Theorem projlem23 5215

Description: Part of Lemma 3.6 of [Beran] p. 101. The hypothesis lets us work with the sequence G which corresponds to Beran's "{||yn-x0||}". Used by projlem25 5217 projlem26 5218.

Hypothesis

Ref	Expression
projlem23.1	⊢ G = {⟨x, y⟩∣(x ∈ ℕ ∧ y = (norm ‘((F ‘x) −v A)))}

Assertion

Ref	Expression
projlem23	⊢ (D ∈ ℕ → (G ‘D) = (norm ‘((F ‘D) −v A)))

Distinct variable group(s):   x,y,D   x,A,y   x,F,y

Proof of Theorem projlem23

Step	Hyp	Ref	Expression
1		fveq2 2832	. . . 4 ⊢ (x = D → (F ‘x) = (F ‘D))
2	1	opreq1d 3012	. . 3 ⊢ (x = D → ((F ‘x) −v A) = ((F ‘D) −v A))
3	2	fveq2d 2836	. 2 ⊢ (x = D → (norm ‘((F ‘x) −v A)) = (norm ‘((F ‘D) −v A)))
4		projlem23.1	. 2 ⊢ G = {⟨x, y⟩∣(x ∈ ℕ ∧ y = (norm ‘((F ‘x) −v A)))}
5		fvex 2838	. 2 ⊢ (norm ‘((F ‘D) −v A)) ∈ V
6	3, 4, 5	fvopab4 2871	1 ⊢ (D ∈ ℕ → (G ‘D) = (norm ‘((F ‘D) −v A)))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  {copab 2055   ‘cfv 2422  (class class class)co 3001  ℕcn 4093   −_v cmv 4962  normcno 4964

This theorem is referenced by:  projlem25 5217  projlem26 5218

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

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  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-id 2125  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-fv 2438  df-opr 3003

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