mmtheorems51 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5001-5100 - Page 51 of 105

Type	Label	Description
Statement
Theorem	1lt2pq 5001	One is less than two (one plus one).
|- 1Q <Q (1Q +Q 1Q)
Theorem	ltaddpq 5002	The sum of two fractions is greater than one of them.
|- A e. V   &   |- B e. V   =>   |- ((A e. Q. /\ B e. Q.) -> A <Q (A +Q B))
Theorem	ltexpq 5003	Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119.
|- A e. V   =>   |- ((A e. Q. /\ B e. Q.) -> (A <Q B <-> E.x(A +Q x) = B))
Theorem	ltexpq2 5004	Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119.
|- A e. V   =>   |- ((A e. Q. /\ B e. Q.) -> (A <Q B <-> E.x(x e. Q. /\ (A +Q x) = B)))
Theorem	halfpq 5005	One-half of any positive fraction exists. Lemma for Proposition 9-2.6(i) of [Gleason] p. 120.
|- (A e. Q. -> E.x(x +Q x) = A)
Theorem	nsmallpq 5006	The is no smallest positive fraction.
|- (A e. Q. -> E.x x <Q A)
Theorem	ltbtwnpq 5007	There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120.
|- A e. V   &   |- B e. V   =>   |- (A <Q B -> E.x(A <Q x /\ x <Q B))
Theorem	ltrpq 5008	Ordering property of reciprocal for positive fractions. Proposition 9-2.6(iv) of [Gleason] p. 120.
|- A e. V   &   |- B e. V   =>   |- (A <Q B -> (*Q` B) <Q (*Q` A))
Definition	df-np 5009	Define the set of positive reals. A "Dedekind cut" is a partition of the positive rational numbers into two classes such that all the numbers of one class are less than all the numbers of the other. A positive real is defined as the lower class of a Dedekind cut. Definition 9-3.1 of [Gleason] p. 121. (Note: This is a "temporary" definition used in the construction of complex numbers df-c 5163, and is intended to be used only by the construction.)
|- P. = {x | (((/) (. x /\ x (. Q.) /\ A.y e. x (A.z(z <Q y -> z e. x) /\ E.z e. x y <Q z))}
Definition	df-1p 5010	Define the positive real 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. Definition of [Gleason] p. 122.
|- 1P = {x | x <Q 1Q}
Definition	df-plp 5011	Define addition on positive reals. 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 9-3.5 of [Gleason] p. 123.
|- +P. = {<.<.x, y>., z>. | ((x e. P. /\ y e. P.) /\ z = {w | E.v e. x E.u e. y w = (v +Q u)})}
Definition	df-mp 5012	Define multiplication on positive reals. 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 9-3.7 of [Gleason] p. 124.
|- .P. = {<.<.x, y>., z>. | ((x e. P. /\ y e. P.) /\ z = {w | E.v e. x E.u e. y w = (v .Q u)})}
Definition	df-ltp 5013	Define ordering on positive reals. 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 9-3.2 of [Gleason] p. 122.
|- <P = {<.x, y>. | ((x e. P. /\ y e. P.) /\ x (. y)}
Theorem	npex 5014	The class of positive reals is a set.
|- P. e. V
Theorem	elnp 5015	Membership in positive reals.
|- (A e. P. <-> (((/) (. A /\ A (. Q.) /\ A.x e. A (A.y(y <Q x -> y e. A) /\ E.y e. A x <Q y)))
Theorem	prn0 5016	A positive real is not empty.
|- (A e. P. -> A =/= (/))
Theorem	prpssnq 5017	A positive real is a subset of the positive fractions.
|- (A e. P. -> A (. Q.)
Theorem	elprpq 5018	A positive real is a set of positive fractions.
|- ((A e. P. /\ B e. A) -> B e. Q.)
Theorem	0npr 5019	The empty set is not a positive real.
|- -. (/) e. P.
Theorem	prcdpq 5020	A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121.
|- ((A e. P. /\ B e. A) -> (C <Q B -> C e. A))
Theorem	prub 5021	A positive fraction not in a positive real is an upper bound. Remark (1) of [Gleason] p. 122.
|- (((A e. P. /\ B e. A) /\ C e. Q.) -> (-. C e. A -> B <Q C))
Theorem	prnmax 5022	A positive real has no largest member. Definition 9-3.1(iii) of [Gleason] p. 121.
|- ((A e. P. /\ B e. A) -> E.x(x e. A /\ B <Q x))
Theorem	prnmadd 5023	A positive real has no largest member. Addition version.
|- B e. V   =>   |- ((A e. P. /\ B e. A) -> E.x(B +Q x) e. A)
Theorem	ltrelpr 5024	Positive real 'less than' is a relation on positive reals.
|- <P (_ (P. X. P.)
Theorem	genpv 5025	Value of general operation (addition or multiplication) on positive reals.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   =>   |- ((A e. P. /\ B e. P.) -> (AFB) = {f | E.gE.h((g e. A /\ h e. B) /\ f = (gGh))})
Theorem	genpelv 5026	Membership in value of general operation (addition or multiplication) on positive reals.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   &   |- C e. V   =>   |- ((A e. P. /\ B e. P.) -> (C e. (AFB) <-> E.fE.g((f e. A /\ g e. B) /\ C = (fGg))))
Theorem	genpprecl 5027	Pre-closure law for general operation on positive reals.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   =>   |- ((A e. P. /\ B e. P.) -> ((C e. A /\ D e. B) -> (CGD) e. (AFB)))
Theorem	genpdm 5028	Domain of general operation on positive reals.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   =>   |- dom F = (P. X. P.)
Theorem	genpn0 5029	The result of an operation on positive reals is not empty.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   =>   |- ((A e. P. /\ B e. P.) -> (/) (. (AFB))
Theorem	genpss 5030	The result of an operation on positive reals is a subset of the positive fractions.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   &   |- ((g e. Q. /\ h e. Q.) -> (gGh) e. Q.)   =>   |- ((A e. P. /\ B e. P.) -> (AFB) (_ Q.)
Theorem	genpnnp 5031	The result of an operation on positive reals is different from the set of positive fractions.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   &   |- ((w e. Q. /\ v e. Q.) -> (wGv) e. Q.)   &   |- (z e. Q. -> (x <Q y <-> (zGx) <Q (zGy)))   &   |- (xGy) = (yGx)   =>   |- ((A e. P. /\ B e. P.) -> -. (AFB) = Q.)
Theorem	genpcd 5032	Downward closure of an operation on positive reals.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   &   |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (gGh) -> x e. (AFB)))   =>   |- ((A e. P. /\ B e. P.) -> (f e. (AFB) -> (x <Q f -> x e. (AFB))))
Theorem	genpnmax 5033	An operation on positive reals has no largest member.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   &   |- (v e. Q. -> (z <Q w <-> (vGz) <Q (vGw)))   &   |- (zGw) = (wGz)   =>   |- ((A e. P. /\ B e. P.) -> (f e. (AFB) -> E.x(x e. (AFB) /\ f <Q x)))
Theorem	genpcl 5034	Closure of an operation on reals.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   &   |- ((x e. Q. /\ y e. Q.) -> (xGy) e. Q.)   &   |- (h e. Q. -> (f <Q g <-> (hGf) <Q (hGg)))   &   |- (xGy) = (yGx)   &   |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (gGh) -> x e. (AFB)))   =>   |- ((A e. P. /\ B e. P.) -> (AFB) e. P.)
Theorem	genpass 5035	Associativity of an operation on reals.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   &   |- B e. V   &   |- C e. V   &   |- dom F = (P.