P. 491 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49001-49100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prstchom 49001	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.)
|- ( ph -> C = (ProsetToCat ` K ) )   &    |- ( ph -> K e. Proset )   &    |- ( ph -> .<_ = ( le ` C ) )   &    |- ( ph -> H = ( Hom ` C ) )   &    |- ( ph -> X e. ( Base ` C ) )   &    |- ( ph -> Y e. ( Base ` C ) )   =>    |- ( ph -> ( X .<_ Y <-> ( X H Y ) =/= (/) ) )
Theorem	prstchom2 49002*	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 49003). 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.)
|- ( ph -> C = (ProsetToCat ` K ) )   &    |- ( ph -> K e. Proset )   &    |- ( ph -> .<_ = ( le ` C ) )   &    |- ( ph -> H = ( Hom ` C ) )   &    |- ( ph -> X e. ( Base ` C ) )   &    |- ( ph -> Y e. ( Base ` C ) )   =>    |- ( ph -> ( X .<_ Y <-> E! f f e. ( X H Y ) ) )
Theorem	prstchom2ALT 49003*	Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 48987. See prstchom2 49002 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.)
|- ( ph -> C = (ProsetToCat ` K ) )   &    |- ( ph -> K e. Proset )   &    |- ( ph -> .<_ = ( le ` C ) )   &    |- ( ph -> H = ( Hom ` C ) )   =>    |- ( ph -> ( X .<_ Y <-> E! f f e. ( X H Y ) ) )
Theorem	oduoppcbas 49004	The dual of a preordered set and the opposite category have the same set of objects. (Contributed by Zhi Wang, 22-Sep-2025.)
|- ( ph -> C = (ProsetToCat ` K ) )   &    |- ( ph -> K e. Proset )   &    |- ( ph -> D = (ProsetToCat ` (ODual ` K ) ) )   &    |- O = (oppCat ` C )   =>    |- ( ph -> ( Base ` D ) = ( Base ` O ) )
Theorem	oduoppcciso 49005	The dual of a preordered set and the opposite category are category-isomorphic. Example 3.6(1) of [Adamek] p. 25. (Contributed by Zhi Wang, 22-Sep-2025.)
|- ( ph -> C = (ProsetToCat ` K ) )   &    |- ( ph -> K e. Proset )   &    |- ( ph -> D = (ProsetToCat ` (ODual ` K ) ) )   &    |- O = (oppCat ` C )   &    |- ( ph -> U e. V )   &    |- ( ph -> D e. U )   &    |- ( ph -> O e. U )   =>    |- ( ph -> D ( ~=c𝑐 ` (CatCat ` U ) ) O )
Theorem	postcpos 49006	The converted category is a poset iff the original proset is a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
|- ( ph -> C = (ProsetToCat ` K ) )   &    |- ( ph -> K e. Proset )   =>    |- ( ph -> ( K e. Poset <-> C e. Poset ) )
Theorem	postcposALT 49007	Alternate proof for postcpos 49006. (Contributed by Zhi Wang, 25-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> C = (ProsetToCat ` K ) )   &    |- ( ph -> K e. Proset )   =>    |- ( ph -> ( K e. Poset <-> C e. Poset ) )
Theorem	postc 49008*	The converted category is a poset iff no distinct objects are isomorphic. (Contributed by Zhi Wang, 25-Sep-2024.)
|- ( ph -> C = (ProsetToCat ` K ) )   &    |- ( ph -> K e. Proset )   &    |- B = ( Base ` C )   =>    |- ( ph -> ( C e. Poset <-> A. x e. B A. y e. B ( x ( ~=c𝑐 ` C ) y -> x = y ) ) )
21.49.15.3  Monoids as categories
Syntax	cmndtc 49009	Class function defining monoids as categories.
class MndToCat
Definition	df-mndtc 49010	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 49012), 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 49013, mndtchom 49016, mndtcco 49017. 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 e. Mnd |-> { <. ( Base ` ndx ) , { m } >. , <. ( Hom ` ndx ) , { <. m , m , ( Base ` m ) >. } >. , <. (comp ` ndx ) , { <. <. m , m , m >. , ( +g ` m ) >. } >. } )
Theorem	mndtcval 49011	Value of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
|- ( ph -> C = (MndToCat ` M ) )   &    |- ( ph -> M e. Mnd )   =>    |- ( ph -> C = { <. ( Base ` ndx ) , { M } >. , <. ( Hom ` ndx ) , { <. M , M , ( Base ` M ) >. } >. , <. (comp ` ndx ) , { <. <. M , M , M >. , ( +g ` M ) >. } >. } )
Theorem	mndtcbasval 49012	The base set of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
|- ( ph -> C = (MndToCat ` M ) )   &    |- ( ph -> M e. Mnd )   &    |- ( ph -> B = ( Base ` C ) )   =>    |- ( ph -> B = { M } )
Theorem	mndtcbas 49013*	The category built from a monoid contains precisely one object. (Contributed by Zhi Wang, 22-Sep-2024.)
