mmtheorems43 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4201-4300 - Page 43 of 123

Type	Label	Description
Statement
Theorem	fparlem3 4201	Lemma for fpar 4203.
Theorem	fparlem4 4202	Lemma for fpar 4203.
Theorem	fpar 4203	Merge two functions in parallel. Use as the second argument of a composition with a (2-place) operation to build compound operations such as z = ((sqr` x) + (abs` y)).
|- H = ((`'(1st |` (V X. V)) o. (F o. (1st |` (V X. V)))) i^i (`'(2nd |` (V X. V)) o. (G o. (2nd |` (V X. V)))))   =>   |- ((F Fn A /\ G Fn B) -> H = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = <.(F` x), (G` y)>.)})
Theorem	fsplit 4204	A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 4203 in order to build compound functions such as y = ((sqr` x) + (abs` x)).
|- `'(1st |` I) = {<.x, y>. | y = <.x, x>.}
Cantor's Theorem
Theorem	canth 4205	No set A is equinumerous to its power set (Cantor's theorem), i.e. no function can map A it onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 4629. Note that A must be a set: this theorem does not hold when A is too large to be a set; see ncanth 4206 for a counterexample. (Use nex 1137 if you want the form -. E.ff:A-onto->P~A.)
|- A e. V   =>   |- -. F:A-onto->P~A
Theorem	ncanth 4206	Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 2787). Specifically, the identity function maps the universe onto its power class. Compare canth 4205 that works for sets. See also the remark in ru 1984 about NF, in which Cantor's theorem fails for sets that are "too large." This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set.
|- I:V-onto->P~V
Miscellaneous ordinal theorems (that depend on functions and relations)
Theorem	iunon 4207	The indexed union of a set of ordinal numbers B(x) is an ordinal number.
|- A e. V   &   |- B e. V   =>   |- (A.x e. A B e. On -> U_x e. A B e. On)
Theorem	iinon 4208	The nonempty indexed intersection of a class of ordinal numbers B(x) is an ordinal number.
|- B e. V   =>   |- ((A.x e. A B e. On /\ A =/= (/)) -> |^|_x e. A B e. On)
Theorem	onfununi 4209	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 -> (F` y) = U_x e. y (F` x))   &   |- ((x e. On /\ y e. On /\ x (_ y) -> (F` x) (_ (F` y))   =>   |- ((S e. T /\ S (_ On /\ S =/= (/)) -> (F` U.S) = U_x e. S (F` x))
Theorem	onopruni 4210	A variant of onfununi 4209 for operations. (Contributed by Eric Schmidt, 26-May-2009.)
|- (Lim y -> (AFy) = U_x e. y (AFx))   &   |- ((x e. On /\ y e. On /\ x (_ y) -> (AFx) (_ (AFy))   =>   |- ((S e. T /\ S (_ On /\ S =/= (/)) -> (AFU.S) = U_x e. S (AFx))
Theorem	onopriun 4211	A variant of onopruni 4210 with indexed unions. (Contributed by Eric Schmidt, 26-May-2009.)
|- (Lim y -> (AFy) = U_x e. y (AFx))   &   |- ((x e. On /\ y e. On /\ x (_ y) -> (AFx) (_ (AFy))   =>   |- ((K e. T /\ A.z e. K L e. On /\ K =/= (/)) -> (AFU_z e. K L) = U_z e. K (AFL))
Transfinite recursion
Theorem	tfrlem1 4212	A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47.
|- (A e. On -> ((F Fn A /\ G Fn A) -> (A.x e. A ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. A (F` x) = (G` x))))
Theorem	tfrlem2 4213	Lemma for transfinite recursion. This provides some messy details needed to link tfrlem1 4212 into the main proof.
Theorem	tfrlem3 4214	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.
Theorem	tfrlem4 4215	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.
Theorem	tfrlem5 4216	Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains.
Theorem	tfrlem6 4217	Lemma for transfinite recursion. The union of all acceptable functions is a relation.
Theorem	tfrlem7 4218	Lemma for transfinite recursion. The union of all acceptable functions is a function.
Theorem	tfrlem8 4219	Lemma for transfinite recursion. The domain of F is ordinal. (The proof was shortened by Alan Sare, 11-Mar-2008.)
