 Home Metamath Proof ExplorerTheorem List (p. 73 of 425) < 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 (1-26947) Hilbert Space Explorer (26948-28472) Users' Mathboxes (28473-42426)

Theorem List for Metamath Proof Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremonfununi 7201* A property of functions on ordinal numbers. Generalization of Theorem Schema 8E of [Enderton] p. 218. (Contributed by Eric Schmidt, 26-May-2009.)
(Lim 𝑦 → (𝐹𝑦) = 𝑥𝑦 (𝐹𝑥))    &   ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → (𝐹𝑥) ⊆ (𝐹𝑦))       ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐹 𝑆) = 𝑥𝑆 (𝐹𝑥))

Theoremonovuni 7202* A variant of onfununi 7201 for operations. (Contributed by Eric Schmidt, 26-May-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
(Lim 𝑦 → (𝐴𝐹𝑦) = 𝑥𝑦 (𝐴𝐹𝑥))    &   ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → (𝐴𝐹𝑥) ⊆ (𝐴𝐹𝑦))       ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐴𝐹 𝑆) = 𝑥𝑆 (𝐴𝐹𝑥))

Theoremonoviun 7203* A variant of onovuni 7202 with indexed unions. (Contributed by Eric Schmidt, 26-May-2009.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
(Lim 𝑦 → (𝐴𝐹𝑦) = 𝑥𝑦 (𝐴𝐹𝑥))    &   ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → (𝐴𝐹𝑥) ⊆ (𝐴𝐹𝑦))       ((𝐾𝑇 ∧ ∀𝑧𝐾 𝐿 ∈ On ∧ 𝐾 ≠ ∅) → (𝐴𝐹 𝑧𝐾 𝐿) = 𝑧𝐾 (𝐴𝐹𝐿))

Theoremonnseq 7204* There are no length ω decreasing sequences in the ordinals. See also noinfep 8316 for a stronger version assuming Regularity. (Contributed by Mario Carneiro, 19-May-2015.)
((𝐹‘∅) ∈ On → ∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹𝑥))

Syntaxwsmo 7205 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.
wff Smo 𝐴

Definitiondf-smo 7206* Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.)
(Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))

Theoremdfsmo2 7207* Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.)
(Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))

Theoremissmo 7208* Conditions for which 𝐴 is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)
𝐴:𝐵⟶On    &   Ord 𝐵    &   ((𝑥𝐵𝑦𝐵) → (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))    &   dom 𝐴 = 𝐵       Smo 𝐴

Theoremissmo2 7209* Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)
(𝐹:𝐴𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)) → Smo 𝐹))

Theoremsmoeq 7210 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
(𝐴 = 𝐵 → (Smo 𝐴 ↔ Smo 𝐵))

Theoremsmodm 7211 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
(Smo 𝐴 → Ord dom 𝐴)

Theoremsmores 7212 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
((Smo 𝐴𝐵 ∈ dom 𝐴) → Smo (𝐴𝐵))

Theoremsmores3 7213 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)
((Smo (𝐴𝐵) ∧ 𝐶 ∈ (dom 𝐴𝐵) ∧ Ord 𝐵) → Smo (𝐴𝐶))

Theoremsmores2 7214 A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)
((Smo 𝐹 ∧ Ord 𝐴) → Smo (𝐹𝐴))

Theoremsmodm2 7215 The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)
((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)

Theoremsmofvon2 7216 The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
(Smo 𝐹 → (𝐹𝐵) ∈ On)

Theoremiordsmo 7217 The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.)
Ord 𝐴       Smo ( I ↾ 𝐴)

Theoremsmo0 7218 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
Smo ∅

Theoremsmofvon 7219 If 𝐵 is a strictly monotone ordinal function, and 𝐴 is in the domain of 𝐵, then the value of the function at 𝐴 is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.)
((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐵𝐴) ∈ On)

Theoremsmoel 7220 If 𝑥 is less than 𝑦 then a strictly monotone function's value will be strictly less at 𝑥 than at 𝑦. (Contributed by Andrew Salmon, 22-Nov-2011.)
((Smo 𝐵𝐴 ∈ dom 𝐵𝐶𝐴) → (𝐵𝐶) ∈ (𝐵𝐴))

Theoremsmoiun 7221* The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)
((Smo 𝐵𝐴 ∈ dom 𝐵) → 𝑥𝐴 (𝐵𝑥) ⊆ (𝐵𝐴))

