mmtheorems50 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4901-5000 - Page 50 of 105

Type	Label	Description
Statement
Syntax	ceq 4901	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 4902	Set of positive fractions.
class Q.
Syntax	c1q 4903	The positive fraction constant 1.
class 1Q
Syntax	cplq 4904	Positive fraction addition.
class +Q
Syntax	cmq 4905	Positive fraction multiplication.
class .Q
Syntax	crq 4906	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 4907	Positive fraction ordering relation.
class <Q
Syntax	cnp 4908	Set of positive reals.
class P.
Syntax	c1p 4909	Positive real constant 1.
class 1P
Syntax	cpp 4910	Positive real addition.
class +P.
Syntax	cmp 4911	Positive real multiplication.
class .P.
Syntax	cltp 4912	Positive real ordering relation.
class <P
Syntax	cplpr 4913	Signed real pre-addition.
class +pR
Syntax	cmpr 4914	Signed real pre-multiplication.
class .pR
Syntax	cer 4915	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 4916	Set of signed reals.
class R.
Syntax	c0r 4917	The signed real constant 0.
class 0R
Syntax	c1r 4918	The signed real constant 1.
class 1R
Syntax	cm1r 4919	The signed real constant -1.
class -1R
Syntax	cplr 4920	Signed real addition.
class +R
Syntax	cmr 4921	Signed real multiplication.
class .R
Syntax	cltr 4922	Signed real ordering relation.
class <R
Definition	df-ni 4923	Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5163, and is intended to be used only by the construction.
|- N. = (om \ {(/)})
Definition	df-pli 4924	Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5163, and is intended to be used only by the construction.
|- +N = ( +o |` (N. X. N.))
Definition	df-mi 4925	Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5163, and is intended to be used only by the construction.
|- .N = ( .o |` (N. X. N.))
Definition	df-lti 4926	Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5163, and is intended to be used only by the construction.
|- <N = (E i^i (N. X. N.))
Theorem	elni 4927	Membership in the class of positive integers.
|- (A e. N. <-> (A e. om /\ A =/= (/)))
Theorem	elni2 4928	Membership in the class of positive integers.
|- (A e. N. <-> (A e. om /\ (/) e. A))
Theorem	pinn 4929	A positive integer is a natural number.
|- (A e. N. -> A e. om)
Theorem	pion 4930	A positive integer is an ordinal number.
|- (A e. N. -> A e. On)
Theorem	piord 4931	A positive integer is ordinal.
|- (A e. N. -> Ord A)
Theorem	niex 4932	The class of positive integers is a set.
|- N. e. V
Theorem	0npi 4933	The empty set is not a positive integer.
|- -. (/) e. N.
Theorem	1pi 4934	Ordinal 'one' is a positive integer.
|- 1o e. N.
Theorem	addpiord 4935	Positive integer addition in terms of ordinal addition.
|- ((A e. N. /\ B e. N.) -> (A +N B) = (A +o B))
Theorem	mulpiord 4936	Positive integer multiplication in terms of ordinal multiplication.
|- ((A e. N. /\ B e. N.) -> (A .N B) = (A .o B))
Theorem	mulidpi 4937	1 is an identity element for multiplication on positive integers.
|- (A e. N. -> (A .N 1o) = A)
Theorem	ltpiord 4938	Positive integer 'less than' in terms of ordinal membership.
|- ((A e. N. /\ B e. N.) -> (A <N B <-> A e. B))
Theorem	ltsopi 4939	Positive integer 'less than' is a strict ordering.
|- <N Or N.
Theorem	ltrelpi 4940	Positive integer 'less than' is a relation on positive integers.
|- <N (_ (N. X. N.)
Theorem	dmaddpi 4941	Domain of addition on positive integers.
|- dom +N = (N. X. N.)
Theorem	dmmulpi 4942	Domain of multiplication on positive integers.
|- dom .N = (N. X. N.)
Theorem	addclpi 4943	Closure of addition of positive integers.
|- ((A e. N. /\ B e. N.) -> (A +N B) e. N.)
Theorem	mulclpi 4944	Closure of multiplication of positive integers.
|- ((A e. N. /\ B e. N.) -> (A .N B) e. N.)
Theorem	addcompi 4945	Addition of positive integers is commutative.
|- A e. V   &   |- B e. V   =>   |- (A +N B) = (B +N A)
Theorem	addasspi 4946	Addition of positive integers is associative.
|- B e. V   &   |- C e. V   =>   |- ((A +N B) +N C) = (A +N (B +N C))
Theorem	mulcompi 4947	Multiplication of positive integers is commutative.
|- A e. V   &   |- B e. V   =>   |- (A .N B) = (B .N A)
Theorem	mulasspi 4948	Multiplication of positive integers is associative.
|- B e. V   &   |- C e. V   =>   |- ((A .N B) .N C) = (A .N (B .N C))
Theorem	distrpi 4949	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 4950	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 4951	There is no identity element for addition on positive integers.
|- B e. V   =>   |- (A e. N. -> -. (A +N B) = A)
Theorem	ltexpi 4952	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 4953	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 4954	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 4955	One is less than two (one plus one).
|- 1o <N (1o +N 1o)
Theorem	nlt1pi 4956	No positive integer is less than one.
|- -. A <N 1o
Theorem	indpi 4957	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 4958	Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5163, 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 4962) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 4960). (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 4959	Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5163, 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 4960	Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5163, 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 4961	Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5163, 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 4962	Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5163, 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 4963	Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5163, 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 4964	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 5163, 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 4965	Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5163, 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 4966	Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 5163, 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 4967	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 4968	Domain of equivalence relation for positive fractions.
|- dom ~Q = (N. X. N.)
Theorem	enqer 4969	The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117.
|- Er ~Q
Theorem	enqeceq 4970	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 4971	The equivalence relation for positive fractions exists.
|- ~Q e. V
Theorem	nqex 4972	The class of positive fractions exists.
|- Q. e. V
Theorem	0npq 4973	The empty set is not a positive fraction.
|- -. (/) e. Q.
Theorem	ltrelpq 4974	Positive fraction 'less than' is a relation on positive fractions.
|- <Q (_ (Q. X. Q.)
Theorem	addcmpblnq 4975	Lemma showing compatibility of addition.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   &   |- F e. V   &   |- G e. V   &   |- R e. V   &   |- S e. V   =>   |- ((((A e. N.