P. 66 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6501-6600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tfrcllembfn 6501*	Lemma for tfrcl 6508. The union of B is a function defined on x. (Contributed by Jim Kingdon, 25-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g : z --> S /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> U. B : D --> S )
Theorem	tfrcllembex 6502*	Lemma for tfrcl 6508. The set B exists. (Contributed by Jim Kingdon, 25-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g : z --> S /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> B e. _V )
Theorem	tfrcllemubacc 6503*	Lemma for tfrcl 6508. The union of B satisfies the recursion rule. (Contributed by Jim Kingdon, 25-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g : z --> S /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) )
Theorem	tfrcllemex 6504*	Lemma for tfrcl 6508. (Contributed by Jim Kingdon, 26-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g : z --> S /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> E. f ( f : D --> S /\ A. u e. D ( f ` u ) = ( G ` ( f |` u ) ) ) )
Theorem	tfrcllemaccex 6505*	We can define an acceptable function on any element of X. As with many of the transfinite recursion theorems, we have hypotheses that state that F is a function and that it is defined up to X. (Contributed by Jim Kingdon, 26-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   =>    |- ( ( ph /\ C e. X ) -> E. g ( g : C --> S /\ A. u e. C ( g ` u ) = ( G ` ( g |` u ) ) ) )
Theorem	tfrcllemres 6506*	Lemma for tfr1on 6494. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> Y e. X )   =>    |- ( ph -> Y C_ dom F )
Theorem	tfrcldm 6507*	Recursion is defined on an ordinal if the characteristic function satisfies a closure hypothesis up to a suitable point. (Contributed by Jim Kingdon, 26-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> Y e. U. X )   =>    |- ( ph -> Y e. dom F )
Theorem	tfrcl 6508*	Closure for transfinite recursion. As with tfr1on 6494, the characteristic function must be defined up to a suitable point, not necessarily on all ordinals. (Contributed by Jim Kingdon, 25-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> Y e. U. X )   =>    |- ( ph -> ( F ` Y ) e. S )
Theorem	tfri1 6509*	Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition. The condition is that G is defined "everywhere", which is stated here as ( G ` x ) e. _V. Alternately, A. x e. On A. f ( f Fn x -> f e. dom G ) would suffice. Given a function G satisfying that condition, we define a class A of all "acceptable" functions. The final function we're interested in is the union F = recs ( G ) of them. F is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of F. In this first part we show that F is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)
|- F = recs ( G )   &    |- ( Fun G /\ ( G ` x ) e. _V )   =>    |- F Fn On
Theorem	tfri2 6510*	Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule G ( as described at tfri1 6509). Here we show that the function F has the property that for any function G satisfying that condition, the "next" value of F is G recursively applied to all "previous" values of F. (Contributed by Jim Kingdon, 4-May-2019.)
|- F = recs ( G )   &    |- ( Fun G /\ ( G ` x ) e. _V )   =>    |- ( A e. On -> ( F ` A ) = ( G ` ( F |` A ) ) )
Theorem	tfri3 6511*	Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule G ( as described at tfri1 6509). Finally, we show that F is unique. We do this by showing that any class B with the same properties of F that we showed in parts 1 and 2 is identical to F. (Contributed by Jim Kingdon, 4-May-2019.)
|- F = recs ( G )   &    |- ( Fun G /\ ( G ` x ) e. _V )   =>    |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> B = F )
Theorem	tfrex 6512*	The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- F = recs ( G )   &    |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) )   =>    |- ( ( ph /\ A e. V ) -> ( F ` A ) e. _V )
2.6.21  Recursive definition generator
Syntax	crdg 6513	Extend class notation with the recursive definition generator, with characteristic function F and initial value I.
class rec ( F , I )
Definition	df-irdg 6514*	Define a recursive definition generator on On (the class of ordinal numbers) with characteristic function F and initial value I. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our rec operation (especially when df-recs 6449 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple. In classical logic it would be easier to divide this definition into cases based on whether the domain of g is zero, a successor, or a limit ordinal. Cases do not (in general) work that way in intuitionistic logic, so instead we choose a definition which takes the union of all the results of the characteristic function for ordinals in the domain of g. This means that this definition has the expected properties for increasing and continuous ordinal functions, which include ordinal addition and multiplication. For finite recursion we also define df-frec 6535 and for suitable characteristic functions df-frec 6535 yields the same result as rec restricted to om, as seen at frecrdg 6552. Note: We introduce rec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Jim Kingdon, 19-May-2019.)
|- rec ( F , I ) = recs ( ( g e. _V |-> ( I u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
Theorem	rdgeq1 6515	Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
|- ( F = G -> rec ( F , A ) = rec ( G , A ) )
Theorem	rdgeq2 6516	Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
|- ( A = B -> rec ( F , A ) = rec ( F , B ) )
Theorem	rdgfun 6517	The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
|- Fun rec ( F , A )
Theorem	rdgtfr 6518*	The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.)
|- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> ( Fun ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) /\ ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` f ) e. _V ) )
Theorem	rdgruledefgg 6519*	The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
|- ( ( F Fn _V /\ A e. V ) -> ( Fun ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) /\ ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` f ) e. _V ) )
Theorem	rdgruledefg 6520*	The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
|- F Fn _V   =>    |- ( A e. V -> ( Fun ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) /\ ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` f ) e. _V ) )
Theorem	rdgexggg 6521	The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
|- ( ( F Fn _V /\ A e. V /\ B e. W ) -> ( rec ( F , A ) ` B ) e. _V )
Theorem	rdgexgg 6522	The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
|- F Fn _V   =>    |- ( ( A e. V /\ B e. W ) -> ( rec ( F , A ) ` B ) e. _V )
Theorem	rdgifnon 6523	The recursive definition generator is a function on ordinal numbers. The F Fn _V condition states that the characteristic function is defined for all sets (being defined for all ordinals might be enough if being used in a manner similar to rdgon 6530; in cases like df-oadd 6564 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.)
|- ( ( F Fn _V /\ A e. V ) -> rec ( F , A ) Fn On )
Theorem	rdgifnon2 6524*	The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.)
|- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> rec ( F , A ) Fn On )
Theorem	rdgivallem 6525*	Value of the recursive definition generator. Lemma for rdgival 6526 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.)
|- ( ( F Fn _V /\ A e. V /\ B e. On ) -> ( rec ( F , A ) ` B ) = ( A u. U_ x e. B ( F ` ( ( rec ( F , A ) |` B ) ` x ) ) ) )
Theorem	rdgival 6526*	Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.)
|- ( ( F Fn _V /\ A e. V /\ B e. On ) -> ( rec ( F , A ) ` B ) = ( A u. U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) ) )
Theorem	rdgss 6527	Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.)
|- ( ph -> F Fn _V )   &    |- ( ph -> I e. V )   &    |- ( ph -> A e. On )   &    |- ( ph -> B e. On )   &    |- ( ph -> A C_ B )   =>    |- ( ph -> ( rec ( F , I ) ` A ) C_ ( rec ( F , I ) ` B ) )
Theorem	rdgisuc1 6528*	One way of describing the value of the recursive definition generator at a successor. There is no condition on the characteristic function F other than F Fn _V. Given that, the resulting expression encompasses both the expected successor term ( F ` ( rec ( F , A ) ` B ) ) but also terms that correspond to the initial value A and to limit ordinals U_ x e. B ( F ` ( rec ( F , A ) ` x ) ). If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 6529. (Contributed by Jim Kingdon, 9-Jun-2019.)
|- ( ph -> F Fn _V )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. On )   =>    |- ( ph -> ( rec ( F , A ) ` suc B ) = ( A u. ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) u. ( F ` ( rec ( F , A ) ` B ) ) ) ) )
Theorem	rdgisucinc 6529*	Value of the recursive definition generator at a successor. This can be thought of as a generalization of oasuc 6608 and omsuc 6616. (Contributed by Jim Kingdon, 29-Aug-2019.)
|- ( ph -> F Fn _V )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. On )   &    |- ( ph -> A. x x C_ ( F ` x ) )   =>    |- ( ph -> ( rec ( F , A ) ` suc B ) = ( F ` ( rec ( F , A ) ` B ) ) )
Theorem	rdgon 6530*	Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.) (Revised by Jim Kingdon, 13-Apr-2022.)
