mmtheorems52 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10766

Statement List for Metamath Proof Explorer - 5101-5200 - Page 52 of 108

Type	Label	Description
Statement
Theorem	ltmpq 5101	Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120.
|- A e. V   &   |- B e. V   =>   |- (C e. Q. -> (A <Q B <-> (C .Q A) <Q (C .Q B)))
Theorem	1lt2pq 5102	One is less than two (one plus one).
|- 1Q <Q (1Q +Q 1Q)
Theorem	ltaddpq 5103	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 5104	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 5105	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 5106	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 5107	The is no smallest positive fraction.
|- (A e. Q. -> E.x x <Q A)
Theorem	ltbtwnpq 5108	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 5109	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 5110	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 5264, 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 5111	Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 5264, and is intended to be used only by the construction. Definition of [Gleason] p. 122.
|- 1P = {x | x <Q 1Q}
Definition	df-plp 5112	Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 5264, 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 5113	Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 5264, 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 5114	Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 5264, 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 5115	The class of positive reals is a set.
|- P. e. V
Theorem	elnp 5116	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 5117	A positive real is not empty.
|- (A e. P. -> A =/= (/))
Theorem	prpssnq 5118	A positive real is a subset of the positive fractions.
|- (A e. P. -> A (. Q.)
Theorem	elprpq 5119	A positive real is a set of positive fractions.
|- ((A e. P. /\ B e. A) -> B e. Q.)
Theorem	0npr 5120	The empty set is not a positive real.
|- -. (/) e. P.
Theorem	prcdpq 5121	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 5122	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 5123	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 5124	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 5125	Positive real 'less than' is a relation on positive reals.
|- <P (_ (P. X. P.)
Theorem	genpv 5126	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 5127	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 5128	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 5129	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 5130	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 5131	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 5132	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 5133	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 5134	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 5135	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 5136	Associativity of an operation on reals.
|- F = {<.<.w, v>.