mmtheorems52 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5101-5200 - Page 52 of 123

Type	Label	Description
Statement
Theorem	axunndlem1 5101	Lemma for the Axiom of Union with no distinct variable conditions.
Theorem	axunnd 5102	A version of the Axiom of Union with no distinct variable conditions.
|- E.xA.y(E.x(y e. x /\ x e. z) -> y e. x)
Theorem	axpowndlem1 5103	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem2 5104	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem3 5105	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem4 5106	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpownd 5107	A version of the Axiom of Power Sets with no distinct variable conditions.
|- (-. x = y -> E.xA.y(A.x(E.z x e. y -> A.y x e. z) -> y e. x))
Theorem	axregndlem1 5108	Lemma for the Axiom of Regularity with no distinct variable conditions.
Theorem	axregndlem2 5109	Lemma for the Axiom of Regularity with no distinct variable conditions.
Theorem	axregnd 5110	A version of the Axiom of Regularity with no distinct variable conditions.
|- (x e. y -> E.x(x e. y /\ A.z(z e. x -> -. z e. y)))
Theorem	axinfndlem1 5111	Lemma for the Axiom of Infinity with no distinct variable conditions.
Theorem	axinfnd 5112	A version of the Axiom of Infinity with no distinct variable conditions.
|- E.x(y e. z -> (y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x))))
Theorem	axacndlem1 5113	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem2 5114	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem3 5115	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem4 5116	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem5 5117	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacnd 5118	A version of the Axiom of Choice with no distinct variable conditions.
|- E.xA.yA.z(A.x(y e. z /\ z e. w) -> E.wA.y(E.w((y e. z /\ z e. w) /\ (y e. w /\ w e. x)) <-> y = w))
Theorem	zfcndext 5119	Axiom of Extensionality ax-ext 1500, reproved from conditionless ZFC version and predicate calculus.
|- (A.z(z e. x <-> z e. y) -> x = y)
Theorem	zfcndrep 5120	Axiom of Replacement ax-rep 2767, reproved from conditionless ZFC axioms.
|- (A.wE.yA.z(A.yph -> z = y) -> E.yA.z(z e. y <-> E.w(w e. x /\ A.yph)))
Theorem	zfcndun 5121	Axiom of Union ax-un 3089, reproved from conditionless ZFC axioms.
|- E.yA.z(E.w(z e. w /\ w e. x) -> z e. y)
Theorem	zfcndpow 5122	Axiom of Power Sets ax-pow 2818, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 2831.
|- E.yA.z(A.w(w e. z -> w e. x) -> z e. y)
Theorem	zfcndreg 5123	Axiom of Regularity ax-reg 4736, reproved from conditionless ZFC axioms.
|- (E.y y e. x -> E.y(y e. x /\ A.z(z e. y -> -. z e. x)))
Theorem	zfcndinf 5124	Axiom of Infinity ax-inf 4767, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 2822 in the proof.
|- E.y(x e. y /\ A.z(z e. y -> E.w(z e. w /\ w e. y)))
Theorem	zfcndac 5125	Axiom of Choice ax-ac 4890, reproved from conditionless ZFC axioms.
|- E.yA.zA.w((z e. w /\ w e. x) -> E.vA.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v))
Real and complex numbers
Dedekind-cut construction of real and complex numbers
Syntax	cnpi 5126	The set of positive integers, which is the set of natural numbers om with 0 removed. Note: This is the start of the Dedekind-cut construction of real and complex numbers. The last lemma of the construction is mulcnsrec 5418. The actual set of Dedekind cuts is defined by df-np 5240.
class N.
Syntax	cpli 5127	Positive integer addition.
class +N
Syntax	cmi 5128	Positive integer multiplication.
class .N
Syntax	clti 5129	Positive integer ordering relation.
class <N
Syntax	cplpq 5130	Positive fraction pre-addition.
class +pQ
Syntax	cmpq 5131	Positive fraction pre-multiplication.
class .pQ
Syntax	ceq 5132	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 5133	Set of positive fractions.
class Q.
Syntax	c1q 5134	The positive fraction constant 1.
class 1Q
Syntax	cplq 5135	Positive fraction addition.
class +Q
Syntax	cmq 5136	Positive fraction multiplication.
class .Q
Syntax	crq 5137	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 5138	Positive fraction ordering relation.
class <Q
Syntax	cnp 5139	Set of positive reals.
class P.
Syntax	c1p 5140	Positive real constant 1.
class 1P
Syntax	cpp 5141	Positive real addition.
class +P.
