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	dmaddpi 5001	Domain of addition on positive integers.
|- dom +N = (N. X. N.)
Theorem	dmmulpi 5002	Domain of multiplication on positive integers.
|- dom .N = (N. X. N.)
Theorem	addclpi 5003	Closure of addition of positive integers.
|- ((A e. N. /\ B e. N.) -> (A +N B) e. N.)
Theorem	mulclpi 5004	Closure of multiplication of positive integers.
|- ((A e. N. /\ B e. N.) -> (A .N B) e. N.)
Theorem	addcompi 5005	Addition of positive integers is commutative.
|- A e. V   &   |- B e. V   =>   |- (A +N B) = (B +N A)
Theorem	addasspi 5006	Addition of positive integers is associative.
|- B e. V   &   |- C e. V   =>   |- ((A +N B) +N C) = (A +N (B +N C))
Theorem	mulcompi 5007	Multiplication of positive integers is commutative.
|- A e. V   &   |- B e. V   =>   |- (A .N B) = (B .N A)
Theorem	mulasspi 5008	Multiplication of positive integers is associative.
|- B e. V   &   |- C e. V   =>   |- ((A .N B) .N C) = (A .N (B .N C))
Theorem	distrpi 5009	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 5010	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 5011	There is no identity element for addition on positive integers.
|- B e. V   =>   |- (A e. N. -> -. (A +N B) = A)
Theorem	ltexpi 5012	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 5013	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 5014	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 5015	One is less than two (one plus one).
|- 1o <N (1o +N 1o)
Theorem	nlt1pi 5016	No positive integer is less than one.
|- -. A <N 1o
Theorem	indpi 5017	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 5018	Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5223, 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 5022) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 5020). (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 5019	Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5223, 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 5020	Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5223, 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 5021	Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5223, 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 5022	Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5223, 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 5023	Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5223, 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 5024	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 5223, 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 5025	Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 5223, 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 5026	Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 5223, 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 5027	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 5028	Domain of equivalence relation for positive fractions.
|- dom ~Q = (N. X. N.)
Theorem	enqer 5029	The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117.
|- Er ~Q
Theorem	enqeceq 5030	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 5031	The equivalence relation for positive fractions exists.
|- ~Q e. V
Theorem	nqex 5032	The class of positive fractions exists.
|- Q. e. V
Theorem	0npq 5033	The empty set is not a positive fraction.
|- -. (/) e. Q.
Theorem	ltrelpq 5034	Positive fraction 'less than' is a relation on positive fractions.
|- <Q (_ (Q. X. Q.)
Theorem	addcmpblnq 5035	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 5036	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 5037	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 5038	Multiplication 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 C), (B .N D)>.] ~Q )
Theorem	ordpipq 5039	Ordering of positive fractions in terms of positive integers.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   =>   |- ([<.A, B>.] ~Q <Q [<.C, D>.] ~Q <-> (A .N D) <N (B .N C))
Theorem	1q 5040	The positive fraction 'one'.
|- 1Q e. Q.
Theorem	addclpq 5041	Closure of addition on positive fractions.
|- ((A e. Q. /\ B e. Q.) -> (A +Q B) e. Q.)
Theorem	dmaddpq 5042	Domain of addition on positive fractions.
|- dom +Q = (Q. X. Q.