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Theorem List for Metamath Proof Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremoelim 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 ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = 𝑥𝐵 (𝐴𝑜 𝑥))

Theoremoacl 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)

Theoremomcl 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)

Theoremoecl 7504 Closure law for ordinal exponentiation. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)

Theoremoa0r 7505 Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)
(𝐴 ∈ On → (∅ +𝑜 𝐴) = 𝐴)

Theoremom0r 7506 Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)
(𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅)

Theoremo1p1e2 7507 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
(1𝑜 +𝑜 1𝑜) = 2𝑜

Theoremo2p2e4 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

Theoremom1 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)

Theoremom1r 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)

Theoremoe1 7511 Ordinal exponentiation with an exponent of 1. (Contributed by NM, 2-Jan-2005.)
(𝐴 ∈ On → (𝐴𝑜 1𝑜) = 𝐴)

Theoremoe1m 7512 Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. (Contributed by NM, 2-Jan-2005.)
(𝐴 ∈ On → (1𝑜𝑜 𝐴) = 1𝑜)

Theoremoaordi 7513 Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)
((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))

Theoremoaord 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) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))

Theoremoacan 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))

Theoremoaword 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) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))

Theoremoawordri 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) → (𝐴𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))

Theoremoaord1 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) → (∅ ∈ 𝐵𝐴 ∈ (𝐴 +𝑜 𝐵)))

Theoremoaword1 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))

Theoremoaword2 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))

Theoremoawordeulem 7521* Lemma for oawordex 7524. (Contributed by NM, 11-Dec-2004.)
𝐴 ∈ On    &   𝐵 ∈ On    &   𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}       (𝐴𝐵 → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)

Theoremoawordeu 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) ∧ 𝐴𝐵) → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)

Theoremoawordexr 7523* Existence theorem for weak ordering of ordinal sum. (Contributed by NM, 12-Dec-2004.)
((𝐴 ∈ On ∧ ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵)

Theoremoawordex 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) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵))

Theoremoaordex 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) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))

Theoremoa00 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 = ∅)))

Theoremoalimcl 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 ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +𝑜 𝐵))

Theoremoaass 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)))

Theoremoarec 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 (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))))

Theoremoaf1o 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))

Theoremoacomf1olem 7531* Lemma for oacomf1o 7532. (Contributed by Mario Carneiro, 30-May-2015.)
𝐹 = (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥))       ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:𝐴1-1-onto→ran 𝐹 ∧ (ran 𝐹𝐵) = ∅))

Theoremoacomf1o 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 = ((𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))       ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(𝐵 +𝑜 𝐴))

Theoremomordi 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) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Theoremomord2 7534 Ordering property of ordinal multiplication. (Contributed by NM, 25-Dec-2004.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Theoremomord 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) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Theoremomcan 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))

Theoremomword 7537 Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))

Theoremomwordi 7538 Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))

Theoremomwordri 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) → (𝐴𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))

Theoremomword1 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))

Theoremomword2 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))

Theoremom00 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 = ∅)))

Theoremom00el 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)))

Theoremomordlim 7544* Ordering involving the product of a limit ordinal. Proposition 8.23 of [TakeutiZaring] p. 64. (Contributed by NM, 25-Dec-2004.)
(((𝐴 ∈ On ∧ (𝐵𝐷 ∧ Lim 𝐵)) ∧ 𝐶 ∈ (𝐴 ·𝑜 𝐵)) → ∃𝑥𝐵 𝐶 ∈ (𝐴 ·𝑜 𝑥))

Theoremomlimcl 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 ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·𝑜 𝐵))

Theoremodi 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)))

Theoremomass 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)))

Theoremoneo 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))

Theoremomeulem1 7549* Lemma for omeu 7552: existence part. (Contributed by Mario Carneiro, 28-Feb-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On ∃𝑦𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)

Theoremomeulem2 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 ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐵𝐷 ∨ (𝐵 = 𝐷𝐶𝐸)) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐶) ∈ ((𝐴 ·𝑜 𝐷) +𝑜 𝐸)))

Theoremomopth2 7551 An ordered pair-like theorem for ordinal multiplication. (Contributed by Mario Carneiro, 29-May-2015.)
(((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → (((𝐴 ·𝑜 𝐵) +𝑜 𝐶) = ((𝐴 ·𝑜 𝐷) +𝑜 𝐸) ↔ (𝐵 = 𝐷𝐶 = 𝐸)))

