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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-prstc 46801 |
Definition of the function converting a preordered set to a category.
Justified by prsthinc 46792.
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 46804, prstchom 46815, and prstcthin 46814. Other important properties include prstcbas 46805, prstcleval 46806, prstcle 46808, prstcocval 46809, prstcoc 46811, prstchom2 46816, and prstcprs 46813. Use those instead. Note that the defining property prstchom 46815 is equivalent to prstchom2 46816 given prstcthin 46814. See thincn0eu 46770 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.) |
β’ ProsetToCat = (π β Proset β¦ ((π sSet β¨(Hom βndx), ((leβπ) Γ {1o})β©) sSet β¨(compβndx), β β©)) | ||
Theorem | prstcval 46802 | Lemma for prstcnidlem 46803 and prstcthin 46814. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) β β’ (π β πΆ = ((πΎ sSet β¨(Hom βndx), ((leβπΎ) Γ {1o})β©) sSet β¨(compβndx), β β©)) | ||
Theorem | prstcnidlem 46803 | Lemma for prstcnid 46804 and prstchomval 46812. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ πΈ = Slot (πΈβndx) & β’ (πΈβndx) β (compβndx) β β’ (π β (πΈβπΆ) = (πΈβ(πΎ sSet β¨(Hom βndx), ((leβπΎ) Γ {1o})β©))) | ||
Theorem | prstcnid 46804 | Components other than Hom and comp are unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ πΈ = Slot (πΈβndx) & β’ (πΈβndx) β (compβndx) & β’ (πΈβndx) β (Hom βndx) β β’ (π β (πΈβπΎ) = (πΈβπΆ)) | ||
Theorem | prstcbas 46805 | The base set is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β π΅ = (BaseβπΎ)) β β’ (π β π΅ = (BaseβπΆ)) | ||
Theorem | prstcleval 46806 | 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.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β€ = (leβπΎ)) β β’ (π β β€ = (leβπΆ)) | ||
Theorem | prstclevalOLD 46807 | Obsolete proof of prstcleval 46806 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.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β€ = (leβπΎ)) β β’ (π β β€ = (leβπΆ)) | ||
Theorem | prstcle 46808 | Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β€ = (leβπΎ)) β β’ (π β (π β€ π β π(leβπΆ)π)) | ||
Theorem | prstcocval 46809 | Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β₯ = (ocβπΎ)) β β’ (π β β₯ = (ocβπΆ)) | ||
Theorem | prstcocvalOLD 46810 | Obsolete proof of prstcocval 46809 as of 12-Nov-2024. Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β₯ = (ocβπΎ)) β β’ (π β β₯ = (ocβπΆ)) | ||
Theorem | prstcoc 46811 | Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β₯ = (ocβπΎ)) β β’ (π β ( β₯ βπ) = ((ocβπΆ)βπ)) | ||
Theorem | prstchomval 46812 | Hom-sets of the constructed category which depend on an arbitrary definition. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β€ = (leβπΆ)) β β’ (π β ( β€ Γ {1o}) = (Hom βπΆ)) | ||
Theorem | prstcprs 46813 | The category is a preordered set. (Contributed by Zhi Wang, 20-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) β β’ (π β πΆ β Proset ) | ||
Theorem | prstcthin 46814 | The preordered set is equipped with a thin category. (Contributed by Zhi Wang, 20-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) β β’ (π β πΆ β ThinCat) | ||
Theorem | prstchom 46815 |
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.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β€ = (leβπΆ)) & β’ (π β π» = (Hom βπΆ)) & β’ (π β π β (BaseβπΆ)) & β’ (π β π β (BaseβπΆ)) β β’ (π β (π β€ π β (ππ»π) β β )) | ||
Theorem | prstchom2 46816* |
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 46817). 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.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β€ = (leβπΆ)) & β’ (π β π» = (Hom βπΆ)) & β’ (π β π β (BaseβπΆ)) & β’ (π β π β (BaseβπΆ)) β β’ (π β (π β€ π β β!π π β (ππ»π))) | ||
Theorem | prstchom2ALT 46817* | Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 46801. See prstchom2 46816 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.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β€ = (leβπΆ)) & β’ (π β π» = (Hom βπΆ)) β β’ (π β (π β€ π β β!π π β (ππ»π))) | ||
Theorem | postcpos 46818 | The converted category is a poset iff the original proset is a poset. (Contributed by Zhi Wang, 26-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) β β’ (π β (πΎ β Poset β πΆ β Poset)) | ||
Theorem | postcposALT 46819 | Alternate proof for postcpos 46818. (Contributed by Zhi Wang, 25-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) β β’ (π β (πΎ β Poset β πΆ β Poset)) | ||
Theorem | postc 46820* | The converted category is a poset iff no distinct objects are isomorphic. (Contributed by Zhi Wang, 25-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ π΅ = (BaseβπΆ) β β’ (π β (πΆ β Poset β βπ₯ β π΅ βπ¦ β π΅ (π₯( βπ βπΆ)π¦ β π₯ = π¦))) | ||
Syntax | cmndtc 46821 | Class function defining monoids as categories. |
class MndToCat | ||
Definition | df-mndtc 46822 |
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 46824) , 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 46825, mndtchom 46828, mndtcco 46829. 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.) |
β’ MndToCat = (π β Mnd β¦ {β¨(Baseβndx), {π}β©, β¨(Hom βndx), {β¨π, π, (Baseβπ)β©}β©, β¨(compβndx), {β¨β¨π, π, πβ©, (+gβπ)β©}β©}) | ||
Theorem | mndtcval 46823 | Value of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) β β’ (π β πΆ = {β¨(Baseβndx), {π}β©, β¨(Hom βndx), {β¨π, π, (Baseβπ)β©}β©, β¨(compβndx), {β¨β¨π, π, πβ©, (+gβπ)β©}β©}) | ||
Theorem | mndtcbasval 46824 | The base set of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) β β’ (π β π΅ = {π}) | ||
