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Theorem List for Intuitionistic Logic Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrdgival 6101* 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 ) ) ) )
 
Theoremrdgss 6102 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 ) )
 
Theoremrdgisuc1 6103* 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 6104. (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 ) ) ) ) )
 
Theoremrdgisucinc 6104* Value of the recursive definition generator at a successor.

This can be thought of as a generalization of oasuc 6179 and omsuc 6187. (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 ) ) )
 
Theoremrdgon 6105* 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.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  A. x  e.  On  ( F `  x )  e. 
 On )   =>    |-  ( ( ph  /\  B  e.  On )  ->  ( rec ( F ,  A ) `  B )  e. 
 On )
 
Theoremrdg0 6106 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
 
Theoremrdg0g 6107 The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
 |-  ( A  e.  C  ->  ( rec ( F ,  A ) `  (/) )  =  A )
 
Theoremrdgexg 6108 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 )
 
2.6.21  Finite recursion
 
Syntaxcfrec 6109 Extend class notation with the finite recursive definition generator, with characteristic function  F and initial value  I.
 class frec ( F ,  I )
 
Definitiondf-frec 6110* 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 6024 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 6116 and frecsuc 6126.

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 4392. 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 6127, this definition and df-irdg 6089 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 )
 
Theoremfreceq1 6111 Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
 |-  ( F  =  G  -> frec ( F ,  A )  = frec ( G ,  A ) )
 
Theoremfreceq2 6112 Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
 |-  ( A  =  B  -> frec ( F ,  A )  = frec ( F ,  B ) )
 
Theoremfrecex 6113 Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.)
 |- frec
 ( F ,  A )  e.  _V
 
Theoremfrecfun 6114 Finite recursion produces a function. See also frecfnom 6120 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 )
 
Theoremnffrec 6115 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 )
 
Theoremfrec0g 6116 The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
 |-  ( A  e.  V  ->  (frec ( F ,  A ) `  (/) )  =  A )
 
Theoremfrecabex 6117* The class abstraction from df-frec 6110 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 )
 
Theoremfrecabcl 6118* The class abstraction from df-frec 6110 exists. Unlike frecabex 6117 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 )
 
Theoremfrectfr 6119* 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 6054 or tfri2d 6055, 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 ) )
 
Theoremfrecfnom 6120* 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 )
 
Theoremfreccllem 6121* Lemma for freccl 6122. 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 )
 
Theoremfreccl 6122* 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 )
 
Theoremfrecfcllem 6123* Lemma for frecfcl 6124. 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 )
 
Theoremfrecfcl 6124* 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 )
 
Theoremfrecsuclem 6125* Lemma for frecsuc 6126. 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 ) ) )
 
Theoremfrecsuc 6126* 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 ) ) )
 
Theoremfrecrdg 6127* Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6110 produces the same results as df-irdg 6089 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 ) )
 
2.6.22  Ordinal arithmetic
 
Syntaxc1o 6128 Extend the definition of a class to include the ordinal number 1.
 class  1o
 
Syntaxc2o 6129 Extend the definition of a class to include the ordinal number 2.
 class  2o
 
Syntaxc3o 6130 Extend the definition of a class to include the ordinal number 3.
 class  3o
 
Syntaxc4o 6131 Extend the definition of a class to include the ordinal number 4.
 class  4o
 
Syntaxcoa 6132 Extend the definition of a class to include the ordinal addition operation.
 class  +o
 
Syntaxcomu 6133 Extend the definition of a class to include the ordinal multiplication operation.
 class  .o
 
Syntaxcoei 6134 Extend the definition of a class to include the ordinal exponentiation operation.
 class𝑜
 
Definitiondf-1o 6135 Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  =  suc  (/)
 
Definitiondf-2o 6136 Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
 |- 
 2o  =  suc  1o
 
Definitiondf-3o 6137 Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |- 
 3o  =  suc  2o
 
Definitiondf-4o 6138 Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |- 
 4o  =  suc  3o
 
Definitiondf-oadd 6139* 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 ) )
 
Definitiondf-omul 6140* 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 ) )
 
Definitiondf-oexpi 6141* 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. (Contributed by Mario Carneiro, 4-Jul-2019.)

 |-𝑜  =  ( x  e.  On ,  y  e.  On  |->  ( rec ( ( z  e.  _V  |->  ( z  .o  x ) ) ,  1o ) `  y ) )
 
Theorem1on 6142 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  e.  On
 
Theorem1oex 6143 Ordinal 1 is a set. (Contributed by BJ, 4-Jul-2022.)
 |- 
 1o  e.  _V
 
Theorem2on 6144 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |- 
 2o  e.  On
 
Theorem2on0 6145 Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
 |- 
 2o  =/=  (/)
 
Theorem3on 6146 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 3o  e.  On
 
Theorem4on 6147 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 4o  e.  On
 
Theoremdf1o2 6148 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
 |- 
 1o  =  { (/) }
 
Theoremdf2o3 6149 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 2o  =  { (/) ,  1o }
 
Theoremdf2o2 6150 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
 |- 
 2o  =  { (/) ,  { (/)
 } }
 
Theorem1n0 6151 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
 |- 
 1o  =/=  (/)
 
Theoremxp01disj 6152 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 }
 ) )  =  (/)
 
