![]() |
Intuitionistic Logic Explorer Theorem List (p. 121 of 135) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-ndx 12001 | Define the structure component index extractor. See theorem ndxarg 12021 to understand its purpose. The restriction to ℕ ensures that ndx is a set. The restriction to some set is necessary since I is a proper class. In principle, we could have chosen ℂ or (if we revise all structure component definitions such as df-base 12004) another set such as the set of finite ordinals ω (df-iom 4513). (Contributed by NM, 4-Sep-2011.) |
⊢ ndx = ( I ↾ ℕ) | ||
Definition | df-slot 12002* |
Define the slot extractor for extensible structures. The class
Slot 𝐴 is a function whose argument can be
any set, although it is
meaningful only if that set is a member of an extensible structure (such
as a partially ordered set or a group).
Note that Slot 𝐴 is implemented as "evaluation at 𝐴". That is, (Slot 𝐴‘𝑆) is defined to be (𝑆‘𝐴), where 𝐴 will typically be a small nonzero natural number. Each extensible structure 𝑆 is a function defined on specific natural number "slots", and this function extracts the value at a particular slot. The special "structure" ndx, defined as the identity function restricted to ℕ, can be used to extract the number 𝐴 from a slot, since (Slot 𝐴‘ndx) = 𝐴 (see ndxarg 12021). This is typically used to refer to the number of a slot when defining structures without having to expose the detail of what that number is (for instance, we use the expression (Base‘ndx) in theorems and proofs instead of its value 1). The class Slot cannot be defined as (𝑥 ∈ V ↦ (𝑓 ∈ V ↦ (𝑓‘𝑥))) because each Slot 𝐴 is a function on the proper class V so is itself a proper class, and the values of functions are sets (fvex 5449). It is necessary to allow proper classes as values of Slot 𝐴 since for instance the class of all (base sets of) groups is proper. (Contributed by Mario Carneiro, 22-Sep-2015.) |
⊢ Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥‘𝐴)) | ||
Theorem | sloteq 12003 | Equality theorem for the Slot construction. The converse holds if 𝐴 (or 𝐵) is a set. (Contributed by BJ, 27-Dec-2021.) |
⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵) | ||
Definition | df-base 12004 | 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.) |
⊢ Base = Slot 1 | ||
Definition | df-sets 12005* | 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-ress 12006 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.) |
⊢ sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})) | ||
Definition | df-ress 12006* |
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 Base set while
leaving operators alone; individual kinds of structures will need to
handle this behavior, by ignoring operators' values outside the range,
defining a function using the base set and applying that, or explicitly
truncating the slot before use.
(Credit for this operator goes to Mario Carneiro.) (Contributed by Stefan O'Rear, 29-Nov-2014.) |
⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉))) | ||
Theorem | brstruct 12007 | The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.) |
⊢ Rel Struct | ||
Theorem | isstruct2im 12008 | The property of being a structure with components in (1st ‘𝑋)...(2nd ‘𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.) |
⊢ (𝐹 Struct 𝑋 → (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))) | ||
Theorem | isstruct2r 12009 | The property of being a structure with components in (1st ‘𝑋)...(2nd ‘𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.) |
⊢ (((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (...‘𝑋))) → 𝐹 Struct 𝑋) | ||
Theorem | structex 12010 | A structure is a set. (Contributed by AV, 10-Nov-2021.) |
⊢ (𝐺 Struct 𝑋 → 𝐺 ∈ V) | ||
Theorem | structn0fun 12011 | A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.) |
⊢ (𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅})) | ||
Theorem | isstructim 12012 | The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.) |
⊢ (𝐹 Struct 〈𝑀, 𝑁〉 → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁))) | ||
Theorem | isstructr 12013 | The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.) |
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 Struct 〈𝑀, 𝑁〉) | ||
Theorem | structcnvcnv 12014 | Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 = (𝐹 ∖ {∅})) | ||
Theorem | structfung 12015 | The converse of the converse of a structure is a function. Closed form of structfun 12016. (Contributed by AV, 12-Nov-2021.) |
⊢ (𝐹 Struct 𝑋 → Fun ◡◡𝐹) | ||
Theorem | structfun 12016 | Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Proof shortened by AV, 12-Nov-2021.) |
⊢ 𝐹 Struct 𝑋 ⇒ ⊢ Fun ◡◡𝐹 | ||
Theorem | structfn 12017 | Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
⊢ 𝐹 Struct 〈𝑀, 𝑁〉 ⇒ ⊢ (Fun ◡◡𝐹 ∧ dom 𝐹 ⊆ (1...𝑁)) | ||
Theorem | strnfvnd 12018 | Deduction version of strnfvn 12019. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 19-Jan-2023.) |
⊢ 𝐸 = Slot 𝑁 & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁)) | ||
Theorem | strnfvn 12019 |
Value of a structure component extractor 𝐸. Normally, 𝐸 is a
defined constant symbol such as Base (df-base 12004) and 𝑁 is a
fixed integer such as 1. 𝑆 is a structure, i.e. a
specific
member of a class of structures.
Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strslfv 12042. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.) |
⊢ 𝑆 ∈ V & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ (𝐸‘𝑆) = (𝑆‘𝑁) | ||
Theorem | strfvssn 12020 | A structure component extractor produces a value which is contained in a set dependent on 𝑆, but not 𝐸. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Jim Kingdon, 19-Jan-2023.) |
⊢ 𝐸 = Slot 𝑁 & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) | ||
Theorem | ndxarg 12021 | Get the numeric argument from a defined structure component extractor such as df-base 12004. (Contributed by Mario Carneiro, 6-Oct-2013.) |
⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ (𝐸‘ndx) = 𝑁 | ||
Theorem | ndxid 12022 |
A structure component extractor is defined by its own index. This
theorem, together with strslfv 12042 below, is useful for avoiding direct
reference to the hard-coded numeric index in component extractor
definitions, such as the 1 in df-base 12004, making it easier to change
should the need arise.
(Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof shortened by BJ, 27-Dec-2021.) |
⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ 𝐸 = Slot (𝐸‘ndx) | ||
Theorem | ndxslid 12023 | 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 12042. (Contributed by Jim Kingdon, 29-Jan-2023.) |
⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) | ||
Theorem | slotslfn 12024 | A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by Jim Kingdon, 10-Feb-2023.) |
⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) ⇒ ⊢ 𝐸 Fn V | ||
Theorem | slotex 12025 | Existence of slot value. A corollary of slotslfn 12024. (Contributed by Jim Kingdon, 12-Feb-2023.) |
⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐸‘𝐴) ∈ V) | ||
Theorem | strndxid 12026 | The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.) |
⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸‘𝑆)) | ||
Theorem | reldmsets 12027 | The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
⊢ Rel dom sSet | ||
Theorem | setsvalg 12028 | Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴})) | ||
Theorem | setsvala 12029 | Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 20-Jan-2023.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) | ||
Theorem | setsex 12030 | Applying the structure replacement function yields a set. (Contributed by Jim Kingdon, 22-Jan-2023.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) ∈ V) | ||
Theorem | strsetsid 12031 | Value of the structure replacement function. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 30-Jan-2023.) |
⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝜑 → 𝑆 Struct 〈𝑀, 𝑁〉) & ⊢ (𝜑 → Fun 𝑆) & ⊢ (𝜑 → (𝐸‘ndx) ∈ dom 𝑆) ⇒ ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉)) | ||
Theorem | fvsetsid 12032 | The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ((𝐹 sSet 〈𝑋, 𝑌〉)‘𝑋) = 𝑌) | ||
Theorem | setsfun 12033 | A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.) |
⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ (𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊)) → Fun (𝐺 sSet 〈𝐼, 𝐸〉)) | ||
Theorem | setsfun0 12034 | 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 12033 is useful for proofs based on isstruct2r 12009 which requires Fun (𝐹 ∖ {∅}) for 𝐹 to be an extensible structure. (Contributed by AV, 7-Jun-2021.) |
⊢ (((𝐺 ∈ 𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊)) → Fun ((𝐺 sSet 〈𝐼, 𝐸〉) ∖ {∅})) | ||
Theorem | setsn0fun 12035 | 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.) |
⊢ (𝜑 → 𝑆 Struct 𝑋) & ⊢ (𝜑 → 𝐼 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) ⇒ ⊢ (𝜑 → Fun ((𝑆 sSet 〈𝐼, 𝐸〉) ∖ {∅})) | ||
Theorem | setsresg 12036 | The structure replacement function does not affect the value of 𝑆 away from 𝐴. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.) |
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝑆 sSet 〈𝐴, 𝐵〉) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴}))) | ||
Theorem | setsabsd 12037 | 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.) |
⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑊) & ⊢ (𝜑 → 𝐵 ∈ 𝑋) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) ⇒ ⊢ (𝜑 → ((𝑆 sSet 〈𝐴, 𝐵〉) sSet 〈𝐴, 𝐶〉) = (𝑆 sSet 〈𝐴, 𝐶〉)) | ||
