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Theorem List for Intuitionistic Logic Explorer - 5901-6000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrecsuclem2 5901* Lemma for frecsuc 5903. (Contributed by Jim Kingdon, 15-Aug-2019.)
𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})       ((𝐹 Fn V A 𝑉 B 𝜔) → ((recs(𝐺) ↾ suc B)‘B) = (frec(𝐹, A)‘B))
 
Theoremfrecsuclem3 5902* Lemma for frecsuc 5903. (Contributed by Jim Kingdon, 15-Aug-2019.)
𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})       ((𝐹 Fn V A 𝑉 B 𝜔) → (frec(𝐹, A)‘suc B) = (𝐹‘(frec(𝐹, A)‘B)))
 
Theoremfrecsuc 5903 The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 15-Aug-2019.)
((𝐹 Fn V A 𝑉 B 𝜔) → (frec(𝐹, A)‘suc B) = (𝐹‘(frec(𝐹, A)‘B)))
 
Theoremfrecrdg 5904* Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 5894 produces the same results as df-irdg 5874 restricted to 𝜔. (Contributed by Jim Kingdon, 29-Aug-2019.)

(φ𝐹 Fn V)    &   (φA 𝑉)    &   (φx x ⊆ (𝐹x))       (φ → frec(𝐹, A) = (rec(𝐹, A) ↾ 𝜔))
 
2.6.22  Ordinal arithmetic
 
Syntaxc1o 5905 Extend the definition of a class to include the ordinal number 1.
class 1𝑜
 
Syntaxc2o 5906 Extend the definition of a class to include the ordinal number 2.
class 2𝑜
 
Syntaxc3o 5907 Extend the definition of a class to include the ordinal number 3.
class 3𝑜
 
Syntaxc4o 5908 Extend the definition of a class to include the ordinal number 4.
class 4𝑜
 
Syntaxcoa 5909 Extend the definition of a class to include the ordinal addition operation.
class +𝑜
 
Syntaxcomu 5910 Extend the definition of a class to include the ordinal multiplication operation.
class ·𝑜
 
Syntaxcoei 5911 Extend the definition of a class to include the ordinal exponentiation operation.
class 𝑜
 
Definitiondf-1o 5912 Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
1𝑜 = suc ∅
 
Definitiondf-2o 5913 Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
2𝑜 = suc 1𝑜
 
Definitiondf-3o 5914 Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
3𝑜 = suc 2𝑜
 
Definitiondf-4o 5915 Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
4𝑜 = suc 3𝑜
 
Definitiondf-oadd 5916* Define the ordinal addition operation. (Contributed by NM, 3-May-1995.)
+𝑜 = (x On, y On ↦ (rec((z V ↦ suc z), x)‘y))
 
Definitiondf-omul 5917* 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 5918* 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 5919 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)
1𝑜 On
 
Theorem2on 5920 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
2𝑜 On
 
Theorem2on0 5921 Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
2𝑜 ≠ ∅
 
Theorem3on 5922 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3𝑜 On
 
Theorem4on 5923 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4𝑜 On
 
Theoremdf1o2 5924 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
1𝑜 = {∅}
 
Theoremdf2o3 5925 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
2𝑜 = {∅, 1𝑜}
 
Theoremdf2o2 5926 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
2𝑜 = {∅, {∅}}
 
Theorem1n0 5927 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
1𝑜 ≠ ∅
 
Theoremxp01disj 5928 Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
((A × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
 
Theoremordgt0ge1 5929 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
(Ord A → (∅ A ↔ 1𝑜A))
 
Theoremordge1n0im 5930 An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.)
(Ord A → (1𝑜AA ≠ ∅))
 
Theoremel1o 5931 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
(A 1𝑜A = ∅)
 
Theoremdif1o 5932 Two ways to say that A is a nonzero number of the set B. (Contributed by Mario Carneiro, 21-May-2015.)
(A (B ∖ 1𝑜) ↔ (A B A ≠ ∅))
 
Theorem2oconcl 5933 Closure of the pair swapping function on 2𝑜. (Contributed by Mario Carneiro, 27-Sep-2015.)
(A 2𝑜 → (1𝑜A) 2𝑜)
 
Theorem0lt1o 5934 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
1𝑜
 
Theoremoafnex 5935 The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.)
(x V ↦ suc x) Fn V
 
Theoremsucinc 5936* Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
𝐹 = (z V ↦ suc z)       x x ⊆ (𝐹x)
 
Theoremsucinc2 5937* Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.)
𝐹 = (z V ↦ suc z)       ((B On A B) → (𝐹A) ⊆ (𝐹B))
 
Theoremfnoa 5938 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 5939 Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((A 𝑉 B 𝑊) → (A +𝑜 B) V)
 
Theoremomfnex 5940* The characteristic function for ordinal multiplication is defined everywhere. (Contributed by Jim Kingdon, 23-Aug-2019.)
(A 𝑉 → (x V ↦ (x +𝑜 A)) Fn V)
 
Theoremfnom 5941 Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.)
·𝑜 Fn (On × On)
 
Theoremomexg 5942 Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((A 𝑉 B 𝑊) → (A ·𝑜 B) V)
 
Theoremfnoei 5943 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 5944 Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
((A 𝑉 B 𝑊) → (A𝑜 B) V)
 
