mmtheorems52 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5101-5200 - Page 52 of 107

Type	Label	Description
Statement
Theorem	addclprlem1 5101	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 5102	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 5103	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 5104	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 5105	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 5106	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 5107	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 5108	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 5109	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 5110	Lemma for distributive law for positive reals.
Theorem	distrlem2pr 5111	Lemma for distributive law for positive reals.
Theorem	distrlem3pr 5112	Lemma for distributive law for positive reals.
Theorem	distrlem4pr 5113	Lemma for distributive law for positive reals.
Theorem	distrlem5pr 5114	Lemma for distributive law for positive reals.
Theorem	distrpr 5115	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 5116	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 5117	Positive real 'less than' in terms of proper subset.
|- ((A e. P. /\ B e. P.) -> (A <P B <-> A (. B))
Theorem	psslinpr 5118	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 5119	Positive real 'less than' is a strict ordering. Part of Proposition 9-3.3 of [Gleason] p. 122.
|- <P Or P.
Theorem	prlem934a 5120	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 5121	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 5122	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 5123	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 5124	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 5125	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem2 5126	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem3 5127	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem4 5128	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem5 5129	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem6 5130	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem7 5131	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexpri 5132	Proposition 9-3.5(iv) of [Gleason] p. 123.
|- B e. V   =>   |- (A <P B -> E.x(x e. P. /\ (A +P. x) = B))
Theorem	ltaprlem 5133	Lemma for Proposition 9-3.5(v) of [Gleason] p. 123.
Theorem	ltapr 5134	Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123.
|- A e. V   &   |- B e. V   =>   |- (C e. P. -> (A <P B <-> (C +P. A) <P (C +P. B)))
Theorem	addcanpr 5135	Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123.
|- B e. V   &   |- C e. V   =>   |- ((A e. P. /\ B e. P.) -> ((A +P. B) = (A +P. C) -> B = C))
Theorem	prlem936a 5136	Sublemma for Lemma 9-3.6 of [Gleason] p. 124. This is a property of positive fractions.
|- ((x e. Q. /\ (z e. Q. /\ y e. Q.)) -> ((y +Q z) <Q (x +Q z) <-> (x +Q z) <Q ((x .Q (*Q` y)) .Q (y +Q z))))
Theorem	prlem936b 5137	Sublemma for Lemma 9-3.6 of [Gleason] p. 124.
|- (((y .Q B) e. A /\ ph) -> (y +Q z) e. A)   &   |- (((A e. P. /\ (y +Q z) e. A) /\ (x e. Q. /\ z e. Q.)) -> (ps -> ch))   &   |- ((x e. Q. /\ (z e. Q. /\ y e. Q.)) -> (ch <-> th))   &   |- ((((1Q <Q B /\ x e. Q.) /\ y e. Q.) /\ ph) -> (th <-> ta))   &   |- ((A e. P. /\ ta) -> (ps -> -. (x .Q B) e. A))   =>   |- (((A e. P. /\ z e. Q.) /\ ((ph /\ y e. Q.) /\ (1Q <Q B /\ (y .Q B) e. A))) -> ((x e. A /\ ps) -> (x e. A /\ -. (x .Q B) e. A)))
Theorem	prlem936 5138	Lemma 9-3.6 of [Gleason] p. 124.
|- B e. V   =>   |- ((A e. P. /\ 1Q <Q B) -> E.x(x e. A /\ -. (x .Q B) e. A))
Theorem	reclem1pr 5139	Lemma for Proposition 9-3.7 of [Gleason] p. 124.
Theorem	reclem2pr 5140	Lemma for Proposition 9-3.7 of [Gleason] p. 124.
Theorem	reclem3pr 5141	Lemma for Proposition 9-3.7(v) of [Gleason] p. 124.
Theorem	reclem4pr 5142	Lemma for Proposition 9-3.7(v) of [Gleason] p. 124.
Theorem	recexpr 5143	The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124.
|- (A e. P. -> E.x(x e. P. /\ (A .P. x) = 1P))
Theorem	suplem1pr 5144	The union of a non-empty, bounded set of positive reals is a positive real. Part of Proposition 9-3.3 of [Gleason] p. 122.
|- (((A (_ P. /\ -. A = (/)) /\ E.x(x e. P. /\ A.y(y e. P. -> (y e. A -> y <P x)))) -> U.A e. P.)
Theorem	suplem2pr 5145	The union of a set of positive reals (if a positive real) is its supremum (least upper bound). Part of Proposition 9-3.3 of [Gleason] p. 122.
|- (A (_ P. -> ((y e. A -> -. U.A <P y) /\ (y <P U.A -> E.z(z e. P. /\ (z e. A /\ y <P z)))))
Theorem	supexpr 5146	The union of a non-empty, bounded set of positive reals has a supremum. Part of Proposition 9-3.3 of [Gleason] p. 122.
|- (((A (_ P. /\ -. A = (/)) /\ E.x(x e. P. /\ A.y(y e. P. -> (y e. A -> y <P x)))) -> E.x(x e. P. /\ A.y(y e. P. -> ((y e. A -> -. x <P y) /\ (y <P x -> E.z(z e. P. /\ (z e. A /\ y <P z)))))))
Definition	df-plpr 5147	Define pre-addition on signed reals. 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 9-4.1 of [Gleason] p. 126.
|- +pR = {<.<.x, y>., z>. | ((x e. (P. X. P.) /\ y e. (P. X. P.)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +P. u), (v +P. f)>.))}
Definition	df-mpr 5148	Define pre-multiplication on signed reals. 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 9-4.1 of [Gleason] p. 126.
|- .pR = {<.<.x, y>., z>. | ((x e. (P. X. P.) /\ y e. (P. X. P.)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.((w .P. u) +P. (v .P. f)), ((w .P. f) +P. (v .P. u))>.))}
Definition	df-enr 5149	Define equivalence relation for signed reals. 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 9-4.1 of [Gleason] p. 126.
|- ~R = {<.x, y>. | ((x e. (P. X. P.) /\ y e. (P. X. P.)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (z +P. u) = (w +P. v)))}
Definition	df-nr 5150	Define class of signed reals. 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 9-4.2 of [Gleason] p. 126.
|- R. = ((P. X. P.)/. ~R )
Definition	df-plr 5151	Define addition on signed reals. 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 9-4.3 of [Gleason] p. 126.
|- +R = {<.<.x, y>., z>. | ((x e. R. /\ y e. R.) /\ E.wE.vE.uE.f((x = [<.w, v>.] ~R /\ y = [<.u, f>.] ~R ) /\ z = [(<.w, v>. +pR <.u, f>.)] ~R ))}
Definition	df-mr 5152	Define multiplication on signed reals. 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 9-4.3 of [Gleason] p. 126.
|- .R = {<.<.x, y>., z>. | ((x e. R. /\ y e. R.) /\