|- ( ph -> C = (MndToCat ` M ) )   &    |- ( ph -> M e. Mnd )   &    |- ( ph -> B = ( Base ` C ) )   =>    |- ( ph -> E! x x e. B )
Theorem	mndtcob 49014	Lemma for mndtchom 49016 and mndtcco 49017. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
|- ( ph -> C = (MndToCat ` M ) )   &    |- ( ph -> M e. Mnd )   &    |- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> X e. B )   =>    |- ( ph -> X = M )
Theorem	mndtcbas2 49015	Two objects in a category built from a monoid are identical. (Contributed by Zhi Wang, 24-Sep-2024.)
|- ( ph -> C = (MndToCat ` M ) )   &    |- ( ph -> M e. Mnd )   &    |- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> X = Y )
Theorem	mndtchom 49016	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.)
|- ( ph -> C = (MndToCat ` M ) )   &    |- ( ph -> M e. Mnd )   &    |- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> H = ( Hom ` C ) )   =>    |- ( ph -> ( X H Y ) = ( Base ` M ) )
Theorem	mndtcco 49017	The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.)
|- ( ph -> C = (MndToCat ` M ) )   &    |- ( ph -> M e. Mnd )   &    |- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> .x. = (comp ` C ) )   =>    |- ( ph -> ( <. X , Y >. .x. Z ) = ( +g ` M ) )
Theorem	mndtcco2 49018	The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.)
|- ( ph -> C = (MndToCat ` M ) )   &    |- ( ph -> M e. Mnd )   &    |- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> .x. = (comp ` C ) )   &    |- ( ph -> .o. = ( <. X , Y >. .x. Z ) )   =>    |- ( ph -> ( G .o. F ) = ( G ( +g ` M ) F ) )
Theorem	mndtccatid 49019*	Lemma for mndtccat 49020 and mndtcid 49021. (Contributed by Zhi Wang, 22-Sep-2024.)
|- ( ph -> C = (MndToCat ` M ) )   &    |- ( ph -> M e. Mnd )   =>    |- ( ph -> ( C e. Cat /\ ( Id ` C ) = ( y e. ( Base ` C ) |-> ( 0g ` M ) ) ) )
Theorem	mndtccat 49020	The function value is a category. (Contributed by Zhi Wang, 22-Sep-2024.)
|- ( ph -> C = (MndToCat ` M ) )   &    |- ( ph -> M e. Mnd )   =>    |- ( ph -> C e. Cat )
Theorem	mndtcid 49021	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.)
|- ( ph -> C = (MndToCat ` M ) )   &    |- ( ph -> M e. Mnd )   &    |- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> X e. B )   &    |- ( ph -> .1. = ( Id ` C ) )   =>    |- ( ph -> ( .1. ` X ) = ( 0g ` M ) )
Theorem	oppgoppchom 49022	The converted opposite monoid has the same hom-set as that of the opposite category. Example 3.6(2) of [Adamek] p. 25. (Contributed by Zhi Wang, 21-Sep-2025.)
|- ( ph -> C = (MndToCat ` M ) )   &    |- ( ph -> M e. Mnd )   &    |- ( ph -> D = (MndToCat ` (oppg ` M ) ) )   &    |- O = (oppCat ` C )   &    |- ( ph -> X e. ( Base ` D ) )   &    |- ( ph -> Y e. ( Base ` O ) )   &    |- ( ph -> H = ( Hom ` D ) )   &    |- ( ph -> J = ( Hom ` O ) )   =>    |- ( ph -> ( X H X ) = ( Y J Y ) )
Theorem	oppgoppcco 49023	The converted opposite monoid has the same composition as that of the opposite category. Example 3.6(2) of [Adamek] p. 25. (Contributed by Zhi Wang, 22-Sep-2025.)
|- ( ph -> C = (MndToCat ` M ) )   &    |- ( ph -> M e. Mnd )   &    |- ( ph -> D = (MndToCat ` (oppg ` M ) ) )   &    |- O = (oppCat ` C )   &    |- ( ph -> X e. ( Base ` D ) )   &    |- ( ph -> Y e. ( Base ` O ) )   &    |- ( ph -> .x. = (comp ` D ) )   &    |- ( ph -> .xb = (comp ` O ) )   =>    |- ( ph -> ( <. X , X >. .x. X ) = ( <. Y , Y >. .xb Y ) )
Theorem	oppgoppcid 49024	The converted opposite monoid has the same identity morphism as that of the opposite category. Example 3.6(2) of [Adamek] p. 25. (Contributed by Zhi Wang, 22-Sep-2025.)