Theoremordgt0ge1 6153 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
 |-  ( Ord  A  ->  ( (/)  e.  A  <->  1o  C_  A ) )
 
Theoremordge1n0im 6154 An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.)
 |-  ( Ord  A  ->  ( 1o  C_  A  ->  A  =/=  (/) ) )
 
Theoremel1o 6155 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
 |-  ( A  e.  1o  <->  A  =  (/) )
 
Theoremdif1o 6156 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  =/=  (/) ) )
 
Theorem2oconcl 6157 Closure of the pair swapping function on  2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( A  e.  2o  ->  ( 1o  \  A )  e.  2o )
 
Theorem0lt1o 6158 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
 |-  (/)  e.  1o
 
Theoremoafnex 6159 The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.)
 |-  ( x  e.  _V  |->  suc  x )  Fn  _V
 
Theoremsucinc 6160* Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
 |-  F  =  ( z  e.  _V  |->  suc  z
 )   =>    |- 
 A. x  x  C_  ( F `  x )
 
Theoremsucinc2 6161* 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 ) )
 
Theoremfnoa 6162 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 6163 Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  +o  B )  e.  _V )
 
Theoremomfnex 6164* 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 )
 
Theoremfnom 6165 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 6166 Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  .o  B )  e.  _V )
 
Theoremfnoei 6167 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 6168 Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A𝑜  B )  e.  _V )
 
Theoremoav 6169* 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 ) )
 
Theoremomv 6170* 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 ) )
 
Theoremoeiv 6171* Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A𝑜  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
 
Theoremoa0 6172 Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
 
Theoremom0 6173 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  e.  On  ->  ( A  .o  (/) )  =  (/) )
 
Theoremoei0 6174 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.)
 |-  ( A  e.  On  ->  ( A𝑜 
 (/) )  =  1o )
 
Theoremoacl 6175 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.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B )  e.  On )
 
Theoremomcl 6176 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.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B )  e.  On )
 
Theoremoeicl 6177 Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A𝑜  B )  e.  On )
 
Theoremoav2 6178* Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B )  =  ( A  u.  U_ x  e.  B  suc  ( A  +o  x ) ) )
 
Theoremoasuc 6179 Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  suc 
 B )  =  suc  ( A  +o  B ) )
 
Theoremomv2 6180* Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B )  =  U_ x  e.  B  ( ( A  .o  x )  +o  A ) )
 
Theoremonasuc 6181 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  +o  suc 
 B )  =  suc  ( A  +o  B ) )
 
Theoremoa1suc 6182 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.)
 |-  ( A  e.  On  ->  ( A  +o  1o )  =  suc  A )
 
Theoremo1p1e2 6183 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
 |-  ( 1o  +o  1o )  =  2o
 
Theoremoawordi 6184 Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  ->  ( C  +o  A )  C_  ( C  +o  B ) ) )
 
Theoremoawordriexmid 6185* A weak ordering property of ordinal addition which implies excluded middle. The property is proposition 8.7 of [TakeutiZaring] p. 59. Compare with oawordi 6184. (Contributed by Jim Kingdon, 15-May-2022.)
 |-  ( ( a  e. 
 On  /\  b  e.  On  /\  c  e.  On )  ->  ( a  C_  b  ->  ( a  +o  c )  C_  ( b  +o  c ) ) )   =>    |-  ( ph  \/  -.  ph )
 
Theoremoaword1 6186 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.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )
 
Theoremomsuc 6187 Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  suc 
 B )  =  ( ( A  .o  B )  +o  A ) )
 
Theoremonmsuc 6188 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  .o  suc 
 B )  =  ( ( A  .o  B )  +o  A ) )
 
2.6.23  Natural number arithmetic
 
Theoremnna0 6189 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
 |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
 
Theoremnnm0 6190 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
 |-  ( A  e.  om  ->  ( A  .o  (/) )  =  (/) )
 
Theoremnnasuc 6191 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  +o  suc 
 B )  =  suc  ( A  +o  B ) )
 
Theoremnnmsuc 6192 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  .o  suc 
 B )  =  ( ( A  .o  B )  +o  A ) )
 
Theoremnna0r 6193 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.)
 |-  ( A  e.  om  ->  ( (/)  +o  A )  =  A )
 
Theoremnnm0r 6194 Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( A  e.  om  ->  ( (/)  .o  A )  =  (/) )
 
Theoremnnacl 6195 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.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  +o  B )  e.  om )
 
Theoremnnmcl 6196 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.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  .o  B )  e.  om )
 
Theoremnnacli 6197  om is closed under addition. Inference form of nnacl 6195. (Contributed by Scott Fenton, 20-Apr-2012.)
 |-  A  e.  om   &    |-  B  e.  om   =>    |-  ( A  +o  B )  e.  om
 
Theoremnnmcli 6198  om is closed under multiplication. Inference form of nnmcl 6196. (Contributed by Scott Fenton, 20-Apr-2012.)
 |-  A  e.  om   &    |-  B  e.  om   =>    |-  ( A  .o  B )  e.  om
 
Theoremnnacom 6199 Addition of natural numbers is commutative. Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 6-May-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  +o  B )  =  ( B  +o  A ) )
 
Theoremnnaass 6200 Addition of natural numbers is associative. Theorem 4K(1) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om  /\  C  e.  om )  ->  ( ( A  +o  B )  +o  C )  =  ( A  +o  ( B  +o  C ) ) )
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