Theorem projlem20 5212

Description: Part of Lemma 3.6 of [Beran] p. 101. Used by projlem27 5219.

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, < )
projlem20.5	⊢ N ∈ ℕ

Assertion

Ref	Expression
projlem20	⊢ (((B ∈ H ∧ C ∈ H) ∧ (D ∈ ℕ ∧ G ∈ ℕ)) → (((norm ‘(B −v A)) < (R + (1 / D)) ∧ (norm ‘(C −v A)) < (R + (1 / G))) → ((N ≤ D ∧ N ≤ G) → (norm ‘(B −v C)) < (√ ‘((4 · ((2 · R) + 1)) / N)))))

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

Proof of Theorem projlem20

Step	Hyp	Ref	Expression
1		opreq1 3006	. . . . . 6 ⊢ (B = if(B ∈ H, B, 0v) → (B −v A) = (if(B ∈ H, B, 0v) −v A))
2	1	fveq2d 2836	. . . . 5 ⊢ (B = if(B ∈ H, B, 0v) → (norm ‘(B −v A)) = (norm ‘(if(B ∈ H, B, 0v) −v A)))
3	2	breq1d 2071	. . . 4 ⊢ (B = if(B ∈ H, B, 0v) → ((norm ‘(B −v A)) < (R + (1 / D)) ↔ (norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / D))))
4	3	anbi1d 469	. . 3 ⊢ (B = if(B ∈ H, B, 0v) → (((norm ‘(B −v A)) < (R + (1 / D)) ∧ (norm ‘(C −v A)) < (R + (1 / G))) ↔ ((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / D)) ∧ (norm ‘(C −v A)) < (R + (1 / G)))))
5		opreq1 3006	. . . . . 6 ⊢ (B = if(B ∈ H, B, 0v) → (B −v C) = (if(B ∈ H, B, 0v) −v C))
6	5	fveq2d 2836	. . . . 5 ⊢ (B = if(B ∈ H, B, 0v) → (norm ‘(B −v C)) = (norm ‘(if(B ∈ H, B, 0v) −v C)))
7	6	breq1d 2071	. . . 4 ⊢ (B = if(B ∈ H, B, 0v) → ((norm ‘(B −v C)) < (√ ‘((4 · ((2 · R) + 1)) / N)) ↔ (norm ‘(if(B ∈ H, B, 0v) −v C)) < (√ ‘((4 · ((2 · R) + 1)) / N))))
8	7	imbi2d 464	. . 3 ⊢ (B = if(B ∈ H, B, 0v) → (((N ≤ D ∧ N ≤ G) → (norm ‘(B −v C)) < (√ ‘((4 · ((2 · R) + 1)) / N))) ↔ ((N ≤ D ∧ N ≤ G) → (norm ‘(if(B ∈ H, B, 0v) −v C)) < (√ ‘((4 · ((2 · R) + 1)) / N)))))
9	4, 8	imbi12d 474	. 2 ⊢ (B = if(B ∈ H, B, 0v) → ((((norm ‘(B −v A)) < (R + (1 / D)) ∧ (norm ‘(C −v A)) < (R + (1 / G))) → ((N ≤ D ∧ N ≤ G) → (norm ‘(B −v C)) < (√ ‘((4 · ((2 · R) + 1)) / N)))) ↔ (((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / D)) ∧ (norm ‘(C −v A)) < (R + (1 / G))) → ((N ≤ D ∧ N ≤ G) → (norm ‘(if(B ∈ H, B, 0v) −v C)) < (√ ‘((4 · ((2 · R) + 1)) / N))))))
10		opreq1 3006	. . . . . 6 ⊢ (C = if(C ∈ H, C, 0v) → (C −v A) = (if(C ∈ H, C, 0v) −v A))
11	10	fveq2d 2836	. . . . 5 ⊢ (C = if(C ∈ H, C, 0v) → (norm ‘(C −v A)) = (norm ‘(if(C ∈ H, C, 0v) −v A)))