|- ( ph -> C = (MndToCat ` M ) )   &    |- ( ph -> M e. Mnd )   &    |- ( ph -> D = (MndToCat ` (oppg ` M ) ) )   &    |- O = (oppCat ` C )   &    |- ( ph -> X e. ( Base ` D ) )   &    |- ( ph -> Y e. ( Base ` O ) )   =>    |- ( ph -> ( ( Id ` D ) ` X ) = ( ( Id ` O ) ` Y ) )
Theorem	grptcmon 49025	All morphisms in a category converted from a group are monomorphisms. (Contributed by Zhi Wang, 23-Sep-2024.)
|- ( ph -> C = (MndToCat ` G ) )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> H = ( Hom ` C ) )   &    |- ( ph -> M = (Mono ` C ) )   =>    |- ( ph -> ( X M Y ) = ( X H Y ) )
Theorem	grptcepi 49026	All morphisms in a category converted from a group are epimorphisms. (Contributed by Zhi Wang, 23-Sep-2024.)
|- ( ph -> C = (MndToCat ` G ) )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> H = ( Hom ` C ) )   &    |- ( ph -> E = (Epi ` C ) )   =>    |- ( ph -> ( X E Y ) = ( X H Y ) )
21.50  Mathbox for Emmett Weisz
21.50.1  Miscellaneous Theorems Some of these theorems are used in the series of lemmas and theorems proving the defining properties of setrecs.
Theorem	nfintd 49027	Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.)
|- ( ph -> F/_ x A )   =>    |- ( ph -> F/_ x |^| A )
Theorem	nfiund 49028*	Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2377. See nfiundg 49029 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.)
|- F/ x ph   &    |- ( ph -> F/_ y A )   &    |- ( ph -> F/_ y B )   =>    |- ( ph -> F/_ y U_ x e. A B )
Theorem	nfiundg 49029	Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2377, see nfiund 49028 for a weaker version that does not require it. (Contributed by Emmett Weisz, 6-Dec-2019.) (New usage is discouraged.)
|- F/ x ph   &    |- ( ph -> F/_ y A )   &    |- ( ph -> F/_ y B )   =>    |- ( ph -> F/_ y U_ x e. A B )
Theorem	iunord 49030*	The indexed union of a collection of ordinal numbers B ( x ) is ordinal. This proof is based on the proof of ssorduni 7805, but does not use it directly, since ssorduni 7805 does not work when B is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.)
|- ( A. x e. A Ord B -> Ord U_ x e. A B )
Theorem	iunordi 49031*	The indexed union of a collection of ordinal numbers B ( x ) is ordinal. (Contributed by Emmett Weisz, 3-Nov-2019.)
|- Ord B   =>    |- Ord U_ x e. A B
Theorem	spd 49032	Specialization deduction, using implicit substitution. Based on the proof of spimed 2393. (Contributed by Emmett Weisz, 17-Jan-2020.)
|- ( ch -> F/ x ps )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( ch -> ( A. x ph -> ps ) )
Theorem	spcdvw 49033*	A version of spcdv 3597 where ps and ch are direct substitutions of each other. This theorem is useful because it does not require ph and x to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.)
|- ( ph -> A e. B )   &    |- ( x = A -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps -> ch ) )
Theorem	tfis2d 49034*	Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.)
|- ( ph -> ( x = y -> ( ps <-> ch ) ) )   &    |- ( ph -> ( x e. On -> ( A. y e. x ch -> ps ) ) )   =>    |- ( ph -> ( x e. On -> ps ) )
Theorem	bnd2d 49035*	Deduction form of bnd2 9940. (Contributed by Emmett Weisz, 19-Jan-2021.)
|- ( ph -> A e. _V )   &    |- ( ph -> A. x e. A E. y e. B ps )   =>    |- ( ph -> E. z ( z C_ B /\ A. x e. A E. y e. z ps ) )
Theorem	dffun3f 49036*	Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.)
|- F/_ x A   &    |- F/_ y A   &    |- F/_ z A   =>    |- ( Fun A <-> ( Rel A /\ A. x E. z A. y ( x A y -> y = z ) ) )
21.50.2  Set Recursion
21.50.2.1  Basic Properties of Set Recursion Symbols in this section: All the symbols used in the definition of setrecs ( F ) are explained in the comment of df-setrecs 49038. The class Y is explained in the comment of setrec1lem1 49041. Glossaries of symbols used in individual proofs, or used differently in different proofs, are in the comments of those proofs.