Syntax	cmp 5142	Positive real multiplication.
class .P.
Syntax	cltp 5143	Positive real ordering relation.
class <P
Syntax	cplpr 5144	Signed real pre-addition.
class +pR
Syntax	cmpr 5145	Signed real pre-multiplication.
class .pR
Syntax	cer 5146	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 5147	Set of signed reals.
class R.
Syntax	c0r 5148	The signed real constant 0.
class 0R
Syntax	c1r 5149	The signed real constant 1.
class 1R
Syntax	cm1r 5150	The signed real constant -1.
class -1R
Syntax	cplr 5151	Signed real addition.
class +R
Syntax	cmr 5152	Signed real multiplication.
class .R
Syntax	cltr 5153	Signed real ordering relation.
class <R
Definition	df-ni 5154	Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5394, and is intended to be used only by the construction.
|- N. = (om \ {(/)})
Definition	df-pli 5155	Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5394, and is intended to be used only by the construction.
|- +N = ( +o |` (N. X. N.))
Definition	df-mi 5156	Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5394, and is intended to be used only by the construction.
|- .N = ( .o |` (N. X. N.))
Definition	df-lti 5157	Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5394, and is intended to be used only by the construction.
|- <N = (E i^i (N. X. N.))
Theorem	elni 5158	Membership in the class of positive integers.
|- (A e. N. <-> (A e. om /\ A =/= (/)))
Theorem	elni2 5159	Membership in the class of positive integers.
|- (A e. N. <-> (A e. om /\ (/) e. A))
Theorem	pinn 5160	A positive integer is a natural number.
|- (A e. N. -> A e. om)
Theorem	pion 5161	A positive integer is an ordinal number.
|- (A e. N. -> A e. On)
Theorem	piord 5162	A positive integer is ordinal.
|- (A e. N. -> Ord A)
Theorem	niex 5163	The class of positive integers is a set.
|- N. e. V
Theorem	0npi 5164	The empty set is not a positive integer.
|- -. (/) e. N.
Theorem	1pi 5165	Ordinal 'one' is a positive integer.
|- 1o e. N.
Theorem	addpiord 5166	Positive integer addition in terms of ordinal addition.
|- ((A e. N. /\ B e. N.) -> (A +N B) = (A +o B))
Theorem	mulpiord 5167	Positive integer multiplication in terms of ordinal multiplication.
|- ((A e. N. /\ B e. N.) -> (A .N B) = (A .o B))
Theorem	mulidpi 5168	1 is an identity element for multiplication on positive integers.
|- (A e. N. -> (A .N 1o) = A)
Theorem	ltpiord 5169	Positive integer 'less than' in terms of ordinal membership.
|- ((A e. N. /\ B e. N.) -> (A <N B <-> A e. B))
Theorem	ltsopi 5170	Positive integer 'less than' is a strict ordering.
|- <N Or N.
Theorem	ltrelpi 5171	Positive integer 'less than' is a relation on positive integers.
|- <N (_ (N. X. N.)
Theorem	dmaddpi 5172	Domain of addition on positive integers.
|- dom +N = (N. X. N.)
Theorem	dmmulpi 5173	Domain of multiplication on positive integers.
|- dom .N = (N. X. N.)
Theorem	addclpi 5174	Closure of addition of positive integers.
|- ((A e. N. /\ B e. N.) -> (A +N B) e. N.)
Theorem	mulclpi 5175	Closure of multiplication of positive integers.
|- ((A e. N. /\ B e. N.) -> (A .N B) e. N.)
Theorem	addcompi 5176	Addition of positive integers is commutative.
|- A e. V   &   |- B e. V   =>   |- (A +N B) = (B +N A)
Theorem	addasspi 5177	Addition of positive integers is associative.
|- B e. V   &   |- C e. V   =>   |- ((A +N B) +N C) = (A +N (B +N C))
Theorem	mulcompi 5178	Multiplication of positive integers is commutative.
|- A e. V   &   |- B e. V   =>   |- (A .N B) = (B .N A)
Theorem	mulasspi 5179	Multiplication of positive integers is associative.
|- B e. V   &   |- C e. V   =>   |- ((A .N B) .N C) = (A .N (B .N C))
Theorem	distrpi 5180	Multiplication of positive integers is distributive.
|- B e. V   &   |- C e. V   =>   |- (A .N (B +N C)) = ((A .N B) +N (A .N C))
Theorem	mulcanpi 5181	Multiplication cancellation law for positive integers.
|- C e. V   =>   |- ((A e. N. /\ B e. N.) -> ((A .N B) = (A .N C) -> B = C))
Theorem	addnidpi 5182	There is no identity element for addition on positive integers.
