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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rdgss 6001 | Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.) |
Theorem | rdgisuc1 6002* |
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 6003. (Contributed by Jim Kingdon, 9-Jun-2019.) |
Theorem | rdgisucinc 6003* |
Value of the recursive definition generator at a successor.
This can be thought of as a generalization of oasuc 6075 and omsuc 6082. (Contributed by Jim Kingdon, 29-Aug-2019.) |
Theorem | rdgon 6004* | 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.) |
Theorem | rdg0 6005 | The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Theorem | rdg0g 6006 | The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.) |
Theorem | rdgexg 6007 | The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Syntax | cfrec 6008 | Extend class notation with the fnite recursive definition generator, with characteristic function and initial value . |
frec | ||
Definition | df-frec 6009* |
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 5951
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 6014 and frecsuc 6022.
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 4355. 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 6023, this definition and df-irdg 5988 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 | ||
Theorem | freceq1 6010 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec frec | ||
Theorem | freceq2 6011 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec frec | ||
Theorem | frecex 6012 | Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.) |
frec | ||
Theorem | nffrec 6013 | Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec | ||
Theorem | frec0g 6014 | The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.) |
frec | ||
Theorem | frecabex 6015* | The class abstraction from df-frec 6009 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.) |
Theorem | frectfr 6016* |
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 5980 or tfri2d 5981,
and this lemma lets us satisfy hypotheses of those theorems.
(Contributed by Jim Kingdon, 15-Aug-2019.) |
Theorem | frecfnom 6017* | The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.) |
frec | ||
Theorem | frecsuclem1 6018* | Lemma for frecsuc 6022. (Contributed by Jim Kingdon, 13-Aug-2019.) |
frec recs | ||
Theorem | frecsuclemdm 6019* | Lemma for frecsuc 6022. (Contributed by Jim Kingdon, 15-Aug-2019.) |
recs | ||
Theorem | frecsuclem2 6020* | Lemma for frecsuc 6022. (Contributed by Jim Kingdon, 15-Aug-2019.) |
recs frec | ||
Theorem | frecsuclem3 6021* | Lemma for frecsuc 6022. (Contributed by Jim Kingdon, 15-Aug-2019.) |
frec frec | ||
Theorem | frecsuc 6022* | The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 15-Aug-2019.) |
frec frec | ||
Theorem | frecrdg 6023* |
Transfinite recursion restricted to omega.
Given a suitable characteristic function, df-frec 6009 produces the same results as df-irdg 5988 restricted to . Presumably the theorem would also hold if were changed to . (Contributed by Jim Kingdon, 29-Aug-2019.) |
frec | ||
Theorem | freccl 6024* | Closure for finite recursion. (Contributed by Jim Kingdon, 25-May-2020.) |
frec | ||
Syntax | c1o 6025 | Extend the definition of a class to include the ordinal number 1. |
Syntax | c2o 6026 | Extend the definition of a class to include the ordinal number 2. |
Syntax | c3o 6027 | Extend the definition of a class to include the ordinal number 3. |
Syntax | c4o 6028 | Extend the definition of a class to include the ordinal number 4. |
Syntax | coa 6029 | Extend the definition of a class to include the ordinal addition operation. |
Syntax | comu 6030 | Extend the definition of a class to include the ordinal multiplication operation. |
Syntax | coei 6031 | Extend the definition of a class to include the ordinal exponentiation operation. |
↑_{𝑜} | ||
Definition | df-1o 6032 | Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.) |
Definition | df-2o 6033 | Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.) |
Definition | df-3o 6034 | Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.) |
Definition | df-4o 6035 | Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.) |
Definition | df-oadd 6036* | Define the ordinal addition operation. (Contributed by NM, 3-May-1995.) |
Definition | df-omul 6037* | Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.) |
Definition | df-oexpi 6038* |
Define the ordinal exponentiation operation.
This definition is similar to a conventional definition of exponentiation except that it defines ↑_{𝑜} to be for all , in order to avoid having different cases for whether the base is or not. (Contributed by Mario Carneiro, 4-Jul-2019.) |
↑_{𝑜} | ||
Theorem | 1on 6039 | Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.) |
Theorem | 2on 6040 | Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Theorem | 2on0 6041 | Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.) |
Theorem | 3on 6042 | Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Theorem | 4on 6043 | Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Theorem | df1o2 6044 | Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.) |
Theorem | df2o3 6045 | Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Theorem | df2o2 6046 | Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.) |
Theorem | 1n0 6047 | Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.) |
Theorem | xp01disj 6048 | Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.) |
Theorem | ordgt0ge1 6049 | Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.) |
Theorem | ordge1n0im 6050 | An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.) |
Theorem | el1o 6051 | Membership in ordinal one. (Contributed by NM, 5-Jan-2005.) |
Theorem | dif1o 6052 | Two ways to say that is a nonzero number of the set . (Contributed by Mario Carneiro, 21-May-2015.) |
Theorem | 2oconcl 6053 | Closure of the pair swapping function on . (Contributed by Mario Carneiro, 27-Sep-2015.) |
Theorem | 0lt1o 6054 | Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.) |
Theorem | oafnex 6055 | The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.) |
Theorem | sucinc 6056* | Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.) |
Theorem | sucinc2 6057* | Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.) |
Theorem | fnoa 6058 | Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Mario Carneiro, 3-Jul-2019.) |
Theorem | oaexg 6059 | Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Theorem | omfnex 6060* | The characteristic function for ordinal multiplication is defined everywhere. (Contributed by Jim Kingdon, 23-Aug-2019.) |
Theorem | fnom 6061 | Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.) |
Theorem | omexg 6062 | Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Theorem | fnoei 6063 | Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by Mario Carneiro, 3-Jul-2019.) |
↑_{𝑜} | ||
Theorem | oeiexg 6064 | Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.) |
↑_{𝑜} | ||
Theorem | oav 6065* | Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | omv 6066* | Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.) |
Theorem | oeiv 6067* | Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.) |
↑_{𝑜} | ||
Theorem | oa0 6068 | Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | om0 6069 | Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | oei0 6070 | 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.) |
↑_{𝑜} | ||
Theorem | oacl 6071 | 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.) |
Theorem | omcl 6072 | 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.) |
Theorem | oeicl 6073 | Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.) |
↑_{𝑜} | ||
Theorem | oav2 6074* | Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.) |
Theorem | oasuc 6075 | Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | omv2 6076* | Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.) |
Theorem | onasuc 6077 | Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Theorem | oa1suc 6078 | 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.) |
Theorem | o1p1e2 6079 | 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.) |
Theorem | oawordi 6080 | Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.) |
Theorem | oaword1 6081 | 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.) |
Theorem | omsuc 6082 | Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | onmsuc 6083 | Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Theorem | nna0 6084 | Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) |
Theorem | nnm0 6085 | Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) |
Theorem | nnasuc 6086 | Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Theorem | nnmsuc 6087 | Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Theorem | nna0r 6088 | 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.) |
Theorem | nnm0r 6089 | Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Theorem | nnacl 6090 | 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.) |
Theorem | nnmcl 6091 | 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.) |
Theorem | nnacli 6092 | is closed under addition. Inference form of nnacl 6090. (Contributed by Scott Fenton, 20-Apr-2012.) |
Theorem | nnmcli 6093 | is closed under multiplication. Inference form of nnmcl 6091. (Contributed by Scott Fenton, 20-Apr-2012.) |
Theorem | nnacom 6094 | 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.) |
Theorem | nnaass 6095 | 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.) |
Theorem | nndi 6096 | Distributive law for natural numbers (left-distributivity). Theorem 4K(3) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Theorem | nnmass 6097 | Multiplication of natural numbers is associative. Theorem 4K(4) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Theorem | nnmsucr 6098 | Multiplication with successor. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | nnmcom 6099 | Multiplication of natural numbers is commutative. Theorem 4K(5) of [Enderton] p. 81. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | nndir 6100 | Distributive law for natural numbers (right-distributivity). (Contributed by Jim Kingdon, 3-Dec-2019.) |
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