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Theorem List for Intuitionistic Logic Explorer - 12301-12400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremqnnen 12301 The rational numbers are countably infinite. Corollary 8.1.23 of [AczelRathjen], p. 75. This is Metamath 100 proof #3. (Contributed by Jim Kingdon, 11-Aug-2023.)
ℚ ≈ ℕ
 
Theoremenctlem 12302* Lemma for enct 12303. One direction of the biconditional. (Contributed by Jim Kingdon, 23-Dec-2023.)
(𝐴𝐵 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)))
 
Theoremenct 12303* Countability is invariant relative to equinumerosity. (Contributed by Jim Kingdon, 23-Dec-2023.)
(𝐴𝐵 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)))
 
Theoremctiunctlemu1st 12304* Lemma for ctiunct 12310. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)    &   (𝜑𝐹:𝑆onto𝐴)    &   ((𝜑𝑥𝐴) → 𝑇 ⊆ ω)    &   ((𝜑𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛𝑇)    &   ((𝜑𝑥𝐴) → 𝐺:𝑇onto𝐵)    &   (𝜑𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}    &   (𝜑𝑁𝑈)       (𝜑 → (1st ‘(𝐽𝑁)) ∈ 𝑆)
 
Theoremctiunctlemu2nd 12305* Lemma for ctiunct 12310. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)    &   (𝜑𝐹:𝑆onto𝐴)    &   ((𝜑𝑥𝐴) → 𝑇 ⊆ ω)    &   ((𝜑𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛𝑇)    &   ((𝜑𝑥𝐴) → 𝐺:𝑇onto𝐵)    &   (𝜑𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}    &   (𝜑𝑁𝑈)       (𝜑 → (2nd ‘(𝐽𝑁)) ∈ (𝐹‘(1st ‘(𝐽𝑁))) / 𝑥𝑇)
 
Theoremctiunctlemuom 12306 Lemma for ctiunct 12310. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)    &   (𝜑𝐹:𝑆onto𝐴)    &   ((𝜑𝑥𝐴) → 𝑇 ⊆ ω)    &   ((𝜑𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛𝑇)    &   ((𝜑𝑥𝐴) → 𝐺:𝑇onto𝐵)    &   (𝜑𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}       (𝜑𝑈 ⊆ ω)
 
Theoremctiunctlemudc 12307* Lemma for ctiunct 12310. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)    &   (𝜑𝐹:𝑆onto𝐴)    &   ((𝜑𝑥𝐴) → 𝑇 ⊆ ω)    &   ((𝜑𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛𝑇)    &   ((𝜑𝑥𝐴) → 𝐺:𝑇onto𝐵)    &   (𝜑𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}       (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑈)
 
Theoremctiunctlemf 12308* Lemma for ctiunct 12310. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)    &   (𝜑𝐹:𝑆onto𝐴)    &   ((𝜑𝑥𝐴) → 𝑇 ⊆ ω)    &   ((𝜑𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛𝑇)    &   ((𝜑𝑥𝐴) → 𝐺:𝑇onto𝐵)    &   (𝜑𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}    &   𝐻 = (𝑛𝑈 ↦ ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))))       (𝜑𝐻:𝑈 𝑥𝐴 𝐵)
 
Theoremctiunctlemfo 12309* Lemma for ctiunct 12310. (Contributed by Jim Kingdon, 28-Oct-2023.)
(𝜑𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)    &   (𝜑𝐹:𝑆onto𝐴)    &   ((𝜑𝑥𝐴) → 𝑇 ⊆ ω)    &   ((𝜑𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛𝑇)    &   ((𝜑𝑥𝐴) → 𝐺:𝑇onto𝐵)    &   (𝜑𝐽:ω–1-1-onto→(ω × ω))    &   𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}    &   𝐻 = (𝑛𝑈 ↦ ((𝐹‘(1st ‘(𝐽𝑛))) / 𝑥𝐺‘(2nd ‘(𝐽𝑛))))    &   𝑥𝐻    &   𝑥𝑈       (𝜑𝐻:𝑈onto 𝑥𝐴 𝐵)
 
Theoremctiunct 12310* A sequence of enumerations gives an enumeration of the union. We refer to "sequence of enumerations" rather than "countably many countable sets" because the hypothesis provides more than countability for each 𝐵(𝑥): it refers to 𝐵(𝑥) together with the 𝐺(𝑥) which enumerates it. Theorem 8.1.19 of [AczelRathjen], p. 74.

