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Theorem List for Metamath Proof Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremomass 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)))
 
Theoremoneo 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))
 
Theoremomeulem1 8603* Lemma for omeu 8606: existence part. (Contributed by Mario Carneiro, 28-Feb-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On ∃𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)
 
Theoremomeulem2 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 ∧ 𝐶𝐴) ∧ (𝐷 ∈ On ∧ 𝐸𝐴)) → ((𝐵𝐷 ∨ (𝐵 = 𝐷𝐶𝐸)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
 
Theoremomopth2 8605 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 8606* 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 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) → ∅ ∈ (𝐴o 𝐵))
 
Theoremoeordi 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)) → (𝐴𝐵 → (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
 
Theoremoeord 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)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
 
Theoremoecan 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) → ((𝐴o 𝐵) = (𝐴o 𝐶) ↔ 𝐵 = 𝐶))
 
Theoremoeword 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)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
 
Theoremoewordi 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) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
 
Theoremoewordri 8613 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 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 ⊆ (𝐴o 𝐵))
 
Theoremoeordsuc 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) → (𝐴𝐵 → (𝐴o suc 𝐶) ∈ (𝐵o suc 𝐶)))
 
Theoremoelim2 8616* 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 8617 Lemma for oeoa 8618. (Contributed by Eric Schmidt, 26-May-2009.)
𝐴 ∈ On    &   ∅ ∈ 𝐴    &   𝐵 ∈ On       (𝐶 ∈ On → (𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))
 
Theoremoeoa 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) → (𝐴o (𝐵 +o 𝐶)) = ((𝐴o 𝐵) ·o (𝐴o 𝐶)))
 
Theoremoeoelem 8619 Lemma for oeoe 8620. (Contributed by Eric Schmidt, 26-May-2009.)
𝐴 ∈ On    &   ∅ ∈ 𝐴       ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶)))
 
Theoremoeoe 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) → ((𝐴o 𝐵) ↑o 𝐶) = (𝐴o (𝐵 ·o 𝐶)))
 
Theoremoelimcl 8621 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 8622* Lemma for oeeu 8624. (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 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 ∈ (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 8624* 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))
 
2.4.24  Natural number arithmetic
 
Theoremnna0 8625 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
 
Theoremnnm0 8626 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
 
Theoremnnasuc 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))
 
Theoremnnmsuc 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))
 
Theoremnnesuc 8629 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 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)
 
Theoremnnm0r 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) = ∅)
 
Theoremnnacl 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)
 
Theoremnnmcl 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)
 
Theoremnnecl 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) → (𝐴o 𝐵) ∈ ω)
 
Theoremnnacli 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
 
Theoremnnmcli 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
 
Theoremnnarcl 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)))
 
Theoremnnacom 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))
 
Theoremnnaordi 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) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
 
Theoremnnaord 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) → (𝐴𝐵 ↔ (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
 
Theoremnnaordr 8641 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 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) → (𝐴𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))
 
Theoremnnaass 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)))
 
Theoremnndi 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)))
 
Theoremnnmass 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)))
 
Theoremnnmsucr 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))
 
Theoremnnmcom 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))
 
Theoremnnaword 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) → (𝐴𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
 
Theoremnnacan 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))
 
Theoremnnaword1 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))
 
Theoremnnaword2 8651 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +o 𝐴))
 
Theoremnnmordi 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) → (𝐴𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
 
Theoremnnmord 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) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
 
Theoremnnmword 8654 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)))
 
Theoremnnmcan 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))
 
Theoremnnmwordi 8656 Weak ordering property of multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
 
Theoremnnmwordri 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) → (𝐴𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶)))
 
Theoremnnawordex 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) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
 
Theoremnnaordex 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) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
 
Theoremnnaordex2 8660* Equivalence for ordering. (Contributed by Scott Fenton, 18-Apr-2025.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o suc 𝑥) = 𝐵))
 
Theorem1onn 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
 
Theorem1onnALT 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
 
Theorem2onn 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
 
Theorem2onnALT 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
 
Theorem3onn 8665 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3o ∈ ω
 
Theorem4onn 8666 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4o ∈ ω
 
Theorem1one2o 8667 Ordinal one is not ordinal two. Analogous to 1ne2 12453. (Contributed by AV, 17-Oct-2023.)
1o ≠ 2o
 
Theoremoaabslem 8668 Lemma for oaabs 8669. (Contributed by NM, 9-Dec-2004.)
((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω)
 
Theoremoaabs 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)
 
Theoremoaabs2 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)
 
Theoremomabslem 8671 Lemma for omabs 8672. (Contributed by Mario Carneiro, 30-May-2015.)
((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) = ω)
 
Theoremomabs 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))
 
Theoremnnm1 8673 Multiply an element of ω by 1o. (Contributed by Mario Carneiro, 17-Nov-2014.)
(𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴)
 
Theoremnnm2 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))
 
Theoremnn2m 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))
 
Theoremnnneo 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))
 
Theoremnneob 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)))
 
Theoremomsmolem 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 (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦))))
 
Theoremomsmo 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 (𝐹𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω–1-1𝐴)
 
Theoremomopthlem1 8680 Lemma for omopthi 8682. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ ω    &   𝐶 ∈ ω       (𝐴𝐶 → ((𝐴 ·o 𝐴) +o (𝐴 ·o 2o)) ∈ (𝐶 ·o 𝐶))
 
Theoremomopthlem2 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))
 
Theoremomopthi 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 = 𝐶𝐵 = 𝐷))
 
Theoremomopth 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 = 𝐶𝐵 = 𝐷)))
 
Theoremnnasmo 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)
 
Theoremeldifsucnn 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
 
Syntaxcnadd 8686 Declare the syntax for natural ordinal addition. See df-nadd 8687.
class +no
 
Definitiondf-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) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))}, (On × On), (𝑧 ∈ V, 𝑎 ∈ V ↦ {𝑤 ∈ On ∣ ((𝑎 “ ({(1st𝑧)} × (2nd𝑧))) ⊆ 𝑤 ∧ (𝑎 “ ((1st𝑧) × {(2nd𝑧)})) ⊆ 𝑤)}))
 
Theoremon2recsfn 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) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))}, (On × On), 𝐺)       𝐹 Fn (On × On)
 
Theoremon2recsov 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) ∧ (((1st𝑥) E (1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥) E (2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))}, (On × On), 𝐺)       ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵𝐺(𝐹 ↾ ((suc 𝐴 × suc 𝐵) ∖ {⟨𝐴, 𝐵⟩}))))
 
Theoremon2ind 8690* Double induction over ordinal numbers. (Contributed by Scott Fenton, 26-Aug-2024.)
(𝑎 = 𝑐 → (𝜑𝜓))    &   (𝑏 = 𝑑 → (𝜓𝜒))    &   (𝑎 = 𝑐 → (𝜃𝜒))    &   (𝑎 = 𝑋 → (𝜑𝜏))    &   (𝑏 = 𝑌 → (𝜏𝜂))    &   ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐𝑎𝑑𝑏 𝜒 ∧ ∀𝑐𝑎 𝜓 ∧ ∀𝑑𝑏 𝜃) → 𝜑))       ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂)
 
Theoremon3ind 8691* Triple induction over ordinals. (Contributed by Scott Fenton, 4-Sep-2024.)
(𝑎 = 𝑑 → (𝜑𝜓))    &   (𝑏 = 𝑒 → (𝜓𝜒))    &   (𝑐 = 𝑓 → (𝜒𝜃))    &   (𝑎 = 𝑑 → (𝜏𝜃))    &   (𝑏 = 𝑒 → (𝜂𝜏))    &   (𝑏 = 𝑒 → (𝜁𝜃))    &   (𝑐 = 𝑓 → (𝜎𝜏))    &   (𝑎 = 𝑋 → (𝜑𝜌))    &   (𝑏 = 𝑌 → (𝜌𝜇))    &   (𝑐 = 𝑍 → (𝜇𝜆))    &   ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (((∀𝑑𝑎𝑒𝑏𝑓𝑐 𝜃 ∧ ∀𝑑𝑎𝑒𝑏 𝜒 ∧ ∀𝑑𝑎𝑓𝑐 𝜁) ∧ (∀𝑑𝑎 𝜓 ∧ ∀𝑒𝑏𝑓𝑐 𝜏 ∧ ∀𝑒𝑏 𝜎) ∧ ∀𝑓𝑐 𝜂) → 𝜑))       ((𝑋 ∈ On ∧ 𝑌 ∈ On ∧ 𝑍 ∈ On) → 𝜆)
 
Theoremcoflton 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.)
(𝜑𝐴 ⊆ On)    &   (𝜑𝐵 ⊆ On)    &   (𝜑𝐶 ⊆ On)    &   (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)    &   (𝜑 → ∀𝑧𝐵𝑤𝐶 𝑧𝑤)       (𝜑 → ∀𝑎𝐴𝑐𝐶 𝑎𝑐)
 
Theoremcofon1 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.)
(𝜑𝐴 ∈ 𝒫 On)    &   (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)    &   (𝜑𝐵 {𝑧 ∈ On ∣ 𝐴𝑧})       (𝜑 {𝑧 ∈ On ∣ 𝐴𝑧} = {𝑤 ∈ On ∣ 𝐵𝑤})
 
Theoremcofon2 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.)
(𝜑𝐴 ∈ 𝒫 On)    &   (𝜑𝐵 ∈ 𝒫 On)    &   (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)    &   (𝜑 → ∀𝑧𝐵𝑤𝐴 𝑧𝑤)       (𝜑 {𝑎 ∈ On ∣ 𝐴𝑎} = {𝑏 ∈ On ∣ 𝐵𝑏})
 
Theoremcofonr 8695* Inverse cofinality law for ordinals. Contrast with cofcutr 27890 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
(𝜑𝐴 ∈ On)    &   (𝜑𝐴 = {𝑥 ∈ On ∣ 𝑋𝑥})       (𝜑 → ∀𝑦𝐴𝑧𝑋 𝑦𝑧)
 
Theoremnaddfn 8696 Natural addition is a function over pairs of ordinals. (Contributed by Scott Fenton, 26-Aug-2024.)
+no Fn (On × On)
 
Theoremnaddcllem 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)}))
 
Theoremnaddcl 8698 Closure law for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) ∈ On)
 
Theoremnaddov 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)})
 
Theoremnaddov2 8700* Alternate expression for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = {𝑥 ∈ On ∣ (∀𝑦𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝐵) ∈ 𝑥)})
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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