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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 0thincg 46601 | Any structure with an empty set of objects is a thin category. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ ThinCat) | ||
Theorem | 0thinc 46602 | The empty category (see 0cat 17472) is thin. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ ∅ ∈ ThinCat | ||
Theorem | indthinc 46603* | An indiscrete category in which all hom-sets have exactly one morphism is a thin category. Constructed here is an indiscrete category where all morphisms are ∅. This is a special case of prsthinc 46605, where ≤ = (𝐵 × 𝐵). This theorem also implies a functor from the category of sets to the category of small categories. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof shortened by Zhi Wang, 19-Sep-2024.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶)) & ⊢ (𝜑 → ∅ = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) | ||
Theorem | indthincALT 46604* | An alternate proof for indthinc 46603 assuming more axioms including ax-pow 5302 and ax-un 7629. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶)) & ⊢ (𝜑 → ∅ = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) | ||
Theorem | prsthinc 46605* | Preordered sets as categories. Similar to example 3.3(4.d) of [Adamek] p. 24, but the hom-sets are not pairwise disjoint. One can define a functor from the category of prosets to the category of small thin categories. See catprs 46562 and catprs2 46563 for inducing a preorder from a category. Example 3.26(2) of [Adamek] p. 33 indicates that it induces a bijection from the equivalence class of isomorphic small thin categories to the equivalence class of order-isomorphic preordered sets. (Contributed by Zhi Wang, 18-Sep-2024.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶)) & ⊢ (𝜑 → ∅ = (comp‘𝐶)) & ⊢ (𝜑 → ≤ = (le‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ Proset ) ⇒ ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) | ||
Theorem | setcthin 46606* | A category of sets all of whose objects contain at most one element is thin. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (SetCat‘𝑈)) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 ∃*𝑝 𝑝 ∈ 𝑥) ⇒ ⊢ (𝜑 → 𝐶 ∈ ThinCat) | ||
Theorem | setc2othin 46607 | The category (SetCat‘2o) is thin. A special case of setcthin 46606. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (SetCat‘2o) ∈ ThinCat | ||
Theorem | thincsect 46608 | In a thin category, one morphism is a section of another iff they are pointing towards each other. (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)))) | ||
Theorem | thincsect2 46609 | In a thin category, 𝐹 is a section of 𝐺 iff 𝐺 is a section of 𝐹. (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑆 = (Sect‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ 𝐺(𝑌𝑆𝑋)𝐹)) | ||
Theorem | thincinv 46610 | In a thin category, 𝐹 is an inverse of 𝐺 iff 𝐹 is a section of 𝐺 (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ 𝑁 = (Inv‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐹(𝑋𝑆𝑌)𝐺)) | ||
Theorem | thinciso 46611 | In a thin category, 𝐹:𝑋⟶𝑌 is an isomorphism iff there is a morphism from 𝑌 to 𝑋. (Contributed by Zhi Wang, 25-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐼 = (Iso‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝑌𝐻𝑋) ≠ ∅)) | ||
Theorem | thinccic 46612 | In a thin category, two objects are isomorphic iff there are morphisms between them in both directions. (Contributed by Zhi Wang, 25-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅))) | ||
Syntax | cprstc 46613 | Class function defining preordered sets as categories. |
class ProsetToCat | ||
Definition | df-prstc 46614 |
Definition of the function converting a preordered set to a category.
Justified by prsthinc 46605.
This definition is somewhat arbitrary. Example 3.3(4.d) of [Adamek] p. 24 demonstrates an alternate definition with pairwise disjoint hom-sets. The behavior of the function is defined entirely, up to isomorphism, by prstcnid 46617, prstchom 46628, and prstcthin 46627. Other important properties include prstcbas 46618, prstcleval 46619, prstcle 46621, prstcocval 46622, prstcoc 46624, prstchom2 46629, and prstcprs 46626. Use those instead. Note that the defining property prstchom 46628 is equivalent to prstchom2 46629 given prstcthin 46627. See thincn0eu 46583 for justification. "ProsetToCat" was taken instead of "ProsetCat" because the latter might mean the category of preordered sets (classes). However, "ProsetToCat" seems too long. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet 〈(Hom ‘ndx), ((le‘𝑘) × {1o})〉) sSet 〈(comp‘ndx), ∅〉)) | ||
Theorem | prstcval 46615 | Lemma for prstcnidlem 46616 and prstcthin 46627. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) ⇒ ⊢ (𝜑 → 𝐶 = ((𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉) sSet 〈(comp‘ndx), ∅〉)) | ||
Theorem | prstcnidlem 46616 | Lemma for prstcnid 46617 and prstchomval 46625. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (comp‘ndx) ⇒ ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉))) | ||
Theorem | prstcnid 46617 | Components other than Hom and comp are unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (comp‘ndx) & ⊢ (𝐸‘ndx) ≠ (Hom ‘ndx) ⇒ ⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶)) | ||
Theorem | prstcbas 46618 | The base set is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | ||
Theorem | prstcleval 46619 | Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐾)) ⇒ ⊢ (𝜑 → ≤ = (le‘𝐶)) | ||
Theorem | prstclevalOLD 46620 | Obsolete proof of prstcleval 46619 as of 12-Nov-2024. Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐾)) ⇒ ⊢ (𝜑 → ≤ = (le‘𝐶)) | ||
Theorem | prstcle 46621 | Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐾)) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ 𝑋(le‘𝐶)𝑌)) | ||
Theorem | prstcocval 46622 | Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ⊥ = (oc‘𝐾)) ⇒ ⊢ (𝜑 → ⊥ = (oc‘𝐶)) | ||
Theorem | prstcocvalOLD 46623 | Obsolete proof of prstcocval 46622 as of 12-Nov-2024. Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ⊥ = (oc‘𝐾)) ⇒ ⊢ (𝜑 → ⊥ = (oc‘𝐶)) | ||
Theorem | prstcoc 46624 | Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ⊥ = (oc‘𝐾)) ⇒ ⊢ (𝜑 → ( ⊥ ‘𝑋) = ((oc‘𝐶)‘𝑋)) | ||
Theorem | prstchomval 46625 | Hom-sets of the constructed category which depend on an arbitrary definition. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐶)) ⇒ ⊢ (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶)) | ||
Theorem | prstcprs 46626 | The category is a preordered set. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) ⇒ ⊢ (𝜑 → 𝐶 ∈ Proset ) | ||
Theorem | prstcthin 46627 | The preordered set is equipped with a thin category. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) ⇒ ⊢ (𝜑 → 𝐶 ∈ ThinCat) | ||
Theorem | prstchom 46628 |
Hom-sets of the constructed category are dependent on the preorder.
Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat. However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)) | ||
Theorem | prstchom2 46629* |
Hom-sets of the constructed category are dependent on the preorder.
Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat ( see prstchom2ALT 46630). However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 21-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) | ||
Theorem | prstchom2ALT 46630* | Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 46614. See prstchom2 46629 for a version that does not depend on the definition. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) | ||
Theorem | postcpos 46631 | The converted category is a poset iff the original proset is a poset. (Contributed by Zhi Wang, 26-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) ⇒ ⊢ (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset)) | ||
Theorem | postcposALT 46632 | Alternate proof for postcpos 46631. (Contributed by Zhi Wang, 25-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) ⇒ ⊢ (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset)) | ||
Theorem | postc 46633* | The converted category is a poset iff no distinct objects are isomorphic. (Contributed by Zhi Wang, 25-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝜑 → (𝐶 ∈ Poset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥( ≃𝑐 ‘𝐶)𝑦 → 𝑥 = 𝑦))) | ||
Syntax | cmndtc 46634 | Class function defining monoids as categories. |
class MndToCat | ||
Definition | df-mndtc 46635 |
Definition of the function converting a monoid to a category. Example
3.3(4.e) of [Adamek] p. 24.
The definition of the base set is arbitrary. The whole extensible structure becomes the object here (see mndtcbasval 46637) , instead of just the base set, as is the case in Example 3.3(4.e) of [Adamek] p. 24. The resulting category is defined entirely, up to isomorphism, by mndtcbas 46638, mndtchom 46641, mndtcco 46642. Use those instead. See example 3.26(3) of [Adamek] p. 33 for more on isomorphism. "MndToCat" was taken instead of "MndCat" because the latter might mean the category of monoids. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
⊢ MndToCat = (𝑚 ∈ Mnd ↦ {〈(Base‘ndx), {𝑚}〉, 〈(Hom ‘ndx), {〈𝑚, 𝑚, (Base‘𝑚)〉}〉, 〈(comp‘ndx), {〈〈𝑚, 𝑚, 𝑚〉, (+g‘𝑚)〉}〉}) | ||
Theorem | mndtcval 46636 | Value of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) ⇒ ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), {𝑀}〉, 〈(Hom ‘ndx), {〈𝑀, 𝑀, (Base‘𝑀)〉}〉, 〈(comp‘ndx), {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}〉}) | ||
Theorem | mndtcbasval 46637 | The base set of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) ⇒ ⊢ (𝜑 → 𝐵 = {𝑀}) | ||
Theorem | mndtcbas 46638* | The category built from a monoid contains precisely one object. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) ⇒ ⊢ (𝜑 → ∃!𝑥 𝑥 ∈ 𝐵) | ||
Theorem | mndtcob 46639 | Lemma for mndtchom 46641 and mndtcco 46642. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 = 𝑀) | ||
Theorem | mndtcbas2 46640 | Two objects in a category built from a monoid are identical. (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 = 𝑌) | ||
Theorem | mndtchom 46641 | The only hom-set of the category built from a monoid is the base set of the monoid. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (Base‘𝑀)) | ||
Theorem | mndtcco 46642 | The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → · = (comp‘𝐶)) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (+g‘𝑀)) | ||
Theorem | mndtcco2 46643 | The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → ⚬ = (〈𝑋, 𝑌〉 · 𝑍)) ⇒ ⊢ (𝜑 → (𝐺 ⚬ 𝐹) = (𝐺(+g‘𝑀)𝐹)) | ||
Theorem | mndtccatid 46644* | Lemma for mndtccat 46645 and mndtcid 46646. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) ⇒ ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ (0g‘𝑀)))) | ||
Theorem | mndtccat 46645 | The function value is a category. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
Theorem | mndtcid 46646 | The identity morphism, or identity arrow, of the category built from a monoid is the identity element of the monoid. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 1 = (Id‘𝐶)) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = (0g‘𝑀)) | ||
Theorem | grptcmon 46647 | All morphisms in a category converted from a group are monomorphisms. (Contributed by Zhi Wang, 23-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺)) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝑀 = (Mono‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌)) | ||
Theorem | grptcepi 46648 | All morphisms in a category converted from a group are epimorphisms. (Contributed by Zhi Wang, 23-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺)) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝐸 = (Epi‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌)) | ||
Some of these theorems are used in the series of lemmas and theorems proving the defining properties of setrecs. | ||
Theorem | nfintd 46649 | Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴) | ||
Theorem | nfiund 46650* | Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2370. See nfiundg 46651 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝐴) & ⊢ (𝜑 → Ⅎ𝑦𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | nfiundg 46651 | Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2370, see nfiund 46650 for a weaker version that does not require it. (Contributed by Emmett Weisz, 6-Dec-2019.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝐴) & ⊢ (𝜑 → Ⅎ𝑦𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | iunord 46652* | The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7670, but does not use it directly, since ssorduni 7670 does not work when 𝐵 is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.) |
⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Ord ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | iunordi 46653* | The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. (Contributed by Emmett Weisz, 3-Nov-2019.) |
