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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | oaword2 7901 | An ordinal is less than or equal to its sum with another. Theorem 21 of [Suppes] p. 209. (Contributed by NM, 7-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐵 +o 𝐴)) | ||
Theorem | oawordeulem 7902* | Lemma for oawordex 7905. (Contributed by NM, 11-Dec-2004.) |
⊢ 𝐴 ∈ On & ⊢ 𝐵 ∈ On & ⊢ 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +o 𝑦)} ⇒ ⊢ (𝐴 ⊆ 𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵) | ||
Theorem | oawordeu 7903* | Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59. (Contributed by NM, 11-Dec-2004.) |
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵) | ||
Theorem | oawordexr 7904* | Existence theorem for weak ordering of ordinal sum. (Contributed by NM, 12-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵) → 𝐴 ⊆ 𝐵) | ||
Theorem | oawordex 7905* | Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 7903 for uniqueness. (Contributed by NM, 12-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)) | ||
Theorem | oaordex 7906* | Existence theorem for ordering of ordinal sum. Similar to Proposition 4.34(f) of [Mendelson] p. 266 and its converse. (Contributed by NM, 12-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))) | ||
Theorem | oa00 7907 | An ordinal sum is zero iff both of its arguments are zero. (Contributed by NM, 6-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅))) | ||
Theorem | oalimcl 7908 | The ordinal sum with a limit ordinal is a limit ordinal. Proposition 8.11 of [TakeutiZaring] p. 60. (Contributed by NM, 8-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +o 𝐵)) | ||
Theorem | oaass 7909 | Ordinal addition is associative. Theorem 25 of [Suppes] p. 211. (Contributed by NM, 10-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶))) | ||
Theorem | oarec 7910* | Recursive definition of ordinal addition. Exercise 25 of [Enderton] p. 240. (Contributed by NM, 26-Dec-2004.) (Revised by Mario Carneiro, 30-May-2015.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))) | ||
Theorem | oaf1o 7911* | Left addition by a constant is a bijection from ordinals to ordinals greater than the constant. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ (𝐴 ∈ On → (𝑥 ∈ On ↦ (𝐴 +o 𝑥)):On–1-1-onto→(On ∖ 𝐴)) | ||
Theorem | oacomf1olem 7912* | Lemma for oacomf1o 7913. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ⇒ ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴–1-1-onto→ran 𝐹 ∧ (ran 𝐹 ∩ 𝐵) = ∅)) | ||
Theorem | oacomf1o 7913* | Define a bijection from 𝐴 +o 𝐵 to 𝐵 +o 𝐴. Thus, the two are equinumerous even if they are not equal (which sometimes occurs, e.g., oancom 8826). (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ 𝐹 = ((𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) ⇒ ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)–1-1-onto→(𝐵 +o 𝐴)) | ||
Theorem | omordi 7914 | Ordering property of ordinal multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.) |
⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))) | ||
Theorem | omord2 7915 | Ordering property of ordinal multiplication. (Contributed by NM, 25-Dec-2004.) |
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))) | ||
Theorem | omord 7916 | Ordering property of ordinal multiplication. Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))) | ||
Theorem | omcan 7917 | Left cancellation law for ordinal multiplication. Proposition 8.20 of [TakeutiZaring] p. 63 and its converse. (Contributed by NM, 14-Dec-2004.) |
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | omword 7918 | Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.) |
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) | ||
Theorem | omwordi 7919 | Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) | ||
Theorem | omwordri 7920 | Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶))) | ||
Theorem | omword1 7921 | An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.) |
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·o 𝐵)) | ||
Theorem | omword2 7922 | An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.) |
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐵 ·o 𝐴)) | ||
