mmtheorems53 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5201-5300 - Page 53 of 123

Type	Label	Description
Statement
Theorem	enqeceq 5201	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 5202	The equivalence relation for positive fractions exists.
|- ~Q e. V
Theorem	nqex 5203	The class of positive fractions exists.
|- Q. e. V
Theorem	0npq 5204	The empty set is not a positive fraction.
|- -. (/) e. Q.
Theorem	ltrelpq 5205	Positive fraction 'less than' is a relation on positive fractions.
|- <Q (_ (Q. X. Q.)
Theorem	addcmpblnq 5206	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 5207	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 5208	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 5209	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 5210	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 5211	The positive fraction 'one'.
|- 1Q e. Q.
Theorem	addclpq 5212	Closure of addition on positive fractions.
|- ((A e. Q. /\ B e. Q.) -> (A +Q B) e. Q.)
Theorem	dmaddpq 5213	Domain of addition on positive fractions.
|- dom +Q = (Q. X. Q.)
Theorem	mulclpq 5214	Closure of multiplication on positive fractions.
|- ((A e. Q. /\ B e. Q.) -> (A .Q B) e. Q.)
Theorem	dmmulpq 5215	Domain of multiplication on positive fractions.
|- dom .Q = (Q. X. Q.)
Theorem	addcompq 5216	Addition of positive fractions is commutative.
|- A e. V   &   |- B e. V   =>   |- (A +Q B) = (B +Q A)
Theorem	addasspq 5217	Addition of positive fractions is associative.
|- B e. V   &   |- C e. V   =>   |- ((A +Q B) +Q C) = (A +Q (B +Q C))
Theorem	mulcompq 5218	Multiplication of positive fractions is commutative.
|- A e. V   &   |- B e. V   =>   |- (A .Q B) = (B .Q A)
Theorem	mulasspq 5219	Multiplication of positive fractions is associative.
|- B e. V   &   |- C e. V   =>   |- ((A .Q B) .Q C) = (A .Q (B .Q C))
Theorem	distrpqlem 5220	Lemma for distributive law: cancellation of common factor.
Theorem	distrpq 5221	Multiplication of positive fractions is distributive.
|- B e. V   &   |- C e. V   =>   |- (A .Q (B +Q C)) = ((A .Q B) +Q (A .Q C))
Theorem	1qec 5222	The equivalence class of ratio 1.
|- A e. V   =>   |- (A e. N. -> 1Q = [<.A, A>.] ~Q )
Theorem	mulidpq 5223	Multiplication identity element for positive fractions.
|- (A e. Q. -> (A .Q 1Q) = A)
Theorem	recmulpq 5224	Relationship between reciprocal and multiplication on positive fractions.
|- B e. V   =>   |- (A e. Q. -> ((*Q` A) = B <-> (A .Q B) = 1Q))
Theorem	recidpq 5225	A positive fraction times its reciprocal is 1.
|- (A e. Q. -> (A .Q (*Q` A)) = 1Q)
Theorem	recclpq 5226	Closure law for positive fraction reciprocal.
|- (A e. Q. -> (*Q` A) e. Q.)
Theorem	recrecpq 5227	Reciprocal of reciprocal of positive fraction.
|- A e. V   =>   |- (A e. Q. -> (*Q` (*Q` A)) = A)
Theorem	dmrecpq 5228	Domain of reciprocal on positive fractions.
|- dom *Q = Q.
Theorem	ltsopq 5229	'Less than' is a strict ordering on positive fractions.
|- <Q Or Q.
Theorem	ltapq 5230	Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120.
|- A e. V   &   |- B e. V   =>   |- (C e. Q. -> (A <Q B <-> (C +Q A) <Q (C +Q B)))
Theorem	ltmpq 5231	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 5232	One is less than two (one plus one).
|- 1Q <Q (1Q +Q 1Q)
Theorem	ltaddpq 5233	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 5234	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 5235	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 5236	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 5237	The is no smallest positive fraction.
|- (A e. Q. -> E.x x <Q A)
Theorem	ltbtwnpq 5238	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 5239	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 5240	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 5394, 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 5241	Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 5394, and is intended to be used only by the construction. Definition of [Gleason] p. 122.
|- 1P = {x | x <Q 1Q}
Definition	df-plp 5242	Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 5394, 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 5243	Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 5394, 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 5244	Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 5394, 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 5245	The class of positive reals is a set.
|- P. e. V
Theorem	elnp 5246	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 5247	A positive real is not empty.
|- (A e. P. -> A =/= (/))
Theorem	prpssnq 5248	A positive real is a subset of the positive fractions.
|- (A e. P. -> A (. Q.)
Theorem	elprpq 5249	A positive real is a set of positive fractions.
|- ((A e. P. /\ B e. A) -> B e. Q.)
Theorem	0npr 5250	The empty set is not a positive real.
|- -. (/) e. P.
Theorem	prcdpq 5251	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 5252	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 5253	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 5254	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 5255	Positive real 'less than' is a relation on positive reals.
|- <P (_ (P. X. P.)
Theorem	genpv 5256	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 5257	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 5258	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 5259	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 5260	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 5261	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 5262	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 5263	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 5264	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 5265	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 5266	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. X. P.)   &   |- ((f e. P. /\ g e. P.) -> (fFg) e. P.)   &   |- ((fGg)Gh) = (fG(gGh))   =>   |- ((AFB)FC) = (AF(BFC))
Theorem	plpv 5267	Value of addition on positive reals.
|- ((A e. P. /\ B e. P.) -> (A +P. B) = {x | E.yE.z((y e. A /\ z e. B) /\ x = (y +Q z))})
Theorem	mpv 5268	Value of multiplication on positive reals.
|- ((A e. P. /\ B e. P.) -> (A .P. B) = {x | E.yE.z((y e. A /\ z e. B) /\ x = (y .Q z))})
Theorem	dmplp 5269	Domain of addition on positive reals.
|- dom +P. = (P. X. P.)
Theorem	dmmp 5270	Domain of multiplication on positive reals.
|- dom .P. = (P. X. P.)
Theorem	1pr 5271	The positive real number 'one'.
|- 1P e. P.
Theorem	addclprlem1 5272	Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123.
|- (((A e. P. /\ g e. A) /\ x e. Q.) -> (x <Q (g +Q h) -> ((x .Q (*Q` (g +Q h))) .Q g) e. A))
Theorem	addclprlem2 5273	Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123.
|- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (g +Q h) -> x e. (A +P. B)))
Theorem	addclpr 5274	Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123.
|- ((A e. P. /\ B e. P.) -> (A +P. B) e. P.)
Theorem	mulclprlem 5275	Lemma to prove downward closure in positive real multiplication. Part of proof of Proposition 9-3.7 of [Gleason] p. 124.
|- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (g .Q h) -> x e. (A .P. B)))
Theorem	mulclpr 5276	Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124.
|- ((A e. P. /\ B e. P.) -> (A .P. B) e. P.)
Theorem	addcompr 5277	Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123.
|- A e. V   &   |- B e. V   =>   |- (A +P. B) = (B +P. A)
Theorem	addasspr 5278	Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123.
|- B e. V   &   |- C e. V   =>   |- ((A +P. B) +P. C) = (A +P. (B +P. C))
Theorem	mulcompr 5279	Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124.
|- A e. V   &   |- B e. V   =>   |- (A .P. B) = (B .P. A)
Theorem	mulasspr 5280	Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124.
|- B e. V   &   |- C e. V   =>   |- ((A .P. B) .P. C) = (A .P. (B .P. C))
Theorem	distrlem1pr 5281	Lemma for distributive law for positive reals.
Theorem	distrlem2pr 5282	Lemma for distributive law for positive reals.
Theorem	distrlem3pr 5283	Lemma for distributive law for positive reals.
Theorem	distrlem4pr 5284	Lemma for distributive law for positive reals.
Theorem	distrlem5pr 5285	Lemma for distributive law for positive reals.
Theorem	distrpr 5286	Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124.
|- B e. V   &   |- C e. V   =>   |- (A .P. (B +P. C)) = ((A .P. B) +P. (A .P. C))
Theorem	1idpr 5287	1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124.
|- (A e. P. -> (A .P. 1P) = A)
Theorem	ltprord 5288	Positive real 'less than' in terms of proper subset.
|- ((A e. P. /\ B e. P.) -> (A <P B <-> A (. B))
Theorem	psslinpr 5289	Proper subset is a linear ordering on positive reals. Part of Proposition 9-3.3 of [Gleason] p. 122.
|- ((A e. P. /\ B e. P.) -> (A (. B \/ A = B \/ B (. A))
Theorem	ltsopr 5290	Positive real 'less than' is a strict ordering. Part of Proposition 9-3.3 of [Gleason] p. 122.
|- <P Or P.
Theorem	prlem934a 5291	Sublemma for Lemma 9-3.4 of [Gleason] p. 122.
|- B e. V   =>   |- (C e. N. -> (((B e. Q. /\ A.x(x e. A -> (x +Q B) e. A)) /\ y e. A) -> (y +Q ([<.C, 1o>.] ~Q .Q B)) e. A))
Theorem	prlem934b 5292	Sublemma for Lemma 9-3.4 of [Gleason] p. 122.
|- (((u e. N. /\ w e. N.) /\ (v e. N. /\ z e. N.)) -> (([<.(w .N v), 1o>.] ~Q .Q [<.z, w>.] ~Q ) = [<.v, u>.] ~Q \/ [<.v, u>.] ~Q <Q ([<.(w .N v), 1o>.] ~Q .Q [<.z, w>.] ~Q )))
Theorem	prlem934 5293	Lemma 9-3.4 of [Gleason] p. 122.
|- ((A e. P. /\ B e. Q.) -> E.x(x e. A /\ -. (x +Q B) e. A))
Theorem	ltaddpr 5294	The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123.
|- ((A e. P. /\ B e. P.) -> A <P (A +P. B))
Theorem	ltaddpr2 5295	The sum of two positive reals is greater than one of them.
|- B e. V   =>   |- (C e. P. -> ((A +P. B) = C -> A <P C))
Theorem	ltexprlem1 5296	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem2 5297	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem3 5298	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem4 5299	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem5 5300	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.