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Theorem List for Metamath Proof Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrdgfnon 7401 The recursive definition generator is a function on ordinal numbers. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
rec(𝐹, 𝐴) Fn On

Theoremrdgvalg 7402* Value of the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘𝐵) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(rec(𝐹, 𝐴) ↾ 𝐵)))

Theoremrdgval 7403* Value of the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐵 ∈ On → (rec(𝐹, 𝐴)‘𝐵) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(rec(𝐹, 𝐴) ↾ 𝐵)))

Theoremrdg0 7404 The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
𝐴 ∈ V       (rec(𝐹, 𝐴)‘∅) = 𝐴

Theoremrdgseg 7405 The initial segments of the recursive definition generator are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
(𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)

Theoremrdgsucg 7406 The value of the recursive definition generator at a successor. (Contributed by NM, 16-Nov-2014.)
(𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))

Theoremrdgsuc 7407 The value of the recursive definition generator at a successor. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
(𝐵 ∈ On → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))

Theoremrdglimg 7408 The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 16-Nov-2014.)
((𝐵 ∈ dom rec(𝐹, 𝐴) ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = (rec(𝐹, 𝐴) “ 𝐵))

Theoremrdglim 7409 The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = (rec(𝐹, 𝐴) “ 𝐵))

Theoremrdg0g 7410 The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
(𝐴𝐶 → (rec(𝐹, 𝐴)‘∅) = 𝐴)

Theoremrdgsucmptf 7411 The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝐷    &   𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)    &   (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)       ((𝐵 ∈ On ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Theoremrdgsucmptnf 7412 The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered-pair class abstraction and where the mapping class 𝐷 is a proper class). This is a technical lemma that can be used together with rdgsucmptf 7411 to help eliminate redundant sethood antecedents. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝐷    &   𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)    &   (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)       𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)

Theoremrdgsucmpt2 7413* This version of rdgsucmpt 7414 avoids the not-free hypothesis of rdgsucmptf 7411 by using two substitutions instead of one. (Contributed by Mario Carneiro, 11-Sep-2015.)
𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)    &   (𝑦 = 𝑥𝐸 = 𝐶)    &   (𝑦 = (𝐹𝐵) → 𝐸 = 𝐷)       ((𝐵 ∈ On ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Theoremrdgsucmpt 7414* The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (Contributed by Mario Carneiro, 9-Sep-2013.)
𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)    &   (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)       ((𝐵 ∈ On ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Theoremrdglim2 7415* The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values. (Contributed by NM, 23-Apr-1995.)
((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})

Theoremrdglim2a 7416* The value of the recursive definition generator at a limit ordinal, in terms of indexed union of all smaller values. (Contributed by NM, 28-Jun-1998.)
((𝐵𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = 𝑥𝐵 (rec(𝐹, 𝐴)‘𝑥))

2.4.17  Finite recursion

Theoremfrfnom 7417 The function generated by finite recursive definition generation is a function on omega. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 14-Nov-2014.)
(rec(𝐹, 𝐴) ↾ ω) Fn ω

Theoremfr0g 7418 The initial value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.)
(𝐴𝐵 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴)

Theoremfrsuc 7419 The successor value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 16-Nov-2014.)
(𝐵 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝐵) = (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝐵)))

Theoremfrsucmpt 7420 The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation). (Contributed by NM, 14-Sep-2003.) (Revised by Scott Fenton, 2-Nov-2011.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝐷    &   𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)    &   (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)       ((𝐵 ∈ ω ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Theoremfrsucmptn 7421 The value of the finite recursive definition generator at a successor (special case where the characteristic function is a mapping abstraction and where the mapping class 𝐷 is a proper class). This is a technical lemma that can be used together with frsucmpt 7420 to help eliminate redundant sethood antecedents. (Contributed by Scott Fenton, 19-Feb-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝐷    &   𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)    &   (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)       𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)

Theoremfrsucmpt2 7422* The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation), using double-substitution instead of a bound variable condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)    &   (𝑦 = 𝑥𝐸 = 𝐶)    &   (𝑦 = (𝐹𝐵) → 𝐸 = 𝐷)       ((𝐵 ∈ ω ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Theoremtz7.48lem 7423* A way of showing an ordinal function is one-to-one. (Contributed by NM, 9-Feb-1997.)
𝐹 Fn On       ((𝐴 ⊆ On ∧ ∀𝑥𝐴𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → Fun (𝐹𝐴))

Theoremtz7.48-2 7424* Proposition 7.48(2) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) (Revised by David Abernethy, 5-May-2013.)
𝐹 Fn On       (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun 𝐹)

Theoremtz7.48-1 7425* Proposition 7.48(1) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
𝐹 Fn On       (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ran 𝐹𝐴)

Theoremtz7.48-3 7426* Proposition 7.48(3) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
𝐹 Fn On       (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ 𝐴 ∈ V)

Theoremtz7.49 7427* Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 10-Jan-2013.)
𝐹 Fn On    &   (𝜑 ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))       ((𝐴𝐵𝜑) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))

