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Theorem List for Intuitionistic Logic Explorer - 6001-6100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrdgss 6001 Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.)
(𝜑𝐹 Fn V)    &   (𝜑𝐼𝑉)    &   (𝜑𝐴 ∈ On)    &   (𝜑𝐵 ∈ On)    &   (𝜑𝐴𝐵)       (𝜑 → (rec(𝐹, 𝐼)‘𝐴) ⊆ (rec(𝐹, 𝐼)‘𝐵))

Theoremrdgisuc1 6002* 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 𝑥𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)).

If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 6003. (Contributed by Jim Kingdon, 9-Jun-2019.)

(𝜑𝐹 Fn V)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵 ∈ On)       (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ ( 𝑥𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))

Theoremrdgisucinc 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.)

(𝜑𝐹 Fn V)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵 ∈ On)    &   (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹𝑥))       (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))

Theoremrdgon 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.)
(𝜑𝐹 Fn V)    &   (𝜑𝐴 ∈ On)    &   (𝜑 → ∀𝑥 ∈ On (𝐹𝑥) ∈ On)       ((𝜑𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)

Theoremrdg0 6005 The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
𝐴 ∈ V       (rec(𝐹, 𝐴)‘∅) = 𝐴

Theoremrdg0g 6006 The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
(𝐴𝐶 → (rec(𝐹, 𝐴)‘∅) = 𝐴)

Theoremrdgexg 6007 The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.)
𝐴 ∈ V    &   𝐹 Fn V       (𝐵𝑉 → (rec(𝐹, 𝐴)‘𝐵) ∈ V)

2.6.21  Finite recursion

Syntaxcfrec 6008 Extend class notation with the fnite recursive definition generator, with characteristic function 𝐹 and initial value 𝐼.
class frec(𝐹, 𝐼)

Definitiondf-frec 6009* Define a recursive definition generator on ω (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 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 ω 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 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))})) ↾ ω)

Theoremfreceq1 6010 Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
(𝐹 = 𝐺 → frec(𝐹, 𝐴) = frec(𝐺, 𝐴))

Theoremfreceq2 6011 Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
(𝐴 = 𝐵 → frec(𝐹, 𝐴) = frec(𝐹, 𝐵))

Theoremfrecex 6012 Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.)
frec(𝐹, 𝐴) ∈ V

Theoremnffrec 6013 Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
𝑥𝐹    &   𝑥𝐴       𝑥frec(𝐹, 𝐴)

Theoremfrec0g 6014 The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
(𝐴𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)

Theoremfrecabex 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.)
(𝜑𝑆𝑉)    &   (𝜑 → ∀𝑦(𝐹𝑦) ∈ V)    &   (𝜑𝐴𝑊)       (𝜑 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚𝑥 ∈ (𝐹‘(𝑆𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥𝐴))} ∈ V)

Theoremfrectfr 6016* Lemma to connect transfinite recursion theorems with finite recursion. That is, given the conditions 𝐹 Fn V and 𝐴𝑉 on frec(𝐹, 𝐴), 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.)

𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})       ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺𝑦) ∈ V))

Theoremfrecfnom 6017* The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.)
((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → frec(𝐹, 𝐴) Fn ω)

Theoremfrecsuclem1 6018* Lemma for frecsuc 6022. (Contributed by Jim Kingdon, 13-Aug-2019.)
𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})       ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐺‘(recs(𝐺) ↾ suc 𝐵)))

Theoremfrecsuclemdm 6019* Lemma for frecsuc 6022. (Contributed by Jim Kingdon, 15-Aug-2019.)
𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})       ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝐵 ∈ ω) → dom (recs(𝐺) ↾ suc 𝐵) = suc 𝐵)

Theoremfrecsuclem2 6020* Lemma for frecsuc 6022. (Contributed by Jim Kingdon, 15-Aug-2019.)
𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})       ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝐵 ∈ ω) → ((recs(𝐺) ↾ suc 𝐵)‘𝐵) = (frec(𝐹, 𝐴)‘𝐵))

Theoremfrecsuclem3 6021* Lemma for frecsuc 6022. (Contributed by Jim Kingdon, 15-Aug-2019.)
𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})       ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))

Theoremfrecsuc 6022* The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 15-Aug-2019.)
((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))

Theoremfrecrdg 6023* Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6009 produces the same results as df-irdg 5988 restricted to ω.

