Theorem occllem4 5183

Description: Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Hypotheses

Ref	Expression
occllem3.1	⊢ G = {⟨x, y⟩∣(x ∈ ℕ ∧ y = ((F ‘x) ·i S))}
occllem4.2	⊢ S ∈ ℋ

Assertion

Ref	Expression
occllem4	⊢ (F:ℕ–→ ℋ → G:ℕ–→ℂ)

Distinct variable group(s):   x,y,F   x,S,y

Proof of Theorem occllem4

Step	Hyp	Ref	Expression
1		ffvrn 2890	. . . . . 6 ⊢ ((F:ℕ–→ ℋ ∧ x ∈ ℕ) → (F ‘x) ∈ ℋ )
2		occllem4.2	. . . . . 6 ⊢ S ∈ ℋ
3	1, 2	jctir 241	. . . . 5 ⊢ ((F:ℕ–→ ℋ ∧ x ∈ ℕ) → ((F ‘x) ∈ ℋ ∧ S ∈ ℋ ))
4		ax-hicl 5043	. . . . 5 ⊢ (((F ‘x) ∈ ℋ ∧ S ∈ ℋ ) → ((F ‘x) ·i S) ∈ ℂ)
5	3, 4	syl 12	. . . 4 ⊢ ((F:ℕ–→ ℋ ∧ x ∈ ℕ) → ((F ‘x) ·i S) ∈ ℂ)
6	5	exp 291	. . 3 ⊢ (F:ℕ–→ ℋ → (x ∈ ℕ → ((F ‘x) ·i S) ∈ ℂ))
7	6	r19.21aiv 1259	. 2 ⊢ (F:ℕ–→ ℋ → ∀x ∈ ℕ ((F ‘x) ·i S) ∈ ℂ)
8		occllem3.1	. . 3 ⊢ G = {⟨x, y⟩∣(x ∈ ℕ ∧ y = ((F ‘x) ·i S))}
9	8	fopab2 2891	. 2 ⊢ (∀x ∈ ℕ ((F ‘x) ·i S) ∈ ℂ ↔ G:ℕ–→ℂ)
10	7, 9	sylib 173	1 ⊢ (F:ℕ–→ ℋ → G:ℕ–→ℂ)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {copab 2055  –→wf 2418   ‘cfv 2422  (class class class)co 3001  ℂcc 4026  ℕcn 4093   ℋ chil 4958   ·_i csp 4963

This theorem is referenced by:  occllem6 5185

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-hicl 5043

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-ral 1205  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-fn 2433  df-f 2434  df-fv 2438

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