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Theorem List for Metamath Proof Explorer - 30801-30900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtrpredeq2 30801 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴 = 𝐵 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐵, 𝑋))

Theoremtrpredeq3 30802 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑋 = 𝑌 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐴, 𝑌))

Theoremtrpredeq1d 30803 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝜑𝑅 = 𝑆)       (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑆, 𝐴, 𝑋))

Theoremtrpredeq2d 30804 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐵, 𝑋))

Theoremtrpredeq3d 30805 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝜑𝑋 = 𝑌)       (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐴, 𝑌))

Theoremeltrpred 30806* A class is a transitive predecessor iff it is in some value of the underlying function. This theorem is not really meant to be used directly: instead refer to trpredpred 30808 and trpredmintr 30811. (Contributed by Scott Fenton, 28-Apr-2012.)
(𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) ↔ ∃𝑖 ∈ ω 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖))

Theoremtrpredlem1 30807* Technical lemma for transitive predecessors properties. All values of the transitive predecessors' underlying function are subsets of the base set. (Contributed by Scott Fenton, 28-Apr-2012.)
(Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)

Theoremtrpredpred 30808 Assuming it exists, the predecessor class is a subset of the transitive predecessors. (Contributed by Scott Fenton, 18-Feb-2011.)
(Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ TrPred(𝑅, 𝐴, 𝑋))

Theoremtrpredss 30809 The transitive predecessors form a subset of the base class. (Contributed by Scott Fenton, 20-Feb-2011.)
(Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → TrPred(𝑅, 𝐴, 𝑋) ⊆ 𝐴)

Theoremtrpredtr 30810 The transitive predecessors are transitive in 𝑅 and 𝐴 (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
((𝑋𝐴𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋)))

Theoremtrpredmintr 30811* The transitive predecessors form the smallest class transitive in 𝑅 and 𝐴. That is, if 𝐵 is another 𝑅, 𝐴 transitive class containing Pred(𝑅, 𝐴, 𝑋), then TrPred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 (Contributed by Scott Fenton, 25-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
(((𝑋𝐴𝑅 Se 𝐴) ∧ (∀𝑦𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → TrPred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)

Theoremtrpredelss 30812 Given a transitive predecessor 𝑌 of 𝑋, the transitive predecessors of 𝑌 are a subset of the transitive predecessors of 𝑋. (Contributed by Scott Fenton, 25-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
((𝑋𝐴𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → TrPred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋)))

Theoremdftrpred3g 30813* The transitive predecessors of 𝑋 are equal to the predecessors of 𝑋 together with their transitive predecessors. (Contributed by Scott Fenton, 26-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
((𝑋𝐴𝑅 Se 𝐴) → TrPred(𝑅, 𝐴, 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∪ 𝑦 ∈ Pred (𝑅, 𝐴, 𝑋)TrPred(𝑅, 𝐴, 𝑦)))

Theoremdftrpred4g 30814* Another recursive expression for the transitive predecessors. (Contributed by Scott Fenton, 27-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
((𝑋𝐴𝑅 Se 𝐴) → TrPred(𝑅, 𝐴, 𝑋) = 𝑦 ∈ Pred (𝑅, 𝐴, 𝑋)({𝑦} ∪ TrPred(𝑅, 𝐴, 𝑦)))

Theoremtrpredpo 30815 If 𝑅 partially orders 𝐴, then the transitive predecessors are the same as the immediate predecessors . (Contributed by Scott Fenton, 28-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
((𝑅 Po 𝐴𝑋𝐴𝑅 Se 𝐴) → TrPred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑋))

Theoremtrpred0 30816 The class of transitive predecessors is empty when 𝐴 is empty. (Contributed by Scott Fenton, 30-Apr-2012.)
TrPred(𝑅, ∅, 𝑋) = ∅

Theoremtrpredex 30817 The transitive predecessors of a relation form a set (NOTE: this is the first theorem in the transitive predecessor series that requires infinity). (Contributed by Scott Fenton, 18-Feb-2011.)
TrPred(𝑅, 𝐴, 𝑋) ∈ V

Theoremtrpredrec 30818* If 𝑌 is an 𝑅, 𝐴 transitive predecessor, then it is either an immediate predecessor or there is a transitive predecessor between 𝑌 and 𝑋. (Contributed by Scott Fenton, 9-May-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
((𝑋𝐴𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))