For "countably many countable sets" the key hypothesis would be (𝜑𝑥𝐴) → ∃𝑔𝑔:ω–onto→(𝐵 ⊔ 1o). This is almost omiunct 12314 (which uses countable choice) although that is for a countably infinite collection not any countable collection.

Compare with the case of two sets instead of countably many, as seen at unct 12312, which says that the union of two countable sets is countable .

The proof proceeds by mapping a natural number to a pair of natural numbers (by xpomen 12265) and using the first number to map to an element 𝑥 of 𝐴 and the second number to map to an element of B(x) . In this way we are able to map to every element of 𝑥𝐴𝐵. Although it would be possible to work directly with countability expressed as 𝐹:ω–onto→(𝐴 ⊔ 1o), we instead use functions from subsets of the natural numbers via ctssdccl 7067 and ctssdc 7069.

(Contributed by Jim Kingdon, 31-Oct-2023.)

(𝜑𝐹:ω–onto→(𝐴 ⊔ 1o))    &   ((𝜑𝑥𝐴) → 𝐺:ω–onto→(𝐵 ⊔ 1o))       (𝜑 → ∃ :ω–onto→( 𝑥𝐴 𝐵 ⊔ 1o))
 
Theoremctiunctal 12311* Variation of ctiunct 12310 which allows 𝑥 to be present in 𝜑. (Contributed by Jim Kingdon, 5-May-2024.)
(𝜑𝐹:ω–onto→(𝐴 ⊔ 1o))    &   (𝜑 → ∀𝑥𝐴 𝐺:ω–onto→(𝐵 ⊔ 1o))       (𝜑 → ∃ :ω–onto→( 𝑥𝐴 𝐵 ⊔ 1o))
 
Theoremunct 12312* The union of two countable sets is countable. Corollary 8.1.20 of [AczelRathjen], p. 75. (Contributed by Jim Kingdon, 1-Nov-2023.)
((∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ∧ ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)) → ∃ :ω–onto→((𝐴𝐵) ⊔ 1o))
 
Theoremomctfn 12313* Using countable choice to find a sequence of enumerations for a collection of countable sets. Lemma 8.1.27 of [AczelRathjen], p. 77. (Contributed by Jim Kingdon, 19-Apr-2024.)
(𝜑CCHOICE)    &   ((𝜑𝑥 ∈ ω) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o))       (𝜑 → ∃𝑓(𝑓 Fn ω ∧ ∀𝑥 ∈ ω (𝑓𝑥):ω–onto→(𝐵 ⊔ 1o)))
 
Theoremomiunct 12314* The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 12310 which has a stronger hypothesis but does not require countable choice. (Contributed by Jim Kingdon, 5-May-2024.)
(𝜑CCHOICE)    &   ((𝜑𝑥 ∈ ω) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o))       (𝜑 → ∃ :ω–onto→( 𝑥 ∈ ω 𝐵 ⊔ 1o))
 
Theoremssomct 12315* A decidable subset of ω is countable. (Contributed by Jim Kingdon, 19-Sep-2024.)
((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω DECID 𝑥𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))
 
Theoremssnnctlemct 12316* Lemma for ssnnct 12317. The result. (Contributed by Jim Kingdon, 29-Sep-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 1)       ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))
 
Theoremssnnct 12317* A decidable subset of is countable. (Contributed by Jim Kingdon, 29-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴) → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))
 
Theoremnnmindc 12318* An inhabited decidable subset of the natural numbers has a minimum. (Contributed by Jim Kingdon, 23-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴 ∧ ∃𝑦 𝑦𝐴) → inf(𝐴, ℝ, < ) ∈ 𝐴)
 
Theoremnnminle 12319* The infimum of a decidable subset of the natural numbers is less than an element of the set. The infimum is also a minimum as shown at nnmindc 12318. (Contributed by Jim Kingdon, 26-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴𝐵𝐴) → inf(𝐴, ℝ, < ) ≤ 𝐵)
 
