Home Metamath Proof ExplorerTheorem List (p. 76 of 419) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27663) Hilbert Space Explorer (27664-29188) Users' Mathboxes (29189-41884)

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

Theoremtz7.49c 7501* Corollary of Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 19-Jan-2013.)
𝐹 Fn On       ((𝐴𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)

Syntaxcseqom 7502 Extend class notation to include index-aware recursive definitions.
class seq𝜔(𝐹, 𝐼)

Definitiondf-seqom 7503* Index-aware recursive definitions over ω. A mashup of df-rdg 7466 and df-seq 12758, this allows for recursive definitions that use an index in the recursion in cases where Infinity is not admitted. (Contributed by Stefan O'Rear, 1-Nov-2014.)
seq𝜔(𝐹, 𝐼) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)

Theoremseqomlem0 7504* Lemma for seq𝜔. Change bound variables. (Contributed by Stefan O'Rear, 1-Nov-2014.)
rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩), ⟨∅, ( I ‘𝐼)⟩)

Theoremseqomlem1 7505* Lemma for seq𝜔. The underlying recursion generates a sequence of pairs with the expected first values. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)       (𝐴 ∈ ω → (𝑄𝐴) = ⟨𝐴, (2nd ‘(𝑄𝐴))⟩)

Theoremseqomlem2 7506* Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)       (𝑄 “ ω) Fn ω

Theoremseqomlem3 7507* Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.)
𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)       ((𝑄 “ ω)‘∅) = ( I ‘𝐼)

Theoremseqomlem4 7508* Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)       (𝐴 ∈ ω → ((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))

Theoremseqomeq12 7509 Equality theorem for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.)
((𝐴 = 𝐵𝐶 = 𝐷) → seq𝜔(𝐴, 𝐶) = seq𝜔(𝐵, 𝐷))

Theoremfnseqom 7510 An index-aware recursive definition defines a function on the natural numbers. (Contributed by Stefan O'Rear, 1-Nov-2014.)
𝐺 = seq𝜔(𝐹, 𝐼)       𝐺 Fn ω

Theoremseqom0g 7511 Value of an index-aware recursive definition at 0. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revise by AV, 17-Sep-2021.)
𝐺 = seq𝜔(𝐹, 𝐼)       (𝐼𝑉 → (𝐺‘∅) = 𝐼)

Theoremseqomsuc 7512 Value of an index-aware recursive definition at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)
𝐺 = seq𝜔(𝐹, 𝐼)       (𝐴 ∈ ω → (𝐺‘suc 𝐴) = (𝐴𝐹(𝐺𝐴)))

2.4.18  Ordinal arithmetic

Syntaxc1o 7513 Extend the definition of a class to include the ordinal number 1.
class 1𝑜

Syntaxc2o 7514 Extend the definition of a class to include the ordinal number 2.
class 2𝑜

Syntaxc3o 7515 Extend the definition of a class to include the ordinal number 3.
class 3𝑜

Syntaxc4o 7516 Extend the definition of a class to include the ordinal number 4.
class 4𝑜

Syntaxcoa 7517 Extend the definition of a class to include the ordinal addition operation.
class +𝑜

Syntaxcomu 7518 Extend the definition of a class to include the ordinal multiplication operation.
class ·𝑜

Syntaxcoe 7519 Extend the definition of a class to include the ordinal exponentiation operation.
class 𝑜

Definitiondf-1o 7520 Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
1𝑜 = suc ∅

Definitiondf-2o 7521 Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
2𝑜 = suc 1𝑜

Definitiondf-3o 7522 Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
3𝑜 = suc 2𝑜

Definitiondf-4o 7523 Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
4𝑜 = suc 3𝑜

+𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))

Definitiondf-omul 7525* Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)
·𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)‘𝑦))

Definitiondf-oexp 7526* Define the ordinal exponentiation operation. (Contributed by NM, 30-Dec-2004.)
𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1𝑜𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)))