⊢ Ord 𝐵 ⇒ ⊢ Ord ∪ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | spd 46654 | Specialization deduction, using implicit substitution. Based on the proof of spimed 2386. (Contributed by Emmett Weisz, 17-Jan-2020.) |
⊢ (𝜒 → Ⅎ𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝜒 → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | spcdvw 46655* | A version of spcdv 3541 where 𝜓 and 𝜒 are direct substitutions of each other. This theorem is useful because it does not require 𝜑 and 𝑥 to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
Theorem | tfis2d 46656* | Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.) |
⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) & ⊢ (𝜑 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜒 → 𝜓))) ⇒ ⊢ (𝜑 → (𝑥 ∈ On → 𝜓)) | ||
Theorem | bnd2d 46657* | Deduction form of bnd2 9728. (Contributed by Emmett Weisz, 19-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) ⇒ ⊢ (𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜓)) | ||
Theorem | dffun3f 46658* | Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑧𝐴 ⇒ ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) | ||
Symbols in this section: All the symbols used in the definition of setrecs(𝐹) are explained in the comment of df-setrecs 46660. The class 𝑌 is explained in the comment of setrec1lem1 46663. Glossaries of symbols used in individual proofs, or used differently in different proofs, are in the comments of those proofs. | ||
Syntax | csetrecs 46659 | Extend class notation to include a set defined by transfinite recursion. |
class setrecs(𝐹) | ||
Definition | df-setrecs 46660* |
Define a class setrecs(𝐹) by transfinite recursion, where
(𝐹‘𝑥) is the set of new elements to add to
the class given the
set 𝑥 of elements in the class so far. We
do not need a base case,
because we can start with the empty set, which is vacuously a subset of
setrecs(𝐹). The goal of this definition is to
construct a class
fulfilling Theorems setrec1 46667 and setrec2v 46672, which give a more
intuitive idea of the meaning of setrecs.
Unlike wrecs,
setrecs is well-defined for any 𝐹 and
meaningful for any
function 𝐹.
For example, see Theorem onsetrec 46683 for how the class On is defined recursively using the successor function. The definition works by building subsets of the desired class and taking the union of those subsets. To find such a collection of subsets, consider an arbitrary set 𝑧, and consider the result when applying 𝐹 to any subset 𝑤 ⊆ 𝑧. Remember that 𝐹 can be any function, and in general we are interested in functions that give outputs that are larger than their inputs, so we have no reason to expect the outputs to be within 𝑧. However, if we restrict the domain of 𝐹 to a given set 𝑦, the resulting range will be a set. Therefore, with this restricted 𝐹, it makes sense to consider sets 𝑧 that are closed under 𝐹 applied to its subsets. Now we can test whether a given set 𝑦 is recursively generated by 𝐹. If every set 𝑧 that is closed under 𝐹 contains 𝑦, that means that every member of 𝑦 must eventually be generated by 𝐹. On the other hand, if some such 𝑧 does not contain a certain element of 𝑦, then that element can be avoided even if we apply 𝐹 in every possible way to previously generated elements. Note that such an omitted element might be eventually recursively generated by 𝐹, but not through the elements of 𝑦. In this case, 𝑦 would fail the condition in the definition, but the omitted element would still be included in some larger 𝑦. For example, if 𝐹 is the successor function, the set {∅, 2o} would fail the condition since 2o is not an element of the successor of ∅ or {∅}. Remember that we are applying 𝐹 to subsets of 𝑦, not elements of 𝑦. In fact, even the set {1o} fails the condition, since the only subset of previously generated elements is ∅, and suc ∅ does not have 1o as an element. However, we can let 𝑦 be any ordinal, since each of its elements is generated by starting with ∅ and repeatedly applying the successor function. A similar definition I initially used for setrecs(𝐹) was setrecs(𝐹) = ∪ ran recs((𝑔 ∈ V ↦ (𝐹‘∪ ran 𝑔))). I had initially tried and failed to find an elementary definition, and I had proven theorems analogous to setrec1 46667 and setrec2v 46672 using the old definition before I found the new one. I decided to change definitions for two reasons. First, as John Horton Conway noted in the Appendix to Part Zero of On Numbers and Games, mathematicians should not be caught up in any particular formalization, such as ZF set theory. Instead, they should work under whatever framework best suits the problem, and the formal bases used for different problems can be shown to be equivalent. Thus, Conway preferred defining surreal numbers as equivalence classes of surreal number forms, rather than sign-expansions. Although sign-expansions are easier to implement in ZF set theory, Conway argued that "formalisation within some particular axiomatic set theory is irrelevant". Furthermore, one of the most remarkable properties of the theory of surreal numbers is that it generates so much from almost nothing. Using sign-expansions as the formal definition destroys the beauty of surreal numbers, because ordinals are already built in. For this reason, I replaced the old definition of setrecs, which also relied heavily on ordinal numbers. On the other hand, both surreal numbers and the elementary definition of setrecs immediately generate the ordinal numbers from a (relatively) very simple set-theoretical basis. Second, although it is still complicated to formalize the theory of recursively generated sets within ZF set theory, it is actually simpler and more natural to do so with set theory directly than with the theory of ordinal numbers. As Conway wrote, indexing the "birthdays" of sets is and should be unnecessary. Using an elementary definition for setrecs removes the reliance on the previously developed theory of ordinal numbers, allowing proofs to be simpler and more