Theoremnninfdclemcl 12320* Lemma for nninfdc 12325. (Contributed by Jim Kingdon, 25-Sep-2024.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛)    &   (𝜑𝑃𝐴)    &   (𝜑𝑄𝐴)       (𝜑 → (𝑃(𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ‘(𝑦 + 1))), ℝ, < ))𝑄) ∈ 𝐴)
 
Theoremnninfdclemf 12321* Lemma for nninfdc 12325. A function from the natural numbers into 𝐴. (Contributed by Jim Kingdon, 23-Sep-2024.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))       (𝜑𝐹:ℕ⟶𝐴)
 
Theoremnninfdclemp1 12322* Lemma for nninfdc 12325. Each element of the sequence 𝐹 is greater than the previous element. (Contributed by Jim Kingdon, 26-Sep-2024.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))    &   (𝜑𝑈 ∈ ℕ)       (𝜑 → (𝐹𝑈) < (𝐹‘(𝑈 + 1)))
 
Theoremnninfdclemlt 12323* Lemma for nninfdc 12325. The function from nninfdclemf 12321 is strictly monotonic. (Contributed by Jim Kingdon, 24-Sep-2024.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))    &   (𝜑𝑈 ∈ ℕ)    &   (𝜑𝑉 ∈ ℕ)    &   (𝜑𝑈 < 𝑉)       (𝜑 → (𝐹𝑈) < (𝐹𝑉))
 
Theoremnninfdclemf1 12324* Lemma for nninfdc 12325. The function from nninfdclemf 12321 is one-to-one. (Contributed by Jim Kingdon, 23-Sep-2024.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑 → ∀𝑥 ∈ ℕ DECID 𝑥𝐴)    &   (𝜑 → ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛)    &   (𝜑 → (𝐽𝐴 ∧ 1 < 𝐽))    &   𝐹 = seq1((𝑦 ∈ ℕ, 𝑧 ∈ ℕ ↦ inf((𝐴 ∩ (ℤ‘(𝑦 + 1))), ℝ, < )), (𝑖 ∈ ℕ ↦ 𝐽))       (𝜑𝐹:ℕ–1-1𝐴)
 
Theoremnninfdc 12325* An unbounded decidable set of positive integers is infinite. (Contributed by Jim Kingdon, 23-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴 ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → ω ≼ 𝐴)
 
Theoremunbendc 12326* An unbounded decidable set of positive integers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Jim Kingdon, 30-Sep-2024.)
((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥𝐴 ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → 𝐴 ≈ ℕ)
 
PART 6  BASIC STRUCTURES
 
6.1  Extensible structures
 
6.1.1  Basic definitions

An "extensible structure" (or "structure" in short, at least in this section) is used to define a specific group, ring, poset, and so on. An extensible structure can contain many components. For example, a group will have at least two components (base set and operation), although it can be further specialized by adding other components such as a multiplicative operation for rings (and still remain a group per our definition). Thus, every ring is also a group. This extensible structure approach allows theorems from more general structures (such as groups) to be reused for more specialized structures (such as rings) without having to reprove anything. Structures are common in mathematics, but in informal (natural language) proofs the details are assumed in ways that we must make explicit.

An extensible structure is implemented as a function (a set of ordered pairs) on a finite (and not necessarily sequential) subset of . The function's argument is the index of a structure component (such as 1 for the base set of a group), and its value is the component (such as the base set). By convention, we normally avoid direct reference to the hard-coded numeric index and instead use structure component extractors such as ndxid 12355 and strslfv 12375. Using extractors makes it easier to change numeric indices and also makes the components' purpose clearer. See the comment of basendx 12385 for more details on numeric indices versus the structure component extractors.

There are many other possible ways to handle structures. We chose this extensible structure approach because this approach (1) results in simpler notation than other approaches we are aware of, and (2) is easier to do proofs with. We cannot use an approach that uses "hidden" arguments; Metamath does not support hidden arguments, and in any case we want nothing hidden. It would be possible to use a categorical approach (e.g., something vaguely similar to Lean's mathlib). However, instances (the chain of proofs that an 𝑋 is a 𝑌 via a bunch of forgetful functors) can cause serious performance problems for automated tooling, and the resulting proofs would be painful to look at directly (in the case of Lean, they are long past the level where people would find it acceptable to look at them directly). Metamath is working under much stricter conditions than this, and it has still managed to achieve about the same level of flexibility through this "extensible structure" approach.

To create a substructure of a given extensible structure, you can simply use the multifunction restriction operator for extensible structures s as defined in df-ress 12339. This 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.

Extensible structures only work well when they represent concrete categories, where there is a "base set", morphisms are functions, and subobjects are subsets with induced operations. In short, they primarily work well for "sets with (some) extra structure". Extensible structures may not suffice for more complicated situations. For example, in manifolds, s would not work. That said, extensible structures are sufficient for many of the structures that set.mm currently considers, and offer a good compromise for a goal-oriented formalization.

 
Syntaxcstr 12327 Extend class notation with the class of structures with components numbered below 𝐴.
class Struct
 
Syntaxcnx 12328 Extend class notation with the structure component index extractor.
class ndx
 
Syntaxcsts 12329 Set components of a structure.
class sSet
 
Syntaxcslot 12330 Extend class notation with the slot function.
class Slot 𝐴
 
Syntaxcbs 12331 Extend class notation with the class of all base set extractors.
class Base
 
Syntaxcress 12332 Extend class notation with the extensible structure builder restriction operator.
class s
 
Definitiondf-struct 12333* Define a structure with components in 𝑀...𝑁. This is not a requirement for groups, posets, etc., but it is a useful assumption for component extraction theorems.

As mentioned in the section header, an "extensible structure should be implemented as a function (a set of ordered pairs)". The current definition, however, is less restrictive: it allows for classes which contain the empty set to be extensible structures. Because of 0nelfun 5200, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 12344: 𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅}).

Allowing an extensible structure to contain the empty set ensures that expressions like {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} are structures without asserting or implying that 𝐴, 𝐵, 𝐶 and 𝐷 are sets (if 𝐴 or 𝐵 is a proper class, then 𝐴, 𝐵⟩ = ∅, see opprc 3773). (Contributed by Mario Carneiro, 29-Aug-2015.)

Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
 
Definitiondf-ndx 12334 Define the structure component index extractor. See Theorem ndxarg 12354 to understand its purpose. The restriction to 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 or (if we revise all structure component definitions such as df-base 12337) another set such as the set of finite ordinals ω (df-iom 4562). (Contributed by NM, 4-Sep-2011.)
ndx = ( I ↾ ℕ)
 
Definitiondf-slot 12335* 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 𝐴𝑆) is defined to be (𝑆𝐴), 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 , can be used to extract the number 𝐴 from a slot, since (Slot 𝐴‘ndx) = 𝐴 (see ndxarg 12354). 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 ↦ (𝑓𝑥))) 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 5500). 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 ↦ (𝑥𝐴))
 
Theoremsloteq 12336 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)
 
Definitiondf-base 12337 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
 
Definitiondf-sets 12338* 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 12339 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}))
 
Definitiondf-ress 12339* 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))⟩)))
 
Theorembrstruct 12340 The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
Rel Struct
 
Theoremisstruct2im 12341 The property of being a structure with components in (1st𝑋)...(2nd𝑋). (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)))
 
Theoremisstruct2r 12342 The property of being a structure with components in (1st𝑋)...(2nd𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹𝑉 ∧ dom 𝐹 ⊆ (...‘𝑋))) → 𝐹 Struct 𝑋)
 
Theoremstructex 12343 A structure is a set. (Contributed by AV, 10-Nov-2021.)
(𝐺 Struct 𝑋𝐺 ∈ V)
 
Theoremstructn0fun 12344 A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.)
(𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅}))
 
Theoremisstructim 12345 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 ∧ 𝑀𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁)))
 
Theoremisstructr 12346 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 ∧ 𝑀𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 Struct ⟨𝑀, 𝑁⟩)
 
