Theorem minveceu 8514

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 3954	. . . . 5 ⊢ (a = b → (AMa) = (AMb))
2	1	fveq2d 3713	. . . 4 ⊢ (a = b → (N ‘(AMa)) = (N ‘(AMb)))
3	2	eqeq1d 1475	. . 3 ⊢ (a = b → ((N ‘(AMa)) = P ↔ (N ‘(AMb)) = P))
4	3	reu4 1924	. 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 2220	. . . . 5 ⊢ ((SubSp ‘U) ∩ CBan) ⊆ (SubSp ‘U)
12	11	sseli 2055	. . . 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 3709	. . . . . 6 ⊢ (j = n → (f ‘j) = (f ‘n))
18	17	opreq2d 3961	. . . . 5 ⊢ (j = n → (AM(f ‘j)) = (AM(f ‘n)))
19	18	fveq2d 3713	. . . 4 ⊢ (j = n → (N ‘(AM(f ‘j))) = (N ‘(AM(f ‘n))))
20		eqid 1468	. . . 4 ⊢ {⟨j, v⟩∣(j ∈ ℕ ⋀ v = (N ‘(AM(f ‘j))))} = {⟨j, v⟩∣(j ∈ ℕ ⋀ v = (N ‘(AM(f ‘j))))}
21		fvex 3717	. . . 4 ⊢ (N ‘(AM(f ‘n))) ∈ V
22	19, 20, 21	fvopab4 3765	. . 3 ⊢ (n ∈ ℕ → ({⟨j, v⟩∣(j ∈ ℕ ⋀ v = (N ‘(AM(f ‘j))))} ‘n) = (N ‘(AM(f ‘n))))
23		eqid 1468	. . 3 ⊢ (IndMet ‘W) = (IndMet ‘W)
24		nnex 5881	. . . 4 ⊢ ℕ ∈ V
25	24	opabex2 3596	. . 3 ⊢ {⟨j, v⟩∣(j ∈ ℕ ⋀ v = (N ‘(AM(f ‘j))))} ∈ V
26		inss2 2221	. . . . 5 ⊢ ((SubSp ‘U) ∩ CBan) ⊆ CBan
27	26	sseli 2055	. . . 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 8509	. 2 ⊢ ∃a ∈ Y (N ‘(AMa)) = P
30		eqid 1468	. . . . . 6 ⊢ ( +v ‘U) = ( +v ‘U)
31		eqid 1468	. . . . . 6 ⊢ ( ·s ‘U) = ( ·s ‘U)
32	9, 30, 7, 31, 8, 14, 6, 15, 13, 16, 5	minveclem38 8513	. . . . 5 ⊢ (((a ∈ Y ⋀ b ∈ Y) ⋀ ((N ‘(AMa)) = P ⋀ (N ‘(AMb)) = P)) → (N ‘(aMb)) ≤ 0)
33	6	phnvi 8406	. . . . . . . . . . 11 ⊢ U ∈ NrmCVec
34	9, 7	nvmcl 8207	. . . . . . . . . . 11 ⊢ ((U ∈ NrmCVec ⋀ a ∈ X ⋀ b ∈ X) → (aMb) ∈ X)
35	33, 34	mp3an1 900	. . . . . . . . . 10 ⊢ ((a ∈ X ⋀ b ∈ X) → (aMb) ∈ X)
36	9, 8	nvge0 8241	. . . . . . . . . . 11 ⊢ ((U ∈ NrmCVec ⋀ (aMb) ∈ X) → 0 ≤ (N ‘(aMb)))
37	33, 36	mpan 693	. . . . . . . . . 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 700	. . . . . . . 8 ⊢ ((a ∈ X ⋀ b ∈ X) → ((N ‘(aMb)) ≤ 0 → ((N ‘(aMb)) ≤ 0 ⋀ 0 ≤ (N ‘(aMb)))))
41		eqid 1468	. . . . . . . . . . . 12 ⊢ (0v ‘U) = (0v ‘U)
42	9, 41, 8	nvz 8236	. . . . . . . . . . 11 ⊢ ((U ∈ NrmCVec ⋀ (aMb) ∈ X) → ((N ‘(aMb)) = 0 ↔ (aMb) = (0v ‘U)))
43	33, 42	mpan 693	. . . . . . . . . 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 8227	. . . . . . . . . . . 12 ⊢ ((U ∈ NrmCVec ⋀ (aMb) ∈ X) → (N ‘(aMb)) ∈ ℝ)
46	33, 45	mpan 693	. . . . . . . . . . 11 ⊢ ((aMb) ∈ X → (N ‘(aMb)) ∈ ℝ)
47	35, 46	syl 10	. . . . . . . . . 10 ⊢ ((a ∈ X ⋀ b ∈ X) → (N ‘(aMb)) ∈ ℝ)
48		0re 5412	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
49		letri3t 5490	. . . . . . . . . . 11 ⊢ (((N ‘(aMb)) ∈ ℝ ⋀ 0 ∈ ℝ) → ((N ‘(aMb)) = 0 ↔ ((N ‘(aMb)) ≤ 0 ⋀ 0 ≤ (N ‘(aMb)))))
50	48, 49	mpan2 694	. . . . . . . . . 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 8224	. . . . . . . . . 10 ⊢ ((U ∈ NrmCVec ⋀ a ∈ X ⋀ b ∈ X) → ((aMb) = (0v ‘U) ↔ a = b))
53	33, 52	mp3an1 900	. . . . . . . . 9 ⊢ ((a ∈ X ⋀ b ∈ X) → ((aMb) = (0v ‘U) ↔ a = b))
54	44, 51, 53	3bitr3d 546	. . . . . . . 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 8478	. . . . . . . 8 ⊢ Y ⊆ X
57	56	sseli 2055	. . . . . . 7 ⊢ (a ∈ Y → a ∈ X)
58	56	sseli 2055	. . . . . . 7 ⊢ (b ∈ Y → b ∈ X)
59	55, 57, 58	syl2an 454	. . . . . 6 ⊢ ((a ∈ Y ⋀ b ∈ Y) → ((N ‘(aMb)) ≤ 0 → a = b))
60	59	adantr 389	. . . . 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 1691	. 2 ⊢ ∀a ∈ Y ∀b ∈ Y (((N ‘(AMa)) = P ⋀ (N ‘(AMb)) = P) → a = b)
64	4, 29, 63	mpbir2an 728	1 ⊢ ∃!a ∈ Y (N ‘(AMa)) = P

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 953   ∈ wcel 955  {cab 1456  ∀wral 1637  ∃wrex 1638  ∃!wreu 1639   ∩ cin 2036   class class class wbr 2609  {copab 2656   ‘cfv 3172  (class class class)co 3948  supcsup 4547  ℝcr 5205  0cc0 5206  -cneg 5265   ≤ cle 5267  ℕcn 5268   < clt 5458  NrmCVeccnv 8141   +_v cpv 8142  Basecba 8143   ·_s cns 8144  0_vcn0v 8145   −_v cnsb 8146  normcnm 8147  IndMetcims 8148  SubSpcss 8314  CPreHilcphl 8402  CBancbn 8453

This theorem is referenced by:  minveccl 8515  minvecdist 8516

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-iin 2559  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  df-r1 4615  df-rank 4616  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-2 5917  df-3 5918  df-4 5919  df-n0 6047  df-z 6083  df-q 6194  df-rp 6219  df-seq1 6245  df-uz 6350  df-exp 6501  df-sqr 6600  df-re 6682  df-im 6683  df-cj 6684  df-abs 6685  df-clim 6913  df-met 7732  df-lm 7860  df-cau 7861  df-cmet 7862  df-grp 7971  df-gid 7972  df-ginv 7973  df-gdiv 7974  df-abl 8036  df-vc 8102  df-nv 8149  df-va 8152  df-ba 8153  df-sm 8154  df-0v 8155  df-vs 8156  df-nm 8157  df-ims 8158  df-ssp 8315  df-ph 8403  df-bn 8454

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