Theoremsmoiso 7222 If 𝐹 is an isomorphism from an ordinal 𝐴 onto 𝐵, which is a subset of the ordinals, then 𝐹 is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)
((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → Smo 𝐹)

Theoremsmoel2 7223 A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.)
(((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐵𝐴𝐶𝐵)) → (𝐹𝐶) ∈ (𝐹𝐵))

Theoremsmo11 7224 A strictly monotone ordinal function is one-to-one. (Contributed by Mario Carneiro, 28-Feb-2013.)
((𝐹:𝐴𝐵 ∧ Smo 𝐹) → 𝐹:𝐴1-1𝐵)

Theoremsmoord 7225 A strictly monotone ordinal function preserves strict ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)
(((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ∈ (𝐹𝐷)))

Theoremsmoword 7226 A strictly monotone ordinal function preserves weak ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)
(((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ⊆ (𝐹𝐷)))

Theoremsmogt 7227 A strictly monotone ordinal function is greater than or equal to its argument. Exercise 1 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 23-Nov-2011.) (Revised by Mario Carneiro, 28-Feb-2013.)
((𝐹 Fn 𝐴 ∧ Smo 𝐹𝐶𝐴) → 𝐶 ⊆ (𝐹𝐶))

Theoremsmorndom 7228 The range of a strictly monotone ordinal function dominates the domain. (Contributed by Mario Carneiro, 13-Mar-2013.)
((𝐹:𝐴𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵) → 𝐴𝐵)

Theoremsmoiso2 7229 The strictly monotone ordinal functions are also epsilon isomorphisms of subclasses of On. (Contributed by Mario Carneiro, 20-Mar-2013.)
((Ord 𝐴𝐵 ⊆ On) → ((𝐹:𝐴onto𝐵 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (𝐴, 𝐵)))

2.4.15  "Strong" transfinite recursion

Syntaxcrecs 7230 Notation for a function defined by strong transfinite recursion.
class recs(𝐹)

Definitiondf-recs 7231 Define a function recs(𝐹) on On, the class of ordinal numbers, by transfinite recursion given a rule 𝐹 which sets the next value given all values so far. See df-rdg 7269 for more details on why this definition is desirable. Unlike df-rdg 7269 which restricts the update rule to use only the previous value, this version allows the update rule to use all previous values, which is why it is described as "strong", although it is actually more primitive. See recsfnon 7262 and recsval 7263 for the primary contract of this definition. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Scott Fenton, 3-Aug-2020.)
recs(𝐹) = wrecs( E , On, 𝐹)

Theoremdfrecs3 7232* The old definition of transfinite recursion. This version is preferred for developement, as it demonstrates the properties of transfinite recursion without relying on well-founded recursion. (Contributed by Scott Fenton, 3-Aug-2020.)
recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}

Theoremrecseq 7233 Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
(𝐹 = 𝐺 → recs(𝐹) = recs(𝐺))

Theoremnfrecs 7234 Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝑥𝐹       𝑥recs(𝐹)

Theoremtfrlem1 7235* A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by Mario Carneiro, 24-May-2019.)
(𝜑𝐴 ∈ On)    &   (𝜑 → (Fun 𝐹𝐴 ⊆ dom 𝐹))    &   (𝜑 → (Fun 𝐺𝐴 ⊆ dom 𝐺))    &   (𝜑 → ∀𝑥𝐴 (𝐹𝑥) = (𝐵‘(𝐹𝑥)))    &   (𝜑 → ∀𝑥𝐴 (𝐺𝑥) = (𝐵‘(𝐺𝑥)))       (𝜑 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))

Theoremtfrlem3a 7236* Lemma for transfinite recursion. Let 𝐴 be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in 𝐴 for later use. (Contributed by NM, 9-Apr-1995.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   𝐺 ∈ V       (𝐺𝐴 ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤))))

Theoremtfrlem3 7237* Lemma for transfinite recursion. Let 𝐴 be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in 𝐴 for later use. (Contributed by NM, 9-Apr-1995.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}       𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))}

Theoremtfrlem4 7238* Lemma for transfinite recursion. 𝐴 is the class of all "acceptable" functions, and 𝐹 is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}       (𝑔𝐴 → Fun 𝑔)

Theoremtfrlem5 7239* Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}       ((𝑔𝐴𝐴) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))

Theoremrecsfval 7240* Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}       recs(𝐹) = 𝐴

Theoremtfrlem6 7241* Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}       Rel recs(𝐹)

Theoremtfrlem7 7242* Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}       Fun recs(𝐹)

Theoremtfrlem8 7243* Lemma for transfinite recursion. The domain of recs is an ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}       Ord dom recs(𝐹)

