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Theorem List for Intuitionistic Logic Explorer - 5901-6000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-smo 5901* Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.)

Theoremdfsmo2 5902* Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.)

Theoremissmo 5903* Conditions for which is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)

Theoremissmo2 5904* Alternative definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremsmoeq 5905 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)

Theoremsmodm 5906 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)

Theoremsmores 5907 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Theoremsmores3 5908 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)

Theoremsmores2 5909 A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)

Theoremsmodm2 5910 The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremsmofvon2dm 5911 The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremiordsmo 5912 The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.)

Theoremsmo0 5913 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)

Theoremsmofvon 5914 If is a strictly monotone ordinal function, and is in the domain of , then the value of the function at is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.)

Theoremsmoel 5915 If is less than then a strictly monotone function's value will be strictly less at than at . (Contributed by Andrew Salmon, 22-Nov-2011.)

Theoremsmoiun 5916* The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)

Theoremsmoiso 5917 If is an isomorphism from an ordinal onto , which is a subset of the ordinals, then is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)

Theoremsmoel2 5918 A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.)

2.6.19  "Strong" transfinite recursion

Syntaxcrecs 5919 Notation for a function defined by strong transfinite recursion.
recs

Definitiondf-recs 5920* Define a function recs on , the class of ordinal numbers, by transfinite recursion given a rule which sets the next value given all values so far. See df-irdg 5957 for more details on why this definition is desirable. Unlike df-irdg 5957 which restricts the update rule to use only the previous value, this version allows the update rule to use all previous values, which is why it is described as "strong", although it is actually more primitive. See tfri1d 5949 and tfri2d 5950 for the primary contract of this definition.

(Contributed by Stefan O'Rear, 18-Jan-2015.)

recs

Theoremrecseq 5921 Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs recs

Theoremnfrecs 5922 Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremtfrlem1 5923* A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by Mario Carneiro, 24-May-2019.)

Theoremtfrlem3ag 5924* Lemma for transfinite recursion. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.)

Theoremtfrlem3a 5925* Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.)

Theoremtfrlem3 5926* Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.)

Theoremtfrlem3-2 5927* Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 17-Apr-2019.)

Theoremtfrlem3-2d 5928* Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.)

Theoremtfrlem4 5929* Lemma for transfinite recursion. is the class of all "acceptable" functions, and is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.)

Theoremtfrlem5 5930* Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)

Theoremrecsfval 5931* Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.)
recs

Theoremtfrlem6 5932* Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremtfrlem7 5933* Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
recs

Theoremtfrlem8 5934* Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.)
recs

Theoremtfrlem9 5935* Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)
recs recs recs

Theoremtfr2a 5936 A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.)
recs

Theoremtfr0 5937 Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.)
recs

Theoremtfrlemisucfn 5938* We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 5946. (Contributed by Jim Kingdon, 2-Jul-2019.)

Theoremtfrlemisucaccv 5939* We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 5946. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Theoremtfrlemibacc 5940* Each element of is an acceptable function. Lemma for tfrlemi1 5946. (Contributed by Jim Kingdon, 14-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Theoremtfrlemibxssdm 5941* The union of is defined on all ordinals. Lemma for tfrlemi1 5946. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Theoremtfrlemibfn 5942* The union of is a function defined on . Lemma for tfrlemi1 5946. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Theoremtfrlemibex 5943* The set exists. Lemma for tfrlemi1 5946. (Contributed by Jim Kingdon, 17-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Theoremtfrlemiubacc 5944* The union of satisfies the recursion rule (lemma for tfrlemi1 5946). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Theoremtfrlemiex 5945* Lemma for tfrlemi1 5946. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Theoremtfrlemi1 5946* We can define an acceptable function on any ordinal.

As with many of the transfinite recursion theorems, we have a hypothesis that states that is a function and that it is defined for all ordinals. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Theoremtfrlemi14d 5947* The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.)
recs

Theoremtfrexlem 5948* The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
recs

Theoremtfri1d 5949* Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition.

The condition is that is defined "everywhere" and here is stated as . Alternatively would suffice.

Given a function satisfying that condition, we define a class of all "acceptable" functions. The final function we're interested in is the union recs of them. is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of . In this first part we show that is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)

recs

Theoremtfri2d 5950* Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 5951). Here we show that the function has the property that for any function satisfying that condition, the "next" value of is recursively applied to all "previous" values of . (Contributed by Jim Kingdon, 4-May-2019.)
recs

Theoremtfri1 5951* Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition.

The condition is that is defined "everywhere" and here is stated as . Alternatively would suffice.

Given a function satisfying that condition, we define a class of all "acceptable" functions. The final function we're interested in is the union recs of them. is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of . In this first part we show that is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)

recs

Theoremtfri2 5952* Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 5951). Here we show that the function has the property that for any function satisfying that condition, the "next" value of is recursively applied to all "previous" values of . (Contributed by Jim Kingdon, 4-May-2019.)
recs

Theoremtfri3 5953* Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 5951). Finally, we show that is unique. We do this by showing that any class with the same properties of that we showed in parts 1 and 2 is identical to . (Contributed by Jim Kingdon, 4-May-2019.)
recs

Theoremtfrex 5954* The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
recs

Theoremtfrfun 5955 Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.)
recs

2.6.20  Recursive definition generator

Syntaxcrdg 5956 Extend class notation with the recursive definition generator, with characteristic function and initial value .

