mmtheorems84 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10682

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

Type	Label	Description
Statement
Theorem	nmcnc 8301	The norm of a normed complex vector space is a continuous function to CC. (For RR, see nmcn 8299.)
|- 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 8302	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 8303	Absolute value is continuous. Alternate proof of abscncf 7228.
|- abs e. (CC-cn->RR)
Theorem	va1cnlem 8304	Lemma for va1cn 8305.
Theorem	va1cn 8305	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 8306	Lemma for sm1cni 8307.
Theorem	sm1cni 8307	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 8308	Extend class notation with the class inner product functions.
class .i
Definition	df-ip 8309	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 8310	Lemma for ipval3 8318.
Theorem	ipfval 8311	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 8312	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 8313	Lemma for ipval3 8318.
Theorem	ipval2lem3 8314	Lemma for ipval3 8318.
Theorem	ipval2lem4 8315	Lemma for ipval3 8318.
Theorem	ipval2 8316	Expansion of the inner product value ipval 8312. 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 8317	Four times the inner product value ipval3 8318, 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 8318	Expansion of the inner product value ipval 8312.
|- 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 8319	Lemma for ipval3 8318.
Theorem	ipval2lem6 8320	Lemma for ipval3 8318.
Theorem	4ipval3 8321	Four times the inner product value ipval3 8318, 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 8322	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 8323	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 8324	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 8325	Mapping for the inner product operation.
|- X = (Base` U)   &   |- P = (.i` U)   =>   |- (U e. NrmCVec -> P:(X X. X)-->CC)
Theorem	ipcj 8326	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 8327	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 8328	Orthogonality (meaning inner product is 0) is commutative.
|- X = (Base` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((APB) = 0 <-> (BPA) = 0))
Theorem	ip0r 8329	Inner product with a zero second argument.
|- X = (Base` U)   &   |- Z = (0v` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (APZ) = 0)
Theorem	ip0l 8330	Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361.
|- X = (Base` U)   &   |- Z = (0v` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> (ZPA) = 0)
Theorem	ipz 8331	The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129.
|- X = (Base` U)   &   |- Z = (0v` U)   &   |- P = (.i` U)   =>   |- ((U e. NrmCVec /\ A e. X) -> ((APA) = 0