Theorem minvecex 9923

Description: Minimizing vector theorem (existence part). There is exactly one vector in a complete subspace W that minimizes the distance to an arbitrary vector A in a parent inner product space. Part of Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. Note that we work with the negative of supremum instead of infimum in order to use theorems we already have available.

Hypotheses

Ref	Expression
minvec10.1	|- R = {x | E.y e. Y x = -u(N` (AMy))}
minvec10.u	|- U e. CPreHil
minvec10.m	|- M = (-v` U)
minvec10.n	|- N = (norm` U)
minvec10.x	|- X = (BaseSet` U)
minvec10.w1	|- W e. (SubSp` U)
minvec10.y	|- Y = (BaseSet` W)
minvec10.a	|- A e. X
minvec33.2	|- P = -usup(R, RR, < )
minvec33.hf	|- (j e. NN -> (F` j) = (N` (AM(f` j))))
minvec33.d	|- D = (IndMet` W)
minvec33.t1	|- F e. _V
minvec34.w2	|- W e. CBan

Assertion

Ref	Expression
minvecex	|- E.a e. Y (N` (AMa)) = P

Distinct variable groups:   f,j,x,y,A   D,a   j,a,F   f,M,j,x,y   f,N,j,x,y   f,a,P,j   R,a   x,U,y   x,W,y   x,a,y,Y,f,j

