Theorem projlem18 5210

Description: Part of Lemma 3.6 of [Beran] p. 101, top. Used by projlem19 5211.

Hypotheses

Ref	Expression
projlem11.1	⊢ A ∈ ℋ
projlem11.2	⊢ H ∈ Cℋ
projlem11.3	⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}
projlem11.4	⊢ R = -sup(S, ℝ, < )
projlem18.5	⊢ B ∈ H
projlem18.6	⊢ C ∈ H

Assertion

Ref	Expression
projlem18	⊢ (4 · (R↑2)) ≤ ((norm ‘((B +v C) −v (2 ·s A)))↑2)

Distinct variable group(s):   v,u,A   u,H,v

Proof of Theorem projlem18

Step	Hyp	Ref	Expression
1		2cn 4471	. . . 4 ⊢ 2 ∈ ℂ
2		projlem11.1	. . . . . 6 ⊢ A ∈ ℋ
3		projlem11.2	. . . . . 6 ⊢ H ∈ Cℋ
4		projlem11.3	. . . . . 6 ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}
5		projlem11.4	. . . . . 6 ⊢ R = -sup(S, ℝ, < )
6	2, 3, 4, 5	projlem11 5203	. . . . 5 ⊢ R ∈ ℝ
7	6	recn 4098	. . . 4 ⊢ R ∈ ℂ
8	1, 7	sqmul 4688	. . 3 ⊢ ((2 · R)↑2) = ((2↑2) · (R↑2))
9		sq2 4710	. . . 4 ⊢ (2↑2) = 4
10	9	opreq1i 3009	. . 3 ⊢ ((2↑2) · (R↑2)) = (4 · (R↑2))
11	8, 10	eqtr2 1120	. 2 ⊢ (4 · (R↑2)) = ((2 · R)↑2)
12		1cn 4101	. . . . . . . 8 ⊢ 1 ∈ ℂ
13		2re 4470	. . . . . . . . 9 ⊢ 2 ∈ ℝ
14		2pos 4479	. . . . . . . . 9 ⊢ 0 < 2
15	13, 14	gt0ne0i 4345	. . . . . . . 8 ⊢ 2 ≠ 0
16	12, 1, 15	divcl 4221	. . . . . . 7 ⊢ (1 / 2) ∈ ℂ
17		projlem18.5	. . . . . . . 8 ⊢ B ∈ H
18		projlem18.6	. . . . . . . 8 ⊢ C ∈ H
19	3	chshi 5132	. . . . . . . . 9 ⊢ H ∈ Sℋ
20		shaddclt 5123	. . . . . . . . 9 ⊢ (H ∈ Sℋ → ((B ∈ H ∧ C ∈ H) → (B +v C) ∈ H))
21	19, 20	ax-mp 6	. . . . . . . 8 ⊢ ((B ∈ H ∧ C ∈ H) → (B +v C) ∈ H)
22	17, 18, 21	mp2an 520	. . . . . . 7 ⊢ (B +v C) ∈ H
23		shmulclt 5124	. . . . . . . 8 ⊢ (H ∈ Sℋ → (((1 / 2) ∈ ℂ ∧ (B +v C) ∈ H) → ((1 / 2) ·s (B +v C)) ∈ H))
24	19, 23	ax-mp 6	. . . . . . 7 ⊢ (((1 / 2) ∈ ℂ ∧ (B +v C) ∈ H) → ((1 / 2) ·s (B +v C)) ∈ H)
25	16, 22, 24	mp2an 520	. . . . . 6 ⊢ ((1 / 2) ·s (B +v C)) ∈ H
26	2, 3, 4, 5	projlem12 5204	. . . . . 6 ⊢ (((1 / 2) ·s (B +v C)) ∈ H → R ≤ (norm ‘(((1 / 2) ·s (B +v C)) −v A)))
27	25, 26	ax-mp 6	. . . . 5 ⊢ R ≤ (norm ‘(((1 / 2) ·s (B +v C)) −v A))
28	3, 17	cheli 5138	. . . . . . . . . . 11 ⊢ B ∈ ℋ
29	3, 18	cheli 5138	. . . . . . . . . . 11 ⊢ C ∈ ℋ
30	28, 29	hvaddcl 4999	. . . . . . . . . 10 ⊢ (B +v C) ∈ ℋ
31	16, 30	hvmulcl 4990	. . . . . . . . 9 ⊢ ((1 / 2) ·s (B +v C)) ∈ ℋ
32	31, 2	hvsubcl 5002	. . . . . . . 8 ⊢ (((1 / 2) ·s (B +v C)) −v A) ∈ ℋ
33	32	normcl 5081	. . . . . . 7 ⊢ (norm ‘(((1 / 2) ·s (B +v C)) −v A)) ∈ ℝ
34	6, 33, 13	lemul2 4396	. . . . . 6 ⊢ (0 < 2 → (R ≤ (norm ‘(((1 / 2) ·s (B +v C)) −v A)) ↔ (2 · R) ≤ (2 · (norm ‘(((1 / 2) ·s (B +v C)) −v A)))))
35	14, 34	ax-mp 6	. . . . 5 ⊢ (R ≤ (norm ‘(((1 / 2) ·s (B +v C)) −v A)) ↔ (2 · R) ≤ (2 · (norm ‘(((1 / 2) ·s (B +v C)) −v A))))
36	27, 35	mpbi 164	. . . 4 ⊢ (2 · R) ≤ (2 · (norm ‘(((1 / 2) ·s (B +v C)) −v A)))
37	1, 32	norm-iii 5087	. . . . 5 ⊢ (norm ‘(2 ·s (((1 / 2) ·s (B +v C)) −v A))) = ((abs ‘2) · (norm ‘(((1 / 2) ·s (B +v C)) −v A)))
38	1, 31, 2	hvsubdistr1 5024	. . . . . . 7 ⊢ (2 ·s (((1 / 2) ·s (B +v C)) −v A)) = ((2 ·s ((1 / 2) ·s (B +v C))) −v (2 ·s A))
39	1, 15	recid 4233	. . . . . . . . . 10 ⊢ (2 · (1 / 2)) = 1
40	39	opreq1i 3009	. . . . . . . . 9 ⊢ ((2 · (1 / 2)) ·s (B +v C)) = (1 ·s (B +v C))
41	1, 16, 30	hvmulass 5020	. . . . . . . . 9 ⊢ ((2 · (1 / 2)) ·s (B +v C)) = (2 ·s ((1 / 2) ·s (B +v C)))
42		ax-hvmulid 4991	. . . . . . . . . 10 ⊢ ((B +v C) ∈ ℋ → (1 ·s (B +v C)) = (B +v C))
43	30, 42	ax-mp 6	. . . . . . . . 9 ⊢ (1 ·s (B +v C)) = (B +v C)
44	40, 41, 43	3eqtr3 1124	. . . . . . . 8 ⊢ (2 ·s ((1 / 2) ·s (B +v C))) = (B +v C)
45	44	opreq1i 3009	. . . . . . 7 ⊢ ((2 ·s ((1 / 2) ·s (B +v C))) −v (2 ·s A)) = ((B +v C) −v (2 ·s A))
46	38, 45	eqtr 1119	. . . . . 6 ⊢ (2 ·s (((1 / 2) ·s (B +v C)) −v A)) = ((B +v C) −v (2 ·s A))
47	46	fveq2i 2835	. . . . 5 ⊢ (norm ‘(2 ·s (((1 / 2) ·s (B +v C)) −v A))) = (norm ‘((B +v C) −v (2 ·s A)))
48		ax0re 4063	. . . . . . . 8 ⊢ 0 ∈ ℝ
49	48, 13, 14	ltlei 4303	. . . . . . 7 ⊢ 0 ≤ 2
50	13	absid 4850	. . . . . . 7 ⊢ (0 ≤ 2 → (abs ‘2) = 2)
51	49, 50	ax-mp 6	. . . . . 6 ⊢ (abs ‘2) = 2
52	51	opreq1i 3009	. . . . 5 ⊢ ((abs ‘2) · (norm ‘(((1 / 2) ·s (B +v C)) −v A))) = (2 · (norm ‘(((1 / 2) ·s (B +v C)) −v A)))
53	37, 47, 52	3eqtr3r 1125	. . . 4 ⊢ (2 · (norm ‘(((1 / 2) ·s (B +v C)) −v A))) = (norm ‘((B +v C) −v (2 ·s A)))
54	36, 53	breqtr 2080	. . 3 ⊢ (2 · R) ≤ (norm ‘((B +v C) −v (2 ·s A)))
55	2, 3, 4, 5	projlem13 5205	. . . . 5 ⊢ 0 ≤ R
56	13, 6	mulge0 4335	. . . . 5 ⊢ ((0 ≤ 2 ∧ 0 ≤ R) → 0 ≤ (2 · R))
57	49, 55, 56	mp2an 520	. . . 4 ⊢ 0 ≤ (2 · R)
58	1, 2	hvmulcl 4990	. . . . . 6 ⊢ (2 ·s A) ∈ ℋ
59	30, 58	hvsubcl 5002	. . . . 5 ⊢ ((B +v C) −v (2 ·s A)) ∈ ℋ
60		normge0t 5077	. . . . 5 ⊢ (((B +v C) −v (2 ·s A)) ∈ ℋ → 0 ≤ (norm ‘((B +v C) −v (2 ·s A))))
61	59, 60	ax-mp 6	. . . 4 ⊢ 0 ≤ (norm ‘((B +v C) −v (2 ·s A)))
62	13, 6	remulcl 4119	. . . . 5 ⊢ (2 · R) ∈ ℝ
63	59	normcl 5081	. . . . 5 ⊢ (norm ‘((B +v C) −v (2 ·s A))) ∈ ℝ
64	62, 63	le2sqe 4701	. . . 4 ⊢ ((0 ≤ (2 · R) ∧ 0 ≤ (norm ‘((B +v C) −v (2 ·s A)))) → ((2 · R) ≤ (norm ‘((B +v C) −v (2 ·s A))) ↔ ((2 · R)↑2) ≤ ((norm ‘((B +v C) −v (2 ·s A)))↑2)))
65	57, 61, 64	mp2an 520	. . 3 ⊢ ((2 · R) ≤ (norm ‘((B +v C) −v (2 ·s A))) ↔ ((2 · R)↑2) ≤ ((norm ‘((B +v C) −v (2 ·s A)))↑2))
66	54, 65	mpbi 164	. 2 ⊢ ((2 · R)↑2) ≤ ((norm ‘((B +v C) −v (2 ·s A)))↑2)
67	11, 66	eqbrtr 2076	1 ⊢ (4 · (R↑2)) ≤ ((norm ‘((B +v C) −v (2 ·s A)))↑2)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {crab 1204   class class class wbr 2054  supcsup 2060   ‘cfv 2422  (class class class)co 3001  ℂcc 4026  ℝcr 4027  0cc0 4028  1c1 4029   · cmulc 4032   < clt 4033  -cneg 4090   / cdiv 4091   ≤ cle 4092  2c2 4454  4c4 4456  ↑cexp 4675  abscabs 4789   ℋ chil 4958   +_v cva 4959   ·_s csm 4960   −_v cmv 4962  normcno 4964   S_ℋ csh 4967   C_ℋ cch 4968

This theorem is referenced by:  projlem19 5211

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-hilex 4983  ax-hvaddcl 4984  ax-hvzercl 4987  ax-hvmulcl 4989  ax-hvmulid 4991  ax-hvmulass 4992  ax-hvdistr1 4993  ax-hvmulzer 4995  ax-hicl 5043  ax-his1 5045  ax-his3 5047  ax-his4 5048

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-sup 2154  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-i 4037  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-3 4463  df-4 4464  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676  df-sqr 4728  df-re 4790  df-im 4791  df-cj 4792  df-abs 4793  df-hvsub 4996  df-hnorm 5074  df-sh 5114  df-ch 5127

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