direct. Formalizing surreal numbers within Metamath is probably still not in the spirit of Conway. He said that "attempts to force arbitrary theories into a single formal straitjacket... produce unnecessarily cumbrous and inelegant contortions." Nevertheless, Metamath has proven to be much more versatile than it seems at first, and I think the theory of surreal numbers can be natural while fitting well into the Metamath framework. The difficulty in writing a definition in Metamath for setrecs(𝐹) is that the necessary properties to prove are self-referential (see setrec1 46667 and setrec2v 46672), so we cannot simply write the properties we want inside a class abstraction as with most definitions. As noted in the comment of df-rdg 8289, this is not actually a requirement of the Metamath language, but we would like to be able to eliminate all definitions by direct mechanical substitution. We cannot define setrecs using a class abstraction directly, because nothing about its individual elements tells us whether they are in the set. We need to know about previous elements first. One way of getting around this problem without indexing is by defining setrecs(𝐹) as a union or intersection of suitable sets. Thus, instead of using a class abstraction for the elements of setrecs(𝐹), which seems to be impossible, we can use a class abstraction for supersets or subsets of setrecs(𝐹), which "know" about multiple individual elements at a time. Note that we cannot define setrecs(𝐹) as an intersection of sets, because in general it is a proper class, so any supersets would also be proper classes. However, a proper class can be a union of sets, as long as the collection of such sets is a proper class. Therefore, it is feasible to define setrecs(𝐹) as a union of a class abstraction. If setrecs(𝐹) = ∪ 𝐴, the elements of A must be subsets of setrecs(𝐹) which together include everything recursively generated by 𝐹. We can do this by letting 𝐴 be the class of sets 𝑥 whose elements are all recursively generated by 𝐹. One necessary condition is that each element of a given 𝑥 ∈ 𝐴 must be generated by 𝐹 when applied to a previous element 𝑦 ∈ 𝐴. In symbols, ∀𝑥 ∈ 𝐴∃𝑦 ∈ 𝐴(𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐹‘𝑦))}. However, this is not sufficient. All fixed points 𝑥 of 𝐹 will satisfy this condition whether they should be in setrecs(𝐹) or not. If we replace the subset relation with the proper subset relation, 𝑥 cannot be the empty set, even though the empty set should be in 𝐴. Therefore this condition cannot be used in the definition, even if we can find a way to avoid making it circular. A better strategy is to find a necessary and sufficient condition for all the elements of a set 𝑦 ∈ 𝐴 to be generated by 𝐹 when applied only to sets of previously generated elements within 𝑦. For example, taking 𝐹 to be the successor function, we can let 𝐴 = On rather than 𝒫 On, and we will still have ∪ 𝐴 = On as required. This gets rid of the circularity of the definition, since we should have a condition to test whether a given set 𝑦 is in 𝐴 without knowing about any of the other elements of 𝐴. The definition I ended up using accomplishes this using induction: 𝐴 is defined as the class of sets 𝑦 for which a sort of induction on the elements of 𝑦 holds. However, when creating a definition for setrecs that did not rely on ordinal numbers, I tried at first to write a definition using the well-founded relation predicate, Fr. I thought that this would be simple to do once I found a suitable definition using induction, just as the least- element principle is equivalent to induction on the positive integers. If we let 𝑅 = {〈𝑎, 𝑏〉 ∣ (𝐹‘𝑎) ⊆ 𝑏}, then (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥∀𝑧 ∈ 𝑥¬ (𝐹‘𝑧) ⊆ 𝑦)). On 22-Jul-2020 I came up with the following definition (Version 1) phrased in terms of induction: ∪ {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ∈ 𝑧 → (𝐹‘𝑤) ∈ 𝑧)) → 𝑦 ∈ 𝑧)} In Aug-2020 I came up with an equivalent definition with the goal of phrasing it in terms of the relation Fr. It is the contrapositive of the previous one with 𝑧 replaced by its complement. ∪ {𝑦 ∣ ∀𝑧 (𝑦 ∈ 𝑧 → ∃𝑤(𝑤 ⊆ 𝑦 ∧ (𝐹‘𝑤) ∈ 𝑧 ∧ ¬ 𝑤 ∈ 𝑧))} These definitions didn't work because the induction didn't "get off the ground." If 𝑧 does not contain the empty set, the condition (∀𝑤...𝑦 ∈ 𝑧 fails, so 𝑦 = ∅ doesn't get included in 𝐴 even though it should. This could be fixed by adding the base case as a separate requirement, but the subtler problem would remain that rather than a set of "acceptable" sets, what we really need is a collection 𝑧 of all individuals that have been generated so far. So one approach is to replace every occurrence of ∈ 𝑧 with ⊆ 𝑧, making 𝑧 a set of individuals rather than a family of sets. That solves this problem, but it complicates the foundedness version of the definition, which looked cleaner in Version 1. There was another problem with Version 1. If we let 𝐹 be the power set function, then the induction in the inductive version works for 𝑧 being the class of transitive sets, restricted to subsets of 𝑦. Therefore, 𝑦 must be transitive by definition of 𝑧. This doesn't affect the union of all such 𝑦, but it may or may not be desirable. The problem is that 𝐹 is only applied to transitive sets, because of the strong requirement 𝑤 ∈ 𝑧, so the definition requires the additional constraint (𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) in order to work. This issue can also be avoided by replacing ∈ 𝑧 with ⊆ 𝑧. The induction version of the result is used in the final definition. Version 2: (18-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧(𝑦 ∩ 𝑧 ≠ ∅ → ∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∩ 𝑧 ≠ ∅))} In the induction version, not only does 𝑧 include all the elements of 𝑦, but it must include the elements of (𝐹‘𝑤) for 𝑤 ⊆ (𝑦 ∩ 𝑧) even if those elements of (𝐹‘𝑤) are not in 𝑦. We shouldn't care about any of the elements of 𝑧 outside 𝑦, but this detail doesn't affect the correctness