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Theorem List for Intuitionistic Logic Explorer - 5901-6000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrdgtfr 5901* The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.)
 F `  _V  V  Fun  _V  |->  u.  U_  dom  F `
  ` 
 _V  |->  u.  U_  dom  F ` 
 `  `  _V
 
Theoremrdgruledefgg 5902* The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
 F  Fn  _V  V  Fun  _V  |->  u.  U_  dom  F `  `  _V  |->  u.  U_  dom  F `
  ` 
 ` 
 _V
 
Theoremrdgruledefg 5903* The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
 F  Fn  _V   =>     V  Fun  _V  |->  u.  U_  dom  F `  `  _V  |->  u.  U_  dom  F `
  ` 
 ` 
 _V
 
Theoremrdgexggg 5904 The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
 F  Fn  _V  V  W  rec F ,  `  _V
 
Theoremrdgexgg 5905 The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
 F  Fn  _V   =>     V  W  rec F ,  `  _V
 
Theoremrdgifnon 5906 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 5913; in cases like df-oadd 5944 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.)
 F  Fn  _V  V  rec F ,  Fn  On
 
Theoremrdgifnon2 5907* The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.)
 F `  _V  V  rec F ,  Fn  On
 
Theoremrdgivallem 5908* Value of the recursive definition generator. Lemma for rdgival 5909 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.)
 F  Fn  _V  V  On  rec F ,  `  u.  U_  F `  rec F ,  |`  `
 
Theoremrdgival 5909* Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.)
 F  Fn  _V  V  On  rec F ,  `  u.  U_  F `  rec F ,  `
 
Theoremrdgss 5910 Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.)
 F  Fn  _V   &     I  V   &     On   &     On   &     C_    =>     rec F ,  I `  C_  rec F ,  I `
 
Theoremrdgisuc1 5911* 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 ,  ` but also terms that correspond to the initial value and to limit ordinals  U_  F `  rec F ,  ` .

If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 5912. (Contributed by Jim Kingdon, 9-Jun-2019.)

 F  Fn  _V   &     V   &     On   =>     rec F ,  `  suc  u.  U_  F `  rec F , 
 `  u.  F `  rec F ,  `
 
Theoremrdgisucinc 5912* Value of the recursive definition generator at a successor.

This can be thought of as a generalization of oasuc 5983 and omsuc 5990. (Contributed by Jim Kingdon, 29-Aug-2019.)

 F  Fn  _V   &     V   &     On   &     C_  F `
    =>     rec F , 
 `  suc  F `  rec F ,  `
 
Theoremrdgon 5913* 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.)
 F  Fn  _V   &     On   &     On  F `
 
 On   =>     On  rec F ,  ` 
 On
 
Theoremrdg0 5914 The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 _V   =>     rec F ,  `  (/)
 
Theoremrdg0g 5915 The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
 C  rec F ,  `  (/)
 
Theoremrdgexg 5916 The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.)
 _V   &     F  Fn  _V   =>     V  rec F ,  `  _V
 
2.6.21  Finite recursion
 
Syntaxcfrec 5917 Extend class notation with the fnite recursive definition generator, with characteristic function  F and initial value  I.
frec F ,  I
 
Definitiondf-frec 5918* 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 5861 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 5922 and frecsuc 5930.

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 4270. 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 5931, this definition and df-irdg 5897 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  _V  |->  {  |  m  om  dom  suc  m  F `  `  m  dom  (/)  I }  |`  om
 
Theoremfreceq1 5919 Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
 F  G frec F , frec G ,
 
Theoremfreceq2 5920 Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
frec F , frec F ,
 
Theoremnffrec 5921 Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
 F/_ F   &     F/_   =>     F/_frec F ,
 
Theoremfrec0g 5922 The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
 V frec F ,  `  (/)
 
Theoremfrecabex 5923* The class abstraction from df-frec 5918 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.)
 S  V   &     F `  _V   &     W   =>     {  |  m 
 om  dom  S  suc  m  F `  S `  m  dom  S  (/)  }  _V
 
Theoremfrectfr 5924* Lemma to connect transfinite recursion theorems with finite recursion. That is, given the conditions  F  Fn  _V and  V on frec F , , we want to be able to apply tfri1d 5890 or tfri2d 5891, and this lemma lets us satisfy hypotheses of those theorems.