Theoremtz7.49c 7428* 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 7429 Extend class notation to include index-aware recursive definitions.
class seq𝜔(𝐹, 𝐼)

Definitiondf-seqom 7430* Index-aware recursive definitions over ω. A mashup of df-rdg 7393 and df-seq 12664, 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 7431* 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 7432* 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 7433* 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 7434* 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 7435* 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 7436 Equality theorem for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.)
((𝐴 = 𝐵𝐶 = 𝐷) → seq𝜔(𝐴, 𝐶) = seq𝜔(𝐵, 𝐷))

Theoremfnseqom 7437 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 7438 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 7439 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 7440 Extend the definition of a class to include the ordinal number 1.
class 1𝑜

Syntaxc2o 7441 Extend the definition of a class to include the ordinal number 2.
class 2𝑜

Syntaxc3o 7442 Extend the definition of a class to include the ordinal number 3.
class 3𝑜

Syntaxc4o 7443 Extend the definition of a class to include the ordinal number 4.
class 4𝑜

Syntaxcoa 7444 Extend the definition of a class to include the ordinal addition operation.
class +𝑜

Syntaxcomu 7445 Extend the definition of a class to include the ordinal multiplication operation.
class ·𝑜

Syntaxcoe 7446 Extend the definition of a class to include the ordinal exponentiation operation.
class 𝑜

Definitiondf-1o 7447 Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
1𝑜 = suc ∅

Definitiondf-2o 7448 Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
2𝑜 = suc 1𝑜

Definitiondf-3o 7449 Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
3𝑜 = suc 2𝑜

Definitiondf-4o 7450 Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
4𝑜 = suc 3𝑜

Definitiondf-oadd 7451* Define the ordinal addition operation. (Contributed by NM, 3-May-1995.)
+𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))

Definitiondf-omul 7452* 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 7453* 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 7454 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)
1𝑜 ∈ On

Theorem2on 7455 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
2𝑜 ∈ On

Theorem2on0 7456 Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
2𝑜 ≠ ∅

Theorem3on 7457 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3𝑜 ∈ On

Theorem4on 7458 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4𝑜 ∈ On

Theoremdf1o2 7459 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
1𝑜 = {∅}

Theoremdf2o3 7460 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
2𝑜 = {∅, 1𝑜}

Theoremdf2o2 7461 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
2𝑜 = {∅, {∅}}

Theorem1n0 7462 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
1𝑜 ≠ ∅

Theoremxp01disj 7463 Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅

Theoremordgt0ge1 7464 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
(Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))

Theoremordge1n0 7465 An ordinal greater than or equal to 1 is nonzero. (Contributed by NM, 21-Dec-2004.)
(Ord 𝐴 → (1𝑜𝐴𝐴 ≠ ∅))

Theoremel1o 7466 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
(𝐴 ∈ 1𝑜𝐴 = ∅)

Theoremdif1o 7467 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 7468 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 7469 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 7470 Closure of the pair swapping function on 2𝑜. (Contributed by Mario Carneiro, 27-Sep-2015.)
(𝐴 ∈ 2𝑜 → (1𝑜𝐴) ∈ 2𝑜)

Theorem0lt1o 7471 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
∅ ∈ 1𝑜

Theoremdif20el 7472 An ordinal greater than one is greater than zero. (Contributed by Mario Carneiro, 25-May-2015.)
(𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)

Theorem0we1 7473 The empty set is a well-ordering of ordinal one. (Contributed by Mario Carneiro, 9-Feb-2015.)
∅ We 1𝑜

Theorembrwitnlem 7474 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 7475 Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
+𝑜 Fn (On × On)

Theoremfnom 7476 Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
·𝑜 Fn (On × On)

Theoremfnoe 7477 Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.)
𝑜 Fn (On × On)

Theoremoav 7478* 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 7479* 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 7480 A helper lemma for oe0 7489 and others. (Contributed by NM, 6-Jan-2005.)
((𝜑𝐴 = ∅) → 𝜓)    &   (((𝐴 ∈ On ∧ 𝜑) ∧ ∅ ∈ 𝐴) → 𝜓)       ((𝐴 ∈ On ∧ 𝜑) → 𝜓)

Theoremoev 7481* 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 7482* 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 7483 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 7484 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 7485 Ordinal exponentiation with zero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 ∈ On → (∅ ↑𝑜 𝐴) = (1𝑜𝐴))

Theoremom0x 7486 Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 7484, 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 7487 Ordinal exponentiation with zero mantissa and zero exponent. Proposition 8.31 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.)
(∅ ↑𝑜 ∅) = 1𝑜

Theoremoe0m1 7488 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 7489 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 7490* Alternate value of ordinal exponentiation. Compare oev 7481. (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 7491 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 7492* Lemma for oesuc 7494. (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 7493 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 7494 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 7495 Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Note that this version of oasuc 7491 does not need Replacement.) (Contributed by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 suc 𝐵) = suc (𝐴 +𝑜 𝐵))

Theoremonmsuc 7496 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 7497 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 7498 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 7499* 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 7500* 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))

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