Theoremoav 5945* 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 5946* 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 5947* 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 5948 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 5949 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 5950 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 5951 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 5952 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 5953 Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)
((A On B On) → (A𝑜 B) On)
 
Theoremoav2 5954* Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.)
((A On B On) → (A +𝑜 B) = (A x B suc (A +𝑜 x)))
 
Theoremoasuc 5955 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 5956* Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.)
((A On B On) → (A ·𝑜 B) = x B ((A ·𝑜 x) +𝑜 A))
 
Theoremonasuc 5957 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 5958 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 5959 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
(1𝑜 +𝑜 1𝑜) = 2𝑜
 
Theoremoawordi 5960 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 5961 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 5962 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 5963 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 5964 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
(A 𝜔 → (A +𝑜 ∅) = A)
 
Theoremnnm0 5965 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
(A 𝜔 → (A ·𝑜 ∅) = ∅)
 
Theoremnnasuc 5966 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 5967 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 5968 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 5969 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 5970 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 5971 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 5972 𝜔 is closed under addition. Inference form of nnacl 5970. (Contributed by Scott Fenton, 20-Apr-2012.)
A 𝜔    &   B 𝜔       (A +𝑜 B) 𝜔
 
Theoremnnmcli 5973 𝜔 is closed under multiplication. Inference form of nnmcl 5971. (Contributed by Scott Fenton, 20-Apr-2012.)
A 𝜔    &   B 𝜔       (A ·𝑜 B) 𝜔
 
Theoremnnacom 5974 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 5975 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 5976 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 5977 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 5978 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 5979 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 5980 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)))
 
Theoremnnsucelsuc 5981 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4179, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4195. (Contributed by Jim Kingdon, 25-Aug-2019.)
(B 𝜔 → (A B ↔ suc A suc B))
 
Theoremnnsucsssuc 5982 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4180, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4192. (Contributed by Jim Kingdon, 25-Aug-2019.)
((A 𝜔 B 𝜔) → (AB ↔ suc A ⊆ suc B))
 
Theoremnntri3or 5983 Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.)
((A 𝜔 B 𝜔) → (A B A = B B A))
 
Theoremnntri2 5984 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
((A 𝜔 B 𝜔) → (A B ↔ ¬ (A = B B A)))
 
Theoremnntri1 5985 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
((A 𝜔 B 𝜔) → (AB ↔ ¬ B A))
 
Theoremnndceq 5986 Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where B is zero, see nndceq0 4262. (Contributed by Jim Kingdon, 31-Aug-2019.)
((A 𝜔 B 𝜔) → DECID A = B)
 
Theoremnndcel 5987 Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.)
((A 𝜔 B 𝜔) → DECID A B)
 
Theoremnnaordi 5988 Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers. (Contributed by NM, 3-Feb-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
((B 𝜔 𝐶 𝜔) → (A B → (𝐶 +𝑜 A) (𝐶 +𝑜 B)))
 
Theoremnnaord 5989 Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers, and its converse. (Contributed by NM, 7-Mar-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔 𝐶 𝜔) → (A B ↔ (𝐶 +𝑜 A) (𝐶 +𝑜 B)))
 
Theoremnnaordr 5990 Ordering property of addition of natural numbers. (Contributed by NM, 9-Nov-2002.)
((A 𝜔 B 𝜔 𝐶 𝜔) → (A B ↔ (A +𝑜 𝐶) (B +𝑜 𝐶)))
 
Theoremnnaword 5991 Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔 𝐶 𝜔) → (AB ↔ (𝐶 +𝑜 A) ⊆ (𝐶 +𝑜 B)))
 
Theoremnnacan 5992 Cancellation law for addition of natural numbers. (Contributed by NM, 27-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔 𝐶 𝜔) → ((A +𝑜 B) = (A +𝑜 𝐶) ↔ B = 𝐶))
 
Theoremnnaword1 5993 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔) → A ⊆ (A +𝑜 B))
 
Theoremnnaword2 5994 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
((A 𝜔 B 𝜔) → A ⊆ (B +𝑜 A))
 
Theoremnnawordi 5995 Adding to both sides of an inequality in 𝜔 (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)
((A 𝜔 B 𝜔 𝐶 𝜔) → (AB → (A +𝑜 𝐶) ⊆ (B +𝑜 𝐶)))
 
Theoremnnmordi 5996 Ordering property of multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 18-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
(((B 𝜔 𝐶 𝜔) 𝐶) → (A B → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))
 
Theoremnnmord 5997 Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 22-Jan-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔 𝐶 𝜔) → ((A B 𝐶) ↔ (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))
 
Theoremnnmword 5998 Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)
(((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → (AB ↔ (𝐶 ·𝑜 A) ⊆ (𝐶 ·𝑜 B)))
 
Theoremnnmcan 5999 Cancellation law for multiplication of natural numbers. (Contributed by NM, 26-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
(((A 𝜔 B 𝜔 𝐶 𝜔) A) → ((A ·𝑜 B) = (A ·𝑜 𝐶) ↔ B = 𝐶))
 
Theorem1onn 6000 One is a natural number. (Contributed by NM, 29-Oct-1995.)
1𝑜 𝜔
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