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

 G  _V  |->  {  |  m 
 om  dom  suc  m  F `  `  m  dom  (/)  }   =>     F `  _V  V  Fun  G  G `  _V
 
Theoremfrecfnom 5925* The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.)
 F `  _V  V frec F ,  Fn  om
 
Theoremfrecsuclem1 5926* Lemma for frecsuc 5930. (Contributed by Jim Kingdon, 13-Aug-2019.)
 G  _V  |->  {  |  m 
 om  dom  suc  m  F `  `  m  dom  (/)  }   =>     F `  _V  V  om frec F ,  ` 
 suc  G ` recs G  |`  suc
 
Theoremfrecsuclemdm 5927* Lemma for frecsuc 5930. (Contributed by Jim Kingdon, 15-Aug-2019.)
 G  _V  |->  {  |  m 
 om  dom  suc  m  F `  `  m  dom  (/)  }   =>     F `  _V  V  om  dom recs G  |`  suc  suc
 
Theoremfrecsuclem2 5928* Lemma for frecsuc 5930. (Contributed by Jim Kingdon, 15-Aug-2019.)
 G  _V  |->  {  |  m 
 om  dom  suc  m  F `  `  m  dom  (/)  }   =>     F `  _V  V  om recs G  |`  suc  ` frec F ,  `
 
Theoremfrecsuclem3 5929* Lemma for frecsuc 5930. (Contributed by Jim Kingdon, 15-Aug-2019.)
 G  _V  |->  {  |  m 
 om  dom  suc  m  F `  `  m  dom  (/)  }   =>     F `  _V  V  om frec F ,  ` 
 suc  F ` frec F ,  `
 
Theoremfrecsuc 5930* The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 15-Aug-2019.)
 F `  _V  V  om frec F ,  ` 
 suc  F ` frec F ,  `
 
Theoremfrecrdg 5931* Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 5918 produces the same results as df-irdg 5897 restricted to  om.

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

 F  Fn  _V   &     V   &     C_  F `
    =>    frec F , 
 rec F ,  |`  om
 
Theoremfreccl 5932* Closure for finite recursion. (Contributed by Jim Kingdon, 25-May-2020.)
 F `  _V   &     S   &     S  F `  S   &     om   =>    frec F , 
 `  S
 
2.6.22  Ordinal arithmetic
 
Syntaxc1o 5933 Extend the definition of a class to include the ordinal number 1.

 1o
 
Syntaxc2o 5934 Extend the definition of a class to include the ordinal number 2.

 2o
 
Syntaxc3o 5935 Extend the definition of a class to include the ordinal number 3.

 3o
 
Syntaxc4o 5936 Extend the definition of a class to include the ordinal number 4.

 4o
 
Syntaxcoa 5937 Extend the definition of a class to include the ordinal addition operation.
 +o
 
Syntaxcomu 5938 Extend the definition of a class to include the ordinal multiplication operation.
 .o
 
Syntaxcoei 5939 Extend the definition of a class to include the ordinal exponentiation operation.
𝑜
 
Definitiondf-1o 5940 Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)

 1o  suc  (/)
 
Definitiondf-2o 5941 Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)

 2o  suc  1o
 
Definitiondf-3o 5942 Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)

 3o  suc  2o
 
Definitiondf-4o 5943 Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)

 4o  suc  3o
 
Definitiondf-oadd 5944* Define the ordinal addition operation. (Contributed by NM, 3-May-1995.)

 +o  On ,  On  |->  rec  _V  |->  suc  ,  `
 
Definitiondf-omul 5945* Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)

 .o  On ,  On  |->  rec  _V  |->  +o  ,  (/) `
 
Definitiondf-oexpi 5946* Define the ordinal exponentiation operation.

This definition is similar to a conventional definition of exponentiation except that it defines  (/)𝑜 to be  1o for all  On, in order to avoid having different cases for whether the base is  (/) or not. (Contributed by Mario Carneiro, 4-Jul-2019.)

𝑜  On ,  On  |->  rec  _V  |-> 
 .o  ,  1o `
 
Theorem1on 5947 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)

 1o  On
 
Theorem2on 5948 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

 2o  On
 
Theorem2on0 5949 Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)

 2o  =/=  (/)
 
Theorem3on 5950 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)

 3o  On
 
Theorem4on 5951 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)

 4o  On
 
Theoremdf1o2 5952 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)

 1o  { (/) }
 
Theoremdf2o3 5953 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)

 2o  { (/) ,  1o }
 
Theoremdf2o2 5954 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)

 2o  { (/) ,  { (/)
 } }
 
Theorem1n0 5955 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)

 1o  =/=  (/)
 
Theoremxp01disj 5956 Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
 X.  { (/) }  i^i  C  X.  { 1o }  (/)
 
