mmtheorems62 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6101-6200 - Page 62 of 175

Type	Label	Description
Statement
Theorem	axpowndlem1 6101	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem2 6102	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem3 6103	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem4 6104	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpownd 6105	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 6106	Lemma for the Axiom of Regularity with no distinct variable conditions.
Theorem	axregndlem2 6107	Lemma for the Axiom of Regularity with no distinct variable conditions.
Theorem	axregnd 6108	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 6109	Lemma for the Axiom of Infinity with no distinct variable conditions.
Theorem	axinfnd 6110	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 6111	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem2 6112	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem3 6113	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem4 6114	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem5 6115	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacnd 6116	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 6117	Axiom of Extensionality ax-ext 1865, reproved from conditionless ZFC version and predicate calculus.
|- (A.z(z e. x <-> z e. y) -> x = y)
Theorem	zfcndrep 6118	Axiom of Replacement ax-rep 3428, 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 6119	Axiom of Union ax-un 3790, reproved from conditionless ZFC axioms.
|- E.yA.z(E.w(z e. w /\ w e. x) -> z e. y)
Theorem	zfcndpow 6120	Axiom of Power Sets ax-pow 3481, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 3498.
|- E.yA.z(A.w(w e. z -> w e. x) -> z e. y)
Theorem	zfcndreg 6121	Axiom of Regularity ax-reg 5695, 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 6122	Axiom of Infinity ax-inf 5728, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 3485 in the proof.
|- E.y(x e. y /\ A.z(z e. y -> E.w(z e. w /\ w e. y)))
Theorem	zfcndac 6123	Axiom of Choice ax-ac 5906, 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 6124	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 6416. The actual set of Dedekind cuts is defined by df-np 6238.
class N.
Syntax	cpli 6125	Positive integer addition.
class +N
Syntax	cmi 6126	Positive integer multiplication.
class .N
Syntax	clti 6127	Positive integer ordering relation.
class <N
Syntax	cplpq 6128	Positive fraction pre-addition.
class +pQ
Syntax	cmpq 6129	Positive fraction pre-multiplication.
class .pQ
Syntax	ceq 6130	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 6131	Set of positive fractions.
class Q.
Syntax	c1q 6132	The positive fraction constant 1.
class 1Q
Syntax	cplq 6133	Positive fraction addition.
class +Q
Syntax	cmq 6134	Positive fraction multiplication.
class .Q
Syntax	crq 6135	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 6136	Positive fraction ordering relation.
class <Q
Syntax	cnp 6137	Set of positive reals.
class P.
Syntax	c1p 6138	Positive real constant 1.
class 1P
Syntax	cpp 6139	Positive real addition.
class +P.
Syntax	cmp 6140	Positive real multiplication.
class .P.
Syntax	cltp 6141	Positive real ordering relation.
class <P
Syntax	cplpr 6142	Signed real pre-addition.
class +pR
Syntax	cmpr 6143	Signed real pre-multiplication.
class .pR
Syntax	cer 6144	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 6145	Set of signed reals.
class R.
Syntax	c0r 6146	The signed real constant 0.
class 0R
Syntax	c1r 6147	The signed real constant 1.
class 1R
Syntax	cm1r 6148	The signed real constant -1.
class -1R
Syntax	cplr 6149	Signed real addition.
class +R
Syntax	cmr 6150	Signed real multiplication.
class .R
Syntax	cltr 6151	Signed real ordering relation.
class <R
Definition	df-ni 6152	Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 6392, and is intended to be used only by the construction.
|- N. = (om \ {(/)})
Definition	df-pli 6153	Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 6392, and is intended to be used only by the construction.
|- +N = ( +o |` (N. X. N.))
Definition	df-mi 6154	Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 6392, and is intended to be used only by the construction.
|- .N = ( .o |` (N. X. N.))
Definition	df-lti 6155	Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 6392, and is intended to be used only by the construction.
|- <N = ( _E i^i (N. X. N.))