Theoremomeu 7552* The division algorithm for ordinal multiplication. (Contributed by Mario Carneiro, 28-Feb-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑧𝑥 ∈ On ∃𝑦𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵))

Theoremoen0 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) → ∅ ∈ (𝐴𝑜 𝐵))

Theoremoeordi 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)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))

Theoremoeord 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)) → (𝐴𝐵 ↔ (𝐶𝑜 𝐴) ∈ (𝐶𝑜 𝐵)))

Theoremoecan 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) → ((𝐴𝑜 𝐵) = (𝐴𝑜 𝐶) ↔ 𝐵 = 𝐶))

Theoremoeword 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)) → (𝐴𝐵 ↔ (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵)))

Theoremoewordi 7558 Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵)))

Theoremoewordri 7559 Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68. (Contributed by NM, 6-Jan-2005.)
((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))

Theoremoeworde 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 ⊆ (𝐴𝑜 𝐵))

Theoremoeordsuc 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) → (𝐴𝐵 → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))

Theoremoelim2 7562* Ordinal exponentiation with a limit exponent. Part of Exercise 4.36 of [Mendelson] p. 250. (Contributed by NM, 6-Jan-2005.)
((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴𝑜 𝐵) = 𝑥 ∈ (𝐵 ∖ 1𝑜)(𝐴𝑜 𝑥))

Theoremoeoalem 7563 Lemma for oeoa 7564. (Contributed by Eric Schmidt, 26-May-2009.)
𝐴 ∈ On    &   ∅ ∈ 𝐴    &   𝐵 ∈ On       (𝐶 ∈ On → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))

Theoremoeoa 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) → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))

Theoremoeoelem 7565 Lemma for oeoe 7566. (Contributed by Eric Schmidt, 26-May-2009.)
𝐴 ∈ On    &   ∅ ∈ 𝐴       ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))

Theoremoeoe 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) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))

Theoremoelimcl 7567 The ordinal exponential with a limit ordinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2015.)
((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim (𝐴𝑜 𝐵))

Theoremoeeulem 7568* Lemma for oeeu 7570. (Contributed by Mario Carneiro, 28-Feb-2013.)
𝑋 = {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑥)}       ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝑋 ∈ On ∧ (𝐴𝑜 𝑋) ⊆ 𝐵𝐵 ∈ (𝐴𝑜 suc 𝑋)))

Theoremoeeui 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 ∈ (𝐴𝑜 𝑥)}    &   𝑃 = (℩𝑤𝑦 ∈ On ∃𝑧 ∈ (𝐴𝑜 𝑋)(𝑤 = ⟨𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑋) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))    &   𝑌 = (1st𝑃)    &   𝑍 = (2nd𝑃)       ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1𝑜) ∧ 𝐸 ∈ (𝐴𝑜 𝐶)) ∧ (((𝐴𝑜 𝐶) ·𝑜 𝐷) +𝑜 𝐸) = 𝐵) ↔ (𝐶 = 𝑋𝐷 = 𝑌𝐸 = 𝑍)))

Theoremoeeu 7570* The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.)
((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃!𝑤𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))

2.4.19  Natural number arithmetic

Theoremnna0 7571 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)

Theoremnnm0 7572 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)

Theoremnnasuc 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))

Theoremnnmsuc 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))

Theoremnnesuc 7575 Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝑜 suc 𝐵) = ((𝐴𝑜 𝐵) ·𝑜 𝐴))

Theoremnna0r 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)

Theoremnnm0r 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) = ∅)

Theoremnnacl 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)

Theoremnnmcl 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)

Theoremnnecl 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) → (𝐴𝑜 𝐵) ∈ ω)

Theoremnnacli 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

Theoremnnmcli 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

Theoremnnarcl 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)))

Theoremnnacom 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))

Theoremnnaordi 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) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))

Theoremnnaord 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) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))

Theoremnnaordr 7587 Ordering property of addition of natural numbers. (Contributed by NM, 9-Nov-2002.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶)))

Theoremnnawordi 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) → (𝐴𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))

Theoremnnaass 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)))

Theoremnndi 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)))

Theoremnnmass 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)))

Theoremnnmsucr 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))

Theoremnnmcom 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))

Theoremnnaword 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) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))

Theoremnnacan 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))

Theoremnnaword1 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))

Theoremnnaword2 7597 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +𝑜 𝐴))

Theoremnnmordi 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) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Theoremnnmord 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) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Theoremnnmword 7600 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)))

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