Theorem | mndtcbas 46825* | The category built from a monoid contains precisely one object. (Contributed by Zhi Wang, 22-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) β β’ (π β β!π₯ π₯ β π΅) | ||
Theorem | mndtcob 46826 | Lemma for mndtchom 46828 and mndtcco 46829. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) β β’ (π β π = π) | ||
Theorem | mndtcbas2 46827 | Two objects in a category built from a monoid are identical. (Contributed by Zhi Wang, 24-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β π = π) | ||
Theorem | mndtchom 46828 | 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.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π» = (Hom βπΆ)) β β’ (π β (ππ»π) = (Baseβπ)) | ||
Theorem | mndtcco 46829 | The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β Β· = (compβπΆ)) β β’ (π β (β¨π, πβ© Β· π) = (+gβπ)) | ||
Theorem | mndtcco2 46830 | The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β Β· = (compβπΆ)) & β’ (π β β¬ = (β¨π, πβ© Β· π)) β β’ (π β (πΊ β¬ πΉ) = (πΊ(+gβπ)πΉ)) | ||
Theorem | mndtccatid 46831* | Lemma for mndtccat 46832 and mndtcid 46833. (Contributed by Zhi Wang, 22-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) β β’ (π β (πΆ β Cat β§ (IdβπΆ) = (π¦ β (BaseβπΆ) β¦ (0gβπ)))) | ||
Theorem | mndtccat 46832 | The function value is a category. (Contributed by Zhi Wang, 22-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) β β’ (π β πΆ β Cat) | ||
Theorem | mndtcid 46833 | 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.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) & β’ (π β 1 = (IdβπΆ)) β β’ (π β ( 1 βπ) = (0gβπ)) | ||
Theorem | grptcmon 46834 | All morphisms in a category converted from a group are monomorphisms. (Contributed by Zhi Wang, 23-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπΊ)) & β’ (π β πΊ β Grp) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π» = (Hom βπΆ)) & β’ (π β π = (MonoβπΆ)) β β’ (π β (πππ) = (ππ»π)) | ||
Theorem | grptcepi 46835 | All morphisms in a category converted from a group are epimorphisms. (Contributed by Zhi Wang, 23-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπΊ)) & β’ (π β πΊ β Grp) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π» = (Hom βπΆ)) & β’ (π β πΈ = (EpiβπΆ)) β β’ (π β (ππΈπ) = (ππ»π)) | ||
Some of these theorems are used in the series of lemmas and theorems proving the defining properties of setrecs. | ||
Theorem | nfintd 46836 | Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.) |
β’ (π β β²π₯π΄) β β’ (π β β²π₯β© π΄) | ||
Theorem | nfiund 46837* | Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2372. See nfiundg 46838 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
β’ β²π₯π & β’ (π β β²π¦π΄) & β’ (π β β²π¦π΅) β β’ (π β β²π¦βͺ π₯ β π΄ π΅) | ||
Theorem | nfiundg 46838 | Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2372, see nfiund 46837 for a weaker version that does not require it. (Contributed by Emmett Weisz, 6-Dec-2019.) (New usage is discouraged.) |
β’ β²π₯π & β’ (π β β²π¦π΄) & β’ (π β β²π¦π΅) β β’ (π β β²π¦βͺ π₯ β π΄ π΅) | ||
Theorem | iunord 46839* | The indexed union of a collection of ordinal numbers π΅(π₯) is ordinal. This proof is based on the proof of ssorduni 7704, but does not use it directly, since ssorduni 7704 does not work when π΅ is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.) |
β’ (βπ₯ β π΄ Ord π΅ β Ord βͺ π₯ β π΄ π΅) | ||
Theorem | iunordi 46840* | The indexed union of a collection of ordinal numbers π΅(π₯) is ordinal. (Contributed by Emmett Weisz, 3-Nov-2019.) |
β’ Ord π΅ β β’ Ord βͺ π₯ β π΄ π΅ | ||
Theorem | spd 46841 | Specialization deduction, using implicit substitution. Based on the proof of spimed 2388. (Contributed by Emmett Weisz, 17-Jan-2020.) |
β’ (π β β²π₯π) & β’ (π₯ = π¦ β (π β π)) β β’ (π β (βπ₯π β π)) | ||
Theorem | spcdvw 46842* | A version of spcdv 3552 where π and π are direct substitutions of each other. This theorem is useful because it does not require π and π₯ to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.) |
β’ (π β π΄ β π΅) & β’ (π₯ = π΄ β (π β π)) β β’ (π β (βπ₯π β π)) | ||
Theorem | tfis2d 46843* | Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.) |
β’ (π β (π₯ = π¦ β (π β π))) & β’ (π β (π₯ β On β (βπ¦ β π₯ π β π))) β β’ (π β (π₯ β On β π)) | ||
Theorem | bnd2d 46844* | Deduction form of bnd2 9763. (Contributed by Emmett Weisz, 19-Jan-2021.) |
β’ (π β π΄ β V) & β’ (π β βπ₯ β π΄ βπ¦ β π΅ π) β β’ (π β βπ§(π§ β π΅ β§ βπ₯ β π΄ βπ¦ β π§ π)) | ||
Theorem | dffun3f 46845* | Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.) |
β’ β²π₯π΄ & β’ β²π¦π΄ & β’ β²π§π΄ β β’ (Fun π΄ β (Rel π΄ β§ βπ₯βπ§βπ¦(π₯π΄π¦ β π¦ = π§))) | ||
Symbols in this section: All the symbols used in the definition of setrecs(πΉ) are explained in the comment of df-setrecs 46847. The class π is explained in the comment of setrec1lem1 46850. Glossaries of symbols used in individual proofs, or used differently in different proofs, are in the comments of those proofs. | ||
Syntax | csetrecs 46846 | Extend class notation to include a set defined by transfinite recursion. |
class setrecs(πΉ) | ||
Definition | df-setrecs 46847* |
Define a class setrecs(πΉ) by transfinite recursion, where
(πΉβπ₯) is the set of new elements to add to
the class given the
set π₯ 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(πΉ). The goal of this definition is to
construct a class
fulfilling Theorems setrec1 46854 and setrec2v 46859, which give a more
intuitive idea of the meaning of setrecs.