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

(𝜑𝐹 Fn V)    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹𝑥))       (𝜑 → frec(𝐹, 𝐴) = (rec(𝐹, 𝐴) ↾ ω))

Theoremfreccl 6024* Closure for finite recursion. (Contributed by Jim Kingdon, 25-May-2020.)
(𝜑 → ∀𝑧(𝐹𝑧) ∈ V)    &   (𝜑𝐴𝑆)    &   ((𝜑𝑧𝑆) → (𝐹𝑧) ∈ 𝑆)    &   (𝜑𝐵 ∈ ω)       (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)

2.6.22  Ordinal arithmetic

Syntaxc1o 6025 Extend the definition of a class to include the ordinal number 1.
class 1𝑜

Syntaxc2o 6026 Extend the definition of a class to include the ordinal number 2.
class 2𝑜

Syntaxc3o 6027 Extend the definition of a class to include the ordinal number 3.
class 3𝑜

Syntaxc4o 6028 Extend the definition of a class to include the ordinal number 4.
class 4𝑜

Syntaxcoa 6029 Extend the definition of a class to include the ordinal addition operation.
class +𝑜

Syntaxcomu 6030 Extend the definition of a class to include the ordinal multiplication operation.
class ·𝑜

Syntaxcoei 6031 Extend the definition of a class to include the ordinal exponentiation operation.
class 𝑜

Definitiondf-1o 6032 Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
1𝑜 = suc ∅

Definitiondf-2o 6033 Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
2𝑜 = suc 1𝑜

Definitiondf-3o 6034 Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
3𝑜 = suc 2𝑜

Definitiondf-4o 6035 Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
4𝑜 = suc 3𝑜

+𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))

Definitiondf-omul 6037* Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)
·𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)‘𝑦))

Definitiondf-oexpi 6038* 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 ∈ On, in order to avoid having different cases for whether the base is or not. (Contributed by Mario Carneiro, 4-Jul-2019.)

𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦))

Theorem1on 6039 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)
1𝑜 ∈ On

Theorem2on 6040 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
2𝑜 ∈ On

Theorem2on0 6041 Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
2𝑜 ≠ ∅

Theorem3on 6042 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3𝑜 ∈ On

Theorem4on 6043 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4𝑜 ∈ On

Theoremdf1o2 6044 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
1𝑜 = {∅}

Theoremdf2o3 6045 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
2𝑜 = {∅, 1𝑜}

Theoremdf2o2 6046 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
2𝑜 = {∅, {∅}}

Theorem1n0 6047 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
1𝑜 ≠ ∅

Theoremxp01disj 6048 Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅

Theoremordgt0ge1 6049 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
(Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))

Theoremordge1n0im 6050 An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.)
(Ord 𝐴 → (1𝑜𝐴𝐴 ≠ ∅))

Theoremel1o 6051 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
(𝐴 ∈ 1𝑜𝐴 = ∅)

Theoremdif1o 6052 Two ways to say that 𝐴 is a nonzero number of the set 𝐵. (Contributed by Mario Carneiro, 21-May-2015.)
(𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ (𝐴𝐵𝐴 ≠ ∅))

Theorem2oconcl 6053 Closure of the pair swapping function on 2𝑜. (Contributed by Mario Carneiro, 27-Sep-2015.)
(𝐴 ∈ 2𝑜 → (1𝑜𝐴) ∈ 2𝑜)