Theoremtfrlem9 7244* Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}       (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))

Theoremtfrlem9a 7245* Lemma for transfinite recursion. Without using ax-rep 4597, show that all the restrictions of recs are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}       (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V)

Theoremtfrlem10 7246* Lemma for transfinite recursion. We define class 𝐶 by extending recs with one ordered pair. We will assume, falsely, that domain of recs is a member of, and thus not equal to, On. Using this assumption we will prove facts about 𝐶 that will lead to a contradiction in tfrlem14 7250, thus showing the domain of recs does in fact equal On. Here we show (under the false assumption) that 𝐶 is a function extending the domain of recs(𝐹) by one. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})       (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))

Theoremtfrlem11 7247* Lemma for transfinite recursion. Compute the value of 𝐶. (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})       (dom recs(𝐹) ∈ On → (𝐵 ∈ suc dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))

Theoremtfrlem12 7248* Lemma for transfinite recursion. Show 𝐶 is an acceptable function. (Contributed by NM, 15-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}    &   𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})       (recs(𝐹) ∈ V → 𝐶𝐴)

Theoremtfrlem13 7249* Lemma for transfinite recursion. If recs is a set function, then 𝐶 is acceptable, and thus a subset of recs, but dom 𝐶 is bigger than dom recs. This is a contradiction, so recs must be a proper class function. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2014.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}        ¬ recs(𝐹) ∈ V

Theoremtfrlem14 7250* Lemma for transfinite recursion. Assuming ax-rep 4597, dom recs ∈ V ↔ recs ∈ V, so since dom recs is an ordinal, it must be equal to On. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}       dom recs(𝐹) = On

Theoremtfrlem15 7251* Lemma for transfinite recursion. Without assuming ax-rep 4597, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 14-Nov-2014.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}       (𝐵 ∈ On → (𝐵 ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ 𝐵) ∈ V))

Theoremtfrlem16 7252* Lemma for finite recursion. Without assuming ax-rep 4597, we can show that the domain of the constructed function is a limit ordinal, and hence contains all the finite ordinals. (Contributed by Mario Carneiro, 14-Nov-2014.)
𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}       Lim dom recs(𝐹)

Theoremtfr1a 7253 A weak version of tfr1 7256 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐹 = recs(𝐺)       (Fun 𝐹 ∧ Lim dom 𝐹)

Theoremtfr2a 7254 A weak version of tfr2 7257 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐹 = recs(𝐺)       (𝐴 ∈ dom 𝐹 → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))

Theoremtfr2b 7255 Without assuming ax-rep 4597, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐹 = recs(𝐺)       (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))

Theoremtfr1 7256 Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47. We start with an arbitrary class 𝐺, normally a function, and define a class 𝐴 of all "acceptable" functions. The final function we're interested in is the union 𝐹 = recs(𝐺) of them. 𝐹 is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of 𝐹. In this first part we show that 𝐹 is a function whose domain is all ordinal numbers. (Contributed by NM, 17-Aug-1994.) (Revised by Mario Carneiro, 18-Jan-2015.)
𝐹 = recs(𝐺)       𝐹 Fn On

Theoremtfr2 7257 Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47. Here we show that the function 𝐹 has the property that for any function 𝐺 whatsoever, the "next" value of 𝐹 is 𝐺 recursively applied to all "previous" values of 𝐹. (Contributed by NM, 9-Apr-1995.) (Revised by Stefan O'Rear, 18-Jan-2015.)
𝐹 = recs(𝐺)       (𝐴 ∈ On → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))

Theoremtfr3 7258* Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47. Finally, we show that 𝐹 is unique. We do this by showing that any class 𝐵 with the same properties of 𝐹 that we showed in parts 1 and 2 is identical to 𝐹. (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs(𝐺)       ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)

Theoremtfr1ALT 7259 Alternate proof of tfr1 7256 using well-founded recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = recs(𝐺)       𝐹 Fn On

Theoremtfr2ALT 7260 Alternate proof of tfr2 7257 using well-founded recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = recs(𝐺)       (𝐴 ∈ On → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))

Theoremtfr3ALT 7261* Alternate proof of tfr3 7258 using well-founded recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = recs(𝐺)       ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)

Theoremrecsfnon 7262 Strong transfinite recursion defines a function on ordinals. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs(𝐹) Fn On