Theorem | setscom 12038 | Component-setting is commutative when the x-values are different. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑊 ∧ 𝐷 ∈ 𝑋)) → ((𝑆 sSet 〈𝐴, 𝐶〉) sSet 〈𝐵, 𝐷〉) = ((𝑆 sSet 〈𝐵, 𝐷〉) sSet 〈𝐴, 𝐶〉)) | ||
Theorem | strslfvd 12039 | Deduction version of strslfv 12042. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 30-Jan-2023.) |
⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → Fun 𝑆) & ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) ⇒ ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) | ||
Theorem | strslfv2d 12040 | Deduction version of strslfv 12042. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.) |
⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → Fun ◡◡𝑆) & ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) | ||
Theorem | strslfv2 12041 | A variation on strslfv 12042 to avoid asserting that 𝑆 itself is a function, which involves sethood of all the ordered pair components of 𝑆. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.) |
⊢ 𝑆 ∈ V & ⊢ Fun ◡◡𝑆 & ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) & ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 ⇒ ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) | ||
Theorem | strslfv 12042 | Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 with a component extractor 𝐸 (such as the base set extractor df-base 12004). By virtue of ndxslid 12023, this can be done without having to refer to the hard-coded numeric index of 𝐸. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 30-Jan-2023.) |
⊢ 𝑆 Struct 𝑋 & ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) & ⊢ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆 ⇒ ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) | ||
Theorem | strslfv3 12043 | Variant on strslfv 12042 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.) |
⊢ (𝜑 → 𝑈 = 𝑆) & ⊢ 𝑆 Struct 𝑋 & ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) & ⊢ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆 & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ 𝐴 = (𝐸‘𝑈) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | strslssd 12044 | Deduction version of strslss 12045. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.) |
⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) & ⊢ (𝜑 → Fun 𝑇) & ⊢ (𝜑 → 𝑆 ⊆ 𝑇) & ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐸‘𝑇) = (𝐸‘𝑆)) | ||
Theorem | strslss 12045 | Propagate component extraction to a structure 𝑇 from a subset structure 𝑆. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Jim Kingdon, 31-Jan-2023.) |
⊢ 𝑇 ∈ V & ⊢ Fun 𝑇 & ⊢ 𝑆 ⊆ 𝑇 & ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) & ⊢ 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 ⇒ ⊢ (𝐸‘𝑇) = (𝐸‘𝑆) | ||
Theorem | strsl0 12046 | All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 31-Jan-2023.) |
⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) ⇒ ⊢ ∅ = (𝐸‘∅) | ||
Theorem | base0 12047 | The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ ∅ = (Base‘∅) | ||
Theorem | setsslid 12048 | Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.) |
⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) ⇒ ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 = (𝐸‘(𝑊 sSet 〈(𝐸‘ndx), 𝐶〉))) | ||
Theorem | setsslnid 12049 | Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.) |
⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) & ⊢ (𝐸‘ndx) ≠ 𝐷 & ⊢ 𝐷 ∈ ℕ ⇒ ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈𝐷, 𝐶〉))) | ||
Theorem | baseval 12050 | Value of the base set extractor. (Normally it is preferred to work with (Base‘ndx) rather than the hard-coded 1 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.) |
⊢ 𝐾 ∈ V ⇒ ⊢ (Base‘𝐾) = (𝐾‘1) | ||
Theorem | baseid 12051 | Utility theorem: index-independent form of df-base 12004. (Contributed by NM, 20-Oct-2012.) |
⊢ Base = Slot (Base‘ndx) | ||
Theorem | basendx 12052 | Index value of the base set extractor. (Normally it is preferred to work with (Base‘ndx) rather than the hard-coded 1 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.) |
⊢ (Base‘ndx) = 1 | ||
Theorem | basendxnn 12053 | 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.) |
⊢ (Base‘ndx) ∈ ℕ | ||
Theorem | baseslid 12054 | The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.) |
