P. 87 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 8601-8700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	omass 8601	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 8602	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 8603*	Lemma for omeu 8606: 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 8604	Lemma for omeu 8606: 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 8605	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 8606*	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 8607	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 8608	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 8609	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 8610	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 8611	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 8612	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 8613	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 8614	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 8615	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 8616*	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 8617	Lemma for oeoa 8618. (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 8618	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 8619	Lemma for oeoe 8620. (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 8620	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 8621	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 8622*	Lemma for oeeu 8624. (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 8623*	The division algorithm for ordinal exponentiation. (This version of oeeu 8624 gives an explicit expression for the unique solution of the equation, in terms of the solution 𝑃 to omeu 8606.) (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 8624*	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 8625	Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
Theorem	nnm0 8626	Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
Theorem	nnasuc 8627	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 8628	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 8629	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 8630	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 8559) so that we can avoid ax-rep 5286, 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 8631	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 8632	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 8633	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 8634	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 8635	ω is closed under addition. Inference form of nnacl 8632. (Contributed by Scott Fenton, 20-Apr-2012.)
⊢ 𝐴 ∈ ω    &   ⊢ 𝐵 ∈ ω    ⇒   ⊢ (𝐴 +o 𝐵) ∈ ω
Theorem	nnmcli 8636	ω is closed under multiplication. Inference form of nnmcl 8633. (Contributed by Scott Fenton, 20-Apr-2012.)
⊢ 𝐴 ∈ ω    &   ⊢ 𝐵 ∈ ω    ⇒   ⊢ (𝐴 ·o 𝐵) ∈ ω
Theorem	nnarcl 8637	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 8638	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 8639	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 8640	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 8641	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 8642	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 8643	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 8644	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 8645	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 8646	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 8647	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 8648	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 8649	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 8650	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 8651	Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +o 𝐴))
Theorem	nnmordi 8652	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 8653	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 8654	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 8655	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 8656	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 8657	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 8658*	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 8659*	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	nnaordex2 8660*	Equivalence for ordering. (Contributed by Scott Fenton, 18-Apr-2025.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))
Theorem	1onn 8661	The ordinal 1 is a natural number. For a shorter proof using Peano's postulates that depends on ax-un 7741, see 1onnALT 8662. Lemma 2.2 of [Schloeder] p. 4. (Contributed by NM, 29-Oct-1995.) Avoid ax-un 7741. (Revised by BTernaryTau, 1-Dec-2024.)
⊢ 1o ∈ ω
Theorem	1onnALT 8662	Shorter proof of 1onn 8661 using Peano's postulates that depends on ax-un 7741. (Contributed by NM, 29-Oct-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 1o ∈ ω
Theorem	2onn 8663	The ordinal 2 is a natural number. For a shorter proof using Peano's postulates that depends on ax-un 7741, see 2onnALT 8664. (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7741. (Revised by BTernaryTau, 1-Dec-2024.)
⊢ 2o ∈ ω
Theorem	2onnALT 8664	Shorter proof of 2onn 8663 using Peano's postulates that depends on ax-un 7741. (Contributed by NM, 28-Sep-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 2o ∈ ω
Theorem	3onn 8665	The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 3o ∈ ω
Theorem	4onn 8666	The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 4o ∈ ω
Theorem	1one2o 8667	Ordinal one is not ordinal two. Analogous to 1ne2 12453. (Contributed by AV, 17-Oct-2023.)
⊢ 1o ≠ 2o
Theorem	oaabslem 8668	Lemma for oaabs 8669. (Contributed by NM, 9-Dec-2004.)
⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω)
Theorem	oaabs 8669	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 8670	The absorption law oaabs 8669 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 8671	Lemma for omabs 8672. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) = ω)
Theorem	omabs 8672	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 8673	Multiply an element of ω by 1o. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴)
Theorem	nnm2 8674	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 8675	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 8676	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 8677*	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)))
Theorem	omsmolem 8678*	Lemma for omsmo 8679. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.)
⊢ (𝑦 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦))))
Theorem	omsmo 8679*	A strictly monotonic ordinal function on the set of natural numbers is one-to-one. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.)
⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω–1-1→𝐴)
Theorem	omopthlem1 8680	Lemma for omopthi 8682. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ 𝐴 ∈ ω    &   ⊢ 𝐶 ∈ ω    ⇒   ⊢ (𝐴 ∈ 𝐶 → ((𝐴 ·o 𝐴) +o (𝐴 ·o 2o)) ∈ (𝐶 ·o 𝐶))
Theorem	omopthlem2 8681	Lemma for omopthi 8682. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ 𝐴 ∈ ω    &   ⊢ 𝐵 ∈ ω    &   ⊢ 𝐶 ∈ ω    &   ⊢ 𝐷 ∈ ω    ⇒   ⊢ ((𝐴 +o 𝐵) ∈ 𝐶 → ¬ ((𝐶 ·o 𝐶) +o 𝐷) = (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵))
Theorem	omopthi 8682	An ordered pair theorem for ω. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 14265. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ 𝐴 ∈ ω    &   ⊢ 𝐵 ∈ ω    &   ⊢ 𝐶 ∈ ω    &   ⊢ 𝐷 ∈ ω    ⇒   ⊢ ((((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)) +o 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	omopth 8683	An ordered pair theorem for finite integers. Analogous to nn0opthi 14265. (Contributed by Scott Fenton, 1-May-2012.)
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ ω)) → ((((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)) +o 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	nnasmo 8684*	There is at most one left additive inverse for natural number addition. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ (𝐴 ∈ ω → ∃*𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
Theorem	eldifsucnn 8685*	Condition for membership in the difference of ω and a nonzero finite ordinal. (Contributed by Scott Fenton, 24-Oct-2024.)
⊢ (𝐴 ∈ ω → (𝐵 ∈ (ω ∖ suc 𝐴) ↔ ∃𝑥 ∈ (ω ∖ 𝐴)𝐵 = suc 𝑥))
2.4.25  Natural addition
Syntax	cnadd 8686	Declare the syntax for natural ordinal addition. See df-nadd 8687.
class +no
Definition	df-nadd 8687*	Define natural ordinal addition. This is a commutative form of addition over the ordinals. (Contributed by Scott Fenton, 26-Aug-2024.)
⊢ +no = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), (𝑧 ∈ V, 𝑎 ∈ V ↦ ∩ {𝑤 ∈ On ∣ ((𝑎 “ ({(1st ‘𝑧)} × (2nd ‘𝑧))) ⊆ 𝑤 ∧ (𝑎 “ ((1st ‘𝑧) × {(2nd ‘𝑧)})) ⊆ 𝑤)}))
Theorem	on2recsfn 8688*	Show that double recursion over ordinals yields a function over pairs of ordinals. (Contributed by Scott Fenton, 3-Sep-2024.)
⊢ 𝐹 = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), 𝐺)    ⇒   ⊢ 𝐹 Fn (On × On)
Theorem	on2recsov 8689*	Calculate the value of the double ordinal recursion operator. (Contributed by Scott Fenton, 3-Sep-2024.)
⊢ 𝐹 = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), 𝐺)    ⇒   ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ ((suc 𝐴 × suc 𝐵) ∖ {⟨𝐴, 𝐵⟩}))))
Theorem	on2ind 8690*	Double induction over ordinal numbers. (Contributed by Scott Fenton, 26-Aug-2024.)
⊢ (𝑎 = 𝑐 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))    &   ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))    &   ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒 ∧ ∀𝑐 ∈ 𝑎 𝜓 ∧ ∀𝑑 ∈ 𝑏 𝜃) → 𝜑))    ⇒   ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂)
Theorem	on3ind 8691*	Triple induction over ordinals. (Contributed by Scott Fenton, 4-Sep-2024.)
⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))    &   ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))    &   ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))    &   ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))    &   ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))    &   ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))    &   ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))    &   ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜃 ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 𝜒 ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 𝜁) ∧ (∀𝑑 ∈ 𝑎 𝜓 ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜏 ∧ ∀𝑒 ∈ 𝑏 𝜎) ∧ ∀𝑓 ∈ 𝑐 𝜂) → 𝜑))    ⇒   ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ On ∧ 𝑍 ∈ On) → 𝜆)
Theorem	coflton 8692*	Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Compare cofsslt 27884 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ (𝜑 → 𝐴 ⊆ On)    &   ⊢ (𝜑 → 𝐵 ⊆ On)    &   ⊢ (𝜑 → 𝐶 ⊆ On)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝑧 ∈ 𝑤)    ⇒   ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑐 ∈ 𝐶 𝑎 ∈ 𝑐)
Theorem	cofon1 8693*	Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐵 and the upper bound of 𝐴 dominates 𝐵, then their upper bounds are equal. Compare with cofcut1 27886 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝒫 On)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)    &   ⊢ (𝜑 → 𝐵 ⊆ ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧})    ⇒   ⊢ (𝜑 → ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} = ∩ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤})
Theorem	cofon2 8694*	Cofinality theorem for ordinals. If 𝐴 and 𝐵 are mutually cofinal, then their upper bounds agree. Compare cofcut2 27888 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝒫 On)    &   ⊢ (𝜑 → 𝐵 ∈ 𝒫 On)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)    ⇒   ⊢ (𝜑 → ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎} = ∩ {𝑏 ∈ On ∣ 𝐵 ⊆ 𝑏})
Theorem	cofonr 8695*	Inverse cofinality law for ordinals. Contrast with cofcutr 27890 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})    ⇒   ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧)
Theorem	naddfn 8696	Natural addition is a function over pairs of ordinals. (Contributed by Scott Fenton, 26-Aug-2024.)
⊢ +no Fn (On × On)
Theorem	naddcllem 8697*	Lemma for ordinal addition closure. (Contributed by Scott Fenton, 26-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +no 𝐵) ∈ On ∧ (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)}))
Theorem	naddcl 8698	Closure law for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) ∈ On)
Theorem	naddov 8699*	The value of natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)})
Theorem	naddov2 8700*	Alternate expression for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥)})