Theoremrecsval 7263 Strong transfinite recursion in terms of all previous values. (Contributed by Stefan O'Rear, 18-Jan-2015.)
(𝐴 ∈ On → (recs(𝐹)‘𝐴) = (𝐹‘(recs(𝐹) ↾ 𝐴)))

Theoremtz7.44lem1 7264* 𝐺 is a function. Lemma for tz7.44-1 7265, tz7.44-2 7266, and tz7.44-3 7267. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
𝐺 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥 dom 𝑥))) ∨ (Lim dom 𝑥𝑦 = ran 𝑥))}       Fun 𝐺

Theoremtz7.44-1 7265* The value of 𝐹 at . Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐻‘(𝑥 dom 𝑥)))))    &   (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))    &   𝐴 ∈ V       (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴)

Theoremtz7.44-2 7266* The value of 𝐹 at a successor ordinal. Part 2 of Theorem 7.44 of [TakeutiZaring] p. 49. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐻‘(𝑥 dom 𝑥)))))    &   (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))    &   (𝑦𝑋 → (𝐹𝑦) ∈ V)    &   𝐹 Fn 𝑋    &   Ord 𝑋       (suc 𝐵𝑋 → (𝐹‘suc 𝐵) = (𝐻‘(𝐹𝐵)))

Theoremtz7.44-3 7267* The value of 𝐹 at a limit ordinal. Part 3 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐻‘(𝑥 dom 𝑥)))))    &   (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))    &   (𝑦𝑋 → (𝐹𝑦) ∈ V)    &   𝐹 Fn 𝑋    &   Ord 𝑋       ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) = (𝐹𝐵))

2.4.16  Recursive definition generator

Syntaxcrdg 7268 Extend class notation with the recursive definition generator, with characteristic function 𝐹 and initial value 𝐼.
class rec(𝐹, 𝐼)

Definitiondf-rdg 7269* Define a recursive definition generator on On (the class of ordinal numbers) with characteristic function 𝐹 and initial value 𝐼. This combines functions 𝐹 in tfr1 7256 and 𝐺 in tz7.44-1 7265 into one definition. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our rec operation (especially when df-recs 7231 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple, as in for example oav 7354, from which we prove the recursive textbook definition as theorems oa0 7359, oasuc 7367, and oalim 7375 (with the help of theorems rdg0 7280, rdgsuc 7283, and rdglim2a 7292). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 7294 and frsuc 7295. Our rec operation apparently does not appear in published literature, although closely related is Definition 25.2 of [Quine] p. 177, which he uses to "turn...a recursion into a genuine or direct definition" (p. 174). Note that the if operations (see df-if 3940) select cases based on whether the domain of 𝑔 is zero, a successor, or a limit ordinal.

An important use of this definition is in the recursive sequence generator df-seq 12532 on the natural numbers (as a subset of the complex numbers), allowing us to define, with direct definitions, recursive infinite sequences such as the factorial function df-fac 12791 and integer powers df-exp 12591.

Note: We introduce rec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))

Theoremrdgeq1 7270 Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
(𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))

Theoremrdgeq2 7271 Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
(𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))

Theoremrdgeq12 7272 Equality theorem for the recursive definition generator. (Contributed by Scott Fenton, 28-Apr-2012.)
((𝐹 = 𝐺𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵))

Theoremnfrdg 7273 Bound-variable hypothesis builder for the recursive definition generator. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
𝑥𝐹    &   𝑥𝐴       𝑥rec(𝐹, 𝐴)

Theoremrdglem1 7274* Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use. (Contributed by NM, 9-Apr-1995.)
{𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔𝑤)))}

Theoremrdgfun 7275 The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
Fun rec(𝐹, 𝐴)

Theoremrdgdmlim 7276 The domain of the recursive definition generator is a limit ordinal. (Contributed by NM, 16-Nov-2014.)
Lim dom rec(𝐹, 𝐴)

Theoremrdgfnon 7277 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 7278* 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 7279* 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 7280 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 7281 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 7282 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 7283 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 7284 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 7285 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 7286 The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
(𝐴𝐶 → (rec(𝐹, 𝐴)‘∅) = 𝐴)

Theoremrdgsucmptf 7287 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 7288 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 7287 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 7289* This version of rdgsucmpt 7290 avoids the not-free hypothesis of rdgsucmptf 7287 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 7290* 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 7291* 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 7292* 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 7293 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 7294 The initial value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.)
(𝐴𝐵 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴)

Theoremfrsuc 7295 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 7296 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 7297 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 7296 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 7298* 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 7299* 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 7300* 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)

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 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
 Copyright terms: Public domain < Previous  Next >