mmtheorems84 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8301-8400 - Page 84 of 108

Type	Label	Description
Statement
Theorem	nvop 8301	A complex inner product space in terms of ordered pair components.
|- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   =>   |- (U e. NrmCVec -> U = <.<.G, S>., N>.)
Theorem	nvoprne 8302	The vector addition and scalar product operations are not identical.
|- (<.<.G, S>., N>. e. NrmCVec -> G =/= S)
Examples of normed complex vector spaces
Theorem	cnnv 8303	The set of complex numbers is a normed complex vector space. The vector operation is +, the scalar product is x., and the norm function is abs. (Contributed by Steve Rodriguez, 3-Dec-2006.)
|- U = <.<. + , x. >., abs>.   =>   |- U e. NrmCVec
Theorem	cnnvg 8304	The vector addition (group) operation of the normed complex vector space of complex numbers.
|- U = <.<. + , x. >., abs>.   =>   |- + = (+v` U)
Theorem	cnnvba 8305	The base set of the normed complex vector space of complex numbers.
|- U = <.<. + , x. >., abs>.   =>   |- CC = (Base` U)
Theorem	cnnvdemo 8306	Derive the associative law for complex number addition axaddass 5289 to demonstrate the use of cnnv 8303, cnnvg 8304, and cnnvba 8305.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) + C) = (A + (B + C)))
Theorem	cnnvs 8307	The scalar product operation of the normed complex vector space of complex numbers.
|- U = <.<. + , x. >., abs>.   =>   |- x. = (.s` U)
Theorem	cnnvnm 8308	The norm operation of the normed complex vector space of complex numbers.
|- U = <.<. + , x. >., abs>.   =>   |- abs = (norm` U)
Theorem	cnnvm 8309	The vector subtraction operation of the normed complex vector space of complex numbers.
|- U = <.<. + , x. >., abs>.   =>   |- - = (-v` U)
Theorem	elimnv 8310	Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem.
|- X = (Base` U)   &   |- Z = (0v` U)   &   |- U e. NrmCVec   =>   |- if(A e. X, A, Z) e. X
Theorem	elimnvu 8311	Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem.
|- if(U e. NrmCVec, U, <.<. + , x. >., abs>.) e. NrmCVec
Induced metric of a normed complex vector space
Theorem	imsval 8312	Value of the induced metric of a normed complex vector space.
|- M = (-v` U)   &   |- N = (norm` U)   &   |- D = (IndMet` U)   =>   |- (U e. NrmCVec -> D = (N o. M))
Theorem	imsdval 8313	Value of the induced metric (distance function) of a normed complex vector space. Equation 1 of [Kreyszig] p. 59.
|- X = (Base` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- D = (IndMet` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (ADB) = (N` (AMB)))
Theorem	imsdval2 8314	Value of the distance function of the induced metric of a normed complex vector space. Equation 1 of [Kreyszig] p. 59.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   &   |- D = (IndMet` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (ADB) = (N` (AG(-u1SB))))
Theorem	nvnd 8315	The norm of a normed complex vector space expressed in terms of the distance function of its induced metric. Problem 1 of [Kreyszig] p. 63.
|- X = (Base` U)   &   |- Z = (0v` U)   &   |- N = (norm` U)   &   |- D = (IndMet` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (N` A) = (ADZ))
Theorem	imsdf 8316	Mapping for the induced metric distance function of a normed complex vector space.
|- X = (Base` U)   &   |- D = (IndMet` U)   =>   |- (U e. NrmCVec -> D:(X X. X)-->RR)
Theorem	imsba 8317	Base set of the induced metric space of a normed complex vector space.
|- X = (Base` U)   &   |- D = (IndMet` U)   =>   |- (U e. NrmCVec -> X = dom dom D)
Theorem	imsbai 8318	Base set of the induced metric space of a normed complex vector space.
|- X = (Base` U)   &   |- D = (IndMet` U)   &   |- U e. NrmCVec   =>   |- X = dom dom D
Theorem	imsmetlem 8319	Lemma for imsmet 8320.
Theorem	imsmet 8320	The induced metric of a normed complex vector space is a metric space. Part of Definition 2.2-1 of [Kreyszig] p. 58.
|- D = (IndMet` U)   =>   |- (U e. NrmCVec -> D e. Met)
Theorem	nvelbl 8321	Membership of a vector in a ball.
|- X = (Base` U)   &   |- M = (-v` U)   &   |- N = (norm` U)   &   |- D = (IndMet` U)   =>   |- (((U e. NrmCVec /\ P e. X /\ A e. X) /\ (R e. RR /\ 0 < R)) -> (A e. (P( ball ` D)R) <-> (N` (AMP)) < R))
Theorem	nvelbl2 8322	Membership of an off-center vector in a ball.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- N = (norm` U)   &   |- D = (IndMet` U)   =>   |- (((U e. NrmCVec /\ P e. X /\ A e. X) /\ (R e. RR /\ 0 < R)) -> ((PGA) e. (P( ball ` D)R) <-> (N` A) < R))
Theorem	nvcnf 8323	A continuous function is an operation (normed complex vector space version of cnf 7759).
|- X = (Base` U)   &   |- Y = (Base` W)   &   |- C = (IndMet` U)   &   |- D = (IndMet` W)   &   |- J = (Open` C)   &   |- K = (Open` D)   =>   |- ((U e. NrmCVec /\ W e. NrmCVec /\ F e. (J Cn K)) -> F:X-->Y)
Theorem	nvcnpf 8324	A continuous function is an operation (normed complex vector space version of cnpf 7760).
|- X = (Base` U)   &   |- Y = (Base` W)   &   |- C = (IndMet` U)   &   |- D = (IndMet` W)   &   |- J = (Open` C)   &   |- K = (Open` D)   =>   |- (((U e. NrmCVec /\ W e. NrmCVec /\ P e. X) /\ F e. ((J CnP K)` P)) -> F:X-->Y)
Theorem	nvcni 8325	Epsilon-delta property of a continuous operator. (Normed complex vector space version of metcni 7891.)
|- X = (Base` U)   &   |- M = (norm` U)   &   |- N = (norm` W)   &   |- R = (-v` U)   &   |- S = (-v` W)   &   |- C = (IndMet` U)   &   |- D = (IndMet` W)   &   |- J = (Open` C)   &   |- K = (Open` D)   =>   |- (((U e. NrmCVec /\ W e. NrmCVec /\ F e. (J Cn K)) /\ (P e. X /\ A e. RR /\ 0 < A)) -> E.x e. RR (0 < x /\ A.y e. X ((M` (PRy)) < x -> (N` ((F` P)S(F` y))) < A)))
Theorem	nvcni2 8326	Epsilon-delta property of a continuous operator. (Normed complex vector space version of metcni 7891.)
|- X = (Base` U)   &   |- M = (norm` U)   &   |- N = (norm` W)   &   |- R = (-v` U)   &   |- S = (-v` W)   &   |- C = (IndMet` U)   &   |- D = (IndMet` W)   &   |- J = (Open` C)   &   |- K = (Open` D)   =>   |- (((U e. NrmCVec /\ W e. NrmCVec /\ F e. (J Cn K)) /\ (P e. X /\ A e. RR /\ 0 < A)) -> E.x e. RR (0 < x /\ A.y e. X ((M` (PRy)) <_ x -> (N` ((F` P)S(F` y))) <_ A)))
Theorem	nvcni3 8327	Epsilon-delta property of a continuous operator. (Normed complex vector space version of metcni 7891.)
|- X = (Base` U)   &   |- M = (norm` U)   &   |- N = (norm` W)   &   |- R = (-v` U)   &   |- S = (-v` W)   &   |- C = (IndMet` U)   &   |- D = (IndMet` W)   &   |- J = (Open` C)   &   |- K = (Open` D)   =>   |- (((U e. NrmCVec /\ W e. NrmCVec /\ F e. (J Cn K)) /\ (P e. X /\ A e. RR /\ 0 < A)) -> E.x e. RR (0 < x /\ A.y e. X ((M` (yRP)) < x -> (N` ((F` y)S(F` P))) < A)))
Theorem	nvlmcl 8328	Closure of the limit of a converging vector sequence.
|- X = (Base` U)   &   |- D = (IndMet` U)   &   |- P e. V   =>   |- ((U e. NrmCVec /\ F(~~>m` D)P) -> P e. X)
Theorem	nvlmle 8329	If the norm of each member of a converging sequence is less than or equal to a given amount, so is the norm of the convergence value.
|- X = (Base` U)   &   |- N = (norm` U)   &   |- D = (IndMet` U)   &   |- P e. V   =>   |- (((U e. NrmCVec /\ F:NN-->X /\ F(~~>m` D)P) /\ (R e. RR /\ A.k e. NN (N` (F` k)) <_ R)) -> (N` P) <_ R)
Theorem	cnims 8330	The metric induced on the complex numbers. cnmet 7901 proves that it is a metric. (Contributed by Steve Rodriguez, 5-Dec-2006; revised by nm 15-Jan-2008.)
|- U = <.<. + , x. >., abs>.   &   |- D = (abs o. - )   =>   |- D = (IndMet` U)
Theorem	sqcn 8331	The square function on complex numbers is continuous.
|- D = (IndMet` <.<. + , x. >., abs>.)   &   |- J = (Open` D)   &   |- F = {<.x, y>. | (x e. CC /\ y = (x^2))}   =>   |- F e. (J Cn J)
Theorem	sqcn2 8332	The square function on complex numbers is continuous.
|- D = (abs o. - )   &   |- J = (Open` D)   &   |- F = {<.w, v>.