Theorem ipfval 9691

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 = (BaseSet` 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 9553	. . . . 5 |- Rel NrmCVec
2		1st2nd 5048	. . . . 5 |- ((Rel NrmCVec /\ U e. NrmCVec) -> U = <.(1st` U), (2nd` U)>.)
3	1, 2	mpan 759	. . . 4 |- (U e. NrmCVec -> U = <.(1st` U), (2nd` U)>.)
4	3	fveq2d 4685	. . 3 |- (U e. NrmCVec -> (.i` U) = (.i` <.(1st` U), (2nd` U)>.))
5		ipfval.7	. . 3 |- P = (.i` U)
6	4, 5	syl5eq 1940	. 2 |- (U e. NrmCVec -> P = (.i` <.(1st` U), (2nd` U)>.))
7		df-opr 4886	. . 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 1963	. . . . 5 |- (U e. NrmCVec -> (U e. NrmCVec <-> <.(1st` U), (2nd` U)>. e. NrmCVec))
10	9	ibi 652	. . . 4 |- (U e. NrmCVec -> <.(1st` U), (2nd` U)>. e. NrmCVec)
11		fvex 4689	. . . . 5 |- (1st` U) e. _V
12		fvex 4689	. . . . 5 |- (2nd` U) e. _V
13		ipfval.6	. . . . . . . . 9 |- N = (norm` U)
14		fvex 4689	. . . . . . . . 9 |- (norm` U) e. _V
15	13, 14	eqeltri 1967	. . . . . . . 8 |- N e. _V
16	15	dmex 4208	. . . . . . 7 |- dom N e. _V
17		eqid 1884	. . . . . . 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 4950	. . . . . 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 2295	. . . . . . . . 9 |- w e. _V
20		visset 2295	. . . . . . . . 9 |- n e. _V
21	19, 20, 12	opth 3532	. . . . . . . 8 |- (<.w, n>. = <.(1st` U), (2nd` U)>. <-> (w = (1st` U) /\ n = (2nd` U)))
22		id 73	. . . . . . . . . . . . . . 15 |- (n = (2nd` U) -> n = (2nd` U))
23	13	nmfval 9558	. . . . . . . . . . . . . . 15 |- N = (2nd` U)
24	22, 23	syl6eqr 1946	. . . . . . . . . . . . . 14 |- (n = (2nd` U) -> n = N)
25	24	dmeqd 4159	. . . . . . . . . . . . 13 |- (n = (2nd` U) -> dom n = dom N)
26	25	eleq2d 1964	. . . . . . . . . . . 12 |- (n = (2nd` U) -> (x e. dom n <-> x e. dom N))
27	25	eleq2d 1964	. . . . . . . . . . . 12 |- (n = (2nd` U) -> (y e. dom n <-> y e. dom N))
28	26, 27	anbi12d 690	. . . . . . . . . . 11 |- (n = (2nd` U) -> ((x e. dom n /\ y e. dom n) <-> (x e. dom N /\ y e. dom N)))
29	28	adantl 424	. . . . . . . . . 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 4683	. . . . . . . . . . . . . . . 16 |- (n = (2nd` U) -> (n` (x(1st` w)((_i^k)(2nd` w)y))) = (N` (x(1st` w)((_i^k)(2nd` w)y))))
31		fveq2 4681	. . . . . . . . . . . . . . . . . . . 20 |- (w = (1st` U) -> (1st` w) = (1st` (1st` U)))
32		ipfval.2	. . . . . . . . . . . . . . . . . . . . 21 |- G = (+v` U)
33	32	vafval 9554	. . . . . . . . . . . . . . . . . . . 20 |- G = (1st` (1st` U))
34	31, 33	syl6eqr 1946	. . . . . . . . . . . . . . . . . . 19 |- (w = (1st` U) -> (1st` w) = G)
35	34	opreqd 4899	. . . . . . . . . . . . . . . . . 18 |- (w = (1st` U) -> (x(1st` w)((_i^k)(2nd` w)y)) = (xG((_i^k)(2nd` w)y)))
36		fveq2 4681	. . . . . . . . . . . . . . . . . . . . 21 |- (w = (1st` U) -> (2nd` w) = (2nd` (1st` U)))
37		ipfval.4	. . . . . . . . . . . . . . . . . . . . . 22 |- S = (.s` U)
38	37	smfval 9556	. . . . . . . . . . . . . . . . . . . . 21 |- S = (2nd` (1st` U))
39	36, 38	syl6eqr 1946	. . . . . . . . . . . . . . . . . . . 20 |- (w = (1st` U) -> (2nd` w) = S)
40	39	opreqd 4899	. . . . . . . . . . . . . . . . . . 19 |- (w = (1st` U) -> ((_i^k)(2nd` w)y) = ((_i^k)Sy))
41	40	opreq2d 4898	. . . . . . . . . . . . . . . . . 18 |- (w = (1st` U) -> (xG((_i^k)(2nd` w)y)) = (xG((_i^k)Sy)))
42	35, 41	eqtrd 1925	. . . . . . . . . . . . . . . . 17 |- (w = (1st` U) -> (x(1st` w)((_i^k)(2nd` w)y)) = (xG((_i^k)Sy)))
43	42	fveq2d 4685	. . . . . . . . . . . . . . . 16 |- (w = (1st` U) -> (N` (x(1st` w)((_i^k)(2nd` w)y))) = (N` (xG((_i^k)Sy))))
44	30, 43	sylan9eqr 1951	. . . . . . . . . . . . . . 15 |- ((w = (1st` U) /\ n = (2nd` U)) -> (n` (x(1st` w)((_i^k)(2nd` w)y))) = (N` (xG((_i^k)Sy))))
45	44	opreq1d 4897	. . . . . . . . . . . . . 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 4898	. . . . . . . . . . . . 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 8253	. . . . . . . . . . . 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 4897	. . . . . . . . . . 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 1895	. . . . . . . . . 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 690	. . . . . . . . 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 4922	. . . . . . . 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)y)))^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))})
52	21, 51	sylbi 216	. . . . . . 7 |- (<.w, n>. = <.(1st` U), (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)y)))^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))})
53		df-ip 9689	. . . . . . 7 |- .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))})}
54	52, 53	oprabval6g 4962	. . . . . 6 |- ((((1st` U) e. _V /\ (2nd` U) e. _V /\ <.(1st` U), (2nd` U)>. e. NrmCVec) /\ {<.<.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) -> ((1st` U).i(2nd` U)) = {<.<.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))})
55	18, 54	mpan2 760	. . . . 5 |- (((1st` U) e. _V /\ (2nd` U) e. _V /\ <.(1st` U), (2nd` U)>. e. NrmCVec) -> ((1st` U).i(2nd` U)) = {<.<.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))})
56	11, 12, 55	mp3an12 1181	. . . 4 |- (<.(1st` U), (2nd` U)>. e. NrmCVec -> ((1st` U).i(2nd` U)) = {<.<.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))})
57	10, 56	syl 12	. . 3 |- (U e. NrmCVec -> ((1st` U).i(2nd` U)) = {<.<.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))})
58		ipfval.1	. . . . . . . . 9 |- X = (BaseSet` U)
59	58, 13	nvf 9618	. . . . . . . 8 |- (U e. NrmCVec -> N:X-->RR)
60		fdm 4567	. . . . . . . 8 |- (N:X-->RR -> dom N = X)
61	59, 60	syl 12	. . . . . . 7 |- (U e. NrmCVec -> dom N = X)
62	61	eleq2d 1964	. . . . . 6 |- (U e. NrmCVec -> (x e. dom N <-> x e. X))
63	61	eleq2d 1964	. . . . . 6 |- (U e. NrmCVec -> (y e. dom N <-> y e. X))
64	62, 63	anbi12d 690	. . . . 5 |- (U e. NrmCVec -> ((x e. dom N /\ y e. dom N) <-> (x e. X /\ y e. X)))
65	64	anbi1d 679	. . . 4 |- (U e. NrmCVec -> (((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 e. X /\ y e. X) /\ z = (sum_k e. (1...4)((_i^k) x. ((N` (xG((_i^k)Sy)))^2)) / 4))))
66	65	oprabbidv 4922	. . 3 |- (U e. NrmCVec -> {<.<.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. X /\ y e. X) /\ z = (sum_k e. (1...4)((_i^k) x. ((N` (xG((_i^k)Sy)))^2)) / 4))})
67	57, 66	eqtrd 1925	. 2 |- (U e. NrmCVec -> ((1st` U).i(2nd` U)) = {<.<.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))})
68	6, 8, 67	3eqtr2d 1932	1 |- (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))})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  <.cop 3046  dom cdm 3986  Rel wrel 3991  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  2ndc2nd 5019  RRcr 6385  1c1 6387  _ici 6388   x. cmul 6391   / cdiv 6447  2c2 7145  4c4 7147  ...cfz 7637  ^cexp 7811  sum_csu 8239  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538  normcnm 9541  .icip 9688

This theorem is referenced by:  ipval 9692  ipf 9705  hhip 10677

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-sum 8240  df-gid 9317  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-nm 9551  df-ip 9689

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