of the definition. If we replaced (𝐹‘𝑤) in the definition by ((𝐹‘𝑤) ∩ 𝑦), we would get the same class for setrecs(𝐹). Suppose we could find a 𝑧 for which the condition fails for a given 𝑦 under the changed definition. Then the antecedent would be true, but 𝑦 ⊆ 𝑧 would be false. We could then simply add all elements of (𝐹‘𝑤) outside of 𝑦 for any 𝑤 ⊆ 𝑦, which we can do because all the classes involved are sets. This is not trivial and requires the axioms of union, power set, and replacement. However, the expanded 𝑧 fails the condition under the Metamath definition. The other direction is easier. If a certain 𝑧 fails the Metamath definition, then all (𝐹‘𝑤) ⊆ 𝑧 for 𝑤 ⊆ (𝑦 ∩ 𝑧), and in particular ((𝐹‘𝑤) ∩ 𝑦) ⊆ 𝑧. The foundedness version is starting to look more like ax-reg 9427! We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝑦 in the foundedness definition. Furthermore, instead of quantifying over 𝑤, quantify over the elements 𝑣 ∈ 𝑧 overlapping with 𝑤. Versions 3, 4, and 5 are all equivalent to Version 2. Version 3 - Foundedness (5-Sep-2020): ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑣 ∈ 𝑧∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ 𝑧 = ∅ ∧ 𝑣 ∈ (𝐹‘𝑤)))} Now, if we replace (𝐹‘𝑤) by ((𝐹‘𝑤) ∩ 𝑦), we do not change the definition. We already know that 𝑣 ∈ 𝑦 since 𝑣 ∈ 𝑧 and 𝑧 ⊆ 𝑦. All we need to show in order to prove that this change leads to an equivalent definition is to find To make our definition look exactly like df-fr 5562, we add another variable 𝑢 representing the nonexistent element of 𝑤 in 𝑧. Version 4 - Foundedness (6-Sep-2020): ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑣 ∈ 𝑧∃𝑤∀𝑢 ∈ 𝑧(𝑤 ⊆ 𝑦 ∧ ¬ 𝑢 ∈ 𝑤 ∧ 𝑣 ∈ (𝐹‘𝑤)) This is so close to df-fr 5562; the only change needed is to switch ∃𝑤 with ∀𝑢 ∈ 𝑧. Unfortunately, I couldn't find any way to switch the quantifiers without interfering with the definition. Maybe there is a definition equivalent to this one that uses Fr, but I couldn't find one. Yet, we can still find a remarkable similarity between Foundedness Version 2 and ax-reg 9427. Rather than a disjoint element of 𝑧, there's a disjoint coverer of an element of 𝑧. Finally, here's a different dead end I followed: To clean up our foundedness definition, we keep 𝑧 as a family of sets 𝑦 but allow 𝑤 to be any subset of ∪ 𝑧 in the induction. With this stronger induction, we can also allow for the stronger requirement 𝒫 𝑦 ⊆ 𝑧 rather than only 𝑦 ∈ 𝑧. This will help improve the foundedness version. Version 1.1 (28-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤 ⊆ 𝑦 → (𝑤 ⊆ ∪ 𝑧 → (𝐹‘𝑤) ∈ 𝑧)) → 𝒫 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧(∃𝑎(𝑎 ⊆ 𝑦 ∧ 𝑎 ∈ 𝑧) → ∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∈ 𝑧))} ( Edit (Aug 31) - this isn't true! Nothing forces the subset of an element of 𝑧 to be in 𝑧. Version 2 does not have this issue. ) Similarly, we could allow 𝑤 to be any subset of any element of 𝑧 rather than any subset of ∪ 𝑧. I think this has the same problem. We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝒫 𝑦 in the foundedness definition: Version 1.2 (31-Aug-2020) Foundedness: ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∈ 𝑧))} Now this looks more like df-fr 5562! The last step necessary to be able to use Fr directly in our definition is to replace (𝐹‘𝑤) with its own setvar variable, corresponding to 𝑦 in df-fr 5562. This definition is incorrect, though, since there's nothing forcing the subset of an element of 𝑧 to be in 𝑧. Version 1.3 (31-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ ∪ 𝑧 → (𝑤 ∈ 𝑧 ∧ (𝐹‘𝑤) ∈ 𝑧))) → 𝒫 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝑤 ∈ 𝑧 ∨ (𝐹‘𝑤) ∈ 𝑧)))} 𝑧 must contain the supersets of each of its elements in the foundedness version, and we can't make any restrictions on 𝑧 or 𝐹, so this doesn't work. Let's try letting R be the covering relation 𝑅 = {〈𝑎, 𝑏〉 ∣ 𝑏 ∈ (𝐹‘𝑎)} to solve the transitivity issue (i.e. that if 𝐹 is the power set relation, 𝐴 consists only of transitive sets). The set (𝐹‘𝑤) corresponds to the variable 𝑦 in df-fr 5562. Thus, in our case, df-fr 5562 is equivalent to (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑤((𝐹‘𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣 ∈ 𝑧𝑣𝑅(𝐹‘𝑤))). Substituting our relation 𝑅 gives (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑤((𝐹‘𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣 ∈ 𝑧(𝐹‘𝑤) ∈ (𝐹‘𝑣))) This doesn't work for non-injective 𝐹 because we need all 𝑧 to be straddlers, but we don't necessarily need all-straddlers; loops within z are fine for non-injective F. Consider the foundedness form of Version 1. We want to show ¬ 𝑤 ∈ 𝑧 ↔ ∀𝑣 ∈ 𝑧¬ 𝑣𝑅(𝐹‘𝑤) so we can replace one with the other. Negate both sides: 𝑤 ∈ 𝑧 ↔ ∃𝑣 ∈ 𝑧𝑣𝑅(𝐹‘𝑤) If 𝐹 is injective, then we should be able to pick a suitable R, being careful about the above problem for some F (for example z = transitivity) when changing the antecedent y e. z' to z =/= (/). If we're clever, we can get rid of the injectivity requirement. The forward direction of the above equivalence always holds, but the key is that although the backwards direction doesn't hold in general, we can always find some z' where it doesn't work for 𝑤 itself. If there exists a z' where the version with the w condition fails, then there exists a z' where the version with the v condition also fails. However, Version 1 is not a correct definition, so this doesn't work either. (Contributed by Emmett Weisz, 18-Aug-2020.) (New usage is discouraged.) |
⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} | ||
Theorem | setrecseq 46661 | Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.) |
⊢ (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺)) | ||
Theorem | nfsetrecs 46662 | Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.) |
⊢ Ⅎ𝑥𝐹 ⇒ ⊢ Ⅎ𝑥setrecs(𝐹) | ||
Theorem | setrec1lem1 46663* |
Lemma for setrec1 46667. This is a utility theorem showing the
equivalence
of the statement 𝑋 ∈ 𝑌 and its expanded form. The proof
uses
elabg 3616 and equivalence theorems.