Proof of Theorem minvecex

Step	Hyp	Ref	Expression
1		minvec10.1	. . . 4 |- R = {x | E.y e. Y x = -u(N` (AMy))}
2		eqeq1 1890	. . . . . . 7 |- (x = u -> (x = -u(N` (AMy)) <-> u = -u(N` (AMy))))
3	2	rexbidv 2124	. . . . . 6 |- (x = u -> (E.y e. Y x = -u(N` (AMy)) <-> E.y e. Y u = -u(N` (AMy))))
4	3	cbvabv 2420	. . . . 5 |- {x | E.y e. Y x = -u(N` (AMy))} = {u | E.y e. Y u = -u(N` (AMy))}
5		opreq2 4890	. . . . . . . . . 10 |- (y = t -> (AMy) = (AMt))
6	5	fveq2d 4685	. . . . . . . . 9 |- (y = t -> (N` (AMy)) = (N` (AMt)))
7	6	negeqd 6516	. . . . . . . 8 |- (y = t -> -u(N` (AMy)) = -u(N` (AMt)))
8	7	eqeq2d 1895	. . . . . . 7 |- (y = t -> (u = -u(N` (AMy)) <-> u = -u(N` (AMt))))
9	8	cbvrexv 2281	. . . . . 6 |- (E.y e. Y u = -u(N` (AMy)) <-> E.t e. Y u = -u(N` (AMt)))
10	9	abbii 2006	. . . . 5 |- {u | E.y e. Y u = -u(N` (AMy))} = {u | E.t e. Y u = -u(N` (AMt))}
11	4, 10	eqtri 1908	. . . 4 |- {x | E.y e. Y x = -u(N` (AMy))} = {u | E.t e. Y u = -u(N` (AMt))}
12	1, 11	eqtri 1908	. . 3 |- R = {u | E.t e. Y u = -u(N` (AMt))}
13		minvec10.u	. . . 4 |- U e. CPreHil
14		minvec10.w1	. . . 4 |- W e. (SubSp` U)
15		minvec10.y	. . . 4 |- Y = (BaseSet` W)
16		minvec10.x	. . . 4 |- X = (BaseSet` U)
17		minvec10.m	. . . 4 |- M = (-v` U)
18		minvec10.n	. . . 4 |- N = (norm` U)
19		minvec10.a	. . . 4 |- A e. X
20	13, 14, 15, 16, 17, 18, 19	minveclem5 9894	. . 3 |- (t e. Y -> (N` (AMt)) e. RR)
21	13	phnvi 9816	. . . . 5 |- U e. NrmCVec
22		eqid 1884	. . . . . 6 |- (SubSp` U) = (SubSp` U)
23	22	sspnv 9724	. . . . 5 |- ((U e. NrmCVec /\ W e. (SubSp` U)) -> W e. NrmCVec)
24	21, 14, 23	mp2an 761	. . . 4 |- W e. NrmCVec
25		eqid 1884	. . . . 5 |- (0v` W) = (0v` W)
26	15, 25	nvzcl 9587	. . . 4 |- (W e. NrmCVec -> (0v` W) e. Y)
27	24, 26	ax-mp 7	. . 3 |- (0v` W) e. Y
28	1, 13, 17, 18, 16, 14, 15, 19	minveclem10 9899	. . 3 |- E.u e. RR A.t e. R t <_ u
29		minvec33.2	. . 3 |- P = -usup(R, RR, < )
30		minvec33.t1	. . 3 |- F e. _V
31		opreq2 4890	. . . 4 |- (t = (f` j) -> (AMt) = (AM(f` j)))
32	31	fveq2d 4685	. . 3 |- (t = (f` j) -> (N` (AMt)) = (N` (AM(f` j))))
33		minvec33.hf	. . 3 |- (j e. NN -> (F` j) = (N` (AM(f` j))))
34	12, 20, 27, 28, 29, 30, 32, 33	infcvgi 8485	. 2 |- E.f(f:NN-->Y /\ F ~~> P)
35		minvec33.d	. . . . 5 |- D = (IndMet` W)
36		minvec34.w2	. . . . 5 |- W e. CBan
37	12, 13, 17, 18, 16, 14, 15, 19, 33, 29, 35, 36	minveclem29 9918	. . . 4 |- ((f:NN-->Y /\ F ~~> P) -> E.a e. Y f(~~>m` D)a)
38	1, 13, 17, 18, 16, 14, 15, 19, 29, 33, 35, 30	minveclem33 9922	. . . . . . . . 9 |- (((f:NN-->Y /\ f(~~>m` D)a) /\ (F ~~> P /\ a e. Y)) -> (N` (AMa)) = P)
39	38	exp43 415	. . . . . . . 8 |- (f:NN-->Y -> (f(~~>m` D)a -> (F ~~> P -> (a e. Y -> (N` (AMa)) = P))))
40	39	com23 36	. . . . . . 7 |- (f:NN-->Y -> (F ~~> P -> (f(~~>m` D)a -> (a e. Y -> (N` (AMa)) = P))))
41	40	com34 40	. . . . . 6 |- (f:NN-->Y -> (F ~~> P -> (a e. Y -> (f(~~>m` D)a -> (N` (AMa)) = P))))
42	41	imp31 389	. . . . 5 |- (((f:NN-->Y /\ F ~~> P) /\ a e. Y) -> (f(~~>m` D)a -> (N` (AMa)) = P))
43	42	reximdva 2203	. . . 4 |- ((f:NN-->Y /\ F ~~> P) -> (E.a e. Y f(~~>m` D)a -> E.a e. Y (N` (AMa)) = P))
44	37, 43	mpd 29	. . 3 |- ((f:NN-->Y /\ F ~~> P) -> E.a e. Y (N` (AMa)) = P)
45	44	19.23aiv 1674	. 2 |- (E.f(f:NN-->Y /\ F ~~> P) -> E.a e. Y (N` (AMa)) = P)
46	34, 45	ax-mp 7	1 |- E.a e. Y (N` (AMa)) = P

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  E.wrex 2106  _Vcvv 2292   class class class wbr 3338  -->wf 3994  ` cfv 3998  (class class class)co 4884  supcsup 5663  RRcr 6385  -ucneg 6446  NNcn 6449   < clt 6653   ~~> cli 8234  ~~>mclm 9197  NrmCVeccnv 9535  BaseSetcba 9537  0vcn0v 9539  -vcnsb 9540  normcnm 9541  IndMetcims 9542  SubSpcss 9719  CPreHilcphl 9812  CBancbn 9864

This theorem is referenced by:  minveceu 9928

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-reg 5695  ax-inf2 5731  ax-ac 5906

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-iin 3258  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-sup 5664  df-r1 5750  df-rank 5751  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-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-rp 7232  df-n0 7309  df-z 7345  df-uz 7587  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-met 9070  df-lm 9200  df-cau 9201  df-cmet 9202  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-vs 9550  df-nm 9551  df-ims 9552  df-ssp 9720  df-ph 9813  df-bn 9865

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