Theorem dipfval 27557

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. (Contributed by NM, 10-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dipfval.1	|- X = ( BaseSet ` U )
dipfval.2	|- G = ( +v ` U )
dipfval.4	|- S = ( .sOLD ` U )
dipfval.6	|- N = ( normCV ` U )
dipfval.7	|- P = ( .iOLD ` U )

Assertion

Ref	Expression
dipfval	|- ( U e. NrmCVec -> P = ( x e. X , y e. X |-> ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( N ` ( x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) ) / 4 ) ) )

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

Allowed substitution hints:   P( x, y, k)

Proof of Theorem dipfval

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dipfval.7	. 2 |- P = ( .iOLD ` U )
2		fveq2 6191	. . . . 5 |- ( u = U -> ( BaseSet ` u ) = ( BaseSet ` U ) )
3		dipfval.1	. . . . 5 |- X = ( BaseSet ` U )
4	2, 3	syl6eqr 2674	. . . 4 |- ( u = U -> ( BaseSet ` u ) = X )
5		fveq2 6191	. . . . . . . . . 10 |- ( u = U -> ( normCV ` u ) = ( normCV ` U ) )
6		dipfval.6	. . . . . . . . . 10 |- N = ( normCV ` U )
7	5, 6	syl6eqr 2674	. . . . . . . . 9 |- ( u = U -> ( normCV ` u ) = N )
8		fveq2 6191	. . . . . . . . . . 11 |- ( u = U -> ( +v ` u ) = ( +v ` U ) )
9		dipfval.2	. . . . . . . . . . 11 |- G = ( +v ` U )
10	8, 9	syl6eqr 2674	. . . . . . . . . 10 |- ( u = U -> ( +v ` u ) = G )
11		eqidd 2623	. . . . . . . . . 10 |- ( u = U -> x = x )
12		fveq2 6191	. . . . . . . . . . . 12 |- ( u = U -> ( .sOLD ` u ) = ( .sOLD ` U ) )
13		dipfval.4	. . . . . . . . . . . 12 |- S = ( .sOLD ` U )
14	12, 13	syl6eqr 2674	. . . . . . . . . . 11 |- ( u = U -> ( .sOLD ` u ) = S )
15	14	oveqd 6667	. . . . . . . . . 10 |- ( u = U -> ( ( _i ^ k ) ( .sOLD ` u ) y ) = ( ( _i ^ k ) S y ) )
16	10, 11, 15	oveq123d 6671	. . . . . . . . 9 |- ( u = U -> ( x ( +v ` u ) ( ( _i ^ k ) ( .sOLD ` u ) y ) ) = ( x G ( ( _i ^ k ) S y ) ) )
17	7, 16	fveq12d 6197	. . . . . . . 8 |- ( u = U -> ( ( normCV ` u ) ` ( x ( +v ` u ) ( ( _i ^ k ) ( .sOLD ` u ) y ) ) ) = ( N ` ( x G ( ( _i ^ k ) S y ) ) ) )
18	17	oveq1d 6665	. . . . . . 7 |- ( u = U -> ( ( ( normCV ` u ) ` ( x ( +v ` u ) ( ( _i ^ k ) ( .sOLD ` u ) y ) ) ) ^ 2 ) = ( ( N ` ( x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) )
19	18	oveq2d 6666	. . . . . 6 |- ( u = U -> ( ( _i ^ k ) x. ( ( ( normCV ` u ) ` ( x ( +v ` u ) ( ( _i ^ k ) ( .sOLD ` u ) y ) ) ) ^ 2 ) ) = ( ( _i ^ k ) x. ( ( N ` ( x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) ) )
20	19	sumeq2sdv 14435	. . . . 5 |- ( u = U -> sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` u ) ` ( x ( +v ` u ) ( ( _i ^ k ) ( .sOLD ` u ) y ) ) ) ^ 2 ) ) = sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( N ` ( x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) ) )
21	20	oveq1d 6665	. . . 4 |- ( u = U -> ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` u ) ` ( x ( +v ` u ) ( ( _i ^ k ) ( .sOLD ` u ) y ) ) ) ^ 2 ) ) / 4 ) = ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( N ` ( x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) ) / 4 ) )
22	4, 4, 21	mpt2eq123dv 6717	. . 3 |- ( u = U -> ( x e. ( BaseSet ` u ) , y e. ( BaseSet ` u ) |-> ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` u ) ` ( x ( +v ` u ) ( ( _i ^ k ) ( .sOLD ` u ) y ) ) ) ^ 2 ) ) / 4 ) ) = ( x e. X , y e. X |-> ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( N ` ( x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) ) / 4 ) ) )
23		df-dip 27556	. . 3 |- .iOLD = ( u e. NrmCVec |-> ( x e. ( BaseSet ` u ) , y e. ( BaseSet ` u ) |-> ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` u ) ` ( x ( +v ` u ) ( ( _i ^ k ) ( .sOLD ` u ) y ) ) ) ^ 2 ) ) / 4 ) ) )
24		fvex 6201	. . . . 5 |- ( BaseSet ` U ) e. _V
25	3, 24	eqeltri 2697	. . . 4 |- X e. _V
26	25, 25	mpt2ex 7247	. . 3 |- ( x e. X , y e. X |-> ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( N ` ( x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) ) / 4 ) ) e. _V
27	22, 23, 26	fvmpt 6282	. 2 |- ( U e. NrmCVec -> ( .iOLD ` U ) = ( x e. X , y e. X |-> ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( N ` ( x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) ) / 4 ) ) )
28	1, 27	syl5eq 2668	1 |- ( U e. NrmCVec -> P = ( x e. X , y e. X |-> ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( N ` ( x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) ) / 4 ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1c1 9937   _ici 9938   x. cmul 9941   / cdiv 10684   2c2 11070   4c4 11072   ...cfz 12326   ^cexp 12860   sum_csu 14416   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   norm_CVcnmcv 27445   .iOLDcdip 27555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-sum 14417  df-dip 27556

This theorem is referenced by:  ipval  27558  ipf  27568  dipcn  27575