20.8.17  Founded Induction

Theoremfrmin 30819* Every (possibly proper) subclass of a class 𝐴 with a founded, set-like relation 𝑅 has a minimal element. Lemma 4.3 of Don Monk's notes for Advanced Set Theory, which can be found at http://euclid.colorado.edu/~monkd/settheory. This is a very strong generalization of tz6.26 5518 and tz7.5 5551. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
(((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)

Theoremfrind 30820* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 30819). This principle states that if 𝐵 is a subclass of a founded class 𝐴 with the property that every element of 𝐵 whose initial segment is included in 𝐴 is itself equal to 𝐴. Compare wfi 5520 and tfi 6826, which are special cases of this theorem that do not require the axiom of infinity to prove. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
(((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐴 = 𝐵)

Theoremfrindi 30821* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 30819). This principle states that if 𝐵 is a subclass of a founded class 𝐴 with the property that every element of 𝐵 whose initial segment is included in 𝐴 is itself equal to 𝐴. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴       ((𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)) → 𝐴 = 𝐵)

Theoremfrinsg 30822* Founded Induction Schema. If a property passes from all elements less than 𝑦 of a founded class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
(𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))       ((𝑅 Fr 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)

Theoremfrins 30823* Founded Induction Schema. If a property passes from all elements less than 𝑦 of a founded class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))       (𝑦𝐴𝜑)

Theoremfrins2fg 30824* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
(𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))    &   𝑦𝜓    &   (𝑦 = 𝑧 → (𝜑𝜓))       ((𝑅 Fr 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)

Theoremfrins2f 30825* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝑦𝜓    &   (𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))       (𝑦𝐴𝜑)

Theoremfrins2g 30826* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 8-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
(𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))    &   (𝑦 = 𝑧 → (𝜑𝜓))       ((𝑅 Fr 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)

Theoremfrins2 30827* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   (𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))       (𝑦𝐴𝜑)

Theoremfrins3 30828* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   (𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜑𝜒))    &   (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))       (𝐵𝐴𝜒)

20.8.18  Ordering Ordinal Sequences

Theoremorderseqlem 30829* Lemma for poseq 30830 and soseq 30831. The function value of a sequene is either in 𝐴 or null. (Contributed by Scott Fenton, 8-Jun-2011.)
𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥𝐴}       (𝐺𝐹 → (𝐺𝑋) ∈ (𝐴 ∪ {∅}))

Theoremposeq 30830* A partial ordering of sequences of ordinals. (Contributed by Scott Fenton, 8-Jun-2011.)
𝑅 Po (𝐴 ∪ {∅})    &   𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥𝐴}    &   𝑆 = {⟨𝑓, 𝑔⟩ ∣ ((𝑓𝐹𝑔𝐹) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥)𝑅(𝑔𝑥)))}       𝑆 Po 𝐹

Theoremsoseq 30831* A linear ordering of sequences of ordinals. (Contributed by Scott Fenton, 8-Jun-2011.)
𝑅 Or (𝐴 ∪ {∅})    &   𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥𝐴}    &   𝑆 = {⟨𝑓, 𝑔⟩ ∣ ((𝑓𝐹𝑔𝐹) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥)𝑅(𝑔𝑥)))}    &    ¬ ∅ ∈ 𝐴       𝑆 Or 𝐹

20.8.19  Well-founded zero, successor, and limits

Syntaxcwsuc 30832 Declare the syntax for well-founded successor.
class wsuc(𝑅, 𝐴, 𝑋)

Syntaxcwlim 30833 Declare the syntax for well-founded limit class.
class WLim(𝑅, 𝐴)

Definitiondf-wsuc 30834 Define the concept of a successor in a well-founded set. (Contributed by Scott Fenton, 13-Jun-2018.)
wsuc(𝑅, 𝐴, 𝑋) = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)

Definitiondf-wlim 30835* Define the class of limit points of a well-founded set. (Contributed by Scott Fenton, 15-Jun-2018.)
WLim(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ sup(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}

Theoremwsuceq123 30836 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌))

Theoremwsuceq1 30837 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 = 𝑆 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐴, 𝑋))

Theoremwsuceq2 30838 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝐴 = 𝐵 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐵, 𝑋))

Theoremwsuceq3 30839 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑋 = 𝑌 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌))

Theoremnfwsuc 30840 Bound-variable hypothesis builder for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
𝑥𝑅    &   𝑥𝐴    &   𝑥𝑋       𝑥wsuc(𝑅, 𝐴, 𝑋)

Theoremwlimeq12 30841 Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
((𝑅 = 𝑆𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵))

Theoremwlimeq1 30842 Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
(𝑅 = 𝑆 → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐴))

