![]() |
Metamath
Proof Explorer Theorem List (p. 85 of 485) | < 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: | ![]() (1-30800) |
![]() (30801-32323) |
![]() (32324-48424) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | tfrlem5 8401* | 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} ⇒ ⊢ ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) | ||
Theorem | recsfval 8402* | 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} ⇒ ⊢ recs(𝐹) = ∪ 𝐴 | ||
Theorem | tfrlem6 8403* | 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} ⇒ ⊢ Rel recs(𝐹) | ||
Theorem | tfrlem7 8404* | 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} ⇒ ⊢ Fun recs(𝐹) | ||
Theorem | tfrlem8 8405* | 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} ⇒ ⊢ Ord dom recs(𝐹) | ||
Theorem | tfrlem9 8406* | 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} ⇒ ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))) | ||
Theorem | tfrlem9a 8407* | Lemma for transfinite recursion. Without using ax-rep 5286, show that all the restrictions of recs are sets. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} ⇒ ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V) | ||
Theorem | tfrlem10 8408* | 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 8412, 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} & ⊢ 𝐶 = (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⇒ ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹)) | ||
Theorem | tfrlem11 8409* | 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} & ⊢ 𝐶 = (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⇒ ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ suc dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)))) | ||
Theorem | tfrlem12 8410* | 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} & ⊢ 𝐶 = (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⇒ ⊢ (recs(𝐹) ∈ V → 𝐶 ∈ 𝐴) | ||
Theorem | tfrlem13 8411* | 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} ⇒ ⊢ ¬ recs(𝐹) ∈ V | ||
Theorem | tfrlem14 8412* | Lemma for transfinite recursion. Assuming ax-rep 5286, 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} ⇒ ⊢ dom recs(𝐹) = On | ||
Theorem | tfrlem15 8413* | Lemma for transfinite recursion. Without assuming ax-rep 5286, 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} ⇒ ⊢ (𝐵 ∈ On → (𝐵 ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ 𝐵) ∈ V)) | ||
Theorem | tfrlem16 8414* | Lemma for finite recursion. Without assuming ax-rep 5286, 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} ⇒ ⊢ Lim dom recs(𝐹) | ||
Theorem | tfr1a 8415 | A weak version of tfr1 8418 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) | ||
Theorem | tfr2a 8416 | A weak version of tfr2 8419 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) ⇒ ⊢ (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) | ||
Theorem | tfr2b 8417 | Without assuming ax-rep 5286, 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 ↔ (𝐹 ↾ 𝐴) ∈ V)) | ||
Theorem | tfr1 8418 | 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 | ||
Theorem | tfr2 8419 | 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 → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) | ||
Theorem | tfr3 8420* | 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 (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹) | ||
Theorem | tfr1ALT 8421 | Alternate proof of tfr1 8418 using well-ordered recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐹 = recs(𝐺) ⇒ ⊢ 𝐹 Fn On | ||
Theorem | tfr2ALT 8422 | Alternate proof of tfr2 8419 using well-ordered recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐹 = recs(𝐺) ⇒ ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) | ||
Theorem | tfr3ALT 8423* | Alternate proof of tfr3 8420 using well-ordered recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐹 = recs(𝐺) ⇒ ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹) | ||
Theorem | recsfnon 8424 | Strong transfinite recursion defines a function on ordinals. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
⊢ recs(𝐹) Fn On | ||
Theorem | recsval 8425 | 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))) | ||
Theorem | tz7.44lem1 8426* | The ordered pair abstraction 𝐺 defined in the hypothesis is a function. This was a lemma for tz7.44-1 8427, tz7.44-2 8428, and tz7.44-3 8429 when they used that definition of 𝐺. Now, they use the maps-to df-mpt 5233 idiom so this lemma is not needed anymore, but is kept in case other applications (for instance in intuitionistic set theory) need it. (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 𝑥 ∧ 𝑦 = ∪ ran 𝑥))} ⇒ ⊢ Fun 𝐺 | ||
Theorem | tz7.44-1 8427* | 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))))) & ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦))) & ⊢ 𝐴 ∈ V ⇒ ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴) | ||
Theorem | tz7.44-2 8428* | The value of 𝐹 at a successor ordinal. Part 2 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 14-Nov-2014.) |
⊢ 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥))))) & ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦))) & ⊢ (𝑦 ∈ 𝑋 → (𝐹 ↾ 𝑦) ∈ V) & ⊢ 𝐹 Fn 𝑋 & ⊢ Ord 𝑋 ⇒ ⊢ (suc 𝐵 ∈ 𝑋 → (𝐹‘suc 𝐵) = (𝐻‘(𝐹‘𝐵))) | ||
Theorem | tz7.44-3 8429* | 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))))) & ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦))) & ⊢ (𝑦 ∈ 𝑋 → (𝐹 ↾ 𝑦) ∈ V) & ⊢ 𝐹 Fn 𝑋 & ⊢ Ord 𝑋 ⇒ ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → (𝐹‘𝐵) = ∪ (𝐹 “ 𝐵)) | ||
