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Definition df-rdg 7772
 Description: 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 7759 and 𝐺 in tz7.44-1 7768 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 7734 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 7858, from which we prove the recursive textbook definition as theorems oa0 7863, oasuc 7871, and oalim 7879 (with the help of theorems rdg0 7783, rdgsuc 7786, and rdglim2a 7795). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 7797 and frsuc 7798. 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 4307) 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 13096 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 13354 and integer powers df-exp 13155. 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.)
Assertion
Ref Expression
df-rdg rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
Distinct variable groups:   𝑔,𝐹   𝑔,𝐼

Detailed syntax breakdown of Definition df-rdg
StepHypRef Expression
1 cF . . 3 class 𝐹
2 cI . . 3 class 𝐼
31, 2crdg 7771 . 2 class rec(𝐹, 𝐼)
4 vg . . . 4 setvar 𝑔
5 cvv 3414 . . . 4 class V
64cv 1655 . . . . . 6 class 𝑔
7 c0 4144 . . . . . 6 class
86, 7wceq 1656 . . . . 5 wff 𝑔 = ∅
96cdm 5342 . . . . . . 7 class dom 𝑔
109wlim 5964 . . . . . 6 wff Lim dom 𝑔
116crn 5343 . . . . . . 7 class ran 𝑔
1211cuni 4658 . . . . . 6 class ran 𝑔
139cuni 4658 . . . . . . . 8 class dom 𝑔
1413, 6cfv 6123 . . . . . . 7 class (𝑔 dom 𝑔)
1514, 1cfv 6123 . . . . . 6 class (𝐹‘(𝑔 dom 𝑔))
1610, 12, 15cif 4306 . . . . 5 class if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
178, 2, 16cif 4306 . . . 4 class if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
184, 5, 17cmpt 4952 . . 3 class (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1918crecs 7733 . 2 class recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
203, 19wceq 1656 1 wff rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
 Colors of variables: wff setvar class This definition is referenced by:  rdgeq1  7773  rdgeq2  7774  nfrdg  7776  rdgfun  7778  rdgdmlim  7779  rdgfnon  7780  rdgvalg  7781  rdgval  7782  rdgseg  7784  dfrdg2  32228  csbrdgg  33714
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