12	11	breq1d 2071	. . . 4 ⊢ (C = if(C ∈ H, C, 0v) → ((norm ‘(C −v A)) < (R + (1 / G)) ↔ (norm ‘(if(C ∈ H, C, 0v) −v A)) < (R + (1 / G))))
13	12	anbi2d 468	. . 3 ⊢ (C = if(C ∈ H, C, 0v) → (((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / D)) ∧ (norm ‘(C −v A)) < (R + (1 / G))) ↔ ((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / D)) ∧ (norm ‘(if(C ∈ H, C, 0v) −v A)) < (R + (1 / G)))))
14		opreq2 3007	. . . . . 6 ⊢ (C = if(C ∈ H, C, 0v) → (if(B ∈ H, B, 0v) −v C) = (if(B ∈ H, B, 0v) −v if(C ∈ H, C, 0v)))
15	14	fveq2d 2836	. . . . 5 ⊢ (C = if(C ∈ H, C, 0v) → (norm ‘(if(B ∈ H, B, 0v) −v C)) = (norm ‘(if(B ∈ H, B, 0v) −v if(C ∈ H, C, 0v))))
16	15	breq1d 2071	. . . 4 ⊢ (C = if(C ∈ H, C, 0v) → ((norm ‘(if(B ∈ H, B, 0v) −v C)) < (√ ‘((4 · ((2 · R) + 1)) / N)) ↔ (norm ‘(if(B ∈ H, B, 0v) −v if(C ∈ H, C, 0v))) < (√ ‘((4 · ((2 · R) + 1)) / N))))
17	16	imbi2d 464	. . 3 ⊢ (C = if(C ∈ H, C, 0v) → (((N ≤ D ∧ N ≤ G) → (norm ‘(if(B ∈ H, B, 0v) −v C)) < (√ ‘((4 · ((2 · R) + 1)) / N))) ↔ ((N ≤ D ∧ N ≤ G) → (norm ‘(if(B ∈ H, B, 0v) −v if(C ∈ H, C, 0v))) < (√ ‘((4 · ((2 · R) + 1)) / N)))))
18	13, 17	imbi12d 474	. 2 ⊢ (C = if(C ∈ H, C, 0v) → ((((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / D)) ∧ (norm ‘(C −v A)) < (R + (1 / G))) → ((N ≤ D ∧ N ≤ G) → (norm ‘(if(B ∈ H, B, 0v) −v C)) < (√ ‘((4 · ((2 · R) + 1)) / N)))) ↔ (((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / D)) ∧ (norm ‘(if(C ∈ H, C, 0v) −v A)) < (R + (1 / G))) → ((N ≤ D ∧ N ≤ G) → (norm ‘(if(B ∈ H, B, 0v) −v if(C ∈ H, C, 0v))) < (√ ‘((4 · ((2 · R) + 1)) / N))))))
19		opreq2 3007	. . . . . 6 ⊢ (D = if(D ∈ ℕ, D, 1) → (1 / D) = (1 / if(D ∈ ℕ, D, 1)))
20	19	opreq2d 3013	. . . . 5 ⊢ (D = if(D ∈ ℕ, D, 1) → (R + (1 / D)) = (R + (1 / if(D ∈ ℕ, D, 1))))
21	20	breq2d 2072	. . . 4 ⊢ (D = if(D ∈ ℕ, D, 1) → ((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / D)) ↔ (norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / if(D ∈ ℕ, D, 1)))))
22	21	anbi1d 469	. . 3 ⊢ (D = if(D ∈ ℕ, D, 1) → (((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / D)) ∧ (norm ‘(if(C ∈ H, C, 0v) −v A)) < (R + (1 / G))) ↔ ((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / if(D ∈ ℕ, D, 1))) ∧ (norm ‘(if(C ∈ H, C, 0v) −v A)) < (R + (1 / G)))))
23		breq2 2066	. . . . 5 ⊢ (D = if(D ∈ ℕ, D, 1) → (N ≤ D ↔ N ≤ if(D ∈ ℕ, D, 1)))
24	23	anbi1d 469	. . . 4 ⊢ (D = if(D ∈ ℕ, D, 1) → ((N ≤ D ∧ N ≤ G) ↔ (N ≤ if(D ∈ ℕ, D, 1) ∧ N ≤ G)))
25	24	imbi1d 465	. . 3 ⊢ (D = if(D ∈ ℕ, D, 1) → (((N ≤ D ∧ N ≤ G) → (norm ‘(if(B ∈ H, B, 0v) −v if(C ∈ H, C, 0v))) < (√ ‘((4 · ((2 · R) + 1)) / N))) ↔ ((N ≤ if(D ∈ ℕ, D, 1) ∧ N ≤ G) → (norm ‘(if(B ∈ H, B, 0v) −v if(C ∈ H, C, 0v))) < (√ ‘((4 · ((2 · R) + 1)) / N)))))
26	22, 25	imbi12d 474	. 2 ⊢ (D = if(D ∈ ℕ, D, 1) → ((((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / D)) ∧ (norm ‘(if(C ∈ H, C, 0v) −v A)) < (R + (1 / G))) → ((N ≤ D ∧ N ≤ G) → (norm ‘(if(B ∈ H, B, 0v) −v if(C ∈ H, C, 0v))) < (√ ‘((4 · ((2 · R) + 1)) / N)))) ↔ (((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / if(D ∈ ℕ, D, 1))) ∧ (norm ‘(if(C ∈ H, C, 0v) −v A)) < (R + (1 / G))) → ((N ≤ if(D ∈ ℕ, D, 1) ∧ N ≤ G) → (norm ‘(if(B ∈ H, B, 0v) −v if(C ∈ H, C, 0v))) < (√ ‘((4 · ((2 · R) + 1)) / N))))))
27		opreq2 3007	. . . . . 6 ⊢ (G = if(G ∈ ℕ, G, 1) → (1 / G) = (1 / if(G ∈ ℕ, G, 1)))
28	27	opreq2d 3013	. . . . 5 ⊢ (G = if(G ∈ ℕ, G, 1) → (R + (1 / G)) = (R + (1 / if(G ∈ ℕ, G, 1))))
29	28	breq2d 2072	. . . 4 ⊢ (G = if(G ∈ ℕ, G, 1) → ((norm ‘(if(C ∈ H, C, 0v) −v A)) < (R + (1 / G)) ↔ (norm ‘(if(C ∈ H, C, 0v) −v A)) < (R + (1 / if(G ∈ ℕ, G, 1)))))
30	29	anbi2d 468	. . 3 ⊢ (G = if(G ∈ ℕ, G, 1) → (((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / if(D ∈ ℕ, D, 1))) ∧ (norm ‘(if(C ∈ H, C, 0v) −v A)) < (R + (1 / G))) ↔ ((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / if(D ∈ ℕ, D, 1))) ∧ (norm ‘(if(C ∈ H, C, 0v) −v A)) < (R + (1 / if(G ∈ ℕ, G, 1))))))
31		breq2 2066	. . . . 5 ⊢ (G = if(G ∈ ℕ, G, 1) → (N ≤ G ↔ N ≤ if(G ∈ ℕ, G, 1)))
32	31	anbi2d 468	. . . 4 ⊢ (G = if(G ∈ ℕ, G, 1) → ((N ≤ if(D ∈ ℕ, D, 1) ∧ N ≤ G) ↔ (N ≤ if(D ∈ ℕ, D, 1) ∧ N ≤ if(G ∈ ℕ, G, 1))))
33	32	imbi1d 465	. . 3 ⊢ (G = if(G ∈ ℕ, G, 1) → (((N ≤ if(D ∈ ℕ, D, 1) ∧ N ≤ G) → (norm ‘(if(B ∈ H, B, 0v) −v if(C ∈ H, C, 0v))) < (√ ‘((4 · ((2 · R) + 1)) / N))) ↔ ((N ≤ if(D ∈ ℕ, D, 1) ∧ N ≤ if(G ∈ ℕ, G, 1)) → (norm ‘(if(B ∈ H, B, 0v) −v if(C ∈ H, C, 0v))) < (√ ‘((4 · ((2 · R) + 1)) / N)))))
34	30, 33	imbi12d 474	. 2 ⊢ (G = if(G ∈ ℕ, G, 1) → ((((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / if(D ∈ ℕ, D, 1))) ∧ (norm ‘(if(C ∈ H, C, 0v) −v A)) < (R + (1 / G))) → ((N ≤ if(D ∈ ℕ, D, 1) ∧ N ≤ G) → (norm ‘(if(B ∈ H, B, 0v) −v if(C ∈ H, C, 0v))) < (√ ‘((4 · ((2 · R) + 1)) / N)))) ↔ (((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / if(D ∈ ℕ, D, 1))) ∧ (norm ‘(if(C ∈ H, C, 0v) −v A)) < (R + (1 / if(G ∈ ℕ, G, 1)))) → ((N ≤ if(D ∈ ℕ, D, 1) ∧ N ≤ if(G ∈ ℕ, G, 1)) → (norm ‘(if(B ∈ H, B, 0v) −v if(C ∈ H, C, 0v))) < (√ ‘((4 · ((2 · R) + 1)) / N))))))
35		projlem11.1	. . 3 ⊢ A ∈ ℋ
36		projlem11.2	. . 3 ⊢ H ∈ Cℋ
37		projlem11.3	. . 3 ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}
38		projlem11.4	. . 3 ⊢ R = -sup(S, ℝ, < )
39		ch0 5133	. . . . 5 ⊢ (H ∈ Cℋ → 0v ∈ H)
40	36, 39	ax-mp 6	. . . 4 ⊢ 0v ∈ H
41	40	elimel 1793	. . 3 ⊢ if(B ∈ H, B, 0v) ∈ H
42	40	elimel 1793	. . 3 ⊢ if(C ∈ H, C, 0v) ∈ H
43		1nn 4432	. . . 4 ⊢ 1 ∈ ℕ
44	43	elimel 1793	. . 3 ⊢ if(D ∈ ℕ, D, 1) ∈ ℕ
45	43	elimel 1793	. . 3 ⊢ if(G ∈ ℕ, G, 1) ∈ ℕ
46		projlem20.5	. . 3 ⊢ N ∈ ℕ
47	35, 36, 37, 38, 41, 42, 44, 45, 46	projlem19 5211	. 2 ⊢ (((norm ‘(if(B ∈ H, B, 0v) −v A)) < (R + (1 / if(D ∈ ℕ, D, 1))) ∧ (norm ‘(if(C ∈ H, C, 0v) −v A)) < (R + (1 / if(G ∈ ℕ, G, 1)))) → ((N ≤ if(D ∈ ℕ, D, 1) ∧ N ≤ if(G ∈ ℕ, G, 1)) → (norm ‘(if(B ∈ H, B, 0v) −v if(C ∈ H, C, 0v))) < (√ ‘((4 · ((2 · R) + 1)) / N))))
48	9, 18, 26, 34, 47	dedth4h 1789	1 ⊢ (((B ∈ H ∧ C ∈ H) ∧ (D ∈ ℕ ∧ G ∈ ℕ)) → (((norm ‘(B −v A)) < (R + (1 / D)) ∧ (norm ‘(C −v A)) < (R + (1 / G))) → ((N ≤ D ∧ N ≤ G) → (norm ‘(B −v C)) < (√ ‘((4 · ((2 · R) + 1)) / N)))))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {crab 1204  ifcif 1776   class class class wbr 2054  supcsup 2060   ‘cfv 2422  (class class class)co 3001  ℝcr 4027  1c1 4029   + caddc 4031   · cmulc 4032   < clt 4033  -cneg 4090   / cdiv 4091   ≤ cle 4092  ℕcn 4093  2c2 4454  4c4 4456  √csqr 4727   ℋ chil 4958  0_vc0v 4961   −_v cmv 4962  normcno 4964   C_ℋ cch 4968

This theorem is referenced by:  projlem27 5219

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-hvcom 4985  ax-hvass 4986  ax-hvzercl 4987  ax-hvaddid 4988  ax-hvmulcl 4989  ax-hvmulid 4991  ax-hvmulass 4992  ax-hvdistr1 4993  ax-hvdistr2 4994  ax-hvmulzer 4995  ax-hicl 5043  ax-his1 5045  ax-his2 5046  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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