Theoremwlimeq2 30843 Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
(𝐴 = 𝐵 → WLim(𝑅, 𝐴) = WLim(𝑅, 𝐵))

Theoremnfwlim 30844 Bound-variable hypothesis builder for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
𝑥𝑅    &   𝑥𝐴       𝑥WLim(𝑅, 𝐴)

Theoremelwlim 30845 Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
(𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋𝐴𝑋 ≠ sup(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))

Theoremwzel 30846 The zero of a well-founded set is a member of that set. (Contributed by Scott Fenton, 13-Jun-2018.)
((𝑅 We 𝐴𝑅 Se 𝐴𝐴 ≠ ∅) → sup(𝐴, 𝐴, 𝑅) ∈ 𝐴)

Theoremwsuclem 30847* Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.)
(𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)    &   (𝜑𝑋𝑉)    &   (𝜑 → ∃𝑤𝐴 𝑋𝑅𝑤)       (𝜑 → ∃𝑥𝐴 (∀𝑦 ∈ Pred (𝑅, 𝐴, 𝑋) ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ Pred (𝑅, 𝐴, 𝑋)𝑦𝑅𝑧)))

Theoremwsucex 30848 Existence theorem for well-founded successor. (Contributed by Scott Fenton, 16-Jun-2018.)
(𝜑𝑅 Or 𝐴)       (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)

Theoremwsuccl 30849* If 𝑋 is a set with an 𝑅 successor in 𝐴, then its well-founded successor is a member of 𝐴. (Contributed by Scott Fenton, 15-Jun-2018.)
(𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)    &   (𝜑𝑋𝑉)    &   (𝜑 → ∃𝑦𝐴 𝑋𝑅𝑦)       (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴)

Theoremwsuclb 30850 A well-founded successor is a lower bound on points after 𝑋. (Contributed by Scott Fenton, 16-Jun-2018.)
(𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝐴)    &   (𝜑𝑋𝑅𝑌)       (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))

Theoremwlimss 30851 The class of limit points is a subclass of the base class. (Contributed by Scott Fenton, 16-Jun-2018.)
WLim(𝑅, 𝐴) ⊆ 𝐴

20.8.20  Founded Recursion

Theoremfrr3g 30852* Functions defined by founded recursion are identical up to relation, domain, and characteristic function. General version of frr3. (Contributed by Scott Fenton, 10-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
(((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺)

Theoremfrrlem1 30853* Lemma for founded recursion. The final item we are interested in is the union of acceptable functions 𝐵. This lemma just changes bound variables for later use. (Contributed by Paul Chapman, 21-Apr-2012.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))}

Theoremfrrlem2 30854* Lemma for founded recursion. An acceptable function is a function. (Contributed by Paul Chapman, 21-Apr-2012.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       (𝑔𝐵 → Fun 𝑔)

Theoremfrrlem3 30855* Lemma for founded recursion. An acceptable function's domain is a subset of 𝐴. (Contributed by Paul Chapman, 21-Apr-2012.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       (𝑔𝐵 → dom 𝑔𝐴)

Theoremfrrlem4 30856* Lemma for founded recursion. Properties of the restriction of an acceptable function to the domain of another acceptable function. (Contributed by Paul Chapman, 21-Apr-2012.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       ((𝑔𝐵𝐵) → ((𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝑎𝐺((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))

Theoremfrrlem5 30857* Lemma for founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Paul Chapman, 21-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       ((𝑔𝐵𝐵) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))

Theoremfrrlem5b 30858* Lemma for founded recursion. The union of a subclass of 𝐵 is a relationship. (Contributed by Paul Chapman, 29-Apr-2012.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       (𝐶𝐵 → Rel 𝐶)

Theoremfrrlem5c 30859* Lemma for founded recursion. The union of a subclass of 𝐵 is a function. (Contributed by Paul Chapman, 29-Apr-2012.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       (𝐶𝐵 → Fun 𝐶)

Theoremfrrlem5d 30860* Lemma for founded recursion. The domain of the union of a subset of 𝐵 is a subset of 𝐴. (Contributed by Paul Chapman, 29-Apr-2012.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       (𝐶𝐵 → dom 𝐶𝐴)

Theoremfrrlem5e 30861* Lemma for founded recursion. The domain of the union of a subset of 𝐵 is closed under predecessors. (Contributed by Paul Chapman, 1-May-2012.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}       (𝐶𝐵 → (𝑋 ∈ dom 𝐶 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))