Variable 𝑌 is the class of sets 𝑦 that are recursively generated by the function 𝐹. In other words, 𝑦 ∈ 𝑌 iff by starting with the empty set and repeatedly applying 𝐹 to subsets 𝑤 of our set, we will eventually generate all the elements of 𝑌. In this theorem, 𝑋 is any element of 𝑌, and 𝑉 is any class. (Contributed by Emmett Weisz, 16-Oct-2020.) (New usage is discouraged.) |
⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))) | ||
Theorem | setrec1lem2 46664* | Lemma for setrec1 46667. If a family of sets are all recursively generated by 𝐹, so is their union. In this theorem, 𝑋 is a family of sets which are all elements of 𝑌, and 𝑉 is any class. Use dfss3 3918, equivalence and equality theorems, and unissb at the end. Sandwich with applications of setrec1lem1. (Contributed by Emmett Weisz, 24-Jan-2021.) (New usage is discouraged.) |
⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ⊆ 𝑌) ⇒ ⊢ (𝜑 → ∪ 𝑋 ∈ 𝑌) | ||
Theorem | setrec1lem3 46665* | Lemma for setrec1 46667. If each element 𝑎 of 𝐴 is covered by a set 𝑥 recursively generated by 𝐹, then there is a single such set covering all of 𝐴. The set is constructed explicitly using setrec1lem2 46664. It turns out that 𝑥 = 𝐴 also works, i.e., given the hypotheses it is possible to prove that 𝐴 ∈ 𝑌. I don't know if proving this fact directly using setrec1lem1 46663 would be any easier than the current proof using setrec1lem2 46664, and it would only slightly simplify the proof of setrec1 46667. Other than the use of bnd2d 46657, this is a purely technical theorem for rearranging notation from that of setrec1lem2 46664 to that of setrec1 46667. (Contributed by Emmett Weisz, 20-Jan-2021.) (New usage is discouraged.) |
⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∃𝑥(𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝑌)) ⇒ ⊢ (𝜑 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ 𝑌)) | ||
Theorem | setrec1lem4 46666* |
Lemma for setrec1 46667. If 𝑋 is recursively generated by 𝐹, then
so is 𝑋 ∪ (𝐹‘𝐴).
In the proof of setrec1 46667, the following is substituted for this theorem's 𝜑: (𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})) Therefore, we cannot declare 𝑧 to be a distinct variable from 𝜑, since we need it to appear as a bound variable in 𝜑. This theorem can be proven without the hypothesis Ⅎ𝑧𝜑, but the proof would be harder to read because theorems in deduction form would be interrupted by theorems like eximi 1836, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 46664 could similarly be made easier to read by adding the hypothesis Ⅎ𝑧𝜑, but I had already finished the proof and decided to leave it as is. (Contributed by Emmett Weisz, 26-Nov-2020.) (New usage is discouraged.) |
⊢ Ⅎ𝑧𝜑 & ⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → 𝑋 ∈ 𝑌) ⇒ ⊢ (𝜑 → (𝑋 ∪ (𝐹‘𝐴)) ∈ 𝑌) | ||
Theorem | setrec1 46667 |
This is the first of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs(𝐹) is closed under 𝐹. This
effectively sets the
actual value of setrecs(𝐹) as a lower bound for
setrecs(𝐹), as it implies that any set
generated by successive
applications of 𝐹 is a member of 𝐵. This
theorem "gets off the
ground" because we can start by letting 𝐴 = ∅, and the
hypotheses
of the theorem will hold trivially.
Variable 𝐵 represents an abbreviation of setrecs(𝐹) or another name of setrecs(𝐹) (for an example of the latter, see theorem setrecon). Proof summary: Assume that 𝐴 ⊆ 𝐵, meaning that all elements of 𝐴 are in some set recursively generated by 𝐹. Then by setrec1lem3 46665, 𝐴 is a subset of some set recursively generated by 𝐹. (It turns out that 𝐴 itself is recursively generated by 𝐹, but we don't need this fact. See the comment to setrec1lem3 46665.) Therefore, by setrec1lem4 46666, (𝐹‘𝐴) is a subset of some set recursively generated by 𝐹. Thus, by ssuni 4877, it is a subset of the union of all sets recursively generated by 𝐹. See df-setrecs 46660 for a detailed description of how the setrecs definition works. (Contributed by Emmett Weisz, 9-Oct-2020.) |
⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) ⊆ 𝐵) | ||
Theorem | setrec2fun 46668* |
This is the second of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs(𝐹) is a subclass of all classes 𝐶 that
are closed
under 𝐹. Taken together, Theorems setrec1 46667 and setrec2v 46672 say
that setrecs(𝐹) is the minimal class closed under
𝐹.