Unlike wrecs,
setrecs is well-defined for any πΉ and
meaningful for any
function πΉ.
For example, see Theorem onsetrec 46870 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 π§, and consider the result when applying πΉ to any subset π€ β π§. Remember that πΉ 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 π§. However, if we restrict the domain of πΉ to a given set π¦, the resulting range will be a set. Therefore, with this restricted πΉ, it makes sense to consider sets π§ that are closed under πΉ applied to its subsets. Now we can test whether a given set π¦ is recursively generated by πΉ. If every set π§ that is closed under πΉ contains π¦, that means that every member of π¦ must eventually be generated by πΉ. On the other hand, if some such π§ does not contain a certain element of π¦, then that element can be avoided even if we apply πΉ in every possible way to previously generated elements. Note that such an omitted element might be eventually recursively generated by πΉ, but not through the elements of π¦. In this case, π¦ would fail the condition in the definition, but the omitted element would still be included in some larger π¦. For example, if πΉ is the successor function, the set {β , 2o} would fail the condition since 2o is not an element of the successor of β or {β }. Remember that we are applying πΉ to subsets of π¦, not elements of π¦. In fact, even the set {1o} fails the condition, since the only subset of previously generated elements is β , and suc β does not have 1o as an element. However, we can let π¦ be any ordinal, since each of its elements is generated by starting with β and repeatedly applying the successor function. A similar definition I initially used for setrecs(πΉ) was setrecs(πΉ) = βͺ ran recs((π β V β¦ (πΉββͺ ran π))). I had initially tried and failed to find an elementary definition, and I had proven theorems analogous to setrec1 46854 and setrec2v 46859 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(πΉ) is that the necessary properties to prove are self-referential (see setrec1 46854 and setrec2v 46859), 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 8324, 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(πΉ) as a union or intersection of suitable sets. Thus, instead of using a class abstraction for the elements of setrecs(πΉ), which seems to be impossible, we can use a class abstraction for supersets or subsets of setrecs(πΉ), which "know" about multiple individual elements at a time. Note that we cannot define setrecs(πΉ) 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(πΉ) as a union of a class abstraction. If setrecs(πΉ) = βͺ π΄, the elements of A must be subsets of setrecs(πΉ) which together include everything recursively generated by πΉ. We can do this by letting π΄ be the class of sets π₯ whose elements are all recursively generated by πΉ. One necessary condition is that each element of a given π₯ β π΄ must be generated by πΉ when applied to a previous element π¦ β π΄. In symbols, βπ₯ β π΄βπ¦ β π΄(π¦ β π₯ β§ π₯ β (πΉβπ¦))}. However, this is not sufficient. All fixed points π₯ of πΉ will satisfy this condition whether they should be in setrecs(πΉ) or not. If we replace the subset relation with the proper subset relation, π₯ cannot be the empty set, even though the empty set should be in π΄. 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 π¦ β π΄ to be generated by πΉ when applied only to sets of previously generated elements within π¦. For example, taking πΉ to be the successor function, we can let π΄ = On rather than π« On, and we will still have βͺ π΄ = On as required. This gets rid of the circularity of the definition, since we should have a condition to test whether a given set π¦ is in π΄ without knowing about any of the other elements of π΄. The definition I ended up using accomplishes this using induction: π΄ is defined as the class of sets π¦ for which a sort of induction on the elements of π¦ 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 π = {β¨π, πβ© β£ (πΉβπ) β π}, then (π Fr π΄ β βπ₯((π₯ β π΄ β§ π₯ β β ) β βπ¦ β π₯βπ§ β π₯Β¬ (πΉβπ§) β π¦)). On 22-Jul-2020 I came up with the following definition (Version 1) phrased in terms of induction: βͺ {π¦ β£ βπ§ (βπ€(π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)} 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 π§ replaced by its complement. βͺ {π¦ β£ βπ§ (π¦ β π§ β βπ€(π€ β π¦ β§ (πΉβπ€) β π§ β§ Β¬ π€ β π§))} These definitions didn't work because the induction didn't "get off the ground." If π§ does not contain the empty set, the condition (βπ€...π¦ β π§ fails, so π¦ = β doesn't get included in π΄ 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 π§ of all individuals that have been generated so far. So one approach is to replace every occurrence of β π§ with β π§, making π§ 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 πΉ be the power set function, then the induction in the inductive version works for π§ being the class of transitive sets, restricted to subsets of π¦. Therefore, π¦ must be transitive by definition of π§. This doesn't affect the union of all such π¦, but it may or may not be desirable. The problem is that πΉ is only applied to transitive sets, because of the strong requirement π€ β π§, so the definition requires the additional constraint (π β π β (πΉβπ) β (πΉβπ)) in order to work. This issue can also be avoided by replacing β π§ with β π§. The induction version of the result is used in the final definition. Version 2: (18-Aug-2020) Induction: βͺ {π¦ β£ βπ§ (βπ€(π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)} Foundedness: βͺ {π¦ β£ βπ§(π¦ β© π§ β β β βπ€(π€ β π¦ β§ π€ β© π§ = β β§ (πΉβπ€) β© π§ β β ))} In the induction version, not only does π§ include all the elements of π¦, but it must include the elements of (πΉβπ€) for π€ β (π¦ β© π§) even if those elements of (πΉβπ€) are not in π¦. We shouldn't care about any of the elements of π§ outside π¦, but this detail doesn't affect the correctness of the definition. If we replaced (πΉβπ€) in the definition by ((πΉβπ€) β© π¦), we would get the same class for setrecs(πΉ). Suppose we could find a π§ for which the condition fails for a given π¦ under the changed definition. Then the antecedent would be true, but π¦ β π§ would be false. We could then simply add all elements of (πΉβπ€) outside of π¦ for any π€ β π¦, 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 π§ fails the condition under the Metamath definition. The other direction is easier. If a certain π§ fails the Metamath definition, then all (πΉβπ€) β π§ for π€ β (π¦ β© π§), and in particular ((πΉβπ€) β© π¦) β π§. The foundedness version is starting to look more like ax-reg 9462! 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 π§ which are subsets of π¦, we can restrict π§ to π¦ in the foundedness definition. Furthermore, instead of quantifying over π€, quantify over the elements π£ β π§ overlapping with π€. Versions 3, 4, and 5 are all equivalent to Version 2. Version 3 - Foundedness (5-Sep-2020): βͺ {π¦ β£ βπ§((π§ β π¦ β§ π§ β β ) β βπ£ β π§βπ€(π€ β π¦ β§ π€ β© π§ = β β§ π£ β (πΉβπ€)))} Now, if we replace (πΉβπ€) by ((πΉβπ€) β© π¦), we do not change the definition. We already know that π£ β π¦ since π£ β π§ and π§ β π¦. 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 5586, we add another variable π’ representing the nonexistent element of π€ in π§. Version 4 - Foundedness (6-Sep-2020): βͺ {π¦ β£ βπ§((π§ β π¦ β§ π§ β β ) β βπ£ β π§βπ€βπ’ β π§(π€ β π¦ β§ Β¬ π’ β π€ β§ π£ β (πΉβπ€)) This is so close to df-fr 5586; the only change needed is to switch βπ€ with βπ’ β π§. 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 9462. Rather than a disjoint element of π§, there's a disjoint coverer of an element of π§. Finally, here's a different dead end I followed: To clean up our foundedness definition, we keep π§ as a family of sets π¦ but allow π€ to be any subset of βͺ π§ in the induction. With this stronger induction, we can also allow for the stronger requirement π« π¦ β π§ rather than only π¦ β π§. This will help improve the foundedness version. Version 1.1 (28-Aug-2020) Induction: βͺ {π¦ β£ βπ§(βπ€ (π€ β π¦ β (π€ β βͺ π§ β (πΉβπ€) β π§)) β π« π¦ β π§)} Foundedness: βͺ {π¦ β£ βπ§(βπ(π β π¦ β§ π β π§) β βπ€(π€ β π¦ β§ π€ β© β© π§ = β β§ (πΉβπ€) β π§))} ( Edit (Aug 31) - this isn't true! Nothing forces the subset of an element of π§ to be in π§. Version 2 does not have this issue. ) Similarly, we could allow π€ to be any subset of any element of π§ rather than any subset of βͺ π§. 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 π§ which are subsets of π¦, we can restrict π§ to π« π¦ in the foundedness definition: Version 1.2 (31-Aug-2020) Foundedness: βͺ {π¦ β£ βπ§((π§ β π« π¦ β§ π§ β β ) β βπ€(π€ β π« π¦ β§ π€ β© β© π§ = β β§ (πΉβπ€) β π§))} Now this looks more like df-fr 5586! The last step necessary to be able to use Fr directly in our definition is to replace (πΉβπ€) with its own setvar variable, corresponding to π¦ in df-fr 5586. This definition is incorrect, though, since there's nothing forcing the subset of an element of π§ to be in π§. Version 1.3 (31-Aug-2020) Induction: βͺ {π¦ β£ βπ§(βπ€(π€ β π¦ β (π€ β βͺ π§ β (π€ β π§ β§ (πΉβπ€) β π§))) β π« π¦ β π§)} Foundedness: βͺ {π¦ β£ βπ§((π§ β π« π¦ β§ π§ β β ) β βπ€(π€ β π« π¦ β§ π€ β© β© π§ = β β§ (π€ β π§ β¨ (πΉβπ€) β π§)))} π§ must contain the supersets of each of its elements in the foundedness version, and we can't make any restrictions on π§ or πΉ, so this doesn't work. Let's