Theorem	tfrlem9 4220	Lemma for transfinite recursion. Here we compute the value of F (the union of all acceptable functions).
Theorem	tfrlem10 4221	Lemma for transfinite recursion. We define class C by extending F with one ordered pair. We will assume, falsely, that domain of F 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 tfrlem13 4224, thus showing the domain of F does in fact equal On. Here we show (under the false assumption) that C is a function extending the domain of F by one. (The proof was shortened by Alan Sare, 20-Feb-2008.)
Theorem	tfrlem11 4222	Lemma for transfinite recursion. Compute the value of C.
Theorem	tfrlem12 4223	Lemma for transfinite recursion. Show C is an acceptable function.
Theorem	tfrlem13 4224	Lemma for transfinite recursion. If dom F is in On, then C is acceptable, and thus a subset of F, but dom C is bigger than dom F. This is a contradiction, so dom F must be On.
Theorem	tfr1 4225	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 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.
|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}   &   |- F = U.A   =>   |- F Fn On
Theorem	tfr2 4226	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.
|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}   &   |- F = U.A   =>   |- (z e. On -> (F` z) = (G` (F |` z)))
Theorem	tfr3 4227	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.
|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}   &   |- F = U.A   =>   |- ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> B = F)
Theorem	tz7.44lem1 4228	G is a function. Lemma for tz7.44-1 4229, tz7.44-2 4230, and tz7.44-3 4231.
|- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}   =>   |- Fun G
Theorem	tz7.44-1 4229	The value of F at (/). Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49.
|- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}   &   |- F Fn On   &   |- (x e. On -> (F` x) = (G` (F |` x)))   &   |- A e. V   =>   |- (F` (/)) = A
Theorem	tz7.44-2 4230	The value of F at a successor ordinal. Part 2 of Theorem 7.44 of [TakeutiZaring] p. 49.
|- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}   &   |- F Fn On   &   |- (x e. On -> (F` x) = (G` (F |` x)))   &   |- B e. On   =>   |- (F` suc B) = (H` (F` B))
Theorem	tz7.44-3 4231	The value of F at a limit ordinal. Part 3 of Theorem 7.44 of [TakeutiZaring] p. 49.
|- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}   &   |- F Fn On   &   |- (x e. On -> (F` x) = (G` (F |` x)))   &   |- B e. On   =>   |- (Lim B -> (F` B) = U.(F"B))
Recursive definition generator
Syntax	crdg 4232	Extend class notation with the recursive definition generator.
class rec(A, B)
Definition	df-rdg 4233	Define a recursive definition generator on On (the class of ordinal numbers) with characteristic function F and initial value A. This combines functions F in tfr1 4225 and G in tz7.44-1 4229 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. But once we get past this hurdle, otherwise recursive definitions become relatively simple, as in for example oav 4286, from which we prove the recursive textbook definition as theorems oa0 4291, oasuc 4299, and oalim 4303 (with the help of theorems rdg0 4242, rdgsuc 4246, and rdglimi 4244). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers om; see fr0g 4253 and frsuc 4254. 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 2416) 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-seq1 6673 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 7135 and integer powers df-exp 6764. 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 recursive definitions in the traditional textbook style.
|- rec(F, A) = U.{f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y)))}
Theorem	dfrdg2 4234	Alternate definition of a recursive definition generator. (This was the original definition, but it was later replaced with the slightly shorter df-rdg 4233.)
|- rec(F, A) = U.{f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (F` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (f |` y)))}
Theorem	rdgeq1 4235	Equality theorem for the recursive definition generator.
|- (F = G -> rec(F, A) = rec(G, A))
Theorem	rdgeq2 4236	Equality theorem for the recursive definition generator.
|- (A = B -> rec(F, A) = rec(F, B))
Theorem	hbrdg 4237	Bound-variable hypothesis builder for the recursive definition generator.
|- (y e. F -> A.x y e. F)   &   |- (y e. A -> A.x y e. A)   =>   |- (y e. rec(F, A) -> A.x y e. rec(F, A))
Theorem	rdglem1 4238	Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use.
|- {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))} = {g | E.z e. On (g Fn z /\ A.w e. z (g` w) = (G` (g |` w)))}
Theorem	rdglem2 4239	Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use.