Theoremstructcnvcnv 12347 Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
(𝐹 Struct 𝑋𝐹 = (𝐹 ∖ {∅}))
 
Theoremstructfung 12348 The converse of the converse of a structure is a function. Closed form of structfun 12349. (Contributed by AV, 12-Nov-2021.)
(𝐹 Struct 𝑋 → Fun 𝐹)
 
Theoremstructfun 12349 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
 
Theoremstructfn 12350 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐹 Struct ⟨𝑀, 𝑁       (Fun 𝐹 ∧ dom 𝐹 ⊆ (1...𝑁))
 
Theoremstrnfvnd 12351 Deduction version of strnfvn 12352. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 19-Jan-2023.)
𝐸 = Slot 𝑁    &   (𝜑𝑆𝑉)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝐸𝑆) = (𝑆𝑁))
 
Theoremstrnfvn 12352 Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 12337) 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 12375. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.)

𝑆 ∈ V    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸𝑆) = (𝑆𝑁)
 
Theoremstrfvssn 12353 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    &   (𝜑𝑆𝑉)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝐸𝑆) ⊆ ran 𝑆)
 
Theoremndxarg 12354 Get the numeric argument from a defined structure component extractor such as df-base 12337. (Contributed by Mario Carneiro, 6-Oct-2013.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸‘ndx) = 𝑁
 
Theoremndxid 12355 A structure component extractor is defined by its own index. This theorem, together with strslfv 12375 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 12337, 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)
 
Theoremndxslid 12356 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 12375. (Contributed by Jim Kingdon, 29-Jan-2023.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
 
Theoremslotslfn 12357 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
 
Theoremslotex 12358 Existence of slot value. A corollary of slotslfn 12357. (Contributed by Jim Kingdon, 12-Feb-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)       (𝐴𝑉 → (𝐸𝐴) ∈ V)
 
Theoremstrndxid 12359 The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.)
(𝜑𝑆𝑉)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸𝑆))
 
Theoremreldmsets 12360 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Rel dom sSet
 
Theoremsetsvalg 12361 Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝑆𝑉𝐴𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
 
Theoremsetsvala 12362 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 20-Jan-2023.)
((𝑆𝑉𝐴𝑋𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
 
Theoremsetsex 12363 Applying the structure replacement function yields a set. (Contributed by Jim Kingdon, 22-Jan-2023.)
((𝑆𝑉𝐴𝑋𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V)
 
Theoremstrsetsid 12364 Value of the structure replacement function. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 30-Jan-2023.)
𝐸 = Slot (𝐸‘ndx)    &   (𝜑𝑆 Struct ⟨𝑀, 𝑁⟩)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → (𝐸‘ndx) ∈ dom 𝑆)       (𝜑𝑆 = (𝑆 sSet ⟨(𝐸‘ndx), (𝐸𝑆)⟩))
 
Theoremfvsetsid 12365 The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.)
((𝐹𝑉𝑋𝑊𝑌𝑈) → ((𝐹 sSet ⟨𝑋, 𝑌⟩)‘𝑋) = 𝑌)
 
Theoremsetsfun 12366 A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.)
(((𝐺𝑉 ∧ Fun 𝐺) ∧ (𝐼𝑈𝐸𝑊)) → Fun (𝐺 sSet ⟨𝐼, 𝐸⟩))
 
Theoremsetsfun0 12367 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 12366 is useful for proofs based on isstruct2r 12342 which requires Fun (𝐹 ∖ {∅}) for 𝐹 to be an extensible structure. (Contributed by AV, 7-Jun-2021.)
(((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼𝑈𝐸𝑊)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
 
Theoremsetsn0fun 12368 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.)
(𝜑𝑆 Struct 𝑋)    &   (𝜑𝐼𝑈)    &   (𝜑𝐸𝑊)       (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
 
Theoremsetsresg 12369 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.)
((𝑆𝑉𝐴𝑊𝐵𝑋) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴})))
 
Theoremsetsabsd 12370 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.)
(𝜑𝑆𝑉)    &   (𝜑𝐴𝑊)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑈)       (𝜑 → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))
 