try letting R be the covering relation π = {β¨π, πβ© β£ π β (πΉβπ)} to solve the transitivity issue (i.e. that if πΉ is the power set relation, π΄ consists only of transitive sets). The set (πΉβπ€) corresponds to the variable π¦ in df-fr 5586. Thus, in our case, df-fr 5586 is equivalent to (π Fr π΄ β βπ§((π§ β π΄ β§ π§ β β ) β βπ€((πΉβπ€) β π§ β§ Β¬ βπ£ β π§π£π (πΉβπ€))). Substituting our relation π gives (π Fr π΄ β βπ§((π§ β π΄ β§ π§ β β ) β βπ€((πΉβπ€) β π§ β§ Β¬ βπ£ β π§(πΉβπ€) β (πΉβπ£))) This doesn't work for non-injective πΉ because we need all π§ 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 Β¬ π€ β π§ β βπ£ β π§Β¬ π£π (πΉβπ€) so we can replace one with the other. Negate both sides: π€ β π§ β βπ£ β π§π£π (πΉβπ€) If πΉ 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 π€ 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.) |
β’ setrecs(πΉ) = βͺ {π¦ β£ βπ§(βπ€(π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)} | ||
Theorem | setrecseq 46848 | Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.) |
β’ (πΉ = πΊ β setrecs(πΉ) = setrecs(πΊ)) | ||
Theorem | nfsetrecs 46849 | Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.) |
β’ β²π₯πΉ β β’ β²π₯setrecs(πΉ) | ||
Theorem | setrec1lem1 46850* |
Lemma for setrec1 46854. This is a utility theorem showing the
equivalence
of the statement π β π and its expanded form. The proof
uses
elabg 3627 and equivalence theorems.
Variable π is the class of sets π¦ that are recursively generated by the function πΉ. In other words, π¦ β π iff by starting with the empty set and repeatedly applying πΉ to subsets π€ of our set, we will eventually generate all the elements of π. In this theorem, π is any element of π, and π is any class. (Contributed by Emmett Weisz, 16-Oct-2020.) (New usage is discouraged.) |
β’ π = {π¦ β£ βπ§(βπ€(π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)} & β’ (π β π β π) β β’ (π β (π β π β βπ§(βπ€(π€ β π β (π€ β π§ β (πΉβπ€) β π§)) β π β π§))) | ||
Theorem | setrec1lem2 46851* | Lemma for setrec1 46854. If a family of sets are all recursively generated by πΉ, so is their union. In this theorem, π is a family of sets which are all elements of π, and π is any class. Use dfss3 3931, 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.) |
β’ π = {π¦ β£ βπ§(βπ€(π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)} & β’ (π β π β π) & β’ (π β π β π) β β’ (π β βͺ π β π) | ||
Theorem | setrec1lem3 46852* | Lemma for setrec1 46854. If each element π of π΄ is covered by a set π₯ recursively generated by πΉ, then there is a single such set covering all of π΄. The set is constructed explicitly using setrec1lem2 46851. It turns out that π₯ = π΄ also works, i.e., given the hypotheses it is possible to prove that π΄ β π. I don't know if proving this fact directly using setrec1lem1 46850 would be any easier than the current proof using setrec1lem2 46851, and it would only slightly simplify the proof of setrec1 46854. Other than the use of bnd2d 46844, this is a purely technical theorem for rearranging notation from that of setrec1lem2 46851 to that of setrec1 46854. (Contributed by Emmett Weisz, 20-Jan-2021.) (New usage is discouraged.) |
β’ π = {π¦ β£ βπ§(βπ€(π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)} & β’ (π β π΄ β V) & β’ (π β βπ β π΄ βπ₯(π β π₯ β§ π₯ β π)) β β’ (π β βπ₯(π΄ β π₯ β§ π₯ β π)) | ||
Theorem | setrec1lem4 46853* |
Lemma for setrec1 46854. If π is recursively generated by πΉ, then
so is π βͺ (πΉβπ΄).
In the proof of setrec1 46854, the following is substituted for this theorem's π: (π β§ (π΄ β π₯ β§ π₯ β {π¦ β£ βπ§(βπ€ (π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)})) Therefore, we cannot declare π§ to be a distinct variable from π, since we need it to appear as a bound variable in π. This theorem can be proven without the hypothesis β²π§π, but the proof would be harder to read because theorems in deduction form would be interrupted by theorems like eximi 1838, making the antecedent of each line something more complicated than π. The proof of setrec1lem2 46851 could similarly be made easier to read by adding the hypothesis β²π§π, 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.) |
β’ β²π§π & β’ π = {π¦ β£ βπ§(βπ€(π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)} & β’ (π β π΄ β V) & β’ (π β π΄ β π) & β’ (π β π β π) β β’ (π β (π βͺ (πΉβπ΄)) β π) | ||
Theorem | setrec1 46854 |
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(πΉ) is closed under πΉ. This
effectively sets the
actual value of setrecs(πΉ) as a lower bound for
setrecs(πΉ), as it implies that any set
generated by successive
applications of πΉ is a member of π΅. This
theorem "gets off the
ground" because we can start by letting π΄ = β
, and the
hypotheses
of the theorem will hold trivially.