Theorem0lt1o 6054 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
∅ ∈ 1𝑜

Theoremoafnex 6055 The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.)
(𝑥 ∈ V ↦ suc 𝑥) Fn V

Theoremsucinc 6056* Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
𝐹 = (𝑧 ∈ V ↦ suc 𝑧)       𝑥 𝑥 ⊆ (𝐹𝑥)

Theoremsucinc2 6057* Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.)
𝐹 = (𝑧 ∈ V ↦ suc 𝑧)       ((𝐵 ∈ On ∧ 𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))

Theoremfnoa 6058 Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Mario Carneiro, 3-Jul-2019.)
+𝑜 Fn (On × On)

Theoremoaexg 6059 Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴 +𝑜 𝐵) ∈ V)

Theoremomfnex 6060* The characteristic function for ordinal multiplication is defined everywhere. (Contributed by Jim Kingdon, 23-Aug-2019.)
(𝐴𝑉 → (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) Fn V)

Theoremfnom 6061 Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.)
·𝑜 Fn (On × On)

Theoremomexg 6062 Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴 ·𝑜 𝐵) ∈ V)

Theoremfnoei 6063 Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
𝑜 Fn (On × On)

Theoremoeiexg 6064 Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑜 𝐵) ∈ V)

Theoremoav 6065* Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))

Theoremomv 6066* Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))

Theoremoeiv 6067* Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))

Theoremoa0 6068 Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)

Theoremom0 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.)
(𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)

Theoremoei0 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.)
(𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)

Theoremoacl 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.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)

Theoremomcl 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.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)

Theoremoeicl 6073 Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)

Theoremoav2 6074* Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (𝐴 𝑥𝐵 suc (𝐴 +𝑜 𝑥)))

Theoremoasuc 6075 Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))

Theoremomv2 6076* Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))

Theoremonasuc 6077 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))

Theoremoa1suc 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.)
(𝐴 ∈ On → (𝐴 +𝑜 1𝑜) = suc 𝐴)

Theoremo1p1e2 6079 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
(1𝑜 +𝑜 1𝑜) = 2𝑜

Theoremoawordi 6080 Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))

Theoremoaword1 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.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +𝑜 𝐵))

Theoremomsuc 6082 Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))

Theoremonmsuc 6083 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))

2.6.23  Natural number arithmetic

Theoremnna0 6084 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)

Theoremnnm0 6085 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)

Theoremnnasuc 6086 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))

Theoremnnmsuc 6087 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))

Theoremnna0r 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.)
(𝐴 ∈ ω → (∅ +𝑜 𝐴) = 𝐴)

Theoremnnm0r 6089 Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
(𝐴 ∈ ω → (∅ ·𝑜 𝐴) = ∅)

Theoremnnacl 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.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈ ω)

Theoremnnmcl 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.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) ∈ ω)

Theoremnnacli 6092 ω is closed under addition. Inference form of nnacl 6090. (Contributed by Scott Fenton, 20-Apr-2012.)
𝐴 ∈ ω    &   𝐵 ∈ ω       (𝐴 +𝑜 𝐵) ∈ ω

Theoremnnmcli 6093 ω is closed under multiplication. Inference form of nnmcl 6091. (Contributed by Scott Fenton, 20-Apr-2012.)
𝐴 ∈ ω    &   𝐵 ∈ ω       (𝐴 ·𝑜 𝐵) ∈ ω

Theoremnnacom 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.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) = (𝐵 +𝑜 𝐴))

Theoremnnaass 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.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))

Theoremnndi 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.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))

Theoremnnmass 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.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))

Theoremnnmsucr 6098 Multiplication with successor. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵))

Theoremnnmcom 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.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴))

Theoremnndir 6100 Distributive law for natural numbers (right-distributivity). (Contributed by Jim Kingdon, 3-Dec-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +𝑜 𝐵) ·𝑜 𝐶) = ((𝐴 ·𝑜 𝐶) +𝑜 (𝐵 ·𝑜 𝐶)))

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