We express this by saying that if 𝐹 respects the ⊆ relation and 𝐶 is closed under 𝐹, then 𝐵 ⊆ 𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7745) to the other class. (Contributed by Emmett Weisz, 15-Feb-2021.) (New usage is discouraged.) |
⊢ Ⅎ𝑎𝐹 & ⊢ 𝐵 = setrecs(𝐹) & ⊢ Fun 𝐹 & ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
Theorem | setrec2lem1 46669* | Lemma for setrec2 46671. The functional part of 𝐹 has the same values as 𝐹. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.) |
⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎) | ||
Theorem | setrec2lem2 46670* | Lemma for setrec2 46671. The functional part of 𝐹 is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.) |
⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) | ||
Theorem | setrec2 46671* |
This is the second of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs(𝐹) is a subclass of all classes 𝐶 that
are closed
under 𝐹. Taken together, Theorems setrec1 46667 and setrec2v 46672
uniquely determine setrecs(𝐹) to be the minimal class closed
under 𝐹.
We express this by saying that if 𝐹 respects the ⊆ relation and 𝐶 is closed under 𝐹, then 𝐵 ⊆ 𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7745) to the other class. (Contributed by Emmett Weisz, 2-Sep-2021.) |
⊢ Ⅎ𝑎𝐹 & ⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
Theorem | setrec2v 46672* | Version of setrec2 46671 with a disjoint variable condition instead of a nonfreeness hypothesis. (Contributed by Emmett Weisz, 6-Mar-2021.) |
⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
Theorem | setis 46673* | Version of setrec2 46671 expressed as an induction schema. This theorem is a generalization of tfis3 7750. (Contributed by Emmett Weisz, 27-Feb-2022.) |
⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝑏 = 𝐴 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → ∀𝑎(∀𝑏 ∈ 𝑎 𝜓 → ∀𝑏 ∈ (𝐹‘𝑎)𝜓)) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝜒)) | ||
Theorem | elsetrecslem 46674* | Lemma for elsetrecs 46675. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 46672. To see why this lemma also requires setrec1 46667, consider what would happen if we replaced 𝐵 with {𝐴}. The antecedent would still hold, but the consequent would fail in general. Consider dispensing with the deduction form. (Contributed by Emmett Weisz, 11-Jul-2021.) (New usage is discouraged.) |
⊢ 𝐵 = setrecs(𝐹) ⇒ ⊢ (𝐴 ∈ 𝐵 → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) | ||
Theorem | elsetrecs 46675* | A set 𝐴 is an element of setrecs(𝐹) iff 𝐴 is generated by some subset of setrecs(𝐹). The proof requires both setrec1 46667 and setrec2 46671, but this theorem is not strong enough to uniquely determine setrecs(𝐹). If 𝐹 respects the subset relation, the theorem still holds if both occurrences of ∈ are replaced by ⊆ for a stronger version of the theorem. (Contributed by Emmett Weisz, 12-Jul-2021.) |
⊢ 𝐵 = setrecs(𝐹) ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) | ||
Theorem | setrecsss 46676 | The setrecs operator respects the subset relation between two functions 𝐹 and 𝐺. (Contributed by Emmett Weisz, 13-Mar-2022.) |
⊢ (𝜑 → Fun 𝐺) & ⊢ (𝜑 → 𝐹 ⊆ 𝐺) ⇒ ⊢ (𝜑 → setrecs(𝐹) ⊆ setrecs(𝐺)) | ||
Theorem | setrecsres 46677 | A recursively generated class is unaffected when its input function is restricted to subsets of the class. (Contributed by Emmett Weisz, 14-Mar-2022.) |
⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → 𝐵 = setrecs((𝐹 ↾ 𝒫 𝐵))) | ||
Theorem | vsetrec 46678 | Construct V using set recursion. The proof indirectly uses trcl 9563, which relies on rec, but theoretically 𝐶 in trcl 9563 could be constructed using setrecs instead. The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable requirement between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 23-Jun-2021.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥) ⇒ ⊢ setrecs(𝐹) = V | ||
Theorem | 0setrec 46679 | If a function sends the empty set to itself, the function will not recursively generate any sets, regardless of its other values. (Contributed by Emmett Weisz, 23-Jun-2021.) |
⊢ (𝜑 → (𝐹‘∅) = ∅) ⇒ ⊢ (𝜑 → setrecs(𝐹) = ∅) | ||
Theorem | onsetreclem1 46680* | Lemma for onsetrec 46683. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) ⇒ ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} | ||
Theorem | onsetreclem2 46681* | Lemma for onsetrec 46683. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) ⇒ ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On) | ||
Theorem | onsetreclem3 46682* | Lemma for onsetrec 46683. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) ⇒ ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎)) | ||
Theorem | onsetrec 46683 |
Construct On using set recursion. When 𝑥 ∈
On, the function
𝐹 constructs the least ordinal greater
than any of the elements of
𝑥, which is ∪ 𝑥 for a limit ordinal and suc ∪ 𝑥 for a
successor ordinal.
