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Type | Label | Description |
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Statement | ||
Theorem | oelim 7501* | Ordinal exponentiation with a limit exponent and nonzero mantissa. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 1-Jan-2005.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ↑𝑜 𝑥)) | ||
Theorem | oacl 7502 | Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On) | ||
Theorem | omcl 7503 | Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On) | ||
Theorem | oecl 7504 | Closure law for ordinal exponentiation. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On) | ||
Theorem | oa0r 7505 | Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.) |
⊢ (𝐴 ∈ On → (∅ +𝑜 𝐴) = 𝐴) | ||
Theorem | om0r 7506 | Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.) |
⊢ (𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅) | ||
Theorem | o1p1e2 7507 | 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.) |
⊢ (1𝑜 +𝑜 1𝑜) = 2𝑜 | ||
Theorem | o2p2e4 7508 | 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 5646. For the usual proof using complex numbers, see 2p2e4 11021. (Contributed by NM, 18-Aug-2021.) |
⊢ (2𝑜 +𝑜 2𝑜) = 4𝑜 | ||
Theorem | om1 7509 | Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 29-Oct-1995.) |
⊢ (𝐴 ∈ On → (𝐴 ·𝑜 1𝑜) = 𝐴) | ||
Theorem | om1r 7510 | Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.) |
⊢ (𝐴 ∈ On → (1𝑜 ·𝑜 𝐴) = 𝐴) | ||
Theorem | oe1 7511 | Ordinal exponentiation with an exponent of 1. (Contributed by NM, 2-Jan-2005.) |
⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 1𝑜) = 𝐴) | ||
Theorem | oe1m 7512 | Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. (Contributed by NM, 2-Jan-2005.) |
⊢ (𝐴 ∈ On → (1𝑜 ↑𝑜 𝐴) = 1𝑜) | ||
Theorem | oaordi 7513 | Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.) |
⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))) | ||
Theorem | oaord 7514 | Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58 and its converse. (Contributed by NM, 5-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 ↔ (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))) | ||
Theorem | oacan 7515 | Left cancellation law for ordinal addition. Corollary 8.5 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +𝑜 𝐵) = (𝐴 +𝑜 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | oaword 7516 | Weak ordering property of ordinal addition. (Contributed by NM, 6-Dec-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵))) | ||
Theorem | oawordri 7517 | Weak ordering property of ordinal addition. Proposition 8.7 of [TakeutiZaring] p. 59. (Contributed by NM, 7-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶))) | ||
Theorem | oaord1 7518 | An ordinal is less than its sum with a nonzero ordinal. Theorem 18 of [Suppes] p. 209 and its converse. (Contributed by NM, 6-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ 𝐴 ∈ (𝐴 +𝑜 𝐵))) | ||
Theorem | oaword1 7519 | An ordinal is less than or equal to its sum with another. Part of Exercise 5 of [TakeutiZaring] p. 62. (For the other part see oaord1 7518.) (Contributed by NM, 6-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +𝑜 𝐵)) | ||
Theorem | oaword2 7520 | 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 7521* | Lemma for oawordex 7524. (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 7522* | 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 7523* | 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 7524* | Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 7522 for uniqueness. (Contributed by NM, 12-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)) | ||
Theorem | oaordex 7525* | 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 7526 | 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 7527 | 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 7528 | 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 7529* | 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 7530* | 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 7531* | Lemma for oacomf1o 7532. (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 7532* | 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 8431). (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 7533 | 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 7534 | 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 7535 | 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 7536 | 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 7537 | 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 7538 | 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 7539 | 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 7540 | 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 7541 | 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 7542 | 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 7543 | 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 7544* | 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 7545 | 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 7546 | 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 7547 | 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 7548 | 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 7549* | Lemma for omeu 7552: 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 7550 | Lemma for omeu 7552: 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 7551 | 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 7552* | 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 7553 | 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 7554 | 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 7555 | 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 7556 | 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 7557 | 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 7558 | 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 7559 | 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 7560 | 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 7561 | 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 7562* | 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 7563 | Lemma for oeoa 7564. (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 7564 | 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 7565 | Lemma for oeoe 7566. (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 7566 | 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 7567 | 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 7568* | Lemma for oeeu 7570. (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 7569* | The division algorithm for ordinal exponentiation. (This version of oeeu 7570 gives an explicit expression for the unique solution of the equation, in terms of the solution 𝑃 to omeu 7552.) (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 7570* | 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 7571 | Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) |
⊢ (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴) | ||
Theorem | nnm0 7572 | Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) |
⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅) | ||
Theorem | nnasuc 7573 | 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 7574 | 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 7575 | 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 7576 | 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 7505) so that we can avoid ax-rep 4699, 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 7577 | 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 7578 | 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 7579 | 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 7580 | 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 7581 | ω is closed under addition. Inference form of nnacl 7578. (Contributed by Scott Fenton, 20-Apr-2012.) |
⊢ 𝐴 ∈ ω & ⊢ 𝐵 ∈ ω ⇒ ⊢ (𝐴 +𝑜 𝐵) ∈ ω | ||
Theorem | nnmcli 7582 | ω is closed under multiplication. Inference form of nnmcl 7579. (Contributed by Scott Fenton, 20-Apr-2012.) |
⊢ 𝐴 ∈ ω & ⊢ 𝐵 ∈ ω ⇒ ⊢ (𝐴 ·𝑜 𝐵) ∈ ω | ||
Theorem | nnarcl 7583 | 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 7584 | 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 7585 | 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 7586 | 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 7587 | 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 7588 | 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 7589 | 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 7590 | 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 7591 | 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 7592 | 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 7593 | 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 7594 | 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 7595 | 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 7596 | 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 7597 | Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +𝑜 𝐴)) | ||
Theorem | nnmordi 7598 | 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 7599 | 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 7600 | 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))) |
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