Theorem ipfval 8314

Description: 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.

Hypotheses

Ref	Expression
ipfval.1	|- X = (Base` U)
ipfval.2	|- G = (+v` U)
ipfval.4	|- S = (.s` U)
ipfval.6	|- N = (norm` U)
ipfval.7	|- P = (.i` U)

Assertion

Ref	Expression
ipfval	|- (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))})

Distinct variable groups:   x,k,y,z,G   k,N,x,y,z   S,k,x,y,z   U,k,x,y,z   x,X,y,z

Proof of Theorem ipfval

Step	Hyp	Ref	Expression
1		nvrel 8185	. . . . 5 |- Rel NrmCVec
2		1st2nd 4101	. . . . 5 |- ((Rel NrmCVec /\ U e. NrmCVec) -> U = <.(1st` U), (2nd` U)>.)
3	1, 2	mpan 694	. . . 4 |- (U e. NrmCVec -> U = <.(1st` U), (2nd` U)>.)
4	3	fveq2d 3723	. . 3 |- (U e. NrmCVec -> (.i` U) = (.i` <.(1st` U), (2nd` U)>.))
5		ipfval.7	. . 3 |- P = (.i` U)
6	4, 5	syl5eq 1517	. 2 |- (U e. NrmCVec -> P = (.i` <.(1st` U), (2nd` U)>.))
7		df-opr 3960	. . 3 |- ((1st` U).i(2nd` U)) = (.i` <.(1st` U), (2nd` U)>.)
8	7	a1i 8	. 2 |- (U e. NrmCVec -> ((1st` U).i(2nd` U)) = (.i` <.(1st` U), (2nd` U)>.))
9	3	eleq1d 1538	. . . . 5 |- (U e. NrmCVec -> (U e. NrmCVec <-> <.(1st` U), (2nd` U)>. e. NrmCVec))
10	9	ibi 591	. . . 4 |- (U e. NrmCVec -> <.(1st` U), (2nd` U)>. e. NrmCVec)
11		fvex 3727	. . . . 5 |- (1st` U) e. V
12		fvex 3727	. . . . 5 |- (2nd` U) e. V
13		ipfval.6	. . . . . . . . 9 |- N = (norm` U)
14		fvex 3727	. . . . . . . . 9 |- (norm` U) e. V
15	13, 14	eqeltr 1542	. . . . . . . 8 |- N e. V
16	15	dmex 3356	. . . . . . 7 |- dom N e. V
17		eqid 1474	. . . . . . 7 |- {<.<.x, y>., z>. | ((x e. dom N /\ y e. dom N) /\ z = (sum_k e. (1...4)((i^k) x. ((N` (xG((i^k)Sy)))^2)) / 4))} = {<.<.x, y>., z>. | ((x e. dom N /\ y e. dom N) /\ z = (sum_k e. (1...4)((i^k) x. ((N` (xG((i^k)Sy)))^2)) / 4))}
18	16, 16, 17	oprabex2 4016	. . . . . 6 |- {<.<.x, y>., z>. | ((x e. dom N /\ y e. dom N) /\ z = (sum_k e. (1...4)((i^k) x. ((N` (xG((i^k)Sy)))^2)) / 4))} e. V
19		visset 1810	. . . . . . . . 9 |- w e. V
20		visset 1810	. . . . . . . . 9 |- n e. V
21	19, 20, 12	opth 2783	. . . . . . . 8 |- (<.w, n>. = <.(1st` U), (2nd` U)>. <-> (w = (1st` U) /\ n = (2nd` U)))
22		id 59	. . . . . . . . . . . . . . 15 |- (n = (2nd` U) -> n = (2nd` U))
23	13	nmfval 8190	. . . . . . . . . . . . . . 15 |- N = (2nd` U)
24	22, 23	syl6eqr 1523	. . . . . . . . . . . . . 14 |- (n = (2nd` U) -> n = N)
25	24	dmeqd 3309	. . . . . . . . . . . . 13 |- (n = (2nd` U) -> dom n = dom N)
26	25	eleq2d 1539	. . . . . . . . . . . 12 |- (n = (2nd` U) -> (x e. dom n <-> x e. dom N))
27	25	eleq2d 1539	. . . . . . . . . . . 12 |- (n = (2nd` U) -> (y e. dom n <-> y e. dom N))
28	26, 27	anbi12d 627	. . . . . . . . . . 11 |- (n = (2nd` U) -> ((x e. dom n /\ y e. dom n) <-> (x e. dom N /\ y e. dom N)))
29	28	adantl 388	. . . . . . . . . 10 |- ((w = (1st` U) /\ n = (2nd` U)) -> ((x e. dom n /\ y e. dom n) <-> (x e. dom N /\ y e. dom N)))
30	24	fveq1d 3721	. . . . . . . . . . . . . . . 16 |- (n = (2nd` U) -> (n` (x(1st` w)((i^k)(2nd` w)y))) = (N` (x(1st` w)((i^k)(2nd` w)y))))
31		fveq2 3719	. . . . . . . . . . . . . . . . . . . 20 |- (w = (1st` U) -> (1st` w) = (1st` (1st` U)))
32		ipfval.2	. . . . . . . . . . . . . . . . . . . . 21 |- G = (+v` U)
33	32	vafval 8186	. . . . . . . . . . . . . . . . . . . 20 |- G = (1st` (1st` U))
34	31, 33	syl6eqr 1523	. . . . . . . . . . . . . . . . . . 19 |- (w = (1st` U) -> (1st` w) = G)
35	34	opreqd 3972	. . . . . . . . . . . . . . . . . 18 |- (w = (1st` U) -> (x(1st` w)((i^k)(2nd` w)y)) = (xG((i^k)(2nd` w)y)))
36		fveq2 3719	. . . . . . . . . . . . . . . . . . . . 21 |- (w = (1st` U) -> (2nd` w) = (2nd` (1st` U)))
37		ipfval.4	. . . . . . . . . . . . . . . . . . . . . 22 |- S = (.s` U)
38	37	smfval 8188	. . . . . . . . . . . . . . . . . . . . 21 |- S = (2nd` (1st` U))
39	36, 38	syl6eqr 1523	. . . . . . . . . . . . . . . . . . . 20 |- (w = (1st` U) -> (2nd` w) = S)
40	39	opreqd 3972	. . . . . . . . . . . . . . . . . . 19 |- (w = (1st` U) -> ((i^k)(2nd` w)y) = ((i^k)Sy))
41	40	opreq2d 3971	. . . . . . . . . . . . . . . . . 18 |- (w = (1st` U) -> (xG((i^k)(2nd` w)y)) = (xG((i^k)Sy)))
42	35, 41	eqtrd 1505	. . . . . . . . . . . . . . . . 17 |- (w = (1st` U) -> (x(1st` w)((i^k)(2nd` w)y)) = (xG((i^k)Sy)))
43	42	fveq2d 3723	. . . . . . . . . . . . . . . 16 |- (w = (1st` U) -> (N` (x(1st` w)((i^k)(2nd` w)y))) = (N` (xG((i^k)Sy))))
44	30, 43	sylan9eqr 1527	. . . . . . . . . . . . . . 15 |- ((w = (1st` U) /\ n = (2nd` U)) -> (n` (x(1st` w)((i^k)(2nd` w)y))) = (N` (xG((i^k)Sy))))
45	44	opreq1d 3970	. . . . . . . . . . . . . 14 |- ((w = (1st` U) /\ n = (2nd` U)) -> ((n` (x(1st` w)((i^k)(2nd` w)y)))^2) = ((N` (xG((i^k)Sy)))^2))
46	45	opreq2d 3971	. . . . . . . . . . . . 13 |- ((w = (1st` U) /\ n = (2nd` U)) -> ((i^k) x. ((n` (x(1st` w)((i^k)(2nd` w)y)))^2)) = ((i^k) x. ((N` (xG((i^k)Sy)))^2)))
47	46	sumeq2sdv 6946	. . . . . . . . . . . 12 |- ((w = (1st` U) /\ n = (2nd` U)) -> sum_k e. (1...4)((i^k) x. ((n` (x(1st` w)((i^k)(2nd` w)y)))^2)) = sum_k e. (1...4)((i^k) x. ((N` (xG((i^k)Sy)))^2)))
48	47	opreq1d 3970	. . . . . . . . . . 11 |- ((w = (1st` U) /\ n = (2nd` U)) -> (sum_k e. (1...4)((i^k) x. ((n` (x(1st` w)((i^k)(2nd` w)y)))^2)) / 4) = (sum_k e. (1...4)((i^k) x. ((N` (xG((i^k)Sy)))^2)) / 4))
49	48	eqeq2d 1484	. . . . . . . . . 10 |- ((w = (1st` U) /\ n = (2nd` U)) -> (z = (sum_k e. (1...4)((i^k) x. ((n` (x(1st` w)((i^k)(2nd` w)y)))^2)) / 4) <-> z = (sum_k e. (1...4)((i^k) x. ((N` (xG((i^k)Sy)))^2)) / 4)))
50	29, 49	anbi12d 627	. . . . . . . . 9 |- ((w = (1st` U) /\ n = (2nd` U)) -> (((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)) <-> ((x e. dom N /\ y e. dom N) /\ z = (sum_k e. (1...4)((i^k) x. ((N` (xG((i^k)Sy)))^2)) / 4))))
51	50	oprabbidv 3991	. . . . . . . 8 |- ((w = (1st` U) /\ n = (2nd` U)) -> {<.<.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)