Theoremsetscom 12371 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       (((𝑆𝑉𝐴𝐵) ∧ (𝐶𝑊𝐷𝑋)) → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
 
Theoremstrslfvd 12372 Deduction version of strslfv 12375. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 30-Jan-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)       (𝜑𝐶 = (𝐸𝑆))
 
Theoremstrslfv2d 12373 Deduction version of strslfv 12375. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
(𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)    &   (𝜑𝐶𝑊)       (𝜑𝐶 = (𝐸𝑆))
 
Theoremstrslfv2 12374 A variation on strslfv 12375 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       (𝐶𝑉𝐶 = (𝐸𝑆))
 
Theoremstrslfv 12375 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 12337). By virtue of ndxslid 12356, 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       (𝐶𝑉𝐶 = (𝐸𝑆))
 
Theoremstrslfv3 12376 Variant on strslfv 12375 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.)
(𝜑𝑈 = 𝑆)    &   𝑆 Struct 𝑋    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆    &   (𝜑𝐶𝑉)    &   𝐴 = (𝐸𝑈)       (𝜑𝐴 = 𝐶)
 
Theoremstrslssd 12377 Deduction version of strslss 12378. (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)    &   (𝜑𝑇𝑉)    &   (𝜑 → Fun 𝑇)    &   (𝜑𝑆𝑇)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)       (𝜑 → (𝐸𝑇) = (𝐸𝑆))
 
Theoremstrslss 12378 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    &   𝑆𝑇    &   (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)    &   ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆       (𝐸𝑇) = (𝐸𝑆)
 
Theoremstrsl0 12379 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‘∅)
 
Theorembase0 12380 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
∅ = (Base‘∅)
 
Theoremsetsslid 12381 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)       ((𝑊𝐴𝐶𝑉) → 𝐶 = (𝐸‘(𝑊 sSet ⟨(𝐸‘ndx), 𝐶⟩)))
 
Theoremsetsslnid 12382 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       ((𝑊𝐴𝐶𝑉) → (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩)))
 
Theorembaseval 12383 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)
 
Theorembaseid 12384 Utility theorem: index-independent form of df-base 12337. (Contributed by NM, 20-Oct-2012.)
Base = Slot (Base‘ndx)
 
Theorembasendx 12385 Index value of the base set extractor.

Use of this theorem is discouraged since the particular value 1 for the index is an implementation detail. It is generally sufficient to work with (Base‘ndx) and use theorems such as baseid 12384 and basendxnn 12386.

The main circumstance in which it is necessary to look at indices directly is when showing that a set of indices are disjoint, in proofs such as lmodstrd 12464. Although we have a few theorems such as basendxnplusgndx 12437, we do not intend to add such theorems for every pair of indices (which would be quadradically many in the number of indices).

(New usage is discouraged.) (Contributed by Mario Carneiro, 2-Aug-2013.)

(Base‘ndx) = 1
 
Theorembasendxnn 12386 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
 
Theorembaseslid 12387 The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.)
(Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ)
 
Theorembasfn 12388 The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
Base Fn V
 
Theoremreldmress 12389 The structure restriction is a proper operator, so it can be used with ovprc1 5869. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Rel dom ↾s
 
Theoremressid2 12390 General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 26-Jan-2023.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = 𝑊)
 
Theoremressval2 12391 Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
 
Theoremressid 12392 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝐵 = (Base‘𝑊)       (𝑊𝑋 → (𝑊s 𝐵) = 𝑊)
 
6.1.2  Slot definitions
 
Syntaxcplusg 12393 Extend class notation with group (addition) operation.
class +g
 
Syntaxcmulr 12394 Extend class notation with ring multiplication.
class .r
 
Syntaxcstv 12395 Extend class notation with involution.
class *𝑟
 
Syntaxcsca 12396 Extend class notation with scalar field.
class Scalar
 
Syntaxcvsca 12397 Extend class notation with scalar product.
class ·𝑠
 
Syntaxcip 12398 Extend class notation with Hermitian form (inner product).
class ·𝑖
 
Syntaxcts 12399 Extend class notation with the topology component of a topological space.
class TopSet
 
Syntaxcple 12400 Extend class notation with "less than or equal to" for posets.
class le
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