mmtheorems83 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8201-8300 - Page 83 of 107

Type	Label	Description
Statement
Theorem	nvgrp 8201	The vector addition operation of a normed complex vector space is a group.
|- G = (+v` U)   =>   |- (U e. NrmCVec -> G e. Grp)
Theorem	nvgf 8202	Mapping for the vector addition operation.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- (U e. NrmCVec -> G:(X X. X)-->X)
Theorem	nvsf 8203	Mapping for the scalar multiplication operation.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- (U e. NrmCVec -> S:(CC X. X)-->X)
Theorem	nvgcl 8204	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 8205	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 8206	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 8207	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 8208	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 8209	Right cancellation law for vector addition.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- ((U e. NrmCVec /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGC) = (BGC) <-> A = B))
Theorem	nvlcan 8210	Left cancellation law for vector addition.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- ((U e. NrmCVec /\ (A e. X /\ B e. X /\ C e. X)) -> ((CGA) = (CGB) <-> A = B))
Theorem	nvadd4 8211	Rearrangement of 4 terms in a vector sum.
|- X = (Base` U)   &   |- G = (+v` U)   =>   |- ((U e. NrmCVec /\ (A e. X /\ B e. X) /\ (C e. X /\ D e. X)) -> ((AGB)G(CGD)) = ((AGC)G(BGD)))
Theorem	nvscl 8212	Closure law for the scalar product operation of a normed complex vector space.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> (ASB) e. X)
Theorem	nvsid 8213	Identity element for the scalar product of a normed complex vector space.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (1SA) = A)
Theorem	nvsass 8214	Associative law for the scalar product of a normed complex vector space.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- ((U e. NrmCVec /\ (A e. CC /\ B e. CC /\ C e. X)) -> ((A x. B)SC) = (AS(BSC)))
Theorem	nvscom 8215	Commutative law for the scalar product of a normed complex vector space.
|- X = (Base` U)   &   |- S = (.s` U)   =>   |- ((U e. NrmCVec /\ (A e. CC /\ B e. CC /\ C e. X)) -> (AS(BSC)) = (BS(ASC)))
Theorem	nvdi 8216	Distributive law for the scalar product of a complex vector space.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   =>   |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> (AS(BGC)) = ((ASB)G(ASC)))
Theorem	nvdir 8217	Distributive law for the scalar product of a complex vector space.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   =>   |- ((U e. NrmCVec /\ (A e. CC /\ B e. CC /\ C e. X)) -> ((A + B)SC) = ((ASC)G(BSC)))
Theorem	nv2 8218	A vector plus itself is two times the vector.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (AGA) = (2SA))
Theorem	vsfval 8219	Value of the function for the vector subtraction operation on a normed complex vector space.
|- G = (+v` U)   &   |- M = (-v` U)   =>   |- M = ( /g ` G)
Theorem	nvzcl 8220	Closure law for the zero vector of a normed complex vector space.
|- X = (Base` U)   &   |- Z = (0v` U)   =>   |- (U e. NrmCVec -> Z e. X)
Theorem	nv0rid 8221	The zero vector is a right identity element.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- Z = (0v` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (AGZ) = A)
Theorem	nv0lid 8222	The zero vector is a left identity element.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- Z = (0v` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (ZGA) = A)
Theorem	nv0 8223	Zero times a vector is the zero vector.
|- X = (Base` U)   &   |- S = (.s` U)   &   |- Z = (0v` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (0SA) = Z)
Theorem	nvsz 8224	Anything times the zero vector is the zero vector.
|- S = (.s` U)   &   |- Z = (0v` U)   =>   |- ((U e. NrmCVec /\ A e. CC) -> (ASZ) = Z)
Theorem	nvinv 8225	Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- M = (inv` G)   =>   |- ((U e. NrmCVec /\ A e. X) -> (-u1SA) = (M` A))
Theorem	invfval 8226	Function for the negative of a vector on a normed complex vector space, in terms of the underlying addition group inverse. (We currently do not have a separate notation for the negative of a vector.)
|- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (S o. `'(2nd |` ({-u1} X. V)))   =>   |- (U e. NrmCVec -> N = (inv` G))
Theorem	nvm 8227	Vector subtraction in terms of group division operation.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- M = (-v` U)   &   |- N = ( /g ` G)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (AMB) = (ANB))
Theorem	nvmval 8228	Value of vector subtraction on a normed complex vector space.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- M = (-v` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (AMB) = (AG(-u1SB)))
Theorem	nvmfval 8229	Value of the function for the vector subtraction operation on a normed complex vector space.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- M = (-v` U)   =>   |- (U e. NrmCVec -> M = {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(-u1Sy)))})
Theorem	nvzs 8230	Two ways to express the negative of a vector.
|- X = (Base` U)   &   |- M = (-v` U)   &   |- S = (.s` U)   &   |- Z = (0v` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (ZMA) = (-u1SA))
Theorem	nvmf 8231	Mapping for the vector subtraction operation.
|- X = (Base` U)   &   |- M = (-v` U)   =>   |- (U e. NrmCVec -> M:(X X. X)-->X)
Theorem	nvmcl 8232	Closure law for the vector subtraction operation of a normed complex vector space.
|- X = (Base` U)   &   |- M = (-v` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (AMB) e. X)
Theorem	nvnnncan1 8233	Vector space analog of nnncan1t 5450.
|- X = (Base` U)   &   |- M = (-v` U)   =>   |- ((U e. NrmCVec /\ (A e. X /\ B e. X /\ C e. X)) -> ((AMB)M(AMC)) = (CMB))
Theorem	nvnnncan2 8234	Vector space analog of nnncan2t 5451.
|- X = (Base` U)   &   |- M = (-v` U)   =>   |- ((U e. NrmCVec /\ (A e. X /\ B e. X /\ C e. X)) -> ((AMC)M(BMC)) = (AMB))
Theorem	nvmdi 8235	Distributive law for scalar product over subtraction.