|- B e. V   =>   |- (A e. N. -> -. (A +N B) = A)
Theorem	ltexpi 5183	Ordering on positive integers in terms of existence of sum.
|- ((A e. N. /\ B e. N.) -> (A <N B <-> E.x(x e. N. /\ (A +N x) = B)))
Theorem	ltapi 5184	Ordering property of addition for positive integers.
|- A e. V   &   |- B e. V   =>   |- (C e. N. -> (A <N B <-> (C +N A) <N (C +N B)))
Theorem	ltmpi 5185	Ordering property of multiplication for positive integers.
|- A e. V   &   |- B e. V   =>   |- (C e. N. -> (A <N B <-> (C .N A) <N (C .N B)))
Theorem	1lt2pi 5186	One is less than two (one plus one).
|- 1o <N (1o +N 1o)
Theorem	nlt1pi 5187	No positive integer is less than one.
|- -. A <N 1o
Theorem	indpi 5188	Principle of Finite Induction on positive integers.
|- (x = 1o -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = (y +N 1o) -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   &   |- ps   &   |- (y e. N. -> (ch -> th))   =>   |- (A e. N. -> ta)
Definition	df-plpq 5189	Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5394, and is intended to be used only by the construction. This "pre-addition" operation works works directly with ordered pairs of integers. The actual positive fraction addition +Q (df-plq 5193) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 5191). (Analogous remarks apply to the other "pre-" operations in the complex number construction that follows.) From Proposition 9-2.3 of [Gleason] p. 117.
|- +pQ = {<.<.x, y>., z>. | ((x e. (N. X. N.) /\ y e. (N. X. N.)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.((w .N f) +N (v .N u)), (v .N f)>.))}
Definition	df-mpq 5190	Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5394, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119.
|- .pQ = {<.<.x, y>., z>. | ((x e. (N. X. N.) /\ y e. (N. X. N.)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w .N u), (v .N f)>.))}
Definition	df-enq 5191	Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5394, and is intended to be used only by the construction. From Proposition 9-2.1 of [Gleason] p. 117.
|- ~Q = {<.x, y>. | ((x e. (N. X. N.) /\ y e. (N. X. N.)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (z .N u) = (w .N v)))}
Definition	df-nq 5192	Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5394, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117.
|- Q. = ((N. X. N.)/. ~Q )
Definition	df-plq 5193	Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5394, and is intended to be used only by the construction. From Proposition 9-2.3 of [Gleason] p. 117.
|- +Q = {<.<.x, y>., z>. | ((x e. Q. /\ y e. Q.) /\ E.wE.vE.uE.f((x = [<.w, v>.] ~Q /\ y = [<.u, f>.] ~Q ) /\ z = [(<.w, v>. +pQ <.u, f>.)] ~Q ))}
Definition	df-mq 5194	Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5394, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119.
|- .Q = {<.<.x, y>., z>. | ((x e. Q. /\ y e. Q.) /\ E.wE.vE.uE.f((x = [<.w, v>.] ~Q /\ y = [<.u, f>.] ~Q ) /\ z = [(<.w, v>. .pQ <.u, f>.)] ~Q ))}
Definition	df-rq 5195	Define reciprocal on positive fractions. It means the same thing as one divided by the argument (although we don't define full division since we will never need it). This is a "temporary" set used in the construction of complex numbers df-c 5394, and is intended to be used only by the construction. From Proposition 9-2.5 of [Gleason] p. 119, who uses an asterisk to denote this unary operation.
|- *Q = {<.x, y>. | (x e. Q. /\ (x .Q y) = 1Q)}
Definition	df-ltq 5196	Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5394, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162.
|- <Q = {<.x, y>. | ((x e. Q. /\ y e. Q.) /\ E.zE.wE.vE.u((x = [<.z, w>.] ~Q /\ y = [<.v, u>.] ~Q ) /\ (z .N u) <N (w .N v)))}
Definition	df-1q 5197	Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 5394, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117.
|- 1Q = [<.1o, 1o>.] ~Q
Theorem	enqbreq 5198	Equivalence relation for positive fractions in terms of positive integers.
|- (((A e. N. /\ B e. N.) /\ (C e. N. /\ D e. N.)) -> (<.A, B>. ~Q <.C, D>. <-> (A .N D) = (B .N C)))
Theorem	dmenq 5199	Domain of equivalence relation for positive fractions.
|- dom ~Q = (N. X. N.)
Theorem	enqer 5200	The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117.
|- Er ~Q