mmtheorems54 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5301-5400 - Page 54 of 123

Type	Label	Description
Statement
Theorem	ltexprlem6 5301	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem7 5302	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexpri 5303	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 5304	Lemma for Proposition 9-3.5(v) of [Gleason] p. 123.
Theorem	ltapr 5305	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 5306	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 5307	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 5308	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 5309	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 5310	Lemma for Proposition 9-3.7 of [Gleason] p. 124.
Theorem	reclem2pr 5311	Lemma for Proposition 9-3.7 of [Gleason] p. 124.
Theorem	reclem3pr 5312	Lemma for Proposition 9-3.7(v) of [Gleason] p. 124.
Theorem	reclem4pr 5313	Lemma for Proposition 9-3.7(v) of [Gleason] p. 124.
Theorem	recexpr 5314	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 5315	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 5316	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 5317	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 5318	Define pre-addition on signed 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 9-4.1 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 5319	Define pre-multiplication on signed 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 9-4.1 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 5320	Define equivalence relation for signed 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 9-4.1 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 5321	Define class of signed 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 9-4.2 of [Gleason] p. 126.
|- R. = ((P. X. P.)/. ~R )
Definition	df-plr 5322	Define addition on signed 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 9-4.3 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 5323	Define multiplication on signed 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 9-4.3 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-ltr 5324	Define ordering relation on signed 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 9-4.4 of [Gleason] p. 127.
|- <R = {<.x, y>. | ((x e. R. /\ y e. R.) /\ E.zE.wE.vE.u((x = [<.z, w>.] ~R /\ y = [<.v, u>.] ~R ) /\ (z +P. u) <P (w +P. v)))}
Definition	df-0r 5325	Define signed real constant 0. 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 9-4.2 of [Gleason] p. 126.
|- 0R = [<.1P, 1P>.] ~R
Definition	df-1r 5326	Define signed 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. From Proposition 9-4.2 of [Gleason] p. 126.
|- 1R = [<.(1P +P. 1P), 1P>.] ~R
Definition	df-m1r 5327	Define signed 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.
|- -1R = [<.1P, (1P +P. 1P)>.] ~R
Theorem	enrbreq 5328	Equivalence relation for signed reals in terms of positive reals.
|- (((A e. P. /\ B e. P.) /\ (C e. P. /\ D e. P.)) -> (<.A, B>. ~R <.C, D>. <-> (A +P. D) = (B +P. C)))
Theorem	dmenr 5329	Domain of equivalence relation for signed reals.
|- dom ~R = (P. X. P.)
Theorem	enrer 5330	The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126.
|- Er ~R
Theorem	enreceq 5331	Equivalence class equality of positive fractions in terms of positive integers.
|- (((A e. P. /\ B e. P.) /\ (C e. P. /\ D e. P.)) -> ([<.A, B>.] ~R = [<.C, D>.] ~R <-> (A +P. D) = (B +P. C)))
Theorem	enrex 5332	The equivalence relation for signed reals exists.
|- ~R e. V
Theorem	srex 5333	The class of signed reals is a set.
|- R. e. V
Theorem	ltrelsr 5334	Signed real 'less than' is a relation on signed reals.
|- <R (_ (R. X. R.)
Theorem	addcmpblnr 5335	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. P. /\ B e. P.) /\ (C e. P. /\ D e. P.)) /\ ((F e. P. /\ G e. P.) /\ (R e. P. /\ S e. P.))) -> (((A +P. D) = (B +P. C) /\ (F +P. S) = (G +P. R)) -> <.(A +P. F), (B +P. G)>. ~R <.(C +P. R), (D +P. S)>.))
Theorem	mulcmpblnrlem 5336	Lemma used in 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 +P. D) = (B +P. C) /\ (F +P. S) = (G +P. R)) -> ((D .P. F) +P. (((A .P. F) +P. (B .P. G)) +P. ((C .P. S) +P. (D .P. R)))) = ((D .P. F) +P. (((A .P. G) +P. (B .P. F)) +P. ((C .P. R) +P. (D .P. S)))))
Theorem	mulcmpblnr 5337	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. P. /\ B e. P.) /\ (C e. P. /\ D e. P.)) /\ ((F e. P. /\ G e. P.) /\ (R e. P. /\ S e. P.))) -> (((A +P. D) = (B +P. C) /\ (F +P. S) = (G +P. R)) -> <.((A .P. F) +P. (B .P. G)), ((A .P. G) +P. (B .P. F))>. ~R <.((C .P. R) +P. (D .P. S)), ((C .P. S) +P. (D .P. R))>.))
Theorem	addsrpr 5338	Addition of signed reals in terms of positive reals.
|- (((A e. P. /\ B e. P.) /\ (C e. P. /\ D e. P.)) -> ([<.A, B>.] ~R +R [<.C, D>.] ~R ) = [<.(A +P. C), (B +P. D)>.] ~R )
Theorem	mulsrpr 5339	Multiplication of signed reals in terms of positive reals.
|- (((A e. P. /\ B e. P.) /\ (C e. P. /\ D e. P.)) -> ([<.A, B>.] ~R .R [<.C, D>.] ~R ) = [<.((A .P. C) +P. (B .P. D)), ((A .P. D) +P. (B .P. C))>.] ~R )
Theorem	ltsrpr 5340	Ordering of signed reals in terms of positive reals.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   =>   |- ([<.A, B>.] ~R <R [<.C, D>.] ~R <-> (A +P. D) <P (B +P. C))
Theorem	gt0srpr 5341	Greater then zero in terms of positive reals.
|- A e. V   &   |- B e. V   =>   |- (0R <R [<.A, B>.] ~R <-> B <P A)
Theorem	0nsr 5342	The empty set is not a signed real.
|- -. (/) e. R.
Theorem	0r 5343	The constant 0R is a signed real.
|- 0R e. R.
Theorem	1r 5344	The constant 1R is a signed real.
|- 1R e. R.
Theorem	m1r 5345	The constant -1R is a signed real.
|- -1R e. R.
Theorem	addclsr 5346	Closure of addition on signed reals.
|- ((A e. R. /\ B e. R.) -> (A +R B) e. R.)
Theorem	mulclsr 5347	Closure of multiplication on signed reals.
|- ((A e. R. /\ B e. R.) -> (A .R B) e. R.)
Theorem	dmaddsr 5348	Domain of addition on signed reals.
|- dom +R = (R. X. R.)
Theorem	dmmulsr 5349	Domain of multiplication on signed reals.
|- dom .R = (R. X. R.)
Theorem	addcomsr 5350	Addition of signed reals is commutative.
|- A e. V   &   |- B e. V   =>   |- (A +R B) = (B +R A)
Theorem	addasssr 5351	Addition of signed reals is associative.
|- B e. V   &   |- C e. V   =>   |- ((A +R B) +R C) = (A +R (B +R C))
Theorem	mulcomsr 5352	Multiplication of signed reals is commutative.
|- A e. V   &   |- B e. V   =>   |- (A .R B) = (B .R A)
Theorem	mulasssr 5353	Multiplication of signed reals is associative.
|- B e. V   &   |- C e. V   =>   |- ((A .R B) .R C) = (A .R (B .R C))
Theorem	distrsr 5354	Multiplication of signed reals is distributive.
|- B e. V   &   |- C e. V   =>   |- (A .R (B +R C)) = ((A .R B) +R (A .R C))
Theorem	m1p1sr 5355	Minus one plus one is zero for signed reals.
|- (-1R +R 1R) = 0R
Theorem	m1m1sr 5356	Minus one times minus one is plus one for signed reals.
|- (-1R .R -1R) = 1R
Theorem	ltsosr 5357	Signed real 'less than' is a strict ordering.
|- <R Or R.
Theorem	0lt1sr 5358	0 is less than 1 for signed reals.
|- 0R <R 1R
Theorem	1ne0sr 5359	1 and 0 are distinct for signed reals.
|- -. 1R = 0R
Theorem	0idsr 5360	The signed real number 0 is an identity element for addition of signed reals.
|- (A e. R. -> (A +R 0R) = A)
Theorem	1idsr 5361	1 is an identity element for multiplication.
|- (A e. R. -> (A .R 1R) = A)
Theorem	00sr 5362	A signed real times 0 is 0.
|- (A e. R. -> (A .R 0R) = 0R)
Theorem	ltasr 5363	Ordering property of addition.
|- A e. V   &   |- B e. V   =>   |- (C e. R. -> (A <R B <-> (C +R A) <R (C +R B)))
Theorem	pn0sr 5364	A signed real plus its negative is zero.
|- (A e. R. -> (A +R (A .R -1R)) = 0R)
Theorem	negexsr 5365	Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126.
|- (A e. R. -> E.x(x e. R. /\ (A +R x) = 0R))
Theorem	recexsrlem 5366	The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126.
|- A e. V   =>   |- (0R <R A -> E.x(x e. R. /\ (A .R x) = 1R))
Theorem	addgt0sr 5367	The sum of two positive signed reals is positive.
|- A e. V   &   |- B e. V   =>   |- ((0R <R A /\ 0R <R B) -> 0R <R (A +R B))
Theorem	mulgt0sr 5368	The product of two positive signed reals is positive.
|- A e. V   &   |- B e. V   =>   |- ((0R <R A /\ 0R <R B) -> 0R <R (A .R B))
Theorem	sqgt0sr 5369	The square of a nonzero signed real is positive.
|- A e. V   =>   |- (A e. R. -> (-. A = 0R -> 0R <R (A .R A)))
Theorem	recexsr 5370	The reciprocal of a nonzero signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126.
|- A e. V   =>   |- (A e. R. -> (-. A = 0R -> E.x(x e. R. /\ (A .R x) = 1R)))
Theorem	ssgt0sr 5371	The sum of squares of signed reals is positive if one is nonzero.
|- A e. V   &   |- B e. V   =>   |- ((A e. R. /\ B e. R.) -> (-. (A = 0R /\ B = 0R) -> 0R <R ((A .R A) +R (B .R B))))
Theorem	mappsrpr 5372	Mapping from positive signed reals to positive reals.
|- A e. V   =>   |- (0R <R [<.(A +P. 1P), 1P>.] ~R <-> A e. P.)
Theorem	ltpsrpr 5373	Mapping of order from positive signed reals to positive reals.
|- A e. V   &   |- B e. V   =>   |- ([<.(A +P. 1P), 1P>.] ~R <R [<.(B +P. 1P), 1P>.] ~R <-> A <P B)
Theorem	map2psrpr 5374	Equivalence for positive signed real.
|- A e. V   =>   |- (0R <R A <-> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A))
Theorem	suppsrlem 5375	Mapping of non-empty subset from positive reals to positive signed reals.
|- B = {w | [<.(w +P. 1P), 1P>.] ~R e. A}   =>   |- ((A.x(x e. A -> 0R <R x) /\ -. A = (/)) -> (B (_ P. /\ -. B = (/)))
Theorem	suppsr 5376	A non-empty, bounded set of positive signed reals has a supremum.
|- B = {w | [<.(w +P. 1P), 1P>.] ~R e. A}   =>   |- (((A.x(x e. A -> 0R <R x) /\ -. A = (/)) /\ E.x(0R <R x /\ A.y(0R <R y -> (y e. A -> y <R x)))) -> E.x(0R <R x /\ A.y(0R <R y -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(0R <R z /\ (z e. A /\ y <R z)))))))
Theorem	suppsr2 5377	A non-empty, bounded set of positive signed reals has a supremum. (Converts quantifier restrictions to all reals.)
|- (((A.x(x e. A -> 0R <R x) /\ -. A = (/)) /\ E.x(x e. R. /\ A.y(y e. R. -> (y e. A -> y <R x)))) -> E.x(x e. R. /\ A.y(y e. R. -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(z e. R. /\ (z e. A /\ y <R z)))))))
Theorem	suppsr3 5378	A non-empty, bounded set with at least one positive real has a supremum.
|- B = {y | (y e. A /\ 0R <R y)}   =>   |- ((E.y(y e. A /\ 0R <R y) /\ E.x(x e. R. /\ A.y(y e. R. -> (y e. A -> y <R x)))) -> E.x(x e. R. /\ A.y(y e. R. -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(z e. R. /\ (z e. A /\ y <R z)))))))
Theorem	supsrlem1 5379	Lemma for supremum theorem.
Theorem	supsrlem2 5380	Lemma for supremum theorem.
Theorem	supsrlem3 5381	Lemma for supremum theorem.
Theorem	supsrlem4 5382	Lemma for supremum theorem.
Theorem	supsrlem5 5383	Lemma for supremum theorem.
Theorem	supsrlem6 5384	Lemma for supremum theorem.
Theorem	supsr 5385	A non-empty, bounded set of signed reals has a supremum.
|- (((A (_ R. /\ -. A = (/)) /\ E.x(x e. R. /\ A.y(y e. R. -> (y e. A -> y <R x)))) -> E.x(x e. R. /\ A.y(y e. R. -> ((y e. A -> -. x <R y) /\ (y <R x -> E.z(z e. R. /\ (z e. A /\ y <R z)))))))
Syntax	cc 5386	Class of complex numbers.
class CC
Syntax	cr 5387	Class of real numbers.
class RR
Syntax	cc0 5388	Extend class notation to include the complex number 0.
class 0
Syntax	c1 5389	Extend class notation to include the complex number 1.
class 1
Syntax	ci 5390	Extend class notation to include the complex number i.
class i
Syntax	caddc 5391	Addition on complex numbers.
class +
Syntax	cltrr 5392	'Less than' predicate (defined over real subset of complex numbers).
class <R
Syntax	cmul 5393	Multiplication on complex numbers. The token x. is a center dot.
class x.
Definition	df-c 5394	Define the set of complex numbers. The 25 axioms for complex numbers start at axcnex 5421.
|- CC = (R. X. R.)
Definition	df-0 5395	Define the complex number 0 (base 10).
|- 0 = <.0R, 0R>.
Definition	df-1 5396	Define the complex number 1 (base 10).
|- 1 = <.1R, 0R>.
Definition	df-i 5397	Define the complex number i (the imaginary unit).
|- i = <.0R, 1R>.
Definition	df-r 5398	Define the set of real numbers.
|- RR = (R. X. {0R})
Definition	df-plus 5399	Define addition over complex numbers.
|- + = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +R u), (v +R f)>.))}
Definition	df-mul 5400	Define multiplication over complex numbers.
|- x. = {<.<.x, y>., z>. | ((x e. CC /\ y e. CC) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.((w .R u) +R (-1R .R (v .R f))), ((v .R u) +R (w .R f))>.))}