndx) ∈ ℕ) |
|
6.1.3 Definition of the structure
product
|
|
Syntax | crest 12492 |
Extend class notation with the function returning a subspace topology.
|
class ↾t |
|
Syntax | ctopn 12493 |
Extend class notation with the topology extractor function.
|
class TopOpen |
|
Definition | df-rest 12494* |
Function returning the subspace topology induced by the topology 𝑦
and the set 𝑥. (Contributed by FL, 20-Sep-2010.)
(Revised by
Mario Carneiro, 1-May-2015.)
|
⊢ ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥))) |
|
Definition | df-topn 12495 |
Define the topology extractor function. This differs from df-tset 12412
when a structure has been restricted using df-ress 12339; 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.)
|
⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t
(Base‘𝑤))) |
|
Theorem | restfn 12496 |
The subspace topology operator is a function on pairs. (Contributed by
Mario Carneiro, 1-May-2015.)
|
⊢ ↾t Fn (V ×
V) |
|
Theorem | topnfn 12497 |
The topology extractor function is a function on the universe.
(Contributed by Mario Carneiro, 13-Aug-2015.)
|
⊢ TopOpen Fn V |
|
Theorem | restval 12498* |
The subspace topology induced by the topology 𝐽 on the set 𝐴.
(Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro,
1-May-2015.)
|
⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))) |
|
Theorem | elrest 12499* |
The predicate "is an open set of a subspace topology". (Contributed
by
FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
|
⊢ ((𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵))) |
|
Theorem | elrestr 12500 |
Sufficient condition for being an open set in a subspace. (Contributed
by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro,
15-Dec-2013.)
|
⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐴 ∈ 𝐽) → (𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆)) |