|- {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))} = {<.z, y>. | ((z = (/) /\ y = A) \/ (-. (z = (/) \/ Lim dom z) /\ y = (H` (z` U.dom z))) \/ (Lim dom z /\ y = U.ran z))}
Theorem	rdgfnon 4240	The recursive definition generator is a function on ordinal numbers.
|- rec(F, A) Fn On
Theorem	rdgval 4241	Value of the recursive definition generator.
|- (g e. On -> (rec(F, A)` g) = ({<.w, z>. | ((w = (/) /\ z = A) \/ (-. (w = (/) \/ Lim dom w) /\ z = (F` (w` U.dom w))) \/ (Lim dom w /\ z = U.ran w))}` (rec(F, A) |` g)))
Theorem	rdg0 4242	The initial value of the recursive definition generator.
|- A e. V   =>   |- (rec(F, A)` (/)) = A
Theorem	rdgsuci 4243	The value of the recursive definition generator at a successor.
|- B e. On   =>   |- (rec(F, A)` suc B) = (F` (rec(F, A)` B))
Theorem	rdglimi 4244	The value of the recursive definition generator at a limit ordinal.
|- B e. On   =>   |- (Lim B -> (rec(F, A)` B) = U.(rec(F, A)"B))
Theorem	rdg0g 4245	The initial value of the recursive definition generator.
|- (A e. C -> (rec(F, A)` (/)) = A)
Theorem	rdgsuc 4246	The value of the recursive definition generator at a successor.
|- (B e. On -> (rec(F, A)` suc B) = (F` (rec(F, A)` B)))
Theorem	rdgsucopab 4247	The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered pair abstraction).
|- (z e. A -> A.x z e. A)   &   |- (z e. B -> A.x z e. B)   &   |- (z e. D -> A.x z e. D)   &   |- F = rec({<.x, y>. | y = C}, A)   &   |- (x = (F` B) -> C = D)   =>   |- ((B e. On /\ D e. R) -> (F` suc B) = D)
Theorem	rdgsucopabn 4248	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 rdgsucopab 4247 to help eliminate redundant sethood antecedents.
|- (z e. A -> A.x z e. A)   &   |- (z e. B -> A.x z e. B)   &   |- (z e. D -> A.x z e. D)   &   |- F = rec({<.x, y>. | y = C}, A)   &   |- (x = (F` B) -> C = D)   =>   |- (-. D e. V -> (F` suc B) = (/))
Theorem	rdglim 4249	The value of the recursive definition generator at a limit ordinal.
|- ((B e. C /\ Lim B) -> (rec(F, A)` B) = U.(rec(F, A)"B))
Theorem	rdglim2 4250	The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values.
|- ((B e. C /\ Lim B) -> (rec(F, A)` B) = U.{y | E.x e. B y = (rec(F, A)` x)})
Theorem	rdglim2a 4251	The value of the recursive definition generator at a limit ordinal, in terms of indexed union of all smaller values.
|- ((B e. C /\ Lim B) -> (rec(F, A)` B) = U_x e. B (rec(F, A)` x))
Finite recursion
Theorem	frfnom 4252	The function generated by finite recursive definition generation is a function on omega.
|- (rec(F, A) |` om) Fn om
Theorem	fr0g 4253	The initial value resulting from finite recursive definition generation.
|- (A e. B -> ((rec(F, A) |` om)` (/)) = A)
Theorem	frsuc 4254	The successor value resulting from finite recursive definition generation.
|- (B e. om -> ((rec(F, A) |` om)` suc B) = (F` ((rec(F, A) |` om)` B)))
Theorem	frsucopab 4255	The successor value resulting from finite recursive definition generation (special case where the generation function is an ordered pair abstraction).
|- (z e. A -> A.x z e. A)   &   |- (z e. B -> A.x z e. B)   &   |- (z e. D -> A.x z e. D)   &   |- F = (rec({<.x, y>. | y = C}, A) |` om)   &   |- (x = (F` B) -> C = D)   =>   |- ((B e. om /\ D e. R) -> (F` suc B) = D)
Theorem	tz7.48lem 4256	A way of showing an ordinal function is one-to-one.
|- F Fn On   =>   |- ((A (_ On /\ A.x e. A A.y e. x -. (F` x) = (F` y)) -> Fun `'(F |` A))
Theorem	tz7.48-1 4257	Proposition 7.48(1) of [TakeutiZaring] p. 51.