Theoremfrrlem6 30862* Lemma for founded recursion. The union of all acceptable functions is a relationship. (Contributed by Paul Chapman, 21-Apr-2012.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}    &   𝐹 = 𝐵       Rel 𝐹

Theoremfrrlem7 30863* Lemma for founded recursion. The domain of 𝐹 is a subclass of 𝐴. (Contributed by Paul Chapman, 21-Apr-2012.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}    &   𝐹 = 𝐵       dom 𝐹𝐴

Theoremfrrlem10 30864* Lemma for founded recursion. The union of all acceptable functions is a function. (Contributed by Paul Chapman, 21-Apr-2012.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}    &   𝐹 = 𝐵       Fun 𝐹

Theoremfrrlem11 30865* Lemma for founded recursion. Here, we calculate the value of 𝐹 (the union of all acceptable functions). (Contributed by Paul Chapman, 21-Apr-2012.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}    &   𝐹 = 𝐵       (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))

20.8.21  Surreal Numbers

Syntaxcsur 30866 Declare the class of all surreal numbers (see df-no 30869).
class No

Syntaxcslt 30867 Declare the less than relationship over surreal numbers (see df-slt 30870).
class <s

Syntaxcbday 30868 Declare the birthday function for surreal numbers (see df-bday 30871).
class bday

Definitiondf-no 30869* Define the class of surreal numbers. The surreal numbers are a proper class of numbers developed by John H. Conway and introduced by Donald Knuth in 1975. They form a proper class into which all ordered fields can be embedded. The approach we take to defining them was first introduced by Hary Goshnor, and is based on the conception of a "sign expansion" of a surreal number. We define the surreals as ordinal-indexed sequences of 1𝑜 and 2𝑜, analagous to Goshnor's ( − ) and ( + ).

After introducing this definition, we will abstract away from it using axioms that Norman Alling developed in "Foundations of Analysis over Surreal Number Fields." This is done in an effort to be agnostic towards the exact implementation of surreals. (Contributed by Scott Fenton, 9-Jun-2011.)

No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1𝑜, 2𝑜}}

Definitiondf-slt 30870* Next, we introduce surreal less-than, a comparison relationship over the surreals by lexicographically ordering them. (Contributed by Scott Fenton, 9-Jun-2011.)
<s = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 No 𝑔 No ) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥)))}

Definitiondf-bday 30871 Finally, we introduce the birthday function. This function maps each surreal to an ordinal. In our implementation, this is the domain of the sign function. The important properties of this function are established later. (Contributed by Scott Fenton, 11-Jun-2011.)
bday = (𝑥 No ↦ dom 𝑥)

Theoremelno 30872* Membership in the surreals. (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.)
(𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜})

Theoremsltval 30873* The value of the surreal less than relationship. (Contributed by Scott Fenton, 14-Jun-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))

Theorembdayval 30874 The value of the birthday function within the surreals. (Contributed by Scott Fenton, 14-Jun-2011.)
(𝐴 No → ( bday 𝐴) = dom 𝐴)

Theoremnofun 30875 A surreal is a function. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → Fun 𝐴)

Theoremnodmon 30876 The domain of a surreal is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → dom 𝐴 ∈ On)

Theoremnorn 30877 The range of a surreal is a subset of the surreal signs. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → ran 𝐴 ⊆ {1𝑜, 2𝑜})

Theoremnofnbday 30878 A surreal is a function over its birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No 𝐴 Fn ( bday 𝐴))

Theoremnodmord 30879 The domain of a surreal has the ordinal property. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → Ord dom 𝐴)

Theoremelno2 30880 An alternative condition for membership in No . (Contributed by Scott Fenton, 21-Mar-2012.)
(𝐴 No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))

Theoremelno3 30881 Another condition for membership in No . (Contributed by Scott Fenton, 14-Apr-2012.)
(𝐴 No ↔ (𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ∧ dom 𝐴 ∈ On))

Theoremsltval2 30882* Alternate expression for surreal less than. Two surreals obey surreal less than iff they obey the sign ordering at the first place they differ. (Contributed by Scott Fenton, 17-Jun-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ (𝐴 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)})))

Theoremnofv 30883 The function value of a surreal is either a sign or the empty set. (Contributed by Scott Fenton, 22-Jun-2011.)
(𝐴 No → ((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜))

Theoremnosgnn0 30884 is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)
¬ ∅ ∈ {1𝑜, 2𝑜}

Theoremnosgnn0i 30885 If 𝑋 is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}       ∅ ≠ 𝑋