Theorem	elni 6156	Membership in the class of positive integers.
|- (A e. N. <-> (A e. om /\ A =/= (/)))
Theorem	elni2 6157	Membership in the class of positive integers.
|- (A e. N. <-> (A e. om /\ (/) e. A))
Theorem	pinn 6158	A positive integer is a natural number.
|- (A e. N. -> A e. om)
Theorem	pion 6159	A positive integer is an ordinal number.
|- (A e. N. -> A e. On)
Theorem	piord 6160	A positive integer is ordinal.
|- (A e. N. -> Ord A)
Theorem	niex 6161	The class of positive integers is a set.
|- N. e. _V
Theorem	0npi 6162	The empty set is not a positive integer.
|- -. (/) e. N.
Theorem	1pi 6163	Ordinal 'one' is a positive integer.
|- 1o e. N.
Theorem	addpiord 6164	Positive integer addition in terms of ordinal addition.
|- ((A e. N. /\ B e. N.) -> (A +N B) = (A +o B))
Theorem	mulpiord 6165	Positive integer multiplication in terms of ordinal multiplication.
|- ((A e. N. /\ B e. N.) -> (A .N B) = (A .o B))
Theorem	mulidpi 6166	1 is an identity element for multiplication on positive integers.
|- (A e. N. -> (A .N 1o) = A)
Theorem	ltpiord 6167	Positive integer 'less than' in terms of ordinal membership.
|- ((A e. N. /\ B e. N.) -> (A <N B <-> A e. B))
Theorem	ltsopi 6168	Positive integer 'less than' is a strict ordering.
|- <N Or N.
Theorem	ltrelpi 6169	Positive integer 'less than' is a relation on positive integers.
|- <N C_ (N. X. N.)
Theorem	dmaddpi 6170	Domain of addition on positive integers.
|- dom +N = (N. X. N.)
Theorem	dmmulpi 6171	Domain of multiplication on positive integers.
|- dom .N = (N. X. N.)
Theorem	addclpi 6172	Closure of addition of positive integers.
|- ((A e. N. /\ B e. N.) -> (A +N B) e. N.)
Theorem	mulclpi 6173	Closure of multiplication of positive integers.
|- ((A e. N. /\ B e. N.) -> (A .N B) e. N.)
Theorem	addcompi 6174	Addition of positive integers is commutative.
|- A e. _V   &   |- B e. _V   =>   |- (A +N B) = (B +N A)
Theorem	addasspi 6175	Addition of positive integers is associative.
|- B e. _V   &   |- C e. _V   =>   |- ((A +N B) +N C) = (A +N (B +N C))
Theorem	mulcompi 6176	Multiplication of positive integers is commutative.
|- A e. _V   &   |- B e. _V   =>   |- (A .N B) = (B .N A)
Theorem	mulasspi 6177	Multiplication of positive integers is associative.
|- B e. _V   &   |- C e. _V   =>   |- ((A .N B) .N C) = (A .N (B .N C))
Theorem	distrpi 6178	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 6179	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 6180	There is no identity element for addition on positive integers.
|- B e. _V   =>   |- (A e. N. -> -. (A +N B) = A)
Theorem	ltexpi 6181	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 6182	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 6183	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 6184	One is less than two (one plus one).
|- 1o <N (1o +N 1o)
Theorem	nlt1pi 6185	No positive integer is less than one.
|- -. A <N 1o
Theorem	indpi 6186	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 6187	Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 6392, 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 6191) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 6189). (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 6188	Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 6392, 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 6189	Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 6392, 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 6190	Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 6392, 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 6191	Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 6392, 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 6192	Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 6392, 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 6193	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 6392, 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 6194	Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 6392, 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 6195	Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 6392, 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 6196	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 6197	Domain of equivalence relation for positive fractions.
|- dom ~Q = (N. X. N.)
Theorem	enqer 6198	The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117.
|- Er ~Q
Theorem	enqeceq 6199	Equivalence class equality of positive fractions in terms of positive integers.
|- (((A e. N. /\ B e. N.) /\ (C e. N. /\ D e. N.)) -> ([<.A, B>.] ~Q = [<.C, D>.] ~Q <-> (A .N D) = (B .N C)))
Theorem	enqex 6200	The equivalence relation for positive fractions exists.
|- ~Q e. _V