Theorem | om00 7923 | The product of two ordinal numbers is zero iff at least one of them is zero. Proposition 8.22 of [TakeutiZaring] p. 64. (Contributed by NM, 21-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅))) | ||
Theorem | om00el 7924 | The product of two nonzero ordinal numbers is nonzero. (Contributed by NM, 28-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ (𝐴 ·o 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))) | ||
Theorem | omordlim 7925* | Ordering involving the product of a limit ordinal. Proposition 8.23 of [TakeutiZaring] p. 64. (Contributed by NM, 25-Dec-2004.) |
⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐷 ∧ Lim 𝐵)) ∧ 𝐶 ∈ (𝐴 ·o 𝐵)) → ∃𝑥 ∈ 𝐵 𝐶 ∈ (𝐴 ·o 𝑥)) | ||
Theorem | omlimcl 7926 | The product of any nonzero ordinal with a limit ordinal is a limit ordinal. Proposition 8.24 of [TakeutiZaring] p. 64. (Contributed by NM, 25-Dec-2004.) |
⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·o 𝐵)) | ||
Theorem | odi 7927 | Distributive law for ordinal arithmetic (left-distributivity). Proposition 8.25 of [TakeutiZaring] p. 64. (Contributed by NM, 26-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))) | ||
Theorem | omass 7928 | Multiplication of ordinal numbers is associative. Theorem 8.26 of [TakeutiZaring] p. 65. (Contributed by NM, 28-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶))) | ||
Theorem | oneo 7929 | If an ordinal number is even, its successor is odd. (Contributed by NM, 26-Jan-2006.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 = (2o ·o 𝐴)) → ¬ suc 𝐶 = (2o ·o 𝐵)) | ||
Theorem | omeulem1 7930* | Lemma for omeu 7933: existence part. (Contributed by Mario Carneiro, 28-Feb-2013.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵) | ||
Theorem | omeulem2 7931 | Lemma for omeu 7933: uniqueness part. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 29-May-2015.) |
⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐵 ∈ 𝐷 ∨ (𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸))) | ||
Theorem | omopth2 7932 | An ordered pair-like theorem for ordinal multiplication. (Contributed by Mario Carneiro, 29-May-2015.) |
⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸) ↔ (𝐵 = 𝐷 ∧ 𝐶 = 𝐸))) | ||
Theorem | omeu 7933* | The division algorithm for ordinal multiplication. (Contributed by Mario Carneiro, 28-Feb-2013.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = 〈𝑥, 𝑦〉 ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) | ||
Theorem | oen0 7934 | Ordinal exponentiation with a nonzero mantissa is nonzero. Proposition 8.32 of [TakeutiZaring] p. 67. (Contributed by NM, 4-Jan-2005.) |
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵)) | ||
Theorem | oeordi 7935 | Ordering law for ordinal exponentiation. Proposition 8.33 of [TakeutiZaring] p. 67. (Contributed by NM, 5-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.) |
⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵))) | ||
Theorem | oeord 7936 | Ordering property of ordinal exponentiation. Corollary 8.34 of [TakeutiZaring] p. 68 and its converse. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 ↔ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵))) | ||
Theorem | oecan 7937 | Left cancellation law for ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.) |
⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | oeword 7938 | Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))) | ||
Theorem | oewordi 7939 | Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) |
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))) | ||
Theorem | oewordri 7940 | Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68. (Contributed by NM, 6-Jan-2005.) |
⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶))) | ||
Theorem | oeworde 7941 | Ordinal exponentiation compared to its exponent. Proposition 8.37 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.) |
⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵)) | ||
Theorem | oeordsuc 7942 | Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.) |
⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o suc 𝐶) ∈ (𝐵 ↑o suc 𝐶))) | ||
Theorem | oelim2 7943* | Ordinal exponentiation with a limit exponent. Part of Exercise 4.36 of [Mendelson] p. 250. (Contributed by NM, 6-Jan-2005.) |
⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥)) | ||
Theorem | oeoalem 7944 | Lemma for oeoa 7945. (Contributed by Eric Schmidt, 26-May-2009.) |
⊢ 𝐴 ∈ On & ⊢ ∅ ∈ 𝐴 & ⊢ 𝐵 ∈ On ⇒ ⊢ (𝐶 ∈ On → (𝐴 ↑o (𝐵 +o 𝐶)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝐶))) | ||
Theorem | oeoa 7945 | Sum of exponents law for ordinal exponentiation. Theorem 8R of [Enderton] p. 238. Also Proposition 8.41 of [TakeutiZaring] p. 69. (Contributed by Eric Schmidt, 26-May-2009.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o (𝐵 +o 𝐶)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝐶))) | ||
Theorem | oeoelem 7946 | Lemma for oeoe 7947. (Contributed by Eric Schmidt, 26-May-2009.) |
⊢ 𝐴 ∈ On & ⊢ ∅ ∈ 𝐴 ⇒ ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶))) | ||
Theorem | oeoe 7947 | Product of exponents law for ordinal exponentiation. Theorem 8S of [Enderton] p. 238. Also Proposition 8.42 of [TakeutiZaring] p. 70. (Contributed by Eric Schmidt, 26-May-2009.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶))) | ||
Theorem | oelimcl 7948 | The ordinal exponential with a limit ordinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2015.) |
⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 ↑o 𝐵)) | ||
Theorem | oeeulem 7949* | Lemma for oeeu 7951. (Contributed by Mario Carneiro, 28-Feb-2013.) |
⊢ 𝑋 = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ⇒ ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝑋 ∈ On ∧ (𝐴 ↑o 𝑋) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑o suc 𝑋))) | ||
Theorem | oeeui 7950* | The division algorithm for ordinal exponentiation. (This version of oeeu 7951 gives an explicit expression for the unique solution of the equation, in terms of the solution 𝑃 to omeu 7933.) (Contributed by Mario Carneiro, 25-May-2015.) |
⊢ 𝑋 = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} & ⊢ 𝑃 = (℩𝑤∃𝑦 ∈ On ∃𝑧 ∈ (𝐴 ↑o 𝑋)(𝑤 = 〈𝑦, 𝑧〉 ∧ (((𝐴 ↑o 𝑋) ·o 𝑦) +o 𝑧) = 𝐵)) & ⊢ 𝑌 = (1st ‘𝑃) & ⊢ 𝑍 = (2nd ‘𝑃) ⇒ ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o) ∧ 𝐸 ∈ (𝐴 ↑o 𝐶)) ∧ (((𝐴 ↑o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵) ↔ (𝐶 = 𝑋 ∧ 𝐷 = 𝑌 ∧ 𝐸 = 𝑍))) | ||
Theorem | oeeu 7951* | The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.) |
⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∃!𝑤∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1o)∃𝑧 ∈ (𝐴 ↑o 𝑥)(𝑤 = 〈𝑥, 𝑦, 𝑧〉 ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) | ||
Theorem | nna0 7952 | Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) |
⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴) | ||
Theorem | nnm0 7953 | Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) |
⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅) | ||
Theorem | nnasuc 7954 | 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)) | ||
Theorem | nnmsuc 7955 | 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)) | ||
Theorem | nnesuc 7956 | Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴)) | ||
Theorem | nna0r 7957 | Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. Note: In this and later theorems, we deliberately avoid the more general ordinal versions of these theorems (in this case oa0r 7886) so that we can avoid ax-rep 4995, which is not needed for finite recursive definitions. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
⊢ (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴) | ||
Theorem | nnm0r 7958 | 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) = ∅) | ||
Theorem | nnacl 7959 | 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) | ||
Theorem | nnmcl 7960 | 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) | ||
Theorem | nnecl 7961 | Closure of exponentiation of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 24-Mar-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ↑o 𝐵) ∈ ω) | ||
Theorem | nnacli 7962 | ω is closed under addition. Inference form of nnacl 7959. (Contributed by Scott Fenton, 20-Apr-2012.) |
⊢ 𝐴 ∈ ω & ⊢ 𝐵 ∈ ω ⇒ ⊢ (𝐴 +o 𝐵) ∈ ω | ||
Theorem | nnmcli 7963 | ω is closed under multiplication. Inference form of nnmcl 7960. (Contributed by Scott Fenton, 20-Apr-2012.) |
⊢ 𝐴 ∈ ω & ⊢ 𝐵 ∈ ω ⇒ ⊢ (𝐴 ·o 𝐵) ∈ ω | ||