|- F Fn On   =>   |- (A.x e. On (F` x) e. (A \ (F"x)) -> ran F (_ A)
Theorem	tz7.48-2 4258	Proposition 7.48(2) of [TakeutiZaring] p. 51.
|- F Fn On   =>   |- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'F)
Theorem	tz7.48-3 4259	Proposition 7.48(3) of [TakeutiZaring] p. 51.
|- F Fn On   =>   |- (A.x e. On (F` x) e. (A \ (F"x)) -> -. A e. V)
Theorem	tz7.49 4260	Proposition 7.49 of [TakeutiZaring] p. 51.
|- F Fn On   &   |- A e. V   =>   |- (A.x e. On ((A \ (F"x)) =/= (/) -> (F` x) e. (A \ (F"x))) -> E.x e. On (A.y e. x (A \ (F"y)) =/= (/) /\ (F"x) = A /\ Fun `'(F |` x)))
Theorem	tz7.49c 4261	Corollary of Proposition 7.49 of [TakeutiZaring] p. 51.
|- F Fn On   &   |- A e. V   =>   |- (A.x e. On ((A \ (F"x)) =/= (/) -> (F` x) e. (A \ (F"x))) -> E.x e. On (F |` x):x-1-1-onto->A)
Abian's "most fundamental" fixed point theorem
Theorem	abianfplem 4262	Lemma for abianfp 4263. We prove by transfinite induction that if F has a fixed point x, then its iterates also equal x. This lemma is used for the "trivial" direction of the main theorem.
Theorem	abianfp 4263	"A most fundamental fixed point theorem" of Alexander Abian (1923-1999), apparently proved in 1998. Let G` 0 = x, G` 1 = F` x, G` 2 = F` (F` x),... be the iterates of F. The theorem reads (using our variable names): "Let F be a mapping from a set A into itself. Then F has a fixed point if and only if: There exists an element x of A such that for every ordinal v , G` v is an element of A, and if G` v is not a fixed point of F then the G` u 's are all distinct for every ordinal u e. v." See df-rdg 4233 for the rec operation. The proof's key idea is to assume that F does not have a fixed point, then use the Axiom of Replacement in the form of f1dmex 3821 to derive that the class of all ordinal numbers exists, contradicting onprc 3143. Our version of this theorem does not require the hypothesis that F be a mapping. Reference: http://us2.metamath.org:88/abian-themostfixed.html. For an application of this theorem, see http://groups.google.com/group/sci.stat.math/msg/1737ee1133c24aeb for its use in a proof of Tarski's fixed point theorem.
|- A e. V   &   |- G = rec({<.z, w>. | w = (F` z)}, x)   =>   |- (E.x e. A (F` x) = x <-> E.x e. A A.v e. On ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))))
Ordinal arithmetic
Syntax	c1o 4264	Extend the definition of a class to include the ordinal number 1.
class 1o
Syntax	c2o 4265	Extend the definition of a class to include the ordinal number 2.
class 2o
Syntax	coa 4266	Extend the definition of a class to include the ordinal addition operation.
class +o
Syntax	comu 4267	Extend the definition of a class to include the ordinal multiplication operation.
class .o
Syntax	coe 4268	Extend the definition of a class to include the ordinal exponentiation operation.
class ^o
Definition	df-1o 4269	Define the ordinal number 1.
|- 1o = suc (/)
Definition	df-2o 4270	Define the ordinal number 2.
|- 2o = suc 1o
Definition	df-oadd 4271	Define the ordinal addition operation.
|- +o = {<.<.x, y>., z>. | ((x e. On /\ y e. On) /\ z = (rec({<.w, v>. | v = suc w}, x)` y))}
Definition	df-omul 4272	Define the ordinal multiplication operation.
|- .o = {<.<.x, y>., z>. | ((x e. On /\ y e. On) /\ z = (rec({<.w, v>. | v = (w +o x)}, (/))` y))}
Definition	df-oexp 4273	Define the ordinal exponentiation operation.
|- ^o = {<.<.x, y>., z>. | ((x e. On /\ y e. On) /\ z = if(x = (/), (1o \ y), (rec({<.w, v>. | v = (w .o x)}, 1o)` y)))}
Theorem	1on 4274	Ordinal 1 is an ordinal number.