|- ( ph -> A e. On )   &    |- ( ph -> A. x e. On ( F ` x ) e. On )   =>    |- ( ( ph /\ B e. On ) -> ( rec ( F , A ) ` B ) e. On )
Theorem	rdg0 6531	The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
|- A e. _V   =>    |- ( rec ( F , A ) ` (/) ) = A
Theorem	rdg0g 6532	The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
|- ( A e. C -> ( rec ( F , A ) ` (/) ) = A )
Theorem	rdgexg 6533	The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- A e. _V   &    |- F Fn _V   =>    |- ( B e. V -> ( rec ( F , A ) ` B ) e. _V )
2.6.22  Finite recursion
Syntax	cfrec 6534	Extend class notation with the finite recursive definition generator, with characteristic function F and initial value I.
class frec ( F , I )
Definition	df-frec 6535*	Define a recursive definition generator on om (the class of finite ordinals) with characteristic function F and initial value I. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our frec operation (especially when df-recs 6449 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple; see frec0g 6541 and frecsuc 6551. Unlike with transfinite recursion, finite recurson can readily divide definitions and proofs into zero and successor cases, because even without excluded middle we have theorems such as nn0suc 4695. The analogous situation with transfinite recursion - being able to say that an ordinal is zero, successor, or limit - is enabled by excluded middle and thus is not available to us. For the characteristic functions which satisfy the conditions given at frecrdg 6552, this definition and df-irdg 6514 restricted to om produce the same result. Note: We introduce frec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Mario Carneiro and Jim Kingdon, 10-Aug-2019.)
|- frec ( F , I ) = (recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. I ) ) } ) ) |` om )
Theorem	freceq1 6536	Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
|- ( F = G -> frec ( F , A ) = frec ( G , A ) )
Theorem	freceq2 6537	Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
|- ( A = B -> frec ( F , A ) = frec ( F , B ) )
Theorem	frecex 6538	Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.)
|- frec ( F , A ) e. _V
Theorem	frecfun 6539	Finite recursion produces a function. See also frecfnom 6545 which also states that the domain of that function is om but which puts conditions on A and F. (Contributed by Jim Kingdon, 13-Feb-2022.)
|- Fun frec ( F , A )
Theorem	nffrec 6540	Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
|- F/_ x F   &    |- F/_ x A   =>    |- F/_ xfrec ( F , A )
Theorem	frec0g 6541	The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
|- ( A e. V -> (frec ( F , A ) ` (/) ) = A )
Theorem	frecabex 6542*	The class abstraction from df-frec 6535 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.)
|- ( ph -> S e. V )   &    |- ( ph -> A. y ( F ` y ) e. _V )   &    |- ( ph -> A e. W )   =>    |- ( ph -> { x | ( E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) \/ ( dom S = (/) /\ x e. A ) ) } e. _V )
Theorem	frecabcl 6543*	The class abstraction from df-frec 6535 exists. Unlike frecabex 6542 the function F only needs to be defined on S, not all sets. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 21-Mar-2022.)
|- ( ph -> N e. om )   &    |- ( ph -> G : N --> S )   &    |- ( ph -> A. y e. S ( F ` y ) e. S )   &    |- ( ph -> A e. S )   =>    |- ( ph -> { x | ( E. m e. om ( dom G = suc m /\ x e. ( F ` ( G ` m ) ) ) \/ ( dom G = (/) /\ x e. A ) ) } e. S )
Theorem	frectfr 6544*	Lemma to connect transfinite recursion theorems with finite recursion. That is, given the conditions F Fn _V and A e. V on frec ( F , A ), we want to be able to apply tfri1d 6479 or tfri2d 6480, and this lemma lets us satisfy hypotheses of those theorems. (Contributed by Jim Kingdon, 15-Aug-2019.)
|- G = ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } )   =>    |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A. y ( Fun G /\ ( G ` y ) e. _V ) )
Theorem	frecfnom 6545*	The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.)
|- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> frec ( F , A ) Fn om )
Theorem	freccllem 6546*	Lemma for freccl 6547. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 27-Mar-2022.)
