Metamath Proof Explorer - mmtheorems39

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Statement List for Metamath Proof Explorer - 3801-3900 - Page 39 of 58

Type	Label	Description
Statement
Theorem	pion 3801	A positive integer is an ordinal number.
|- (A e. N. -> A e. On)
Theorem	piord 3802	A positive integer is ordinal.
|- (A e. N. -> Ord A)
Theorem	niex 3803	The class of positive integers is a set.
|- N. e. V
Theorem	0npi 3804	The empty set is not a positive integer.
|- -. (/) e. N.
Theorem	1pi 3805	Ordinal 'one' is a positive integer.
|- 1o e. N.
Theorem	addpiord 3806	Positive integer addition in terms of ordinal addition.
|- ((A e. N. /\ B e. N.) -> (A +N B) = (A +o B))
Theorem	mulpiord 3807	Positive integer multiplication in terms of ordinal multiplication.
|- ((A e. N. /\ B e. N.) -> (A .N B) = (A .o B))
Theorem	mulidpi 3808	1 is an identity element for multiplication on positive integers.
|- (A e. N. -> (A .N 1o) = A)
Theorem	ltpiord 3809	Positive integer 'less than' in terms of ordinal membership.
|- ((A e. N. /\ B e. N.) -> (A <N B <-> A e. B))
Theorem	ltsopi 3810	Positive integer 'less than' is a strict ordering.
|- <N Or N.
Theorem	ltrelpi 3811	Positive integer 'less than' is a relation on positive integers.
|- <N (_ (N. X. N.)
Theorem	dmaddpi 3812	Domain of addition on positive integers.
|- dom +N = (N. X. N.)
Theorem	dmmulpi 3813	Domain of multiplication on positive integers.
|- dom .N = (N. X. N.)
Theorem	addclpi 3814	Closure of addition of positive integers.
|- ((A e. N. /\ B e. N.) -> (A +N B) e. N.)
Theorem	mulclpi 3815	Closure of multiplication of positive integers.
|- ((A e. N. /\ B e. N.) -> (A .N B) e. N.)
Theorem	addcompi 3816	Addition of positive integers is commutative.
|- A e. V   &   |- B e. V   =>   |- (A +N B) = (B +N A)
Theorem	addasspi 3817	Addition of positive integers is associative.
|- B e. V   &   |- C e. V   =>   |- ((A +N B) +N C) = (A +N (B +N C))
Theorem	mulcompi 3818	Multiplication of positive integers is commutative.
|- A e. V   &   |- B e. V   =>   |- (A .N B) = (B .N A)
Theorem	mulasspi 3819	Multiplication of positive integers is associative.
|- B e. V   &   |- C e. V   =>   |- ((A .N B) .N C) = (A .N (B .N C))
Theorem	distrpi 3820	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 3821	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 3822	There is no identity element for addition on positive integers.
|- B e. V   =>   |- (A e. N. -> -. (A +N B) = A)
Theorem	ltexpi 3823	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 3824	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	ltmpi 3825	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 3826	One is less than two (one plus one).
|- 1o <N (1o +N 1o)
Theorem	nlt1pi 3827	No positive integer is less than one.
|- -. A <N 1o
Theorem	indpi 3828	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 3829	Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. 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 3830	Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 4034, 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 3831	Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 4034, 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 3832	Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 4034, 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 3833	Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 4034, 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 3834	Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 4034, 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 3835	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 4034, 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 3836	Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 4034, 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 3837	Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 4034, 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 3838	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 3839	Domain of equivalence relation for positive fractions.
|- dom ~Q = (N. X. N.)
Theorem	enqer 3840	The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117.
|- Er ~Q
Theorem	enqeceq 3841	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 3842	The equivalence relation for positive fractions exists.
|- ~Q e. V
Theorem	nqex 3843	The class of positive fractions exists.
|- Q. e. V
Theorem	0npq 3844	The empty set is not a positive fraction.
|- -. (/) e. Q.
Theorem	ltrelpq 3845	Positive fraction 'less than' is a relation on positive fractions.
|- <Q (_ (Q. X. Q.)
Theorem	addcmpblnq 3846	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 3847	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 3848	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 3849	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 3850	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