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	genpcl 5101	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 5102	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 5103	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 5104	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 5105	Domain of addition on positive reals.
|- dom +P. = (P. X. P.)
Theorem	dmmp 5106	Domain of multiplication on positive reals.
|- dom .P. = (P. X. P.)
Theorem	1pr 5107	The positive real number 'one'.
|- 1P e. P.
Theorem	addclprlem1 5108	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 5109	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 5110	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 5111	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 5112	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 5113	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 5114	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 5115	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 5116	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 5117	Lemma for distributive law for positive reals.
Theorem	distrlem2pr 5118	Lemma for distributive law for positive reals.
Theorem	distrlem3pr 5119	Lemma for distributive law for positive reals.
Theorem	distrlem4pr 5120	Lemma for distributive law for positive reals.
Theorem	distrlem5pr 5121	Lemma for distributive law for positive reals.
Theorem	distrpr 5122	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 5123	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 5124	Positive real 'less than' in terms of proper subset.
|- ((A e. P. /\ B e. P.) -> (A <P B <-> A (. B))
Theorem	psslinpr 5125	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 5126	Positive real 'less than' is a strict ordering. Part of Proposition 9-3.3 of [Gleason] p. 122.
|- <P Or P.
Theorem	prlem934a 5127	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 5128	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 5129	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 5130	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 5131	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 5132	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem2 5133	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem3 5134	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem4 5135	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem5 5136	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem6 5137	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem7 5138	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexpri 5139	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 5140	Lemma for Proposition 9-3.5(v) of [Gleason] p. 123.
Theorem	ltapr 5141	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 5142	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 5143	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 5144	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 5145	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 5146	Lemma for Proposition 9-3.7 of [Gleason] p. 124.
Theorem	reclem2pr 5147	Lemma for Proposition 9-3.7 of [Gleason] p. 124.
Theorem	reclem3pr 5148	Lemma for Proposition 9-3.7(v) of [Gleason] p. 124.
Theorem	reclem4pr 5149	Lemma for Proposition 9-3.7(v) of [Gleason] p. 124.
Theorem	recexpr 5150	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 5151	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 5152	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 5153	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. -> ((