For example, (𝐹‘{1o, 2o}) = {∪ {1o, 2o}, suc ∪ {1o, 2o}} = {2o, 3o} which contains 3o, and (𝐹‘ω) = {∪ ω, suc ∪ ω} = {ω, ω +o 1o}, which contains ω. If we start with the empty set and keep applying 𝐹 transfinitely many times, all ordinal numbers will be generated. Any function 𝐹 fulfilling lemmas onsetreclem2 46681 and onsetreclem3 46682 will recursively generate On; for example, 𝐹 = (𝑥 ∈ V ↦ suc suc ∪ 𝑥}) also works. Whether this function or the function in the theorem is used, taking this theorem as a definition of On is unsatisfying because it relies on the different properties of limit and successor ordinals. A different approach could be to let 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ 𝒫 𝑥 ∣ Tr 𝑦}), based on dfon2 33895. The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable condition between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 22-Jun-2021.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) ⇒ ⊢ setrecs(𝐹) = On | ||
Model organization after organization of reals - see TOC | ||
Syntax | cpg 46684 | Extend class notation to include the class of partisan game forms. |
class Pg | ||
Definition | df-pg 46685 | Define the class of partisan games. More precisely, this is the class of partisan game forms, many of which represent equal partisan games. In Metamath, equality between partisan games is represented by a different equivalence relation than class equality. (Contributed by Emmett Weisz, 22-Aug-2021.) |
⊢ Pg = setrecs((𝑥 ∈ V ↦ (𝒫 𝑥 × 𝒫 𝑥))) | ||
Theorem | elpglem1 46686* | Lemma for elpg 46689. (Contributed by Emmett Weisz, 28-Aug-2021.) |
⊢ (∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)) → ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)) | ||
Theorem | elpglem2 46687* | Lemma for elpg 46689. (Contributed by Emmett Weisz, 28-Aug-2021.) |
⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))) | ||
Theorem | elpglem3 46688* | Lemma for elpg 46689. (Contributed by Emmett Weisz, 28-Aug-2021.) |
⊢ (∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))) | ||
Theorem | elpg 46689 | Membership in the class of partisan games. In John Horton Conway's On Numbers and Games, this is stated as "If 𝐿 and 𝑅 are any two sets of games, then there is a game {𝐿 ∣ 𝑅}. All games are constructed in this way." The first sentence corresponds to the backward direction of our theorem, and the second to the forward direction. (Contributed by Emmett Weisz, 27-Aug-2021.) |
⊢ (𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)) | ||
This is the mathbox of David A. Wheeler, dwheeler at dwheeler dot com. Among other things, I have added a number of formal definitions for widely-used functions, e.g., those defined in ISO 80000-2:2009(E) Quantities and units - Part 2: Mathematical signs and symbols used in the natural sciences and technology and the NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/. | ||
Theorem | sbidd 46690 | An identity theorem for substitution. See sbid 2247. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.) |
⊢ (𝜑 → [𝑥 / 𝑥]𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | sbidd-misc 46691 | An identity theorem for substitution. See sbid 2247. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.) |
⊢ ((𝜑 → [𝑥 / 𝑥]𝜓) ↔ (𝜑 → 𝜓)) | ||
As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Here we formally define the widely-used relations 'greater than' and 'greater than or equal to', so that we have formal definitions of them, as well as a few related theorems. | ||
Syntax | cge-real 46692 | Extend wff notation to include the 'greater than or equal to' relation, see df-gte 46694. |
class ≥ | ||
Syntax | cgt 46693 | Extend wff notation to include the 'greater than' relation, see df-gt 46695. |
class > | ||
Definition | df-gte 46694 |
Define the 'greater than or equal' predicate over the reals. Defined in
ISO 80000-2:2009(E) operation 2-7.10. It is used as a primitive in the
"NIST Digital Library of Mathematical Functions" , front
introduction,
"Common Notations and Definitions" section at
http://dlmf.nist.gov/front/introduction#Sx4.
This relation is merely
the converse of the 'less than or equal to' relation defined by df-le 11094.
We do not write this as (𝑥 ≥ 𝑦 ↔ 𝑦 ≤ 𝑥), and similarly we do not write ` > ` as (𝑥 > 𝑦 ↔ 𝑦 < 𝑥), because these are not definitional axioms as understood by mmj2 (those definitions will be flagged as being "potentially non-conservative"). We could write them this way: ⊢ > = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑦 < 𝑥)} and ⊢ ≥ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑦 ≤ 𝑥)} but these are very complicated. This definition of ≥, and the similar one for > (df-gt 46695), are a bit strange when you see them for the first time, but these definitions are much simpler for us to process and are clearly conservative definitions. (My thanks to Mario Carneiro for pointing out this simpler approach.) See gte-lte 46696 for a more conventional expression of the relationship between < and >. As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
⊢ ≥ = ◡ ≤ | ||
Definition | df-gt 46695 |
The 'greater than' relation is merely the converse of the 'less than or
equal to' relation defined by df-lt 10963. Defined in ISO 80000-2:2009(E)
operation 2-7.12. See df-gte 46694 for a discussion on why this approach is
used for the definition. See gt-lt 46697 and gt-lth 46699 for more conventional
expression of the relationship between < and
>.
As a stylistic issue, set.mm prefers 'less than or equal' instead of 'greater than or equal' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
⊢ > = ◡ < | ||
Theorem | gte-lte 46696 | Simple relationship between ≤ and ≥. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴)) | ||
Theorem | gt-lt 46697 | Simple relationship between < and >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵 ↔ 𝐵 < 𝐴)) | ||
Theorem | gte-lteh 46698 | Relationship between ≤ and ≥ using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴) | ||
Theorem | gt-lth 46699 | Relationship between < and > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 > 𝐵 ↔ 𝐵 < 𝐴) | ||
Theorem | ex-gt 46700 | Simple example of >, in this case, 0 is not greater than 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.) |
⊢ ¬ 0 > 0 |
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