mmtheorems42 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4101-4200 - Page 42 of 107

Type	Label	Description
Statement
Theorem	2ndrn 4101	The second ordered pair component of a member of a relation belongs to the range of the relation.
|- ((Rel R /\ A e. R) -> (2nd` A) e. ran R)
Theorem	sbcopeq1a 4102	Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 1940, that avoids the existential quantifiers of copsexg 2787).
|- (<.x, y>. = A -> (ph <-> [(1st` A) / x][(2nd` A) / y]ph))
Theorem	csbopeq1a 4103	Equality theorem for substitution of a class A for an ordered pair <.x, y>. in B (analog of csbeq1a 2002).
|- (<.x, y>. = A -> B = [_(1st` A) / x]_[_(2nd` A) / y]_B)
Theorem	dfopab2 4104	A way to define an ordered-pair class abstraction without using existential quantifiers.
|- {<.x, y>. | ph} = {z | (z e. (V X. V) /\ [(1st` z) / x][(2nd` z) / y]ph)}
Theorem	dfoprab3 4105	A way to define an operation class abstraction without using existential quantifiers.
|- {<.<.x, y>., z>. | ph} = {<.w, z>. | (w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph)}
Theorem	dfoprab5 4106	A way to define an operation class abstraction without using existential quantifiers.
|- (<.x, y>. = w -> C = D)   =>   |- {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} = {<.w, z>. | (w e. (A X. B) /\ z = D)}
Theorem	dfoprab4 4107	A way to define an operation class abstraction without using existential quantifiers.
|- ((x = (1st` w) /\ y = (2nd` w)) -> C = D)   =>   |- {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} = {<.w, z>. | (w e. (A X. B) /\ z = D)}
Theorem	elopabi 4108	A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors.
|- (x = (1st` A) -> (ph <-> ps))   &   |- (y = (2nd` A) -> (ps <-> ch))   =>   |- (A e. {<.x, y>. | ph} -> ch)
Theorem	eloprabi 4109	A consequence of membership in an operation class abstraction, using ordered pair extractors.
|- (x = (1st` (1st` A)) -> (ph <-> ps))   &   |- (y = (2nd` (1st` A)) -> (ps <-> ch))   &   |- (z = (2nd` A) -> (ch <-> th))   =>   |- (A e. {<.<.x, y>., z>. | ph} -> th)
Theorem	foprab2 4110	Mapping of an operation class abstraction.
|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- (A.x e. A A.y e. B C e. D <-> F:(A X. B)-->D)
Theorem	foprab 4111	Mapping of an operation class abstraction.
|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   &   |- ((x e. A /\ y e. B) -> C e. D)   =>   |- F:(A X. B)-->D
Theorem	fnoprab2g 4112	Functionality and domain of an operation class abstraction.
|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- (A.x e. A A.y e. B C e. V <-> F Fn (A X. B))
Theorem	fnoprab2 4113	Functionality and domain of an operation class abstraction.
|- C e. V   &   |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- F Fn (A X. B)
Theorem	dmoprab2 4114	Domain of an operation class abstraction.
|- C e. V   &   |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- dom F = (A X. B)
Theorem	elrnoprabg 4115	Membership in the range of an operation class abstraction.
|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- (A.x e. A A.y e. B C e. R -> (D e. ran F <-> E.x e. A E.y e. B D = C))
Theorem	elrnoprab 4116	Membership in the range of an operation class abstraction.
|- C e. V   &   |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- (D e. ran F <-> E.x e. A E.y e. B D = C)
Theorem	df1st2 4117	An alternate possible definition of the 1st function.
|- {<.<.x, y>., z>. | z = x} = (1st |` (V X. V))
Theorem	df2nd2 4118	An alternate possible definition of the 2nd function.
|- {<.<.x, y>., z>. | z = y} = (2nd |` (V X. V))
Ordinal arithmetic
Syntax	c1o 4119	Extend the definition of a class to include the ordinal number 1.
class 1o
Syntax	c2o 4120	Extend the definition of a class to include the ordinal number 2.
class 2o
Syntax	coa 4121	Extend the definition of a class to include the ordinal addition operation.
class +o
Syntax	comu 4122	Extend the definition of a class to include the ordinal multiplication operation.
class .o
Syntax	coe 4123	Extend the definition of a class to include the ordinal exponentiation operation.
class ^o
Definition	df-1o 4124	Define the ordinal number 1.
|- 1o = suc (/)
Definition	df-2o 4125	Define the ordinal number 2.
|- 2o = suc 1o
Definition	df-oadd 4126	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 4127	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 4128	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 4129	Ordinal 1 is an ordinal number.
|- 1o e. On
Theorem	2on 4130	Ordinal 2 is an ordinal number.
|- 2o e. On
Theorem	df1o2 4131	Expanded value of the ordinal number 1.
|- 1o = {(/)}
Theorem	df2o2 4132	Expanded value of the ordinal number 2.
|- 2o = {(/), {(/)}}
Theorem	1ne0 4133	Ordinal one is not equal to ordinal zero.
|- 1o =/= (/)
Theorem	xp01disj 4134	Cross products with the singletons of ordinals 0 and 1 are disjoint.
|- ((A X. {(/)}) i^i (C X. {1o})) = (/)
Theorem	ordgt0ge1 4135	Two ways to express that an ordinal class is positive.
|- (Ord A -> ((/) e. A <-> 1o (_ A))
Theorem	ordge1n0 4136	An ordinal greater than or equal to 1 is nonzero.
|- (Ord A -> (1o (_ A <-> A =/= (/)))
Theorem	el1o 4137	Membership in ordinal one.
|- (A e. 1o <-> A = (/))
Theorem	0lt1o 4138	Ordinal zero is less than ordinal one.
|- (/) e. 1o
Theorem	fnoa 4139	Functionality and domain of ordinal addition.
|- +o Fn (On X. On)
Theorem	fnom 4140	Functionality and domain of ordinal multiplication.
|- .o Fn (On X. On)
Theorem	oav 4141	Value of ordinal addition.
|- ((A e. On /\ B e. On) -> (A +o B) = (rec({<.x, y>. | y = suc x}, A)` B))
Theorem	omv 4142	Value of ordinal multiplication.
|- ((A e. On /\ B e. On) -> (A .o B) = (rec({<.x, y>. | y = (x +o A)}, (/))` B))
Theorem	oe0lem 4143	A helper lemma for oe0 4152 and others.
|- ((ph /\ A = (/)) -> ps)   &   |- (((A e. On /\ ph) /\ (/) e. A) -> ps)   =>   |- ((A e. On /\ ph) -> ps)
Theorem	oev 4144	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 4145	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 4146	Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57.
|- (A e. On -> (A +o (/)) = A)
Theorem	om0 4147	Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62.
|- (A e. On -> (A .o (/)) = (/))
Theorem	oe0m 4148	Ordinal exponentiation with zero mantissa.
|- (A e. On -> ((/) ^o A) = (1o \ A))
Theorem	om0x 4149	Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 4147, 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 4150	Ordinal exponentiation with zero mantissa and zero exponent. Proposition 8.31 of [TakeutiZaring] p. 67.
|- ((/) ^o (/)) = 1o
Theorem	oe0m1 4151	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 4152	Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67.
|- (A e. On -> (A ^o (/)) = 1o)
Theorem	oev2 4153	Alternate value of ordinal exponentiation. Compare oev 4144.
|- ((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 4154	Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56.
|- ((A e. On /\ B e. On) -> (