Syntax | crdg 8430 | Extend class notation with the recursive definition generator, with characteristic function 𝐹 and initial value 𝐼. |
class rec(𝐹, 𝐼) | ||
Definition | df-rdg 8431* |
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 8418
and 𝐺 in tz7.44-1 8427 into one
definition. This rather amazing operation allows 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 8392 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 8532,
from which
we prove the recursive textbook definition as Theorems oa0 8537,
oasuc 8545,
and oalim 8553 (with the help of Theorems rdg0 8442,
rdgsuc 8445, and
rdglim2a 8454). We can also restrict the rec operation to define
otherwise recursive functions on the natural numbers ω; see
fr0g 8457 and frsuc 8458. 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 4531) 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 14003 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 14269 and integer powers df-exp 14063. 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)))))) | ||
Theorem | rdgeq1 8432 | 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)) | ||
Theorem | rdgeq2 8433 | 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)) | ||
Theorem | rdgeq12 8434 | Equality theorem for the recursive definition generator. (Contributed by Scott Fenton, 28-Apr-2012.) |
⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵)) | ||
Theorem | nfrdg 8435 | Bound-variable hypothesis builder for the recursive definition generator. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥rec(𝐹, 𝐴) | ||
Theorem | rdglem1 8436* | 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))} | ||
Theorem | rdgfun 8437 | The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ Fun rec(𝐹, 𝐴) | ||
Theorem | rdgdmlim 8438 | The domain of the recursive definition generator is a limit ordinal. (Contributed by NM, 16-Nov-2014.) |
⊢ Lim dom rec(𝐹, 𝐴) | ||
Theorem | rdgfnon 8439 | 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 | ||
Theorem | rdgvalg 8440* | 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))) | ||
Theorem | rdgval 8441* | 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))) | ||
Theorem | rdg0 8442 | 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 | ||
Theorem | rdgseg 8443 | 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) | ||
Theorem | rdgsucg 8444 | 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))) | ||
Theorem | rdgsuc 8445 | 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))) | ||
Theorem | rdglimg 8446 | 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)) | ||
Theorem | rdglim 8447 | The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ (rec(𝐹, 𝐴) “ 𝐵)) | ||
Theorem | rdg0g 8448 | The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.) |
⊢ (𝐴 ∈ 𝐶 → (rec(𝐹, 𝐴)‘∅) = 𝐴) | ||
Theorem | rdgsucmptf 8449 | 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.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝐷 & ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴) & ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) ⇒ ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) | ||
Theorem | rdgsucmptnf 8450 | 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 8449 to help eliminate redundant sethood antecedents. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝐷 & ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴) & ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) ⇒ ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅) | ||
Theorem | rdgsucmpt2 8451* | This version of rdgsucmpt 8452 avoids the not-free hypothesis of rdgsucmptf 8449 by using two substitutions instead of one. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴) & ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶) & ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷) ⇒ ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) | ||
Theorem | rdgsucmpt 8452* | 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 = (𝐹‘𝐵) → 𝐶 = 𝐷) ⇒ ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) | ||
Theorem | rdglim2 8453* | 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.) |
⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)}) | ||
Theorem | rdglim2a 8454* | 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.) |
⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑥)) | ||
Theorem | rdg0n 8455 | If 𝐴 is a proper class, then the recursive function generator at ∅ is the empty set. (Contributed by Scott Fenton, 31-Oct-2024.) |
⊢ (¬ 𝐴 ∈ V → (rec(𝐹, 𝐴)‘∅) = ∅) | ||
Theorem | frfnom 8456 | 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 | ||
Theorem | fr0g 8457 | The initial value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.) |
⊢ (𝐴 ∈ 𝐵 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴) | ||
Theorem | frsuc 8458 | 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))) | ||
Theorem | frsucmpt 8459 | 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.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝐷 & ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) & ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) ⇒ ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) | ||
Theorem | frsucmptn 8460 | 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 8459 to help eliminate redundant sethood antecedents. (Contributed by Scott Fenton, 19-Feb-2011.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝐷 & ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) & ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) ⇒ ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅) | ||
Theorem | frsucmpt2 8461* | 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 = 𝑥 → 𝐸 = 𝐶) & ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷) ⇒ ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) | ||
Theorem | tz7.48lem 8462* | A way of showing an ordinal function is one-to-one. (Contributed by NM, 9-Feb-1997.) |
⊢ 𝐹 Fn On ⇒ ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ 𝐴)) | ||
Theorem | tz7.48-2 8463* | 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 (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹) | ||
Theorem | tz7.48-1 8464* | Proposition 7.48(1) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) |
⊢ 𝐹 Fn On ⇒ ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴) | ||
Theorem | tz7.48-3 8465* | Proposition 7.48(3) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) |
⊢ 𝐹 Fn On ⇒ ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 𝐴 ∈ V) | ||
Theorem | tz7.49 8466* | 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 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜑) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥))) | ||
Theorem | tz7.49c 8467* | 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 ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴) | ||
Syntax | cseqom 8468 | Extend class notation to include index-aware recursive definitions. |
class seqω(𝐹, 𝐼) | ||
Definition | df-seqom 8469* | Index-aware recursive definitions over ω. A mashup of df-rdg 8431 and df-seq 14003, 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) | ||
Theorem | seqomlem0 8470* | 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)〉) | ||
Theorem | seqomlem1 8471* | 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 → (𝑄‘𝐴) = 〈𝐴, (2nd ‘(𝑄‘𝐴))〉) | ||
Theorem | seqomlem2 8472* | 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 | ||
Theorem | seqomlem3 8473* | 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) | ||
Theorem | seqomlem4 8474* | 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))) | ||
Theorem | seqomeq12 8475 | Equality theorem for seqω. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷)) | ||
Theorem | fnseqom 8476 | 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 | ||
Theorem | seqom0g 8477 | Value of an index-aware recursive definition at 0. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by AV, 17-Sep-2021.) |
⊢ 𝐺 = seqω(𝐹, 𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → (𝐺‘∅) = 𝐼) | ||
Theorem | seqomsuc 8478 | 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(𝐺‘𝐴))) | ||
Theorem | omsucelsucb 8479 | Membership is inherited by successors for natural numbers. (Contributed by AV, 15-Sep-2023.) |
⊢ (𝑁 ∈ ω ↔ suc 𝑁 ∈ suc ω) | ||
Syntax | c1o 8480 | Extend the definition of a class to include the ordinal number 1. |
class 1o | ||
Syntax | c2o 8481 | Extend the definition of a class to include the ordinal number 2. |
class 2o | ||
Syntax | c3o 8482 | Extend the definition of a class to include the ordinal number 3. |
class 3o | ||
Syntax | c4o 8483 | Extend the definition of a class to include the ordinal number 4. |
class 4o | ||
Syntax | coa 8484 | Extend the definition of a class to include the ordinal addition operation. |
class +o | ||
Syntax | comu 8485 | Extend the definition of a class to include the ordinal multiplication operation. |
class ·o | ||
Syntax | coe 8486 | Extend the definition of a class to include the ordinal exponentiation operation. |
class ↑o | ||
Definition | df-1o 8487 | Define the ordinal number 1. Definition 2.1 of [Schloeder] p. 4. (Contributed by NM, 29-Oct-1995.) |
⊢ 1o = suc ∅ | ||
Definition | df-2o 8488 | Define the ordinal number 2. Lemma 3.17 of [Schloeder] p. 10. (Contributed by NM, 18-Feb-2004.) |
⊢ 2o = suc 1o | ||
Definition | df-3o 8489 | Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ 3o = suc 2o | ||
Definition | df-4o 8490 | Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ 4o = suc 3o | ||
Definition | df-oadd 8491* | Define the ordinal addition operation. (Contributed by NM, 3-May-1995.) |
⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦)) | ||
Definition | df-omul 8492* | Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.) |
⊢ ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦)) | ||
Definition | df-oexp 8493* | Define the ordinal exponentiation operation. (Contributed by NM, 30-Dec-2004.) |
⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))) | ||
Theorem | df1o2 8494 | Expanded value of the ordinal number 1. Definition 2.1 of [Schloeder] p. 4. (Contributed by NM, 4-Nov-2002.) |
⊢ 1o = {∅} | ||
Theorem | df2o3 8495 | Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ 2o = {∅, 1o} | ||
Theorem | df2o2 8496 | Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.) |
⊢ 2o = {∅, {∅}} | ||
Theorem | 1oex 8497 | Ordinal 1 is a set. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by AV, 1-Jul-2022.) Remove dependency on ax-10 2129, ax-11 2146, ax-12 2166, ax-un 7741. (Revised by Zhi Wang, 19-Sep-2024.) |
⊢ 1o ∈ V | ||
Theorem | 2oex 8498 | 2o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2129, ax-11 2146, ax-12 2166, ax-un 7741. (Proof shortened by Zhi Wang, 19-Sep-2024.) |
⊢ 2o ∈ V | ||
Theorem | 1on 8499 | Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.) Avoid ax-un 7741. (Revised by BTernaryTau, 30-Nov-2024.) |
⊢ 1o ∈ On | ||
Theorem | 1onOLD 8500 | Obsolete version of 1on 8499 as of 30-Nov-2024. (Contributed by NM, 29-Oct-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 1o ∈ On |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |