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Theorem List for Metamath Proof Explorer - 31701-31800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtrpredlem1 31701* 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 31702 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 31703 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 31704 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 31705* 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 31706 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 31707* 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 31708* 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 31709 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 31710 The class of transitive predecessors is empty when 𝐴 is empty. (Contributed by Scott Fenton, 30-Apr-2012.)
TrPred(𝑅, ∅, 𝑋) = ∅

Theoremtrpredex 31711 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 31712* 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 31713* 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 5699 and tz7.5 5732. (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 31714* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 31713). 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 5701 and tfi 7038, 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 31715* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 31713). 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 31716* 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 31717* 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 31718* 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 31719* 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 31720* 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 31721* 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 31722* 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 31723* Lemma for poseq 31724 and soseq 31725. 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 31724* 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 31725* 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 31726 Declare the syntax for well-founded successor.
class wsuc(𝑅, 𝐴, 𝑋)

SyntaxcwsucOLD 31727 Declare the syntax for well-founded successor. (New usage is discouraged.)
class wsucOLD(𝑅, 𝐴, 𝑋)

Syntaxcwlim 31728 Declare the syntax for well-founded limit class.
class WLim(𝑅, 𝐴)

SyntaxcwlimOLD 31729 Declare the syntax for well-founded limit class. (New usage is discouraged.)
class WLimOLD(𝑅, 𝐴)

Definitiondf-wsuc 31730 Define the concept of a successor in a well-founded set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.)
wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)

Definitiondf-wsucOLD 31731 Define the concept of a successor in a well-founded set. (Contributed by Scott Fenton, 13-Jun-2018.) Obsolete version of df-wsuc 31730 as of 10-Oct-2021. (New usage is discouraged.)
wsucOLD(𝑅, 𝐴, 𝑋) = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)

Definitiondf-wlim 31732* Define the class of limit points of a well-founded set. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
WLim(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}

Definitiondf-wlimOLD 31733* Define the class of limit points of a well-founded set. (Contributed by Scott Fenton, 15-Jun-2018.) Obsolete version of df-wlim 31732 as of 10-Oct-2021. (New usage is discouraged.)
WLimOLD(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ sup(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}

Theoremwsuceq123 31734 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌))

Theoremwsuceq1 31735 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 = 𝑆 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐴, 𝑋))

Theoremwsuceq2 31736 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝐴 = 𝐵 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐵, 𝑋))

Theoremwsuceq3 31737 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑋 = 𝑌 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌))

Theoremnfwsuc 31738 Bound-variable hypothesis builder for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
𝑥𝑅    &   𝑥𝐴    &   𝑥𝑋       𝑥wsuc(𝑅, 𝐴, 𝑋)

Theoremwlimeq12 31739 Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
((𝑅 = 𝑆𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵))

Theoremwlimeq1 31740 Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
(𝑅 = 𝑆 → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐴))

Theoremwlimeq2 31741 Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
(𝐴 = 𝐵 → WLim(𝑅, 𝐴) = WLim(𝑅, 𝐵))

Theoremnfwlim 31742 Bound-variable hypothesis builder for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
𝑥𝑅    &   𝑥𝐴       𝑥WLim(𝑅, 𝐴)

Theoremelwlim 31743 Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
(𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋𝐴𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))

TheoremelwlimOLD 31744 Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) Obsolete version of elwlim 31743 as of 10-Oct-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(𝑋 ∈ WLimOLD(𝑅, 𝐴) ↔ (𝑋𝐴𝑋 ≠ sup(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))

Theoremwzel 31745 The zero of a well-founded set is a member of that set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.)
((𝑅 We 𝐴𝑅 Se 𝐴𝐴 ≠ ∅) → inf(𝐴, 𝐴, 𝑅) ∈ 𝐴)

TheoremwzelOLD 31746 The zero of a well-founded set is a member of that set. (Contributed by Scott Fenton, 13-Jun-2018.) Obsolete version of wzel 31745 as of 10-Oct-2021. (New usage is discouraged.) (Proof modification is discouraged.)
((𝑅 We 𝐴𝑅 Se 𝐴𝐴 ≠ ∅) → sup(𝐴, 𝐴, 𝑅) ∈ 𝐴)

Theoremwsuclem 31747* Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
(𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)    &   (𝜑𝑋𝑉)    &   (𝜑 → ∃𝑤𝐴 𝑋𝑅𝑤)       (𝜑 → ∃𝑥𝐴 (∀𝑦 ∈ Pred (𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)))

TheoremwsuclemOLD 31748* Obsolete version of wsuclem 31747 as of 10-Oct-2021. (Contributed by Scott Fenton, 15-Jun-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(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 31749 Existence theorem for well-founded successor. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
(𝜑𝑅 Or 𝐴)       (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V)

Theoremwsuccl 31750* 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.) (Proof shortened by AV, 10-Oct-2021.)
(𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)    &   (𝜑𝑋𝑉)    &   (𝜑 → ∃𝑦𝐴 𝑋𝑅𝑦)       (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴)

Theoremwsuclb 31751 A well-founded successor is a lower bound on points after 𝑋. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
(𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝐴)    &   (𝜑𝑋𝑅𝑌)       (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋))

Theoremwlimss 31752 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 31753* 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 31754* 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 31755* 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 31756* 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 31757* 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 31758* 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 31759* 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 31760* 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 31761* 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 31762* 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 31763* 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 31764* 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 31765* 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 31766* 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 31767 Declare the class of all surreal numbers (see df-no 31770).
class No

Syntaxcslt 31768 Declare the less than relationship over surreal numbers (see df-slt 31771).
class <s

Syntaxcbday 31769 Declare the birthday function for surreal numbers (see df-bday 31772).
class bday

Definitiondf-no 31770* 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 31771* 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 31772 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 31773* 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 31774* 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 31775 The value of the birthday function within the surreals. (Contributed by Scott Fenton, 14-Jun-2011.)
(𝐴 No → ( bday 𝐴) = dom 𝐴)

Theoremnofun 31776 A surreal is a function. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → Fun 𝐴)

Theoremnodmon 31777 The domain of a surreal is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → dom 𝐴 ∈ On)

Theoremnorn 31778 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 31779 A surreal is a function over its birthday. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No 𝐴 Fn ( bday 𝐴))

Theoremnodmord 31780 The domain of a surreal has the ordinal property. (Contributed by Scott Fenton, 16-Jun-2011.)
(𝐴 No → Ord dom 𝐴)

Theoremelno2 31781 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 31782 Another condition for membership in No . (Contributed by Scott Fenton, 14-Apr-2012.)
(𝐴 No ↔ (𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ∧ dom 𝐴 ∈ On))

Theoremsltval2 31783* 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 31784 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 31785 is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)
¬ ∅ ∈ {1𝑜, 2𝑜}

Theoremnosgnn0i 31786 If 𝑋 is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.)
𝑋 ∈ {1𝑜, 2𝑜}       ∅ ≠ 𝑋

Theoremnoreson 31787 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 )

Theoremsltintdifex 31788* 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 31789 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))

Theoremnoxp1o 31790 The Cartesian product of an ordinal and {1𝑜} is a surreal. (Contributed by Scott Fenton, 12-Jun-2011.)
(𝐴 ∈ On → (𝐴 × {1𝑜}) ∈ No )

Theoremnoseponlem 31791* Lemma for nosepon 31792. 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 31792* 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)

Theoremnoextend 31793 Extending a surreal by one sign value results in a new surreal. (Contributed by Scott Fenton, 22-Nov-2021.)
𝑋 ∈ {1𝑜, 2𝑜}       (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ No )

Theoremnoextendseq 31794 Extend a surreal by a sequence of ordinals. (Contributed by Scott Fenton, 30-Nov-2021.)
𝑋 ∈ {1𝑜, 2𝑜}       ((𝐴 No 𝐵 ∈ On) → (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No )

Theoremnoextenddif 31795* Calculate the place where a surreal and its extension differ. (Contributed by Scott Fenton, 22-Nov-2021.)
𝑋 ∈ {1𝑜, 2𝑜}       (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} = dom 𝐴)

Theoremnoextendlt 31796 Extending a surreal with a negative sign results in a smaller surreal. (Contributed by Scott Fenton, 22-Nov-2021.)
(𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) <s 𝐴)

Theoremnoextendgt 31797 Extending a surreal with a positive sign results in a bigger surreal. (Contributed by Scott Fenton, 22-Nov-2021.)
(𝐴 No 𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩}))

Theoremnolesgn2o 31798 Given 𝐴 less than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 2𝑜, then 𝐵(𝑋) = 2𝑜. (Contributed by Scott Fenton, 6-Dec-2021.)
(((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (𝐵𝑋) = 2𝑜)

Theoremnolesgn2ores 31799 Given 𝐴 less than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 2𝑜, then (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋). (Contributed by Scott Fenton, 6-Dec-2021.)
(((𝐴 No 𝐵 No 𝑋 ∈ On) ∧ ((𝐴𝑋) = (𝐵𝑋) ∧ (𝐴𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋))

20.8.22  Surreal Numbers: Ordering

Theoremsltsolem1 31800 Lemma for sltso 31801. The sign expansion relationship totally orders the surreal signs. (Contributed by Scott Fenton, 8-Jun-2011.)
{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} Or ({1𝑜, 2𝑜} ∪ {∅})

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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 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