P. 86 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 8501-8600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	oelim 8501*	Ordinal exponentiation with a limit exponent and nonzero base. Definition 8.30 of [TakeutiZaring] p. 67. Definition 2.6 of [Schloeder] p. 4. (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 8502	Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. Remark 2.8 of [Schloeder] p. 5. (Contributed by NM, 5-May-1995.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
Theorem	omcl 8503	Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. Remark 2.8 of [Schloeder] p. 5. (Contributed by NM, 3-Aug-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
Theorem	oecl 8504	Closure law for ordinal exponentiation. Remark 2.8 of [Schloeder] p. 5. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
Theorem	oa0r 8505	Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. Lemma 2.14 of [Schloeder] p. 5. (Contributed by NM, 5-May-1995.)
⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴)
Theorem	om0r 8506	Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)
⊢ (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)
Theorem	o1p1e2 8507	1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
⊢ (1o +o 1o) = 2o
Theorem	o2p2e4 8508	2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6341. For the usual proof using complex numbers, see 2p2e4 12323. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5237, from a comment by Sophie. (Revised by SN, 23-Mar-2024.)
⊢ (2o +o 2o) = 4o
Theorem	om1 8509	Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. Lemma 2.15 of [Schloeder] p. 5. (Contributed by NM, 29-Oct-1995.)
⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)
Theorem	om1r 8510	Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. Lemma 2.15 of [Schloeder] p. 5. (Contributed by NM, 3-Aug-2004.)
⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)
Theorem	oe1 8511	Ordinal exponentiation with an exponent of 1. Lemma 2.16 of [Schloeder] p. 6. (Contributed by NM, 2-Jan-2005.)
⊢ (𝐴 ∈ On → (𝐴 ↑o 1o) = 𝐴)
Theorem	oe1m 8512	Ordinal exponentiation with a base of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. Lemma 2.17 of [Schloeder] p. 6. (Contributed by NM, 2-Jan-2005.)
⊢ (𝐴 ∈ On → (1o ↑o 𝐴) = 1o)
Theorem	oaordi 8513	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 8514	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 8515	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 8516	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 8517	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 8518	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 8519	An ordinal is less than or equal to its sum with another. Part of Exercise 5 of [TakeutiZaring] p. 62. Lemma 3.2 of [Schloeder] p. 7. (For the other part see oaord1 8518.) (Contributed by NM, 6-Dec-2004.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +o 𝐵))
Theorem	oaword2 8520	An ordinal is less than or equal to its sum with another. Theorem 21 of [Suppes] p. 209. Lemma 3.3 of [Schloeder] p. 7. (Contributed by NM, 7-Dec-2004.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐵 +o 𝐴))
Theorem	oawordeulem 8521*	Lemma for oawordex 8524. (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 8522*	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 8523*	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 8524*	Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 8522 for uniqueness. (Contributed by NM, 12-Dec-2004.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
Theorem	oaordex 8525*	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 8526	An ordinal sum is zero iff both of its arguments are zero. Lemma 3.10 of [Schloeder] p. 8. (Contributed by NM, 6-Dec-2004.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
Theorem	oalimcl 8527	The ordinal sum with a limit ordinal is a limit ordinal. Proposition 8.11 of [TakeutiZaring] p. 60. Lemma 3.4 of [Schloeder] p. 7. (Contributed by NM, 8-Dec-2004.)
⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 +o 𝐵))
Theorem	oaass 8528	Ordinal addition is associative. Theorem 25 of [Suppes] p. 211. Theorem 4.2 of [Schloeder] p. 11. (Contributed by NM, 10-Dec-2004.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶)))
Theorem	oarec 8529*	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 8530*	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 8531*	Lemma for oacomf1o 8532. (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 8532*	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 9611). (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 8533	Ordering property of ordinal multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63. Lemma 3.15 of [Schloeder] p. 9. (Contributed by NM, 14-Dec-2004.)
⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
Theorem	omord2 8534	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 8535	Ordering property of ordinal multiplication. Proposition 8.19 of [TakeutiZaring] p. 63. Theorem 3.16 of [Schloeder] p. 9. (Contributed by NM, 14-Dec-2004.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
Theorem	omcan 8536	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 8537	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 8538	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 8539	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 8540	An ordinal is less than or equal to its product with another. Lemma 3.11 of [Schloeder] p. 8. (Contributed by NM, 21-Dec-2004.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·o 𝐵))
Theorem	omword2 8541	An ordinal is less than or equal to its product with another. Lemma 3.12 of [Schloeder] p. 9. (Contributed by NM, 21-Dec-2004.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐵 ·o 𝐴))
Theorem	om00 8542	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 8543	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 8544*	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 8545	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 8546	Distributive law for ordinal arithmetic (left-distributivity). Proposition 8.25 of [TakeutiZaring] p. 64. Theorem 4.3 of [Schloeder] p. 12. (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 8547	Multiplication of ordinal numbers is associative. Theorem 8.26 of [TakeutiZaring] p. 65. Theorem 4.4 of [Schloeder] p. 13. (Contributed by NM, 28-Dec-2004.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))
Theorem	oneo 8548	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 8549*	Lemma for omeu 8552: 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 8550	Lemma for omeu 8552: 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 8551	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 8552*	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 8553	Ordinal exponentiation with a nonzero base 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 8554	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 8555	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 8556	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 8557	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 8558	Weak ordering property of ordinal exponentiation. Lemma 3.19 of [Schloeder] p. 10. (Contributed by NM, 6-Jan-2005.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
Theorem	oewordri 8559	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 8560	Ordinal exponentiation compared to its exponent. Proposition 8.37 of [TakeutiZaring] p. 68. Lemma 3.20 of [Schloeder] p. 10. (Contributed by NM, 7-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))
Theorem	oeordsuc 8561	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 8562*	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 8563	Lemma for oeoa 8564. (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 8564	Sum of exponents law for ordinal exponentiation. Theorem 8R of [Enderton] p. 238. Also Proposition 8.41 of [TakeutiZaring] p. 69. Theorem 4.7 of [Schloeder] p. 14. (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 8565	Lemma for oeoe 8566. (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 8566	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 8567	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 8568*	Lemma for oeeu 8570. (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 8569*	The division algorithm for ordinal exponentiation. (This version of oeeu 8570 gives an explicit expression for the unique solution of the equation, in terms of the solution 𝑃 to omeu 8552.) (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 8570*	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))
2.4.24  Natural number arithmetic
Theorem	nna0 8571	Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
Theorem	nnm0 8572	Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
Theorem	nnasuc 8573	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 8574	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 8575	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 8576	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 8505) so that we can avoid ax-rep 5237, 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 8577	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 8578	Closure of addition of natural numbers. Proposition 8.9 of [TakeutiZaring] p. 59. Theorem 2.20 of [Schloeder] p. 6. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω)
Theorem	nnmcl 8579	Closure of multiplication of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. Theorem 2.20 of [Schloeder] p. 6. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω)
Theorem	nnecl 8580	Closure of exponentiation of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. Theorem 2.20 of [Schloeder] p. 6. (Contributed by NM, 24-Mar-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ↑o 𝐵) ∈ ω)
Theorem	nnacli 8581	ω is closed under addition. Inference form of nnacl 8578. (Contributed by Scott Fenton, 20-Apr-2012.)
⊢ 𝐴 ∈ ω    &   ⊢ 𝐵 ∈ ω    ⇒   ⊢ (𝐴 +o 𝐵) ∈ ω
Theorem	nnmcli 8582	ω is closed under multiplication. Inference form of nnmcl 8579. (Contributed by Scott Fenton, 20-Apr-2012.)
⊢ 𝐴 ∈ ω    &   ⊢ 𝐵 ∈ ω    ⇒   ⊢ (𝐴 ·o 𝐵) ∈ ω
Theorem	nnarcl 8583	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 8584	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 8585	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 8586	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 8587	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 8588	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 8589	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 8590	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 8591	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 8592	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 8593	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 8594	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 8595	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 8596	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 8597	Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +o 𝐴))
Theorem	nnmordi 8598	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 8599	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 8600	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)))