⊢ (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ) | ||
Theorem | basfn 12055 | The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.) |
⊢ Base Fn V | ||
Theorem | reldmress 12056 | The structure restriction is a proper operator, so it can be used with ovprc1 5815. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
⊢ Rel dom ↾s | ||
Theorem | ressid2 12057 | General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 26-Jan-2023.) |
⊢ 𝑅 = (𝑊 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊) | ||
Theorem | ressval2 12058 | Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
⊢ 𝑅 = (𝑊 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉)) | ||
Theorem | ressid 12059 | Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s 𝐵) = 𝑊) | ||
Syntax | cplusg 12060 | Extend class notation with group (addition) operation. |
class +g | ||
Syntax | cmulr 12061 | Extend class notation with ring multiplication. |
class .r | ||
Syntax | cstv 12062 | Extend class notation with involution. |
class *𝑟 | ||
Syntax | csca 12063 | Extend class notation with scalar field. |
class Scalar | ||
Syntax | cvsca 12064 | Extend class notation with scalar product. |
class ·𝑠 | ||
Syntax | cip 12065 | Extend class notation with Hermitian form (inner product). |
class ·𝑖 | ||
Syntax | cts 12066 | Extend class notation with the topology component of a topological space. |
class TopSet | ||
Syntax | cple 12067 | Extend class notation with "less than or equal to" for posets. |
class le | ||
Syntax | coc 12068 | Extend class notation with the class of orthocomplementation extractors. |
class oc | ||
Syntax | cds 12069 | Extend class notation with the metric space distance function. |
class dist | ||
Syntax | cunif 12070 | Extend class notation with the uniform structure. |
class UnifSet | ||
Syntax | chom 12071 | Extend class notation with the hom-set structure. |
class Hom | ||
Syntax | cco 12072 | Extend class notation with the composition operation. |
class comp | ||
Definition | df-plusg 12073 | Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ +g = Slot 2 | ||
Definition | df-mulr 12074 | Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ .r = Slot 3 | ||
Definition | df-starv 12075 | 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 12076 | 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 12077 | Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ ·𝑠 = Slot 6 | ||
Definition | df-ip 12078 | Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ ·𝑖 = Slot 8 | ||
Definition | df-tset 12079 | 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 12080 | 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 12081 | 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 12082 | 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 12083 | Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ UnifSet = Slot ;13 | ||
Definition | df-hom 12084 | Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ Hom = Slot ;14 | ||
Definition | df-cco 12085 | Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ comp = Slot ;15 | ||
Theorem | strleund 12086 | 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 12087 | 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 12088 | 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 12089 | 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 12090 | 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 12091 | Index value of the df-plusg 12073 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (+g‘ndx) = 2 | ||
Theorem | plusgid 12092 | Utility theorem: index-independent form of df-plusg 12073. (Contributed by NM, 20-Oct-2012.) |
⊢ +g = Slot (+g‘ndx) | ||
Theorem | plusgslid 12093 | Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.) |
⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ) | ||
Theorem | opelstrsl 12094 | 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 12095 | 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 12096 | 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 12097 | 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 12098 | 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 12099 | 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 12100 | 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)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |