Theorem projlem22 5214

Description: Part of Lemma 3.6 of [Beran] p. 100. Here we show a member of the vector sequence is bounded. Used by projlem27 5219.

Hypothesis

Ref	Expression
projlem21.1	⊢ (φ ↔ (F:ℕ–→H ∧ ∀w ∈ ℕ ((R − (1 / w)) < (norm ‘((F ‘w) −v A)) ∧ (norm ‘((F ‘w) −v A)) < (R + (1 / w)))))

Assertion

Ref	Expression
projlem22	⊢ (φ → (D ∈ ℕ → (norm ‘((F ‘D) −v A)) < (R + (1 / D))))

Distinct variable group(s):   w,A   w,D   w,F   w,R

Proof of Theorem projlem22

Step	Hyp	Ref	Expression
1		projlem21.1	. . 3 ⊢ (φ ↔ (F:ℕ–→H ∧ ∀w ∈ ℕ ((R − (1 / w)) < (norm ‘((F ‘w) −v A)) ∧ (norm ‘((F ‘w) −v A)) < (R + (1 / w)))))
2	1	pm3.27bd 263	. 2 ⊢ (φ → ∀w ∈ ℕ ((R − (1 / w)) < (norm ‘((F ‘w) −v A)) ∧ (norm ‘((F ‘w) −v A)) < (R + (1 / w))))
3		pm3.27 260	. . 3 ⊢ (((R − (1 / w)) < (norm ‘((F ‘w) −v A)) ∧ (norm ‘((F ‘w) −v A)) < (R + (1 / w))) → (norm ‘((F ‘w) −v A)) < (R + (1 / w)))
4	3	r19.20si 1254	. 2 ⊢ (∀w ∈ ℕ ((R − (1 / w)) < (norm ‘((F ‘w) −v A)) ∧ (norm ‘((F ‘w) −v A)) < (R + (1 / w))) → ∀w ∈ ℕ (norm ‘((F ‘w) −v A)) < (R + (1 / w)))
5		fveq2 2832	. . . . . 6 ⊢ (w = D → (F ‘w) = (F ‘D))
6	5	opreq1d 3012	. . . . 5 ⊢ (w = D → ((F ‘w) −v A) = ((F ‘D) −v A))
7	6	fveq2d 2836	. . . 4 ⊢ (w = D → (norm ‘((F ‘w) −v A)) = (norm ‘((F ‘D) −v A)))
8		opreq2 3007	. . . . 5 ⊢ (w = D → (1 / w) = (1 / D))
9	8	opreq2d 3013	. . . 4 ⊢ (w = D → (R + (1 / w)) = (R + (1 / D)))
10	7, 9	breq12d 2073	. . 3 ⊢ (w = D → ((norm ‘((F ‘w) −v A)) < (R + (1 / w)) ↔ (norm ‘((F ‘D) −v A)) < (R + (1 / D))))
11	10	rcla4v 1402	. 2 ⊢ (∀w ∈ ℕ (norm ‘((F ‘w) −v A)) < (R + (1 / w)) → (D ∈ ℕ → (norm ‘((F ‘D) −v A)) < (R + (1 / D))))
12	2, 4, 11	3syl 21	1 ⊢ (φ → (D ∈ ℕ → (norm ‘((F ‘D) −v A)) < (R + (1 / D))))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201   class class class wbr 2054  –→wf 2418   ‘cfv 2422  (class class class)co 3001  1c1 4029   + caddc 4031   < clt 4033   − cmin 4089   / cdiv 4091  ℕcn 4093   −_v cmv 4962  normcno 4964

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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