Definitiondf-irdg 5957* Define a recursive definition generator on (the class of ordinal numbers) with characteristic function and initial value . 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 operation (especially when df-recs 5920 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 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 . 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 5978 and for suitable characteristic functions df-frec 5978 yields the same result as restricted to , as seen at frecrdg 5992.

Note: We introduce 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.)

recs

Theoremrdgeq1 5958 Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

Theoremrdgeq2 5959 Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

Theoremrdgfun 5960 The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremrdgtfr 5961* The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.)

Theoremrdgruledefgg 5962* The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)

Theoremrdgruledefg 5963* The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)

Theoremrdgexggg 5964 The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)

Theoremrdgexgg 5965 The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)

Theoremrdgifnon 5966 The recursive definition generator is a function on ordinal numbers. The 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 5973; in cases like df-oadd 6005 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.)

Theoremrdgifnon2 5967* The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.)

Theoremrdgivallem 5968* Value of the recursive definition generator. Lemma for rdgival 5969 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.)

Theoremrdgival 5969* Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.)

Theoremrdgss 5970 Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.)

Theoremrdgisuc1 5971* One way of describing the value of the recursive definition generator at a successor. There is no condition on the characteristic function other than . Given that, the resulting expression encompasses both the expected successor term but also terms that correspond to the initial value and to limit ordinals .

If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 5972. (Contributed by Jim Kingdon, 9-Jun-2019.)

Theoremrdgisucinc 5972* Value of the recursive definition generator at a successor.

This can be thought of as a generalization of oasuc 6044 and omsuc 6051. (Contributed by Jim Kingdon, 29-Aug-2019.)

Theoremrdgon 5973* 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.)

Theoremrdg0 5974 The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremrdg0g 5975 The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)

Theoremrdgexg 5976 The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.)

2.6.21  Finite recursion

Syntaxcfrec 5977 Extend class notation with the fnite recursive definition generator, with characteristic function and initial value .
frec

Definitiondf-frec 5978* Define a recursive definition generator on (the class of finite ordinals) with characteristic function and initial value . 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 5920 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 5983 and frecsuc 5991.

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 4327. 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 5992, this definition and df-irdg 5957 restricted to 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 recs

Theoremfreceq1 5979 Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
frec frec

Theoremfreceq2 5980 Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
frec frec

Theoremfrecex 5981 Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.)
frec

Theoremnffrec 5982 Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
frec

Theoremfrec0g 5983 The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
frec

Theoremfrecabex 5984* The class abstraction from df-frec 5978 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.)

Theoremfrectfr 5985* Lemma to connect transfinite recursion theorems with finite recursion. That is, given the conditions and on frec , we want to be able to apply tfri1d 5949 or tfri2d 5950, and this lemma lets us satisfy hypotheses of those theorems.

(Contributed by Jim Kingdon, 15-Aug-2019.)

Theoremfrecfnom 5986* The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.)
frec

Theoremfrecsuclem1 5987* Lemma for frecsuc 5991. (Contributed by Jim Kingdon, 13-Aug-2019.)
frec recs

Theoremfrecsuclemdm 5988* Lemma for frecsuc 5991. (Contributed by Jim Kingdon, 15-Aug-2019.)
recs

Theoremfrecsuclem2 5989* Lemma for frecsuc 5991. (Contributed by Jim Kingdon, 15-Aug-2019.)
recs frec

Theoremfrecsuclem3 5990* Lemma for frecsuc 5991. (Contributed by Jim Kingdon, 15-Aug-2019.)
frec frec

Theoremfrecsuc 5991* The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 15-Aug-2019.)
frec frec

Theoremfrecrdg 5992* Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 5978 produces the same results as df-irdg 5957 restricted to .

Presumably the theorem would also hold if were changed to . (Contributed by Jim Kingdon, 29-Aug-2019.)

frec

Theoremfreccl 5993* Closure for finite recursion. (Contributed by Jim Kingdon, 25-May-2020.)
frec

2.6.22  Ordinal arithmetic

Syntaxc1o 5994 Extend the definition of a class to include the ordinal number 1.

Syntaxc2o 5995 Extend the definition of a class to include the ordinal number 2.

Syntaxc3o 5996 Extend the definition of a class to include the ordinal number 3.

Syntaxc4o 5997 Extend the definition of a class to include the ordinal number 4.

Syntaxcoa 5998 Extend the definition of a class to include the ordinal addition operation.

Syntaxcomu 5999 Extend the definition of a class to include the ordinal multiplication operation.

Syntaxcoei 6000 Extend the definition of a class to include the ordinal exponentiation operation.
𝑜

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