Theoremnoreson 30886 The restriction of a surreal to an ordinal is still a surreal. (Contributed by Scott Fenton, 4-Sep-2011.)
((𝐴 No 𝐵 ∈ On) → (𝐴𝐵) ∈ No )

Theoremsltsgn1 30887* If 𝐴 <s 𝐵, then the sign of 𝐴 at the first place they differ is either undefined or 1𝑜. (Contributed by Scott Fenton, 4-Sep-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → ((𝐴 {𝑘 ∈ On ∣ (𝐴𝑘) ≠ (𝐵𝑘)}) = ∅ ∨ (𝐴 {𝑘 ∈ On ∣ (𝐴𝑘) ≠ (𝐵𝑘)}) = 1𝑜)))

Theoremsltsgn2 30888* If 𝐴 <s 𝐵, then the sign of 𝐵 at the first place they differ is either undefined or 2𝑜. (Contributed by Scott Fenton, 4-Sep-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → ((𝐵 {𝑘 ∈ On ∣ (𝐴𝑘) ≠ (𝐵𝑘)}) = ∅ ∨ (𝐵 {𝑘 ∈ On ∣ (𝐴𝑘) ≠ (𝐵𝑘)}) = 2𝑜)))

Theoremsltintdifex 30889* If 𝐴 <s 𝐵, then the intersection of all the ordinals that have differing signs in 𝐴 and 𝐵 exists. (Contributed by Scott Fenton, 22-Feb-2012.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 {𝑎 ∈ On ∣ (𝐴𝑎) ≠ (𝐵𝑎)} ∈ V))

Theoremsltres 30890 If the restrictions of two surreals to a given ordinal obey surreal less than, then so do the two surreals themselves. (Contributed by Scott Fenton, 4-Sep-2011.)
((𝐴 No 𝐵 No 𝑋 ∈ On) → ((𝐴𝑋) <s (𝐵𝑋) → 𝐴 <s 𝐵))

Theoremnoxpsgn 30891 The Cartesian product of an ordinal and the singleton of a sign is a surreal. (Contributed by Scott Fenton, 21-Jun-2011.)
𝑋 ∈ {1𝑜, 2𝑜}       (𝐴 ∈ On → (𝐴 × {𝑋}) ∈ No )

Theoremnoxp1o 30892 The Cartesian product of an ordinal and {1𝑜} is a surreal. (Contributed by Scott Fenton, 12-Jun-2011.)
(𝐴 ∈ On → (𝐴 × {1𝑜}) ∈ No )

Theoremnoxp2o 30893 The Cartesian product of an ordinal and {2𝑜} is a surreal. (Contributed by Scott Fenton, 12-Jun-2011.)
(𝐴 ∈ On → (𝐴 × {2𝑜}) ∈ No )

Theoremnoseponlem 30894* Lemma for nosepon 30895. Consider a case of proper subset domain. (Contributed by Scott Fenton, 21-Sep-2020.)
((𝐴 No 𝐵 No ∧ dom 𝐴 ∈ dom 𝐵) → ¬ ∀𝑥 ∈ On (𝐴𝑥) = (𝐵𝑥))

Theoremnosepon 30895* Given two unequal surreals, the minimal ordinal at which they differ is an ordinal. (Contributed by Scott Fenton, 21-Sep-2020.)
((𝐴 No 𝐵 No 𝐴𝐵) → {𝑥 ∈ On ∣ (𝐴𝑥) ≠ (𝐵𝑥)} ∈ On)

20.8.22  Surreal Numbers: Ordering

Theoremsltsolem1 30896 Lemma for sltso 30897. The sign expansion relationship totally orders the surreal signs. (Contributed by Scott Fenton, 8-Jun-2011.)
{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} Or ({1𝑜, 2𝑜} ∪ {∅})

Theoremsltso 30897 Surreal less than totally orders the surreals. Alling's axiom (O). (Contributed by Scott Fenton, 9-Jun-2011.)
<s Or No

Theoremsltirr 30898 Surreal less than is irreflexive. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → ¬ 𝐴 <s 𝐴)

Theoremslttr 30899 Surreal less than is transitive. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No 𝐶 No ) → ((𝐴 <s 𝐵𝐵 <s 𝐶) → 𝐴 <s 𝐶))

Theoremsltasym 30900 Surreal less than is asymmetric. (Contributed by Scott Fenton, 16-Jun-2011.)
((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → ¬ 𝐵 <s 𝐴))

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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 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