mmtheorems84 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8301-8400 - Page 84 of 107

Type	Label	Description
Statement
Theorem	nmcnilem 8301	Lemma for nmcni 8302.
Theorem	nmcni 8302	The norm of a normed complex vector space is a continuous function.
|- N = (norm` U)   &   |- C = (IndMet` U)   &   |- D = ((abs o. - ) |` (RR X. RR))   &   |- J = (Open` C)   &   |- K = (Open` D)   &   |- U e. NrmCVec   =>   |- N e. (J Cn K)
Theorem	nmcn 8303	The norm of a normed complex vector space is a continuous function.
|- N = (norm` U)   &   |- C = (IndMet` U)   &   |- D = ((abs o. - ) |` (RR X. RR))   &   |- J = (Open` C)   &   |- K = (Open` D)   =>   |- (U e. NrmCVec -> N e. (J Cn K))
Theorem	nmcn2 8304	The norm of a normed complex vector space is a continuous function to RR.
|- N = (norm` U)   &   |- D = (IndMet` U)   &   |- J = (Open` D)   =>   |- (U e. NrmCVec -> N e. (J Cn (topGen` ran (,))))
Theorem	nmcnc 8305	The norm of a normed complex vector space is a continuous function to CC. (For RR, see nmcn 8303.)
|- N = (norm` U)   &   |- C = (IndMet` U)   &   |- D = (abs o. - )   &   |- J = (Open` C)   &   |- K = (Open` D)   =>   |- (U e. NrmCVec -> N e. (J Cn K))
Theorem	abscn 8306	The absolute value function on complex numbers is continuous.
|- C = (abs o. - )   &   |- R = ((abs o. - ) |` (RR X. RR))   &   |- J = (Open` C)   &   |- K = (Open` R)   =>   |- abs e. (J Cn K)
Theorem	abscncfALT 8307	Absolute value is continuous. Alternate proof of abscncf 7228.
|- abs e. (CC-cn->RR)
Theorem	va1cnlem 8308	Lemma for va1cn 8309.
Theorem	va1cn 8309	Vector addition is continuous in its first operand.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- D = (IndMet` U)   &   |- J = (Open` D)   &   |- F = {<.w, v>. | (w e. X /\ v = (wGA))}   &   |- U e. NrmCVec   =>   |- (A e. X -> F e. (J Cn J))
Theorem	sm1cnilem 8310	Lemma for sm1cni 8311.
Theorem	sm1cni 8311	Scalar multiplication is continuous in its first operand.
|- X = (Base` U)   &   |- S = (.s` U)   &   |- C = (abs o. - )   &   |- D = (IndMet` U)   &   |- J = (Open` C)   &   |- K = (Open` D)   &   |- F = {<.w, v>. | (w e. CC /\ v = (wSA))}   &   |- U e. NrmCVec   &   |- A e. X   =>   |- F e. (J Cn K)
Inner product
Syntax	cip 8312	Extend class notation with the class inner product functions.
class .i
Definition	df-ip 8313	Define a function that maps a complex normed vector space to its inner product operation in case its norm satisfies the parallelogram identity (otherwise the operation is still defined, but not meaningful). Based on Exercise 4(a) of [ReedSimon] p. 63 and Theorem 6.44 of [Ponnusamy] p. 361. Vector addition is (1st` w), the scalar product is (2nd` w), and the norm is n.
|- .i = {<.<.w, n>., p>. | (<.w, n>. e. NrmCVec /\ p = {<.<.x, y>., z>. | ((x e. dom n /\ y e. dom n) /\ z = (sum_k e. (1...4)((i^k) x. ((n` (x(1st` w)((i^k)(2nd` w)y)))^2)) / 4))})}
Theorem	ipval2lem1 8314	Lemma for ipval3 8322.
Theorem	ipfval 8315	The inner product function on a normed complex vector space. The definition is meaningful for vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   =>   |- (U e. NrmCVec -> P = {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (sum_k e. (1...4)((i^k) x. ((N` (xG((i^k)Sy)))^2)) / 4))})
Theorem	ipval 8316	Value of the inner product. The definition is meaningful for normed complex vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law, although for convenience we define it for any normed complex vector space. The vector (group) addition operation is G, the scalar product is S, the norm is N, and the set of vectors is X. Equation 6.45 of [Ponnusamy] p. 361.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (APB) = (sum_k e. (1...4)((i^k) x. ((N` (AG((i^k)SB)))^2)) / 4))
Theorem	ipval2lem2 8317	Lemma for ipval3 8322.
Theorem	ipval2lem3 8318	Lemma for ipval3 8322.
Theorem	ipval2lem4 8319	Lemma for ipval3 8322.
Theorem	ipval2 8320	Expansion of the inner product value ipval 8316. Warning: The HTML proof page is 0.5MB in size.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (APB) = (((((N` (AGB))^2) - ((N` (AG(-u1SB)))^2)) + (i x. (((N` (AG(iSB)))^2) - ((N` (AG(-uiSB)))^2)))) / 4))
Theorem	4ipval2 8321	Four times the inner product value ipval3 8322, useful for simplifying certain proofs.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (4 x. (APB)) = ((((N` (AGB))^2) - ((N` (AG(-u1SB)))^2)) + (i x. (((N` (AG(iSB)))^2) - ((N` (AG(-uiSB)))^2)))))
Theorem	ipval3 8322	Expansion of the inner product value ipval 8316.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   &   |- M = (-v` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (APB) = (((((N` (AGB))^2) - ((N` (AMB))^2)) + (i x. (((N` (AG(iSB)))^2) - ((N` (AM(iSB)))^2)))) / 4))
Theorem	ipval2lem5 8323	Lemma for ipval3 8322.
Theorem	ipval2lem6 8324	Lemma for ipval3 8322.
Theorem	4ipval3 8325	Four times the inner product value ipval3 8322, useful for simplifying certain proofs.
|- X = (Base` U)   &   |- G = (+v` U)   &   |- S = (.s` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   &   |- M = (-v` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (4 x. (APB)) = ((((N` (AGB))^2) - ((N` (AMB))^2)) + (i x. (((N` (AG(iSB)))^2) - ((N` (AM(iSB)))^2)))))
Theorem	ipid 8326	The inner product of a vector with itself is the square of the vector's norm. Equation I4 of [Ponnusamy] p. 362.
|- X = (Base` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (APA) = ((N` A)^2))
Theorem	ipnm 8327	Norm expressed in terms of inner product.
|- X = (Base` U)   &   |- N = (norm` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (N` A) = (sqr` (APA)))
Theorem	ipcl 8328	An inner product is a complex number.
|- X = (Base` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (APB) e. CC)
Theorem	ipf 8329	Mapping for the inner product operation.
|- X = (Base` U)   &   |- P = (.i` U)   =>   |- (U e. NrmCVec -> P:(X X. X)-->CC)
Theorem	ipcj 8330	The complex conjugate of an inner product reverses its arguments. Equation I1 of [Ponnusamy] p. 362.
|- X = (Base` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (*` (APB)) = (BPA))
Theorem	ipipcj 8331	An inner product times its conjugate.
|- X = (Base` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((APB) x. (BPA)) = ((abs` (APB))^2))
Theorem	iporthcom 8332	Orthogonality (meaning inner product is 0) is commutative.
|- X = (Base` U)   &   |- P = (