Theorem | nnarcl 7964 | Reverse closure law for addition of natural numbers. Exercise 1 of [TakeutiZaring] p. 62 and its converse. (Contributed by NM, 12-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝐵 ∈ ω))) | ||
Theorem | nnacom 7965 | 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)) | ||
Theorem | nnaordi 7966 | 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))) | ||
Theorem | nnaord 7967 | 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))) | ||
Theorem | nnaordr 7968 | 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))) | ||
Theorem | nnawordi 7969 | 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) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶))) | ||
Theorem | nnaass 7970 | 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))) | ||
Theorem | nndi 7971 | 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))) | ||
Theorem | nnmass 7972 | 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))) | ||
Theorem | nnmsucr 7973 | 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)) | ||
Theorem | nnmcom 7974 | 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)) | ||
Theorem | nnaword 7975 | Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))) | ||
Theorem | nnacan 7976 | 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)) | ||
Theorem | nnaword1 7977 | 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)) | ||
Theorem | nnaword2 7978 | Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +o 𝐴)) | ||
Theorem | nnmordi 7979 | 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))) | ||
Theorem | nnmord 7980 | 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))) | ||
Theorem | nnmword 7981 | Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) | ||
Theorem | nnmcan 7982 | 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)) | ||
Theorem | nnmwordi 7983 | Weak ordering property of multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) | ||
Theorem | nnmwordri 7984 | Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶))) | ||
Theorem | nnawordex 7985* | Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)) | ||
Theorem | nnaordex 7986* | Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))) | ||
Theorem | 1onn 7987 | One is a natural number. (Contributed by NM, 29-Oct-1995.) |
⊢ 1o ∈ ω | ||
Theorem | 2onn 7988 | The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.) |
⊢ 2o ∈ ω | ||
Theorem | 3onn 7989 | The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ 3o ∈ ω | ||
Theorem | 4onn 7990 | The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ 4o ∈ ω | ||
Theorem | oaabslem 7991 | Lemma for oaabs 7992. (Contributed by NM, 9-Dec-2004.) |
⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω) | ||
Theorem | oaabs 7992 | Ordinal addition absorbs a natural number added to the left of a transfinite number. Proposition 8.10 of [TakeutiZaring] p. 59. (Contributed by NM, 9-Dec-2004.) (Proof shortened by Mario Carneiro, 29-May-2015.) |
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +o 𝐵) = 𝐵) | ||
Theorem | oaabs2 7993 | The absorption law oaabs 7992 is also a property of higher powers of ω. (Contributed by Mario Carneiro, 29-May-2015.) |
⊢ (((𝐴 ∈ (ω ↑o 𝐶) ∧ 𝐵 ∈ On) ∧ (ω ↑o 𝐶) ⊆ 𝐵) → (𝐴 +o 𝐵) = 𝐵) | ||
Theorem | omabslem 7994 | Lemma for omabs 7995. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) = ω) | ||
Theorem | omabs 7995 | Ordinal multiplication is also absorbed by powers of ω. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵)) | ||
Theorem | nnm1 7996 | Multiply an element of ω by 1o. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴) | ||
Theorem | nnm2 7997 | Multiply an element of ω by 2o. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ω → (𝐴 ·o 2o) = (𝐴 +o 𝐴)) | ||
Theorem | nn2m 7998 | Multiply an element of ω by 2o. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ω → (2o ·o 𝐴) = (𝐴 +o 𝐴)) | ||
Theorem | nnneo 7999 | If a natural number is even, its successor is odd. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → ¬ suc 𝐶 = (2o ·o 𝐵)) | ||
Theorem | nneob 8000* | A natural number is even iff its successor is odd. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))) |
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