Theorem minveceu 8579

Description: Minimizing vector theorem. 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. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. Note that we work with the negative of the supremum of negatives instead of infimum in order to use theorems we already have available.

Hypotheses

Ref	Expression
minvec.x	⊢ X = (Base ‘U)
minvec.m	⊢ M = ( −v ‘U)
minvec.n	⊢ N = (norm ‘U)
minvec.y	⊢ Y = (Base ‘W)
minvec.1	⊢ R = {x∣∃y ∈ Y x = -(N ‘(AMy))}
minvec.2	⊢ P = -sup(R, ℝ, < )
minvec.u	⊢ U ∈ CPreHil
minvec.w	⊢ W ∈ ((SubSp ‘U) ∩ CBan)
minvec.a	⊢ A ∈ X

Assertion

Ref	Expression
minveceu	⊢ ∃!a ∈ Y (N ‘(AMa)) = P

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

Proof of Theorem minveceu

Step	Hyp	Ref	Expression
1		opreq2 3975	. . . . 5 ⊢ (a = b → (AMa) = (AMb))
2	1	fveq2d 3734	. . . 4 ⊢ (a = b → (N ‘(AMa)) = (N ‘(AMb)))
3	2	eqeq1d 1486	. . 3 ⊢ (a = b → ((N ‘(AMa)) = P ↔ (N ‘(AMb)) = P))
4	3	reu4 1937	. 2 ⊢ (∃!a ∈ Y (N ‘(AMa)) = P ↔ (∃a ∈ Y (N ‘(AMa)) = P ⋀ ∀a ∈ Y ∀b ∈ Y (((N ‘(AMa)) = P ⋀ (N ‘(AMb)) = P) → a = b)))
5		minvec.1	. . 3 ⊢ R = {x∣∃y ∈ Y x = -(N ‘(AMy))}
6		minvec.u	. . 3 ⊢ U ∈ CPreHil
7		minvec.m	. . 3 ⊢ M = ( −v ‘U)
8		minvec.n	. . 3 ⊢ N = (norm ‘U)
9		minvec.x	. . 3 ⊢ X = (Base ‘U)
10		minvec.w	. . . 4 ⊢ W ∈ ((SubSp ‘U) ∩ CBan)
11		inss1 2233	. . . . 5 ⊢ ((SubSp ‘U) ∩ CBan) ⊆ (SubSp ‘U)
12	11	sseli 2068	. . . 4 ⊢ (W ∈ ((SubSp ‘U) ∩ CBan) → W ∈ (SubSp ‘U))
13	10, 12	ax-mp 7	. . 3 ⊢ W ∈ (SubSp ‘U)
14		minvec.y	. . 3 ⊢ Y = (Base ‘W)
15		minvec.a	. . 3 ⊢ A ∈ X
16		minvec.2	. . 3 ⊢ P = -sup(R, ℝ, < )
17		fveq2 3730	. . . . . 6 ⊢ (j = n → (f ‘j) = (f ‘n))
18	17	opreq2d 3982	. . . . 5 ⊢ (j = n → (AM(f ‘j)) = (AM(f ‘n)))
19	18	fveq2d 3734	. . . 4 ⊢ (j = n → (N ‘(AM(f ‘j))) = (N ‘(AM(f ‘n))))
20		eqid 1478	. . . 4 ⊢ {⟨j, v⟩∣(j ∈ ℕ ⋀ v = (N ‘(AM(f ‘j))))} = {⟨j, v⟩∣(j ∈ ℕ ⋀ v = (N ‘(AM(f ‘j))))}
21		fvex 3738	. . . 4 ⊢ (N ‘(AM(f ‘n))) ∈ V
22	19, 20, 21	fvopab4 3786	. . 3 ⊢ (n ∈ ℕ → ({⟨j, v⟩∣(j ∈ ℕ ⋀ v = (N ‘(AM(f ‘j))))} ‘n) = (N ‘(AM(f ‘n))))
23		eqid 1478	. . 3 ⊢ (IndMet ‘W) = (IndMet ‘W)
24		nnex 5935	. . . 4 ⊢ ℕ ∈ V
25	24	opabex2 3616	. . 3 ⊢ {⟨j, v⟩∣(j ∈ ℕ ⋀ v = (N ‘(AM(f ‘j))))} ∈ V
26		inss2 2234	. . . . 5 ⊢ ((SubSp ‘U) ∩ CBan) ⊆ CBan
27	26	sseli 2068	. . . 4 ⊢ (W ∈ ((SubSp ‘U) ∩ CBan) → W ∈ CBan)
28	10, 27	ax-mp 7	. . 3 ⊢ W ∈ CBan
29	5, 6, 7, 8, 9, 13, 14, 15, 16, 22, 23, 25, 28	minvecex 8574	. 2 ⊢ ∃a ∈ Y (N ‘(AMa)) = P
30		eqid 1478	. . . . . 6 ⊢ ( +v ‘U) = ( +v ‘U)
31		eqid 1478	. . . . . 6 ⊢ ( ·s ‘U) = ( ·s ‘U)
32	9, 30, 7, 31, 8, 14, 6, 15, 13, 16, 5	minveclem38 8578	. . . . 5 ⊢ (((a ∈ Y ⋀ b ∈ Y) ⋀ ((N ‘(AMa)) = P ⋀ (N ‘(AMb)) = P)) → (N ‘(aMb)) ≤ 0)
33	6	phnvi 8471	. . . . . . . . . . 11 ⊢ U ∈ NrmCVec
34	9, 7	nvmcl 8263	. . . . . . . . . . 11 ⊢ ((U ∈ NrmCVec ⋀ a ∈ X ⋀ b ∈ X) → (aMb) ∈ X)
35	33, 34	mp3an1 905	. . . . . . . . . 10 ⊢ ((a ∈ X ⋀ b ∈ X) → (aMb) ∈ X)
36	9, 8	nvge0 8298	. . . . . . . . . . 11 ⊢ ((U ∈ NrmCVec ⋀ (aMb) ∈ X) → 0 ≤ (N ‘(aMb)))
37	33, 36	mpan 697	. . . . . . . . . 10 ⊢ ((aMb) ∈ X → 0 ≤ (N ‘(aMb)))
38	35, 37	syl 10	. . . . . . . . 9 ⊢ ((a ∈ X ⋀ b ∈ X) → 0 ≤ (N ‘(aMb)))
39		idd 61	. . . . . . . . 9 ⊢ ((a ∈ X ⋀ b ∈ X) → (((N ‘(aMb)) ≤ 0 ⋀ 0 ≤ (N ‘(aMb))) → ((N ‘(aMb)) ≤ 0 ⋀ 0 ≤ (N ‘(aMb)))))
40	38, 39	mpan2d 704	. . . . . . . 8 ⊢ ((a ∈ X ⋀ b ∈ X) → ((N ‘(aMb)) ≤ 0 → ((N ‘(aMb)) ≤ 0 ⋀ 0 ≤ (N ‘(aMb)))))
41		eqid 1478	. . . . . . . . . . . 12 ⊢ (0v ‘U) = (0v ‘U)
42	9, 41, 8	nvz 8293	. . . . . . . . . . 11 ⊢ ((U ∈ NrmCVec ⋀ (aMb) ∈ X) → ((N ‘(aMb)) = 0 ↔ (aMb) = (0v ‘U)))
43	33, 42	mpan 697	. . . . . . . . . 10 ⊢ ((aMb) ∈ X → ((N ‘(aMb)) = 0 ↔ (aMb) = (0v ‘U)))
44	35, 43	syl 10	. . . . . . . . 9 ⊢ ((a ∈ X ⋀ b ∈ X) → ((N ‘(aMb)) = 0 ↔ (aMb) = (0v ‘U)))
45	9, 8	nvcl 8283	. . . . . . . . . . . 12 ⊢ ((U ∈ NrmCVec ⋀ (aMb) ∈ X) → (N ‘(aMb)) ∈ ℝ)
46	33, 45	mpan 697	. . . . . . . . . . 11 ⊢ ((aMb) ∈ X → (N ‘(aMb)) ∈ ℝ)
47	35, 46	syl 10	. . . . . . . . . 10 ⊢ ((a ∈ X ⋀ b ∈ X) → (N ‘(aMb)) ∈ ℝ)
48		0re 5452	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
49		letri3t 5529	. . . . . . . . . . 11 ⊢ (((N ‘(aMb)) ∈ ℝ ⋀ 0 ∈ ℝ) → ((N ‘(aMb)) = 0 ↔ ((N ‘(aMb)) ≤ 0 ⋀ 0 ≤ (N ‘(aMb)))))
50	48, 49	mpan2 698	. . . . . . . . . 10 ⊢ ((N ‘(aMb)) ∈ ℝ → ((N ‘(aMb)) = 0 ↔ ((N ‘(aMb)) ≤ 0 ⋀ 0 ≤ (N ‘(aMb)))))
51	47, 50	syl 10	. . . . . . . . 9 ⊢ ((a ∈ X ⋀ b ∈ X) → ((N ‘(aMb)) = 0 ↔ ((N ‘(aMb)) ≤ 0 ⋀ 0 ≤ (N ‘(aMb)))))
52	9, 7, 41	nvmeq0 8280	. . . . . . . . . 10 ⊢ ((U ∈ NrmCVec ⋀ a ∈ X ⋀ b ∈ X) → ((aMb) = (0v ‘U) ↔ a = b))
53	33, 52	mp3an1 905	. . . . . . . . 9 ⊢ ((a ∈ X ⋀ b ∈ X) → ((aMb) = (0v ‘U) ↔ a = b))
54	44, 51, 53	3bitr3d 550	. . . . . . . 8 ⊢ ((a ∈ X ⋀ b ∈ X) → (((N ‘(aMb)) ≤ 0 ⋀ 0 ≤ (N ‘(aMb))) ↔ a = b))
55	40, 54	sylibd 202	. . . . . . 7 ⊢ ((a ∈ X ⋀ b ∈ X) → ((N ‘(aMb)) ≤ 0 → a = b))
56	6, 13, 14, 9	minveclem3 8543	. . . . . . . 8 ⊢ Y ⊆ X
57	56	sseli 2068	. . . . . . 7 ⊢ (a ∈ Y → a ∈ X)
58	56	sseli 2068	. . . . . . 7 ⊢ (b ∈ Y → b ∈ X)
59	55, 57, 58	syl2an 456	. . . . . 6 ⊢ ((a ∈ Y ⋀ b ∈ Y) → ((N ‘(aMb)) ≤ 0 → a = b))
60	59	adantr 391	. . . . 5 ⊢ (((a ∈ Y ⋀ b ∈ Y) ⋀ ((N ‘(AMa)) = P ⋀ (N ‘(AMb)) = P)) → ((N ‘(aMb)) ≤ 0 → a = b))
61	32, 60	mpd 26	. . . 4 ⊢ (((a ∈ Y ⋀ b ∈ Y) ⋀ ((N ‘(AMa)) = P ⋀ (N ‘(AMb)) = P)) → a = b)
62	61	ex 373	. . 3 ⊢ ((a ∈ Y ⋀ b ∈ Y) → (((N ‘(AMa)) = P ⋀ (N ‘(AMb)) = P) → a = b))
63	62	rgen2a 1702	. 2 ⊢ ∀a ∈ Y ∀b ∈ Y (((N ‘(AMa)) = P ⋀ (N ‘(AMb)) = P) → a = b)
64	4, 29, 63	mpbir2an 732	1 ⊢ ∃!a ∈ Y (N ‘(AMa)) = P

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  {cab 1466  ∀wral 1648  ∃wrex 1649  ∃!wreu 1650   ∩ cin 2049   class class class wbr 2624  {copab 2671   ‘cfv 3188  (class class class)co 3969  supcsup 4582  ℝcr 5245  0cc0 5246  -cneg 5305   ≤ cle 5307  ℕcn 5308   < clt 5498  NrmCVeccnv 8199   +_v cpv 8200  Basecba 8201   ·_s cns 8202  0_vcn0v 8203   −_v cnsb 8204  normcnm 8205  IndMetcims 8206  SubSpcss 8376  CPreHilcphl 8467  CBancbn 8518

This theorem is referenced by:  minveccl 8580  minvecdist 8581

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-iin 2573  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-r1 4653  df-rank 4654  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-3 5973  df-4 5974  df-n0 6102  df-z 6138  df-rp 6282  df-seq1 6309  df-uz 6419  df-exp 6570  df-sqr 6671  df-re 6752  df-im 6753  df-cj 6754  df-abs 6755  df-clim 6975  df-met 7790  df-lm 7919  df-cau 7920  df-cmet 7921  df-grp 8034  df-gid 8035  df-ginv 8036  df-gdiv 8037  df-abl 8096  df-vc 8161  df-nv 8207  df-va 8210  df-ba 8211  df-sm 8212  df-0v 8213  df-vs 8214  df-nm 8215  df-ims 8216  df-ssp 8377  df-ph 8468  df-bn 8519

Copyright terms: Public domain