Variable π΅ represents an abbreviation of setrecs(πΉ) or another name of setrecs(πΉ) (for an example of the latter, see theorem setrecon). Proof summary: Assume that π΄ β π΅, meaning that all elements of π΄ are in some set recursively generated by πΉ. Then by setrec1lem3 46852, π΄ is a subset of some set recursively generated by πΉ. (It turns out that π΄ itself is recursively generated by πΉ, but we don't need this fact. See the comment to setrec1lem3 46852.) Therefore, by setrec1lem4 46853, (πΉβπ΄) is a subset of some set recursively generated by πΉ. Thus, by ssuni 4892, it is a subset of the union of all sets recursively generated by πΉ. See df-setrecs 46847 for a detailed description of how the setrecs definition works. (Contributed by Emmett Weisz, 9-Oct-2020.) |
β’ π΅ = setrecs(πΉ) & β’ (π β π΄ β V) & β’ (π β π΄ β π΅) β β’ (π β (πΉβπ΄) β π΅) | ||
Theorem | setrec2fun 46855* |
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(πΉ) is a subclass of all classes πΆ that
are closed
under πΉ. Taken together, Theorems setrec1 46854 and setrec2v 46859 say
that setrecs(πΉ) is the minimal class closed under
πΉ.
We express this by saying that if πΉ respects the β relation and πΆ is closed under πΉ, then π΅ β πΆ. By substituting strategically constructed classes for πΆ, we can easily prove many useful properties. Although this theorem cannot show equality between π΅ and πΆ, if we intend to prove equality between π΅ and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7780) to the other class. (Contributed by Emmett Weisz, 15-Feb-2021.) (New usage is discouraged.) |
β’ β²ππΉ & β’ π΅ = setrecs(πΉ) & β’ Fun πΉ & β’ (π β βπ(π β πΆ β (πΉβπ) β πΆ)) β β’ (π β π΅ β πΆ) | ||
Theorem | setrec2lem1 46856* | Lemma for setrec2 46858. The functional part of πΉ has the same values as πΉ. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.) |
β’ ((πΉ βΎ {π₯ β£ β!π¦ π₯πΉπ¦})βπ) = (πΉβπ) | ||
Theorem | setrec2lem2 46857* | Lemma for setrec2 46858. The functional part of πΉ is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.) |
β’ Fun (πΉ βΎ {π₯ β£ β!π¦ π₯πΉπ¦}) | ||
Theorem | setrec2 46858* |
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(πΉ) is a subclass of all classes πΆ that
are closed
under πΉ. Taken together, Theorems setrec1 46854 and setrec2v 46859
uniquely determine setrecs(πΉ) to be the minimal class closed
under πΉ.
We express this by saying that if πΉ respects the β relation and πΆ is closed under πΉ, then π΅ β πΆ. By substituting strategically constructed classes for πΆ, we can easily prove many useful properties. Although this theorem cannot show equality between π΅ and πΆ, if we intend to prove equality between π΅ and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7780) to the other class. (Contributed by Emmett Weisz, 2-Sep-2021.) |
β’ β²ππΉ & β’ π΅ = setrecs(πΉ) & β’ (π β βπ(π β πΆ β (πΉβπ) β πΆ)) β β’ (π β π΅ β πΆ) | ||
Theorem | setrec2v 46859* | Version of setrec2 46858 with a disjoint variable condition instead of a nonfreeness hypothesis. (Contributed by Emmett Weisz, 6-Mar-2021.) |
β’ π΅ = setrecs(πΉ) & β’ (π β βπ(π β πΆ β (πΉβπ) β πΆ)) β β’ (π β π΅ β πΆ) | ||
Theorem | setis 46860* | Version of setrec2 46858 expressed as an induction schema. This theorem is a generalization of tfis3 7785. (Contributed by Emmett Weisz, 27-Feb-2022.) |
β’ π΅ = setrecs(πΉ) & β’ (π = π΄ β (π β π)) & β’ (π β βπ(βπ β π π β βπ β (πΉβπ)π)) β β’ (π β (π΄ β π΅ β π)) | ||
Theorem | elsetrecslem 46861* | Lemma for elsetrecs 46862. Any element of setrecs(πΉ) is generated by some subset of setrecs(πΉ). This is much weaker than setrec2v 46859. To see why this lemma also requires setrec1 46854, consider what would happen if we replaced π΅ with {π΄}. 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.) |
β’ π΅ = setrecs(πΉ) β β’ (π΄ β π΅ β βπ₯(π₯ β π΅ β§ π΄ β (πΉβπ₯))) | ||
Theorem | elsetrecs 46862* | A set π΄ is an element of setrecs(πΉ) iff π΄ is generated by some subset of setrecs(πΉ). The proof requires both setrec1 46854 and setrec2 46858, but this theorem is not strong enough to uniquely determine setrecs(πΉ). If πΉ respects the subset relation, the theorem still holds if both occurrences of β are replaced by β for a stronger version of the theorem. (Contributed by Emmett Weisz, 12-Jul-2021.) |
β’ π΅ = setrecs(πΉ) β β’ (π΄ β π΅ β βπ₯(π₯ β π΅ β§ π΄ β (πΉβπ₯))) | ||
Theorem | setrecsss 46863 | The setrecs operator respects the subset relation between two functions πΉ and πΊ. (Contributed by Emmett Weisz, 13-Mar-2022.) |
β’ (π β Fun πΊ) & β’ (π β πΉ β πΊ) β β’ (π β setrecs(πΉ) β setrecs(πΊ)) | ||
Theorem | setrecsres 46864 | A recursively generated class is unaffected when its input function is restricted to subsets of the class. (Contributed by Emmett Weisz, 14-Mar-2022.) |
β’ π΅ = setrecs(πΉ) & β’ (π β Fun πΉ) β β’ (π β π΅ = setrecs((πΉ βΎ π« π΅))) | ||
Theorem | vsetrec 46865 | Construct V using set recursion. The proof indirectly uses trcl 9598, which relies on rec, but theoretically πΆ in trcl 9598 could be constructed using setrecs instead. The proof of this theorem uses the dummy variable π rather than π₯ to avoid a distinct variable requirement between πΉ and π₯. (Contributed by Emmett Weisz, 23-Jun-2021.) |
β’ πΉ = (π₯ β V β¦ π« π₯) β β’ setrecs(πΉ) = V | ||
Theorem | 0setrec 46866 | 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.) |
β’ (π β (πΉββ ) = β ) β β’ (π β setrecs(πΉ) = β ) | ||
Theorem | onsetreclem1 46867* | Lemma for onsetrec 46870. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
β’ πΉ = (π₯ β V β¦ {βͺ π₯, suc βͺ π₯}) β β’ (πΉβπ) = {βͺ π, suc βͺ π} | ||
Theorem | onsetreclem2 46868* | Lemma for onsetrec 46870. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
β’ πΉ = (π₯ β V β¦ {βͺ π₯, suc βͺ π₯}) β β’ (π β On β (πΉβπ) β On) | ||
Theorem | onsetreclem3 46869* | Lemma for onsetrec 46870. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
β’ πΉ = (π₯ β V β¦ {βͺ π₯, suc βͺ π₯}) β β’ (π β On β π β (πΉβπ)) | ||
Theorem | onsetrec 46870 |
Construct On using set recursion. When π₯ β
On, the function
πΉ constructs the least ordinal greater
than any of the elements of
π₯, which is βͺ π₯ for a limit ordinal and suc βͺ π₯ for a
successor ordinal.
For example, (πΉβ{1o, 2o}) = {βͺ {1o, 2o}, suc βͺ {1o, 2o}} = {2o, 3o} which contains 3o, and (πΉβΟ) = {βͺ Ο, suc βͺ Ο} = {Ο, Ο +o 1o}, which contains Ο. If we start with the empty set and keep applying πΉ transfinitely many times, all ordinal numbers will be generated. Any function πΉ fulfilling lemmas onsetreclem2 46868 and onsetreclem3 46869 will recursively generate On; for example, πΉ = (π₯ β V β¦ suc suc βͺ π₯}) 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 πΉ = (π₯ β V β¦ {π¦ β π« π₯ β£ Tr π¦}), based on dfon2 34145. The proof of this theorem uses the dummy variable π rather than π₯ to avoid a distinct variable condition between πΉ and π₯. (Contributed by Emmett Weisz, 22-Jun-2021.) |
β’ πΉ = (π₯ β V β¦ {βͺ π₯, suc βͺ π₯}) β β’ setrecs(πΉ) = On | ||
Model organization after organization of reals - see TOC | ||
Syntax | cpg 46871 | Extend class notation to include the class of partisan game forms. |
class Pg | ||
Definition | df-pg 46872 | 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.) |
β’ Pg = setrecs((π₯ β V β¦ (π« π₯ Γ π« π₯))) | ||
Theorem | elpglem1 46873* | Lemma for elpg 46876. (Contributed by Emmett Weisz, 28-Aug-2021.) |
β’ (βπ₯(π₯ β Pg β§ ((1st βπ΄) β π« π₯ β§ (2nd βπ΄) β π« π₯)) β ((1st βπ΄) β Pg β§ (2nd βπ΄) β Pg)) | ||
Theorem | elpglem2 46874* | Lemma for elpg 46876. (Contributed by Emmett Weisz, 28-Aug-2021.) |
β’ (((1st βπ΄) β Pg β§ (2nd βπ΄) β Pg) β βπ₯(π₯ β Pg β§ ((1st βπ΄) β π« π₯ β§ (2nd βπ΄) β π« π₯))) | ||
Theorem | elpglem3 46875* | Lemma for elpg 46876. (Contributed by Emmett Weisz, 28-Aug-2021.) |
β’ (βπ₯(π₯ β Pg β§ π΄ β ((π¦ β V β¦ (π« π¦ Γ π« π¦))βπ₯)) β (π΄ β (V Γ V) β§ βπ₯(π₯ β Pg β§ ((1st βπ΄) β π« π₯ β§ (2nd βπ΄) β π« π₯)))) | ||
Theorem | elpg 46876 | Membership in the class of partisan games. In John Horton Conway's On Numbers and Games, this is stated as "If πΏ and π are any two sets of games, then there is a game {πΏ β£ π }. 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.) |
β’ (π΄ β Pg β (π΄ β (V Γ V) β§ (1st βπ΄) β Pg β§ (2nd βπ΄) β 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 46877 | An identity theorem for substitution. See sbid 2249. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.) |
β’ (π β [π₯ / π₯]π) β β’ (π β π) | ||
Theorem | sbidd-misc 46878 | An identity theorem for substitution. See sbid 2249. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.) |
β’ ((π β [π₯ / π₯]π) β (π β π)) | ||
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 46879 | Extend wff notation to include the 'greater than or equal to' relation, see df-gte 46881. |
class β₯ | ||
Syntax | cgt 46880 | Extend wff notation to include the 'greater than' relation, see df-gt 46882. |
class > | ||
Definition | df-gte 46881 |
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 11129.
We do not write this as (π₯ β₯ π¦ β π¦ β€ π₯), and similarly we do not write ` > ` as (π₯ > π¦ β π¦ < π₯), 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: β’ > = {β¨π₯, π¦β© β£ ((π₯ β β* β§ π¦ β β*) β§ π¦ < π₯)} and β’ β₯ = {β¨π₯, π¦β© β£ ((π₯ β β* β§ π¦ β β*) β§ π¦ β€ π₯)} but these are very complicated. This definition of β₯, and the similar one for > (df-gt 46882), 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 46883 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.) |
β’ β₯ = β‘ β€ | ||
Definition | df-gt 46882 |
The 'greater than' relation is merely the converse of the 'less than or
equal to' relation defined by df-lt 10998. Defined in ISO 80000-2:2009(E)
operation 2-7.12. See df-gte 46881 for a discussion on why this approach is
used for the definition. See gt-lt 46884 and gt-lth 46886 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.) |
β’ > = β‘ < | ||
Theorem | gte-lte 46883 | Simple relationship between β€ and β₯. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
β’ ((π΄ β V β§ π΅ β V) β (π΄ β₯ π΅ β π΅ β€ π΄)) | ||
Theorem | gt-lt 46884 | Simple relationship between < and >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
β’ ((π΄ β V β§ π΅ β V) β (π΄ > π΅ β π΅ < π΄)) | ||
Theorem | gte-lteh 46885 | Relationship between β€ and β₯ using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.) |
β’ π΄ β V & β’ π΅ β V β β’ (π΄ β₯ π΅ β π΅ β€ π΄) | ||
Theorem | gt-lth 46886 | Relationship between < and > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.) |
β’ π΄ β V & β’ π΅ β V β β’ (π΄ > π΅ β π΅ < π΄) | ||
Theorem | ex-gt 46887 | 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.) |
β’ Β¬ 0 > 0 | ||
Theorem | ex-gte 46888 | Simple example of β₯, in this case, 0 is greater than or equal to 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.) |
β’ 0 β₯ 0 | ||
It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as (cosβ(i Β· π₯)). However, I believe it's important to give formal definitions for these conventional functions as they are typically used, so here they are. A few related identities are also proved. | ||
Syntax | csinh 46889 | Extend class notation to include the hyperbolic sine function, see df-sinh 46892. |
class sinh | ||
Syntax | ccosh 46890 | Extend class notation to include the hyperbolic cosine function. see df-cosh 46893. |
class cosh | ||
Syntax | ctanh 46891 | Extend class notation to include the hyperbolic tangent function, see df-tanh 46894. |
class tanh | ||
Definition | df-sinh 46892 | Define the hyperbolic sine function (sinh). We define it this way for cmpt 5187, which requires the form (π₯ β π΄ β¦ π΅). See sinhval-named 46895 for a simple way to evaluate it. We define this function by dividing by i, which uses fewer operations than many conventional definitions (and thus is more convenient to use in set.mm). See sinh-conventional 46898 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.) |
β’ sinh = (π₯ β β β¦ ((sinβ(i Β· π₯)) / i)) | ||
Definition | df-cosh 46893 | Define the hyperbolic cosine function (cosh). We define it this way for cmpt 5187, which requires the form (π₯ β π΄ β¦ π΅). (Contributed by David A. Wheeler, 10-May-2015.) |
β’ cosh = (π₯ β β β¦ (cosβ(i Β· π₯))) | ||
Definition | df-tanh 46894 | Define the hyperbolic tangent function (tanh). We define it this way for cmpt 5187, which requires the form (π₯ β π΄ β¦ π΅). (Contributed by David A. Wheeler, 10-May-2015.) |
β’ tanh = (π₯ β (β‘cosh β (β β {0})) β¦ ((tanβ(i Β· π₯)) / i)) | ||
Theorem | sinhval-named 46895 | Value of the named sinh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-sinh 46892. See sinhval 15971 for a theorem to convert this further. See sinh-conventional 46898 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.) |
β’ (π΄ β β β (sinhβπ΄) = ((sinβ(i Β· π΄)) / i)) | ||
Theorem | coshval-named 46896 | Value of the named cosh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-cosh 46893. See coshval 15972 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.) |
β’ (π΄ β β β (coshβπ΄) = (cosβ(i Β· π΄))) | ||
Theorem | tanhval-named 46897 | Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh 46894. (Contributed by David A. Wheeler, 10-May-2015.) |
β’ (π΄ β (β‘cosh β (β β {0})) β (tanhβπ΄) = ((tanβ(i Β· π΄)) / i)) | ||
Theorem | sinh-conventional 46898 | Conventional definition of sinh. Here we show that the sinh definition we're using has the same meaning as the conventional definition used in some other sources. We choose a slightly different definition of sinh because it has fewer operations, and thus is more convenient to manipulate using set.mm. (Contributed by David A. Wheeler, 10-May-2015.) |
β’ (π΄ β β β (sinhβπ΄) = (-i Β· (sinβ(i Β· π΄)))) | ||
Theorem | sinhpcosh 46899 | Prove that (sinhβπ΄) + (coshβπ΄) = (expβπ΄) using the conventional hyperbolic trigonometric functions. (Contributed by David A. Wheeler, 27-May-2015.) |
β’ (π΄ β β β ((sinhβπ΄) + (coshβπ΄)) = (expβπ΄)) | ||
Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them. | ||
Syntax | csec 46900 | Extend class notation to include the secant function, see df-sec 46903. |
class sec |
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