mmtheorems83 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8201-8300 - Page 83 of 108

Type	Label	Description
Statement
Theorem	isvclem 8201	Lemma for isvc 8205.
Theorem	vcoprnelem 8202	Lemma for vcoprne 8203.
Theorem	vcoprne 8203	The operations of a complex vector space cannot be identical.
|- (<.G, S>. e. CVec -> G =/= S)
Theorem	vcex 8204	The components of a complex vector space are sets.
|- (<.G, S>. e. CVec -> (G e. V /\ S e. V))
Theorem	isvc 8205	The predicate "is a complex vector space."
|- X = ran G   =>   |- (<.G, S>. e. CVec <-> (G e. Abel /\ S:(CC X. X)-->X /\ A.x e. X ((1Sx) = x /\ A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx)))))))
Theorem	isvci 8206	Properties that determine a complex vector space.
|- G e. Abel   &   |- dom G = (X X. X)   &   |- S:(CC X. X)-->X   &   |- (x e. X -> (1Sx) = x)   &   |- ((y e. CC /\ x e. X /\ z e. X) -> (yS(xGz)) = ((ySx)G(ySz)))   &   |- ((y e. CC /\ z e. CC /\ x e. X) -> ((y + z)Sx) = ((ySx)G(zSx)))   &   |- ((y e. CC /\ z e. CC /\ x e. X) -> ((y x. z)Sx) = (yS(zSx)))   &   |- W = <.G, S>.   =>   |- W e. CVec
Examples of complex vector spaces
Theorem	cnvc 8207	The set of complex numbers is a complex vector space. The vector operation is +, and the scalar product is x..
|- <. + , x. >. e. CVec
Normed complex vector spaces
Definition and basic properties
Syntax	cnv 8208	Extend class notation with the class of all normed complex vector spaces.
class NrmCVec
Syntax	cpv 8209	Extend class notation with vector addition in a normed complex vector space. In the literature, the subscript "v" is omitted, but we need it to avoid ambiguity with complex number addition + caddc 5239.
class +v
Syntax	cba 8210	Extend class notation with the base set of a normed complex vector space. (Note that Base is capitalized because, once it is fixed for a particular vector space U, it is not a function, unlike e.g. norm. This is our typical convention.)
class Base
Syntax	cns 8211	Extend class notation with scalar multiplication in a normed complex vector space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity.
class .s
Syntax	cn0v 8212	Extend class notation with zero vector in a normed complex vector space.
class 0v
Syntax	cnsb 8213	Extend class notation with vector subtraction in a normed complex vector space.
class -v
Syntax	cnm 8214	Extend class notation with the norm function in a normed complex vector space. In the literature, the norm of A is usually written "|| A ||", but we use function notation to take advantage of our existing theorems about functions.
class norm
Syntax	cims 8215	Extend class notation with the class of the induced metrics on normed complex vector spaces.
class IndMet
Definition	df-nv 8216	Define the class of all normed complex vector spaces.
|- NrmCVec = {<.<.g, s>., n>. | (<.g, s>. e. CVec /\ n:ran g-->RR /\ A.x e. ran g(((n` x) = 0 -> x = (Id` g)) /\ A.y e. CC (n` (ysx)) = ((abs` y) x. (n` x)) /\ A.y e. ran g(n` (xgy)) <_ ((n` x) + (n` y))))}
Theorem	nvss 8217	Structure of the class of all normed complex vectors spaces.
|- NrmCVec (_ ((V X. V) X. V)
Theorem	nvvcop 8218	A normed complex vector space is a vector space.
|- (<.<.G, S>., N>. e. NrmCVec -> <.G, S>. e. CVec)
Definition	df-va 8219	Define vector addition on a normed complex vector space.
|- +v = (1st o. 1st)
Definition	df-ba 8220	Define the base set of a normed complex vector space.
|- Base = {<.x, y>. | y = ran (+v` x)}
Definition	df-sm 8221	Define scalar multiplication on a normed complex vector space.
|- .s = (2nd o. 1st)
Definition	df-0v 8222	Define the zero vector in a normed complex vector space.
|- 0v = (Id o. +v)
Definition	df-vs 8223	Define vector subtraction on a normed complex vector space.
|- -v = ( /g o. +v)
Definition	df-nm 8224	Define the norm function in a normed complex vector space.
|- norm = 2nd
Definition	df-ims 8225	Define the induced metric on a normed complex vector space.
|- IndMet = {<.u, d>. | (u e. NrmCVec /\ d = ((norm` u) o. (-v` u)))}
Theorem	nvrel 8226	The class of all normed complex vectors spaces is a relation.
|- Rel NrmCVec
Theorem	vafval 8227	Value of the function for the vector addition (group) operation on a normed complex vector space.
|- G = (+v` U)   =>   |- G = (1st` (1st` U))
Theorem	bafval 8228	Value of the function for the base set of a normed complex vector space.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- X = ran G
Theorem	smfval 8229	Value of the function for the scalar multiplication operation on a normed complex vector space.
|- S = (.s` U)   =>   |- S = (2nd` (1st` U))
Theorem	0vfval 8230	Value of the function for the zero vector on a normed complex vector space.
|- G = (+v` U)   &   |- Z = (0v` U)   =>   |- Z = (Id` G)
Theorem	nmfval 8231	Value of the norm function in a normed complex vector space.
|- N = (norm` U)   =>   |- N = (2nd` U)
Theorem	nvop2 8232	A normed complex vector space is an ordered pair of a vector space and a norm operation.
|- W = (1st` U)   &   |- N = (norm` U)   =>   |- (U e. NrmCVec -> U = <.W, N>.)
Theorem	nvvop 8233	The vector space component of a normed complex vector space is an ordered pair of the underlying group and a scalar product.
|- W = (1st` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   =>   |- (U e. NrmCVec -> W = <.G, S>.)
Theorem	isnvlem 8234	Lemma for isnv 8236.
Theorem	nvex 8235	The components of a normed complex vector space are sets.
|- (<.<.G, S>., N>. e. NrmCVec -> (G e. V /\ S e. V /\ N e. V))
Theorem	isnv 8236	The predicate "is a normed complex vector space."
|- X = ran G   &   |- Z = (Id` G)   =>   |- (<.<.G, S>., N>. e. NrmCVec <-> (<.G, S>. e. CVec /\ N:X-->RR /\ A.x e. X (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y)))))
Theorem	isnvi 8237	Properties that determine a normed complex vector space.
|- X = ran G   &   |- Z = (Id` G)   &   |- <.G, S>. e. CVec   &   |- N:X-->RR   &   |- ((x e. X /\ (N` x) = 0) -> x = Z)   &   |- ((y e. CC /\ x e. X) -> (N` (ySx)) = ((abs` y) x. (N` x)))   &   |- ((x e. X /\ y e. X) -> (N` (xGy)) <_ ((N` x) + (N` y)))   &   |- U = <.<.G, S>., N>.   =>   |- U e. NrmCVec
Theorem	nvi 8238	The properties of a normed complex vector space, which is a vector space accompanied by a norm.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- Z = (0v` U)   &   |- N = (norm` U)   =>   |- (U e. NrmCVec -> (<.G, S>. e. CVec /\ N:X-->RR /\ A.x e. X (((N` x) = 0 -> x = Z) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y)))))
Theorem	nvvc 8239	The vector space component of a normed complex vector space.
|- W = (1st` U)   =>   |- (U e. NrmCVec -> W e. CVec)
Theorem	nvabl 8240	The vector addition operation of a normed complex vector space is an Abelian group.
|- G = (+v` U)   =>   |- (U e. NrmCVec -> G e. Abel)
Theorem	nvgrp 8241	The vector addition operation of a normed complex vector space is a group.
|- G = (+v` U)   =>   |- (U e. NrmCVec -> G e. Grp)
Theorem	nvgf 8242	Mapping for the vector addition operation.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- (U e. NrmCVec -> G:(X X. X)-->X)
Theorem	nvsf 8243	Mapping for the scalar multiplication operation.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- (U e. NrmCVec -> S:(CC X. X)-->X)
Theorem	nvgcl 8244	Closure law for the vector addition (group) operation of a normed complex vector space.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (AGB) e. X)
Theorem	nvcom 8245	The vector addition (group) operation is commutative.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (AGB) = (BGA))
Theorem	nvass 8246	The vector addition (group) operation is associative.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- ((U e. NrmCVec /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = (AG(BGC)))
Theorem	nvadd12 8247	Commutative/associative law for vector addition.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- ((U e. NrmCVec /\ (A e. X /\ B e. X /\ C e. X)) -> (AG(BGC)) = (BG(AGC)))
Theorem	nvadd23 8248	Commutative/associative law for vector addition.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- ((U e. NrmCVec /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = ((AGC)GB))
Theorem	nvrcan 8249	Right cancellation law for vector addition.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- ((U e. NrmCVec /\ (A e. X /\