Theorem1on 7527 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)
1𝑜 ∈ On

Theorem2on 7528 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
2𝑜 ∈ On

Theorem2on0 7529 Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
2𝑜 ≠ ∅

Theorem3on 7530 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3𝑜 ∈ On

Theorem4on 7531 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4𝑜 ∈ On

Theoremdf1o2 7532 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
1𝑜 = {∅}

Theoremdf2o3 7533 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
2𝑜 = {∅, 1𝑜}

Theoremdf2o2 7534 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
2𝑜 = {∅, {∅}}

Theorem1n0 7535 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
1𝑜 ≠ ∅

Theoremxp01disj 7536 Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅

Theoremordgt0ge1 7537 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
(Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))

Theoremordge1n0 7538 An ordinal greater than or equal to 1 is nonzero. (Contributed by NM, 21-Dec-2004.)
(Ord 𝐴 → (1𝑜𝐴𝐴 ≠ ∅))

Theoremel1o 7539 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
(𝐴 ∈ 1𝑜𝐴 = ∅)

Theoremdif1o 7540 Two ways to say that 𝐴 is a nonzero number of the set 𝐵. (Contributed by Mario Carneiro, 21-May-2015.)
(𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ (𝐴𝐵𝐴 ≠ ∅))

Theoremondif1 7541 Two ways to say that 𝐴 is a nonzero ordinal number. (Contributed by Mario Carneiro, 21-May-2015.)
(𝐴 ∈ (On ∖ 1𝑜) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))

Theoremondif2 7542 Two ways to say that 𝐴 is an ordinal greater than one. (Contributed by Mario Carneiro, 21-May-2015.)
(𝐴 ∈ (On ∖ 2𝑜) ↔ (𝐴 ∈ On ∧ 1𝑜𝐴))

Theorem2oconcl 7543 Closure of the pair swapping function on 2𝑜. (Contributed by Mario Carneiro, 27-Sep-2015.)
(𝐴 ∈ 2𝑜 → (1𝑜𝐴) ∈ 2𝑜)

Theorem0lt1o 7544 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
∅ ∈ 1𝑜

Theoremdif20el 7545 An ordinal greater than one is greater than zero. (Contributed by Mario Carneiro, 25-May-2015.)
(𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)

Theorem0we1 7546 The empty set is a well-ordering of ordinal one. (Contributed by Mario Carneiro, 9-Feb-2015.)
∅ We 1𝑜

Theorembrwitnlem 7547 Lemma for relations which assert the existence of a witness in a two-parameter set. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 23-Aug-2015.)
𝑅 = (𝑂 “ (V ∖ 1𝑜))    &   𝑂 Fn 𝑋       (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅)

Theoremfnoa 7548 Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
+𝑜 Fn (On × On)

Theoremfnom 7549 Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
·𝑜 Fn (On × On)

Theoremfnoe 7550 Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.)
𝑜 Fn (On × On)

Theoremoav 7551* Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐴)‘𝐵))

Theoremomv 7552* Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))

Theoremoe0lem 7553 A helper lemma for oe0 7562 and others. (Contributed by NM, 6-Jan-2005.)
((𝜑𝐴 = ∅) → 𝜓)    &   (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓)       ((𝐴 ∈ On ∧ 𝜑) → 𝜓)

Theoremoev 7554* Value of ordinal exponentiation. (Contributed by NM, 30-Dec-2004.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) = if(𝐴 = ∅, (1𝑜𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)))

Theoremoevn0 7555* Value of ordinal exponentiation at a nonzero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))

Theoremoa0 7556 Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)

Theoremom0 7557 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)

Theoremoe0m 7558 Ordinal exponentiation with zero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 ∈ On → (∅ ↑𝑜 𝐴) = (1𝑜𝐴))

Theoremom0x 7559 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 7557, this version works whether or not 𝐴 is an ordinal. However, since it is an artifact of our particular function value definition outside the domain, we will not use it in order to be conventional and present it only as a curiosity. (Contributed by NM, 1-Feb-1996.)
(𝐴 ·𝑜 ∅) = ∅

Theoremoe0m0 7560 Ordinal exponentiation with zero mantissa and zero exponent. Proposition 8.31 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.)
(∅ ↑𝑜 ∅) = 1𝑜

Theoremoe0m1 7561 Ordinal exponentiation with zero mantissa and nonzero exponent. Proposition 8.31(2) of [TakeutiZaring] p. 67 and its converse. (Contributed by NM, 5-Jan-2005.)
(𝐴 ∈ On → (∅ ∈ 𝐴 ↔ (∅ ↑𝑜 𝐴) = ∅))

Theoremoe0 7562 Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)

Theoremoev2 7563* Alternate value of ordinal exponentiation. Compare oev 7554. (Contributed by NM, 2-Jan-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))

Theoremoasuc 7564 Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))

Theoremoesuclem 7565* Lemma for oesuc 7567. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Lim 𝑋    &   (𝐵𝑋 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)))       ((𝐴 ∈ On ∧ 𝐵𝑋) → (𝐴𝑜 suc 𝐵) = ((𝐴𝑜 𝐵) ·𝑜 𝐴))

Theoremomsuc 7566 Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))

Theoremoesuc 7567 Ordinal exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 suc 𝐵) = ((𝐴𝑜 𝐵) ·𝑜 𝐴))

Theoremonasuc 7568 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 7564 does not need Replacement.) (Contributed by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))

Theoremonmsuc 7569 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))

Theoremonesuc 7570 Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝑜 suc 𝐵) = ((𝐴𝑜 𝐵) ·𝑜 𝐴))

Theoremoa1suc 7571 Addition with 1 is same as successor. Proposition 4.34(a) of [Mendelson] p. 266. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 16-Nov-2014.)
(𝐴 ∈ On → (𝐴 +𝑜 1𝑜) = suc 𝐴)

Theoremoalim 7572* Ordinal addition with a limit ordinal. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 +𝑜 𝐵) = 𝑥𝐵 (𝐴 +𝑜 𝑥))

Theoremomlim 7573* Ordinal multiplication with a limit ordinal. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 ·𝑜 𝐵) = 𝑥𝐵 (𝐴 ·𝑜 𝑥))

Theoremoelim 7574* 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 7575 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 7576 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 7577 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 7578 Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)
(𝐴 ∈ On → (∅ +𝑜 𝐴) = 𝐴)

Theoremom0r 7579 Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)
(𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅)

Theoremo1p1e2 7580 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.)
(1𝑜 +𝑜 1𝑜) = 2𝑜

Theoremo2p2e4 7581 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 5698. For the usual proof using complex numbers, see 2p2e4 11104. (Contributed by NM, 18-Aug-2021.)
(2𝑜 +𝑜 2𝑜) = 4𝑜

Theoremom1 7582 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 7583 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 7584 Ordinal exponentiation with an exponent of 1. (Contributed by NM, 2-Jan-2005.)
(𝐴 ∈ On → (𝐴𝑜 1𝑜) = 𝐴)

Theoremoe1m 7585 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 7586 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 7587 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 7588 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 7589 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 7590 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 7591 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 7592 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 7591.) (Contributed by NM, 6-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +𝑜 𝐵))

Theoremoaword2 7593 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 7594* Lemma for oawordex 7597. (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 7595* 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 7596* Existence theorem for weak ordering of ordinal sum. (Contributed by NM, 12-Dec-2004.)
((𝐴 ∈ On ∧ ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵)

Theoremoawordex 7597* Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 7595 for uniqueness. (Contributed by NM, 12-Dec-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵))

Theoremoaordex 7598* 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 7599 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 7600 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))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41884
 Copyright terms: Public domain < Previous  Next >