Theoremordgt0ge1 5957 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
 Ord  (/)  1o  C_
 
Theoremordge1n0im 5958 An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.)
 Ord  1o  C_  =/=  (/)
 
Theoremel1o 5959 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
 1o  (/)
 
Theoremdif1o 5960 Two ways to say that is a nonzero number of the set . (Contributed by Mario Carneiro, 21-May-2015.)
 \  1o  =/= 
 (/)
 
Theorem2oconcl 5961 Closure of the pair swapping function on  2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
 2o  1o  \  2o
 
Theorem0lt1o 5962 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
 (/)  1o
 
Theoremoafnex 5963 The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.)
 _V  |->  suc  Fn  _V
 
Theoremsucinc 5964* Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
 F  _V  |->  suc    =>     C_  F `
 
Theoremsucinc2 5965* Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.)
 F  _V  |->  suc    =>     On  F `  C_  F `
 
Theoremfnoa 5966 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
 
Theoremoaexg 5967 Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
 V  W  +o  _V
 
Theoremomfnex 5968* The characteristic function for ordinal multiplication is defined everywhere. (Contributed by Jim Kingdon, 23-Aug-2019.)
 V  _V  |->  +o  Fn  _V
 
Theoremfnom 5969 Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.)

 .o  Fn  On  X. 
 On
 
Theoremomexg 5970 Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
 V  W  .o  _V
 
Theoremfnoei 5971 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
 
Theoremoeiexg 5972 Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
 V  W 𝑜  _V
 
Theoremoav 5973* Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 On  On  +o  rec  _V 
 |->  suc  ,  `
 
Theoremomv 5974* Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)
 On  On  .o  rec  _V 
 |->  +o  ,  (/) `
 
Theoremoeiv 5975* Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.)
 On  On 𝑜 
 rec  _V  |-> 
 .o  ,  1o `
 
Theoremoa0 5976 Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 On  +o  (/)
 
Theoremom0 5977 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 On  .o  (/)  (/)
 
Theoremoei0 5978 Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 On 𝑜 
 (/)  1o
 
Theoremoacl 5979 Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.) (Constructive proof by Jim Kingdon, 26-Jul-2019.)
 On  On  +o  On
 
Theoremomcl 5980 Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Constructive proof by Jim Kingdon, 26-Jul-2019.)
 On  On  .o  On
 
Theoremoeicl 5981 Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)
 On  On 𝑜  On
 
Theoremoav2 5982* Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.)
 On  On  +o  u.  U_  suc  +o
 
Theoremoasuc 5983 Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 On  On  +o  suc  suc  +o
 
Theoremomv2 5984* Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.)
 On  On  .o  U_ 
 .o  +o
 
Theoremonasuc 5985 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by Mario Carneiro, 16-Nov-2014.)
 On  om  +o  suc  suc  +o
 
Theoremoa1suc 5986 Addition with 1 is same as successor. Proposition 4.34(a) of [Mendelson] p. 266. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 16-Nov-2014.)
 On  +o  1o  suc
 
Theoremo1p1e2 5987 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
 1o  +o  1o  2o
 
Theoremoawordi 5988 Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.)
 On  On  C  On  C_  C  +o  C_  C  +o
 
Theoremoaword1 5989 An ordinal is less than or equal to its sum with another. Part of Exercise 5 of [TakeutiZaring] p. 62. (Contributed by NM, 6-Dec-2004.)
 On  On  C_  +o
 
Theoremomsuc 5990 Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 On  On  .o  suc  .o 
 +o
 
Theoremonmsuc 5991 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 On  om  .o  suc  .o 
 +o
 
2.6.23  Natural number arithmetic
 
Theoremnna0 5992 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
 om  +o  (/)
 
Theoremnnm0 5993 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
 om  .o  (/)  (/)
 
Theoremnnasuc 5994 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 om  om  +o  suc  suc  +o
 
Theoremnnmsuc 5995 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 om  om  .o  suc  .o 
 +o
 
Theoremnna0r 5996 Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 om  (/)  +o
 
Theoremnnm0r 5997 Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 om  (/)  .o  (/)
 
Theoremnnacl 5998 Closure of addition of natural numbers. Proposition 8.9 of [TakeutiZaring] p. 59. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 om  om  +o  om
 
Theoremnnmcl 5999 Closure of multiplication of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 om  om  .o  om
 
Theoremnnacli 6000  om is closed under addition. Inference form of nnacl 5998. (Contributed by Scott Fenton, 20-Apr-2012.)
 om   &     om   =>     +o  om
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