|- ( ph -> A e. S )   &    |- ( ( ph /\ z e. S ) -> ( F ` z ) e. S )   &    |- ( ph -> B e. om )   &    |- G = recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) )   =>    |- ( ph -> (frec ( F , A ) ` B ) e. S )
Theorem	freccl 6547*	Closure for finite recursion. (Contributed by Jim Kingdon, 27-Mar-2022.)
|- ( ph -> A e. S )   &    |- ( ( ph /\ z e. S ) -> ( F ` z ) e. S )   &    |- ( ph -> B e. om )   =>    |- ( ph -> (frec ( F , A ) ` B ) e. S )
Theorem	frecfcllem 6548*	Lemma for frecfcl 6549. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 30-Mar-2022.)
|- G = recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) )   =>    |- ( ( A. z e. S ( F ` z ) e. S /\ A e. S ) -> frec ( F , A ) : om --> S )
Theorem	frecfcl 6549*	Finite recursion yields a function on the natural numbers. (Contributed by Jim Kingdon, 30-Mar-2022.)
|- ( ( A. z e. S ( F ` z ) e. S /\ A e. S ) -> frec ( F , A ) : om --> S )
Theorem	frecsuclem 6550*	Lemma for frecsuc 6551. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 29-Mar-2022.)
|- G = ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } )   =>    |- ( ( A. z e. S ( F ` z ) e. S /\ A e. S /\ B e. om ) -> (frec ( F , A ) ` suc B ) = ( F ` (frec ( F , A ) ` B ) ) )
Theorem	frecsuc 6551*	The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 31-Mar-2022.)
|- ( ( A. z e. S ( F ` z ) e. S /\ A e. S /\ B e. om ) -> (frec ( F , A ) ` suc B ) = ( F ` (frec ( F , A ) ` B ) ) )
Theorem	frecrdg 6552*	Transfinite recursion restricted to omega. Given a suitable characteristic function, df-frec 6535 produces the same results as df-irdg 6514 restricted to om. Presumably the theorem would also hold if F Fn _V were changed to A. z ( F ` z ) e. _V. (Contributed by Jim Kingdon, 29-Aug-2019.)
|- ( ph -> F Fn _V )   &    |- ( ph -> A e. V )   &    |- ( ph -> A. x x C_ ( F ` x ) )   =>    |- ( ph -> frec ( F , A ) = ( rec ( F , A ) |` om ) )
2.6.23  Ordinal arithmetic
Syntax	c1o 6553	Extend the definition of a class to include the ordinal number 1.
class 1o
Syntax	c2o 6554	Extend the definition of a class to include the ordinal number 2.
class 2o
Syntax	c3o 6555	Extend the definition of a class to include the ordinal number 3.
class 3o
Syntax	c4o 6556	Extend the definition of a class to include the ordinal number 4.
class 4o
Syntax	coa 6557	Extend the definition of a class to include the ordinal addition operation.
class +o
Syntax	comu 6558	Extend the definition of a class to include the ordinal multiplication operation.
class .o
Syntax	coei 6559	Extend the definition of a class to include the ordinal exponentiation operation.
class ↑o
Definition	df-1o 6560	Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
|- 1o = suc (/)
Definition	df-2o 6561	Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
|- 2o = suc 1o
Definition	df-3o 6562	Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
|- 3o = suc 2o
Definition	df-4o 6563	Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
|- 4o = suc 3o
Definition	df-oadd 6564*	Define the ordinal addition operation. (Contributed by NM, 3-May-1995.)
|- +o = ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> suc z ) , x ) ` y ) )
Definition	df-omul 6565*	Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)
|- .o = ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z +o x ) ) , (/) ) ` y ) )
Definition	df-oexpi 6566*	Define the ordinal exponentiation operation. This definition is similar to a conventional definition of exponentiation except that it defines (/) ↑o A to be 1o for all A e. On, in order to avoid having different cases for whether the base is (/) or not. We do not yet have an extensive development of ordinal exponentiation. For background on ordinal exponentiation without excluded middle, see Tom de Jong, Nicolai Kraus, Fredrik Nordvall Forsberg, and Chuangjie Xu (2025), "Ordinal Exponentiation in Homotopy Type Theory", arXiv:2501.14542 , https://arxiv.org/abs/2501.14542 which is formalized in the TypeTopology proof library at https://ordinal-exponentiation-hott.github.io/. (Contributed by Mario Carneiro, 4-Jul-2019.)
|- ↑o = ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) )
Theorem	1on 6567	Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)
|- 1o e. On
Theorem	1oex 6568	Ordinal 1 is a set. (Contributed by BJ, 4-Jul-2022.)
|- 1o e. _V
Theorem	2on 6569	Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|- 2o e. On
Theorem	2on0 6570	Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
|- 2o =/= (/)
Theorem	3on 6571	Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- 3o e. On
Theorem	4on 6572	Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- 4o e. On
Theorem	df1o2 6573	Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
|- 1o = { (/) }
Theorem	df2o3 6574	Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- 2o = { (/) , 1o }
Theorem	df2o2 6575	Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
|- 2o = { (/) , { (/) } }
Theorem	1n0 6576	Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
|- 1o =/= (/)
Theorem	xp01disj 6577	Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
|- ( ( A X. { (/) } ) i^i ( C X. { 1o } ) ) = (/)
Theorem	xp01disjl 6578	Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
|- ( ( { (/) } X. A ) i^i ( { 1o } X. C ) ) = (/)
Theorem	ordgt0ge1 6579	Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
|- ( Ord A -> ( (/) e. A <-> 1o C_ A ) )
Theorem	ordge1n0im 6580	An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.)
|- ( Ord A -> ( 1o C_ A -> A =/= (/) ) )
Theorem	el1o 6581	Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
|- ( A e. 1o <-> A = (/) )
Theorem	dif1o 6582	Two ways to say that A is a nonzero number of the set B. (Contributed by Mario Carneiro, 21-May-2015.)
|- ( A e. ( B \ 1o ) <-> ( A e. B /\ A =/= (/) ) )
Theorem	2oconcl 6583	Closure of the pair swapping function on 2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
|- ( A e. 2o -> ( 1o \ A ) e. 2o )
Theorem	0lt1o 6584	Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
|- (/) e. 1o
Theorem	0lt2o 6585	Ordinal zero is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
|- (/) e. 2o
Theorem	1lt2o 6586	Ordinal one is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
|- 1o e. 2o
Theorem	el2oss1o 6587	Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 16313. (Contributed by Jim Kingdon, 8-Aug-2022.)
|- ( A e. 2o -> A C_ 1o )
Theorem	oafnex 6588	The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.)
|- ( x e. _V |-> suc x ) Fn _V
Theorem	sucinc 6589*	Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
|- F = ( z e. _V |-> suc z )   =>    |- A. x x C_ ( F ` x )
Theorem	sucinc2 6590*	Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.)
|- F = ( z e. _V |-> suc z )   =>    |- ( ( B e. On /\ A e. B ) -> ( F ` A ) C_ ( F ` B ) )
Theorem	fnoa 6591	Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Mario Carneiro, 3-Jul-2019.)
|- +o Fn ( On X. On )
Theorem	oaexg 6592	Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- ( ( A e. V /\ B e. W ) -> ( A +o B ) e. _V )
Theorem	omfnex 6593*	The characteristic function for ordinal multiplication is defined everywhere. (Contributed by Jim Kingdon, 23-Aug-2019.)
|- ( A e. V -> ( x e. _V |-> ( x +o A ) ) Fn _V )
Theorem	fnom 6594	Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.)
|- .o Fn ( On X. On )
Theorem	omexg 6595	Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- ( ( A e. V /\ B e. W ) -> ( A .o B ) e. _V )
Theorem	fnoei 6596	Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
|- ↑o Fn ( On X. On )
Theorem	oeiexg 6597	Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- ( ( A e. V /\ B e. W ) -> ( A ↑o B ) e. _V )
Theorem	oav 6598*	Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( ( A e. On /\ B e. On ) -> ( A +o B ) = ( rec ( ( x e. _V |-> suc x ) , A ) ` B ) )
Theorem	omv 6599*	Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)
|- ( ( A e. On /\ B e. On ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
Theorem	oeiv 6600*	Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.)
|- ( ( A e. On /\ B e. On ) -> ( A ↑o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )