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Theorem List for Metamath Proof Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoaordex 7401* 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 7402 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 7403 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 7404 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 7405* 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 7406* 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 7407* Lemma for oacomf1o 7408. (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 7408* 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 8307). (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 7409 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 7410 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 7411 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 7412 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 7413 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 7414 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 7415 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 7416 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 7417 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 7418 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 7419 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 7420* 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 7421 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 7422 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 7423 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 7424 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 7425* Lemma for omeu 7428: existence part. (Contributed by Mario Carneiro, 28-Feb-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On ∃𝑦𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝐵)
 
Theoremomeulem2 7426 Lemma for omeu 7428: 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 7427 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 7428* 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 7429 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 7430 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 7431 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 7432 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 7433 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 7434 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 7435 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 7436 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 7437 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 7438* 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 7439 Lemma for oeoa 7440. (Contributed by Eric Schmidt, 26-May-2009.)
𝐴 ∈ On    &   ∅ ∈ 𝐴    &   𝐵 ∈ On       (𝐶 ∈ On → (𝐴𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴𝑜 𝐵) ·𝑜 (𝐴𝑜 𝐶)))
 
Theoremoeoa 7440 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 7441 Lemma for oeoe 7442. (Contributed by Eric Schmidt, 26-May-2009.)
𝐴 ∈ On    &   ∅ ∈ 𝐴       ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐵) ↑𝑜 𝐶) = (𝐴𝑜 (𝐵 ·𝑜 𝐶)))
 
Theoremoeoe 7442 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 7443 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 7444* Lemma for oeeu 7446. (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 7445* The division algorithm for ordinal exponentiation. (This version of oeeu 7446 gives an explicit expression for the unique solution of the equation, in terms of the solution 𝑃 to omeu 7428.) (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 7446* 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 7447 Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
 
Theoremnnm0 7448 Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.)
(𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
 
Theoremnnasuc 7449 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 7450 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 7451 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 7452 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 7381) so that we can avoid ax-rep 4597, 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 7453 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 7454 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 7455 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 7456 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 7457 ω is closed under addition. Inference form of nnacl 7454. (Contributed by Scott Fenton, 20-Apr-2012.)
𝐴 ∈ ω    &   𝐵 ∈ ω       (𝐴 +𝑜 𝐵) ∈ ω
 
Theoremnnmcli 7458 ω is closed under multiplication. Inference form of nnmcl 7455. (Contributed by Scott Fenton, 20-Apr-2012.)
𝐴 ∈ ω    &   𝐵 ∈ ω       (𝐴 ·𝑜 𝐵) ∈ ω
 
Theoremnnarcl 7459 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 7460 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 7461 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 7462 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 7463 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 7464 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 7465 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 7466 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 7467 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 7468 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 7469 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 7470 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 7471 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 7472 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 7473 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +𝑜 𝐴))
 
Theoremnnmordi 7474 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 7475 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 7476 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 7477 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 7478 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 7479 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 7480* 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 7481* 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)))
 
Theorem1onn 7482 One is a natural number. (Contributed by NM, 29-Oct-1995.)
1𝑜 ∈ ω
 
Theorem2onn 7483 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
2𝑜 ∈ ω
 
Theorem3onn 7484 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3𝑜 ∈ ω
 
Theorem4onn 7485 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4𝑜 ∈ ω
 
Theoremoaabslem 7486 Lemma for oaabs 7487. (Contributed by NM, 9-Dec-2004.)
((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 ω) = ω)
 
Theoremoaabs 7487 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 7488 The absorption law oaabs 7487 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 7489 Lemma for omabs 7490. (Contributed by Mario Carneiro, 30-May-2015.)
((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·𝑜 ω) = ω)
 
Theoremomabs 7490 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 7491 Multiply an element of ω by 1𝑜. (Contributed by Mario Carneiro, 17-Nov-2014.)
(𝐴 ∈ ω → (𝐴 ·𝑜 1𝑜) = 𝐴)
 
Theoremnnm2 7492 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 7493 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 7494 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 7495* 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 7496* Lemma for omsmo 7497. (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 7497* 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 7498 Lemma for omopthi 7500. (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 7499 Lemma for omopthi 7500. (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 7500 An ordered pair theorem for ω. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 12787. (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))
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