mmtheorems51 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5001-5100 - Page 51 of 107

Type	Label	Description
Statement
Theorem	0npi 5001	The empty set is not a positive integer.
|- -. (/) e. N.
Theorem	1pi 5002	Ordinal 'one' is a positive integer.
|- 1o e. N.
Theorem	addpiord 5003	Positive integer addition in terms of ordinal addition.
|- ((A e. N. /\ B e. N.) -> (A +N B) = (A +o B))
Theorem	mulpiord 5004	Positive integer multiplication in terms of ordinal multiplication.
|- ((A e. N. /\ B e. N.) -> (A .N B) = (A .o B))
Theorem	mulidpi 5005	1 is an identity element for multiplication on positive integers.
|- (A e. N. -> (A .N 1o) = A)
Theorem	ltpiord 5006	Positive integer 'less than' in terms of ordinal membership.
|- ((A e. N. /\ B e. N.) -> (A <N B <-> A e. B))
Theorem	ltsopi 5007	Positive integer 'less than' is a strict ordering.
|- <N Or N.
Theorem	ltrelpi 5008	Positive integer 'less than' is a relation on positive integers.
|- <N (_ (N. X. N.)
Theorem	dmaddpi 5009	Domain of addition on positive integers.
|- dom +N = (N. X. N.)
Theorem	dmmulpi 5010	Domain of multiplication on positive integers.
|- dom .N = (N. X. N.)
Theorem	addclpi 5011	Closure of addition of positive integers.
|- ((A e. N. /\ B e. N.) -> (A +N B) e. N.)
Theorem	mulclpi 5012	Closure of multiplication of positive integers.
|- ((A e. N. /\ B e. N.) -> (A .N B) e. N.)
Theorem	addcompi 5013	Addition of positive integers is commutative.
|- A e. V   &   |- B e. V   =>   |- (A +N B) = (B +N A)
Theorem	addasspi 5014	Addition of positive integers is associative.
|- B e. V   &   |- C e. V   =>   |- ((A +N B) +N C) = (A +N (B +N C))
Theorem	mulcompi 5015	Multiplication of positive integers is commutative.
|- A e. V   &   |- B e. V   =>   |- (A .N B) = (B .N A)
Theorem	mulasspi 5016	Multiplication of positive integers is associative.
|- B e. V   &   |- C e. V   =>   |- ((A .N B) .N C) = (A .N (B .N C))
Theorem	distrpi 5017	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 5018	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 5019	There is no identity element for addition on positive integers.
|- B e. V   =>   |- (A e. N. -> -. (A +N B) = A)
Theorem	ltexpi 5020	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 5021	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 5022	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 5023	One is less than two (one plus one).
|- 1o <N (1o +N 1o)
Theorem	nlt1pi 5024	No positive integer is less than one.
|- -. A <N 1o
Theorem	indpi 5025	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 5026	Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5231, 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 5030) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 5028). (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 5027	Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5231, 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 5028	Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5231, 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 5029	Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5231, 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 5030	Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5231, 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 5031	Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5231, 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 5032	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 5231, 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 5033	Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5231, 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 5034	Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 5231, 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 5035	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 5036	Domain of equivalence relation for positive fractions.
|- dom ~Q = (N. X. N.)
Theorem	enqer 5037	The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117.
|- Er ~Q
Theorem	enqeceq 5038	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 5039	The equivalence relation for positive fractions exists.
|- ~Q e. V
Theorem	nqex 5040	The class of positive fractions exists.
|- Q. e. V
Theorem	0npq 5041	The empty set is not a positive fraction.
|- -. (/) e. Q.
Theorem	ltrelpq 5042	Positive fraction 'less than' is a relation on positive fractions.
|- <Q (_ (Q. X. Q.)
Theorem	addcmpblnq 5043	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. /\ B e. N.) /\ (C e. N. /\ D e. N.)) /\ ((F e. N. /\ G e. N.) /\ (R e. N. /\ S e. N.))) -> (((A .N D) = (B .N C) /\ (F .N S) = (G .N R)) -> <.((A .N G) +N (B .N F)), (B .N G)>. ~Q <.((C .N S) +N (D .N R)), (D .N S)>.))
Theorem	mulcmpblnq 5044	Lemma showing compatibility of multiplication.
|- 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. /\ B e. N.) /\ (C e. N. /\ D e. N.)) /\ ((F e. N. /\ G e. N.) /\ (R e. N. /\ S e. N.))) -> (((A .N D) = (B .N C) /\ (F .N S) = (G .N R)) -> <.(A .N F), (B .N G)>. ~Q <.(C .N R), (D .N S)>.))
Theorem	addpipq 5045	Addition of positive fractions in terms of positive integers.
|- (((A e. N. /\ B e. N.) /\ (C e. N. /\ D e. N.)) -> ([<.A, B>.] ~Q +Q [<.C, D>.] ~Q ) = [<.((A .N D) +N (B .N C)), (B .N D)>.] ~Q )
Theorem	mulpipq 5046	Multiplication of positive fractions in terms of positive integers.
|- (((A e. N. /\ B e. N.) /\ (C e. N. /\