|- 1o e. On
Theorem	2on 4275	Ordinal 2 is an ordinal number.
|- 2o e. On
Theorem	df1o2 4276	Expanded value of the ordinal number 1.
|- 1o = {(/)}
Theorem	df2o2 4277	Expanded value of the ordinal number 2.
|- 2o = {(/), {(/)}}
Theorem	1n0 4278	Ordinal one is not equal to ordinal zero.
|- 1o =/= (/)
Theorem	xp01disj 4279	Cross products with the singletons of ordinals 0 and 1 are disjoint.
|- ((A X. {(/)}) i^i (C X. {1o})) = (/)
Theorem	ordgt0ge1 4280	Two ways to express that an ordinal class is positive.
|- (Ord A -> ((/) e. A <-> 1o (_ A))
Theorem	ordge1n0 4281	An ordinal greater than or equal to 1 is nonzero.
|- (Ord A -> (1o (_ A <-> A =/= (/)))
Theorem	el1o 4282	Membership in ordinal one.
|- (A e. 1o <-> A = (/))
Theorem	0lt1o 4283	Ordinal zero is less than ordinal one.
|- (/) e. 1o
Theorem	fnoa 4284	Functionality and domain of ordinal addition.
|- +o Fn (On X. On)
Theorem	fnom 4285	Functionality and domain of ordinal multiplication.
|- .o Fn (On X. On)
Theorem	oav 4286	Value of ordinal addition.
|- ((A e. On /\ B e. On) -> (A +o B) = (rec({<.x, y>. | y = suc x}, A)` B))
Theorem	omv 4287	Value of ordinal multiplication.
|- ((A e. On /\ B e. On) -> (A .o B) = (rec({<.x, y>. | y = (x +o A)}, (/))` B))
Theorem	oe0lem 4288	A helper lemma for oe0 4297 and others.
|- ((ph /\ A = (/)) -> ps)   &   |- (((A e. On /\ ph) /\ (/) e. A) -> ps)   =>   |- ((A e. On /\ ph) -> ps)
Theorem	oev 4289	Value of ordinal exponentiation.
|- ((A e. On /\ B e. On) -> (A ^o B) = if(A = (/), (1o \ B), (rec({<.x, y>. | y = (x .o A)}, 1o)` B)))
Theorem	oevn0 4290	Value of ordinal exponentiation at a nonzero mantissa.
|- (((A e. On /\ B e. On) /\ (/) e. A) -> (A ^o B) = (rec({<.x, y>. | y = (x .o A)}, 1o)` B))
Theorem	oa0 4291	Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57.
|- (A e. On -> (A +o (/)) = A)
Theorem	om0 4292	Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62.
|- (A e. On -> (A .o (/)) = (/))
Theorem	oe0m 4293	Ordinal exponentiation with zero mantissa.
|- (A e. On -> ((/) ^o A) = (1o \ A))
Theorem	om0x 4294	Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 4292, this version works whether or not A is an ordinal. However, since it is an artifact of our particular function value definition outside the domain, we will not use it in order to be conventional and present it only as a curiosity.
|- (A .o (/)) = (/)
Theorem	oe0m0 4295	Ordinal exponentiation with zero mantissa and zero exponent. Proposition 8.31 of [TakeutiZaring] p. 67.
|- ((/) ^o (/)) = 1o
Theorem	oe0m1 4296	Ordinal exponentiation with zero mantissa and nonzero exponent. Proposition 8.31(2) of [TakeutiZaring] p. 67 and its converse.
|- (A e. On -> ((/) e. A <-> ((/) ^o A) = (/)))
Theorem	oe0 4297	Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67.
|- (A e. On -> (A ^o (/)) = 1o)
Theorem	oev2 4298	Alternate value of ordinal exponentiation. Compare oev 4289.
|- ((A e. On /\ B e. On) -> (A ^o B) = ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i ((V \ |^|A) u. |^|B)))
Theorem	oasuc 4299	Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56.
|- ((A e. On /\ B e. On) -> (A +o suc B) = suc (A +o B))
Theorem	oa1suc 4300	Addition with 1 is same as successor. Proposition 4.34(a) of [Mendelson] p. 266.
|- (A e. On -> (A +o 1o) = suc A)