Syntax	csetrecs 49037	Extend class notation to include a set defined by transfinite recursion.
class setrecs ( F )
Definition	df-setrecs 49038*	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 49045 and setrec2v 49050, 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 49062 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 C_ 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 ) = U. ran recs ( ( g e. _V |-> ( F ` U. ran g ) ) ). I had initially tried and failed to find an elementary definition, and I had proven theorems analogous to setrec1 49045 and setrec2v 49050 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 49045 and setrec2v 49050), 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 8458, 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 ) = U. 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 e. A must be generated by F when applied to a previous element y e. A. In symbols, A. x e. A E. y e. A ( y C_ x /\ x C_ ( 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 e. 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 U. 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 ) C_ b }, then ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x A. z e. x -. ( F ` z ) C_ y ) ). On 22-Jul-2020 I came up with the following definition (Version 1) phrased in terms of induction: U. { y | A. z ( A. w ( w C_ y -> ( w e. z -> ( F ` w ) e. z ) ) -> y e. 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. U. { y | A. z ( y e. z -> E. w ( w C_ y /\ ( F ` w ) e. z /\ -. w e. 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 ( A. w ... y e. 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 e. z with C_ 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 e. z, so the definition requires the additional constraint ( a C_ b -> ( F ` a ) C_ ( F ` b ) ) in order to work. This issue can also be avoided by replacing e. z with C_ z. The induction version of the result is used in the final definition. Version 2: (18-Aug-2020) Induction: U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } Foundedness: U. { y | A. z ( y i^i z =/= (/) -> E. w ( w C_ y /\ w i^i z = (/) /\ ( F ` w ) i^i 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 C_ ( y i^i 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 ) i^i 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 C_ z would be false. We could then simply add all elements of ( F ` w ) outside of y for any w C_ 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 ) C_ z for w C_ ( y i^i z ), and in particular ( ( F ` w ) i^i y ) C_ z. The foundedness version is starting to look more like ax-reg 9639! 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 e. z overlapping with w. Versions 3, 4, and 5 are all equivalent to Version 2. Version 3 - Foundedness (5-Sep-2020): U. { y | A. z ( ( z C_ y /\ z =/= (/) ) -> E. v e. z E. w ( w C_ y /\ w i^i z = (/) /\ v e. ( F ` w ) ) ) } Now, if we replace ( F ` w ) by ( ( F ` w ) i^i y ), we do not change the definition. We already know that v e. y since v e. z and z C_ 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 5645, we add another variable u representing the nonexistent element of w in z. Version 4 - Foundedness (6-Sep-2020): U. { y | A. z ( ( z C_ y /\ z =/= (/) ) -> E. v e. z E. w A. u e. z ( w C_ y /\ -. u e. w /\ v e. ( F ` w ) ) This is so close to df-fr 5645; the only change needed is to switch E. w with A. u e. 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 9639. 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 U. z in the induction. With this stronger induction, we can also allow for the stronger requirement ~P y C_ z rather than only y e. z. This will help improve the foundedness version. Version 1.1 (28-Aug-2020) Induction: U. { y | A. z ( A. w ( w C_ y -> ( w C_ U. z -> ( F ` w ) e. z ) ) -> ~P y C_ z ) } Foundedness: U. { y | A. z ( E. a ( a C_ y /\ a e. z ) -> E. w ( w C_ y /\ w i^i |^| z = (/) /\ ( F ` w ) e. 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 U. 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: U. { y | A. z ( ( z C_ ~P y /\ z =/= (/) ) -> E. w ( w e. ~P y /\ w i^i |^| z = (/) /\ ( F ` w ) e. z ) ) } Now this looks more like df-fr 5645! 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 5645. 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: U. { y | A. z ( A. w ( w C_ y -> ( w C_ U. z -> ( w e. z /\ ( F ` w ) e. z ) ) ) -> ~P y C_ z ) } Foundedness: U. { y | A. z ( ( z C_ ~P y /\ z =/= (/) ) -> E. w ( w e. ~P y /\ w i^i |^| z = (/) /\ ( w e. z \/ ( F ` w ) e. 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 e. ( 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 5645. Thus, in our case, df-fr 5645 is equivalent to ( R Fr A <-> A. z ( ( z C_ A /\ z =/= (/) ) -> E. w ( ( F ` w ) e. z /\ -. E. v e. z v R ( F ` w ) ) ). Substituting our relation R gives ( R Fr A <-> A. z ( ( z C_ A /\ z =/= (/) ) -> E. w ( ( F ` w ) e. z /\ -. E. v e. z ( F ` w ) e. ( 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 e. z <-> A. v e. z -. v R ( F ` w ) so we can replace one with the other. Negate both sides: w e. z <-> E. v e. z v R ( 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 ) = U. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }
Theorem	setrecseq 49039	Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.)
|- ( F = G -> setrecs ( F ) = setrecs ( G ) )
Theorem	nfsetrecs 49040	Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.)
|- F/_ x F   =>    |- F/_ xsetrecs ( F )
Theorem	setrec1lem1 49041*	Lemma for setrec1 49045. This is a utility theorem showing the equivalence of the statement X e. Y and its expanded form. The proof uses elabg 3680 and equivalence theorems. Variable Y is the class of sets y that are recursively generated by the function F. In other words, y e. 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 | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }   &    |- ( ph -> X e. V )   =>    |- ( ph -> ( X e. Y <-> A. z ( A. w ( w C_ X -> ( w C_ z -> ( F ` w ) C_ z ) ) -> X C_ z ) ) )
Theorem	setrec1lem2 49042*	Lemma for setrec1 49045. 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 3987, 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 | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }   &    |- ( ph -> X e. V )   &    |- ( ph -> X C_ Y )   =>    |- ( ph -> U. X e. Y )
Theorem	setrec1lem3 49043*	Lemma for setrec1 49045. 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 49042. It turns out that x = A also works, i.e., given the hypotheses it is possible to prove that A e. Y. I don't know if proving this fact directly using setrec1lem1 49041 would be any easier than the current proof using setrec1lem2 49042, and it would only slightly simplify the proof of setrec1 49045. Other than the use of bnd2d 49035, this is a purely technical theorem for rearranging notation from that of setrec1lem2 49042 to that of setrec1 49045. (Contributed by Emmett Weisz, 20-Jan-2021.) (New usage is discouraged.)
|- Y = { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }   &    |- ( ph -> A e. _V )   &    |- ( ph -> A. a e. A E. x ( a e. x /\ x e. Y ) )   =>    |- ( ph -> E. x ( A C_ x /\ x e. Y ) )
Theorem	setrec1lem4 49044*	Lemma for setrec1 49045. If X is recursively generated by F, then so is X u. ( F ` A ). In the proof of setrec1 49045, the following is substituted for this theorem's ph: ( ph /\ ( A C_ x /\ x e. { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) } ) ) Therefore, we cannot declare z to be a distinct variable from ph, since we need it to appear as a bound variable in ph. This theorem can be proven without the hypothesis F/ z ph, but the proof would be harder to read because theorems in deduction form would be interrupted by theorems like eximi 1834, making the antecedent of each line something more complicated than ph. The proof of setrec1lem2 49042 could similarly be made easier to read by adding the hypothesis F/ z ph, 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.)
|- F/ z ph   &    |- Y = { y | A. z ( A. w ( w C_ y -> ( w C_ z -> ( F ` w ) C_ z ) ) -> y C_ z ) }   &    |- ( ph -> A e. _V )   &    |- ( ph -> A C_ X )   &    |- ( ph -> X e. Y )   =>    |- ( ph -> ( X u. ( F ` A ) ) e. Y )
Theorem	setrec1 49045	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 C_ B, meaning that all elements of A are in some set recursively generated by F. Then by setrec1lem3 49043, 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 49043.) Therefore, by setrec1lem4 49044, ( F ` A ) is a subset of some set recursively generated by F. Thus, by ssuni 4940, it is a subset of the union of all sets recursively generated by F. See df-setrecs 49038 for a detailed description of how the setrecs definition works. (Contributed by Emmett Weisz, 9-Oct-2020.)
|- B = setrecs ( F )   &    |- ( ph -> A e. _V )   &    |- ( ph -> A C_ B )   =>    |- ( ph -> ( F ` A ) C_ B )
Theorem	setrec2fun 49046*	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 49045 and setrec2v 49050 say that setrecs ( F ) is the minimal class closed under F. We express this by saying that if F respects the C_ relation and C is closed under F, then B C_ 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 7881) to the other class. (Contributed by Emmett Weisz, 15-Feb-2021.) (New usage is discouraged.)
|- F/_ a F   &    |- B = setrecs ( F )   &    |- Fun F   &    |- ( ph -> A. a ( a C_ C -> ( F ` a ) C_ C ) )   =>    |- ( ph -> B C_ C )
Theorem	setrec2lem1 49047*	Lemma for setrec2 49049. The functional part of F has the same values as F. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.)
|- ( ( F |` { x | E! y x F y } ) ` a ) = ( F ` a )
Theorem	setrec2lem2 49048*	Lemma for setrec2 49049. The functional part of F is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.)
|- Fun ( F |` { x | E! y x F y } )
Theorem	setrec2 49049*	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 49045 and setrec2v 49050 uniquely determine setrecs ( F ) to be the minimal class closed under F. We express this by saying that if F respects the C_ relation and C is closed under F, then B C_ 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 7881) to the other class. (Contributed by Emmett Weisz, 2-Sep-2021.)
|- F/_ a F   &    |- B = setrecs ( F )   &    |- ( ph -> A. a ( a C_ C -> ( F ` a ) C_ C ) )   =>    |- ( ph -> B C_ C )
Theorem	setrec2v 49050*	Version of setrec2 49049 with a disjoint variable condition instead of a nonfreeness hypothesis. (Contributed by Emmett Weisz, 6-Mar-2021.)
|- B = setrecs ( F )   &    |- ( ph -> A. a ( a C_ C -> ( F ` a ) C_ C ) )   =>    |- ( ph -> B C_ C )
Theorem	setrec2mpt 49051*	Version of setrec2 49049 where F is defined using maps-to notation. Deduction form is omitted in the second hypothesis for simplicity. In practice, nothing important is lost since we are only interested in one choice of A, S, and V at a time. However, we are interested in what happens when C varies, so deduction form is used in the third hypothesis. (Contributed by Emmett Weisz, 4-Jun-2024.)
|- B = setrecs ( ( a e. A |-> S ) )   &    |- ( a e. A -> S e. V )   &    |- ( ph -> A. a ( a C_ C -> S C_ C ) )   =>    |- ( ph -> B C_ C )
Theorem	setis 49052*	Version of setrec2 49049 expressed as an induction schema. This theorem is a generalization of tfis3 7886. (Contributed by Emmett Weisz, 27-Feb-2022.)
|- B = setrecs ( F )   &    |- ( b = A -> ( ps <-> ch ) )   &    |- ( ph -> A. a ( A. b e. a ps -> A. b e. ( F ` a ) ps ) )   =>    |- ( ph -> ( A e. B -> ch ) )
21.50.2.2  Examples and properties of set recursion
Theorem	elsetrecslem 49053*	Lemma for elsetrecs 49054. Any element of setrecs ( F ) is generated by some subset of setrecs ( F ). This is much weaker than setrec2v 49050. To see why this lemma also requires setrec1 49045, 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 e. B -> E. x ( x C_ B /\ A e. ( F ` x ) ) )
Theorem	elsetrecs 49054*	A set A is an element of setrecs ( F ) iff A is generated by some subset of setrecs ( F ). The proof requires both setrec1 49045 and setrec2 49049, 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 e. are replaced by C_ for a stronger version of the theorem. (Contributed by Emmett Weisz, 12-Jul-2021.)
|- B = setrecs ( F )   =>    |- ( A e. B <-> E. x ( x C_ B /\ A e. ( F ` x ) ) )
Theorem	setrecsss 49055	The setrecs operator respects the subset relation between two functions F and G. (Contributed by Emmett Weisz, 13-Mar-2022.)
|- ( ph -> Fun G )   &    |- ( ph -> F C_ G )   =>    |- ( ph -> setrecs ( F ) C_ setrecs ( G ) )
Theorem	setrecsres 49056	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 )   &    |- ( ph -> Fun F )   =>    |- ( ph -> B = setrecs ( ( F |` ~P B ) ) )
Theorem	vsetrec 49057	Construct _V using set recursion. The proof indirectly uses trcl 9775, which relies on rec, but theoretically C in trcl 9775 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 e. _V |-> ~P x )   =>    |- setrecs ( F ) = _V
Theorem	0setrec 49058	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.)
|- ( ph -> ( F ` (/) ) = (/) )   =>    |- ( ph -> setrecs ( F ) = (/) )
Theorem	onsetreclem1 49059*	Lemma for onsetrec 49062. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
|- F = ( x e. _V |-> { U. x , suc U. x } )   =>    |- ( F ` a ) = { U. a , suc U. a }
Theorem	onsetreclem2 49060*	Lemma for onsetrec 49062. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
|- F = ( x e. _V |-> { U. x , suc U. x } )   =>    |- ( a C_ On -> ( F ` a ) C_ On )
Theorem	onsetreclem3 49061*	Lemma for onsetrec 49062. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
|- F = ( x e. _V |-> { U. x , suc U. x } )   =>    |- ( a e. On -> a e. ( F ` a ) )
Theorem	onsetrec 49062	Construct On using set recursion. When x e. On, the function F constructs the least ordinal greater than any of the elements of x, which is U. x for a limit ordinal and suc U. x for a successor ordinal. For example, ( F ` { 1o , 2o } ) = { U. { 1o , 2o } , suc U. { 1o , 2o } } = { 2o , 3o } which contains 3o, and ( F ` om ) = { U. om , suc U. om } = { om , om +o 1o }, which contains om. 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 49060 and onsetreclem3 49061 will recursively generate On; for example, F = ( x e. _V |-> suc suc U. 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 e. _V |-> { y e. ~P x | Tr y } ), based on dfon2 35787. 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 e. _V |-> { U. x , suc U. x } )   =>    |- setrecs ( F ) = On
21.50.3  Construction of Games and Surreal Numbers Model organization after organization of reals - see TOC
Syntax	cpg 49063	Extend class notation to include the class of partisan game forms.
class Pg
Definition	df-pg 49064	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 e. _V |-> ( ~P x X. ~P x ) ) )
Theorem	elpglem1 49065*	Lemma for elpg 49068. (Contributed by Emmett Weisz, 28-Aug-2021.)
|- ( E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) -> ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) )
Theorem	elpglem2 49066*	Lemma for elpg 49068. (Contributed by Emmett Weisz, 28-Aug-2021.)
|- ( ( ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) -> E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) )
Theorem	elpglem3 49067*	Lemma for elpg 49068. (Contributed by Emmett Weisz, 28-Aug-2021.)
|- ( E. x ( x C_ Pg /\ A e. ( ( y e. _V |-> ( ~P y X. ~P y ) ) ` x ) ) <-> ( A e. ( _V X. _V ) /\ E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) )
Theorem	elpg 49068	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 e. Pg <-> ( A e. ( _V X. _V ) /\ ( 1st ` A ) C_ Pg /\ ( 2nd ` A ) C_ Pg ) )
Theorem	pgindlem 49069	Lemma for pgind 49071. (Contributed by Emmett Weisz, 27-May-2024.) (New usage is discouraged.)
|- ( x e. ( ~P z X. ~P z ) -> ( ( 1st ` x ) u. ( 2nd ` x ) ) C_ z )
Theorem	pgindnf 49070*	Version of pgind 49071 with extraneous not-free requirements. (Contributed by Emmett Weisz, 27-May-2024.) (New usage is discouraged.)
|- F/ x ph   &    |- F/ y ph   &    |- ( x = y -> ( ps <-> ch ) )   &    |- ( y = A -> ( ch <-> th ) )   &    |- ( ph -> A. x ( A. y e. ( ( 1st ` x ) u. ( 2nd ` x ) ) ch -> ps ) )   =>    |- ( ph -> ( A e. Pg -> th ) )
Theorem	pgind 49071*	Induction on partizan games. (Contributed by Emmett Weisz, 27-May-2024.)
|- ( x = y -> ( ps <-> ch ) )   &    |- ( y = A -> ( ch <-> th ) )   &    |- ( ph -> A. x ( A. y e. ( ( 1st ` x ) u. ( 2nd ` x ) ) ch -> ps ) )   =>    |- ( ph -> ( A e. Pg -> th ) )
21.51  Mathbox for David A. Wheeler 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/.
21.51.1  Natural deduction
Theorem	sbidd 49072	An identity theorem for substitution. See sbid 2255. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
|- ( ph -> [ x / x ] ps )   =>    |- ( ph -> ps )
Theorem	sbidd-misc 49073	An identity theorem for substitution. See sbid 2255. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
|- ( ( ph -> [ x / x ] ps ) <-> ( ph -> ps ) )
21.51.2  Greater than, greater than or equal to. 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 49074	Extend wff notation to include the 'greater than or equal to' relation, see df-gte 49076.
class >_
Syntax	cgt 49075	Extend wff notation to include the 'greater than' relation, see df-gt 49077.
class >
Definition	df-gte 49076	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 11308. 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 e. RR* /\ y e. RR* ) /\ y < x ) } and |- >_ = { <. x , y >. | ( ( x e. RR* /\ y e. RR* ) /\ y <_ x ) } but these are very complicated. This definition of >_, and the similar one for > (df-gt 49077), 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 49078 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 49077	The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 11175. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 49076 for a discussion on why this approach is used for the definition. See gt-lt 49079 and gt-lth 49081 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 49078	Simple relationship between <_ and >_. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
|- ( ( A e. _V /\ B e. _V ) -> ( A >_ B <-> B <_ A ) )
Theorem	gt-lt 49079	Simple relationship between < and >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
|- ( ( A e. _V /\ B e. _V ) -> ( A > B <-> B < A ) )
Theorem	gte-lteh 49080	Relationship between <_ and >_ using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
|- A e. _V   &    |- B e. _V   =>    |- ( A >_ B <-> B <_ A )
Theorem	gt-lth 49081	Relationship between < and > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
|- A e. _V   &    |- B e. _V   =>    |- ( A > B <-> B < A )
Theorem	ex-gt 49082	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
Theorem	ex-gte 49083	Simple example of >_, 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.)
|- 0 >_ 0
21.51.3  Hyperbolic trigonometric functions It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as ( cos ` ( _i x. x ) ). 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 49084	Extend class notation to include the hyperbolic sine function, see df-sinh 49087.
class sinh
Syntax	ccosh 49085	Extend class notation to include the hyperbolic cosine function. see df-cosh 49088.
class cosh
Syntax	ctanh 49086	Extend class notation to include the hyperbolic tangent function, see df-tanh 49089.
class tanh
Definition	df-sinh 49087	Define the hyperbolic sine function (sinh). We define it this way for cmpt 5234, which requires the form ( x e. A |-> B ). See sinhval-named 49090 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 49093 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.)
|- sinh = ( x e. CC |-> ( ( sin ` ( _i x. x ) ) / _i ) )
Definition	df-cosh 49088	Define the hyperbolic cosine function (cosh). We define it this way for cmpt 5234, which requires the form ( x e. A |-> B ). (Contributed by David A. Wheeler, 10-May-2015.)
|- cosh = ( x e. CC |-> ( cos ` ( _i x. x ) ) )
Definition	df-tanh 49089	Define the hyperbolic tangent function (tanh). We define it this way for cmpt 5234, which requires the form ( x e. A |-> B ). (Contributed by David A. Wheeler, 10-May-2015.)
|- tanh = ( x e. ( `'cosh " ( CC \ { 0 } ) ) |-> ( ( tan ` ( _i x. x ) ) / _i ) )
Theorem	sinhval-named 49090	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 49087. See sinhval 16196 for a theorem to convert this further. See sinh-conventional 49093 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.)
|- ( A e. CC -> (sinh ` A ) = ( ( sin ` ( _i x. A ) ) / _i ) )
Theorem	coshval-named 49091	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 49088. See coshval 16197 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.)
|- ( A e. CC -> (cosh ` A ) = ( cos ` ( _i x. A ) ) )
Theorem	tanhval-named 49092	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 49089. (Contributed by David A. Wheeler, 10-May-2015.)
|- ( A e. ( `'cosh " ( CC \ { 0 } ) ) -> (tanh ` A ) = ( ( tan ` ( _i x. A ) ) / _i ) )
Theorem	sinh-conventional 49093	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.)
|- ( A e. CC -> (sinh ` A ) = ( -u _i x. ( sin ` ( _i x. A ) ) ) )
Theorem	sinhpcosh 49094	Prove that (sinh ` A ) + (cosh ` A ) = ( exp ` A ) using the conventional hyperbolic trigonometric functions. (Contributed by David A. Wheeler, 27-May-2015.)
|- ( A e. CC -> ( (sinh ` A ) + (cosh ` A ) ) = ( exp ` A ) )
21.51.4  Reciprocal trigonometric functions (sec, csc, cot) Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them.
Syntax	csec 49095	Extend class notation to include the secant function, see df-sec 49098.
class sec
Syntax	ccsc 49096	Extend class notation to include the cosecant function, see df-csc 49099.
class csc
Syntax	ccot 49097	Extend class notation to include the cotangent function, see df-cot 49100.
class cot
Definition	df-sec 49098*	Define the secant function. We define it this way for cmpt 5234, which requires the form ( x e. A |-> B ). The sec function is defined in ISO 80000-2:2009(E) operation 2-13.6 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 5234. (Contributed by David A. Wheeler, 14-Mar-2014.)
|- sec = ( x e. { y e. CC | ( cos ` y ) =/= 0 } |-> ( 1 / ( cos ` x ) ) )
Definition	df-csc 49099*	Define the cosecant function. We define it this way for cmpt 5234, which requires the form ( x e. A |-> B ). The csc function is defined in ISO 80000-2:2009(E) operation 2-13.7 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 5234. (Contributed by David A. Wheeler, 14-Mar-2014.)
|- csc = ( x e. { y e. CC | ( sin ` y ) =/= 0 } |-> ( 1 / ( sin ` x ) ) )
Definition	df-cot 49100*	Define the cotangent function. We define it this way for cmpt 5234, which requires the form ( x e. A |-> B ). The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 5234. (Contributed by David A. Wheeler, 14-Mar-2014.)
|- cot = ( x e. { y e. CC | ( sin ` y ) =/= 0 } |-> ( ( cos ` x ) / ( sin ` x ) ) )