Definition df-hcau 8781

Description: Define the set of Cauchy sequences on a Hilbert space. See hcau 8990 for its membership relation. Note that f:ℕ–→ ℋ is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of Cauchy sequence in [Beran] p. 96.

Assertion

Ref	Expression
df-hcau	⊢ Cauchy = {f∣(f:ℕ–→ ℋ ⋀ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ⋀ y ≤ w) → (normh ‘((f ‘z) −h (f ‘w))) < x)))}

Distinct variable group:   x,y,z,w,f

Detailed syntax breakdown of Definition df-hcau

Step	Hyp	Ref	Expression
1		ccau 8734	. 2 class Cauchy
2		cn 5276	. . . . 5 class ℕ
3		chil 8727	. . . . 5 class ℋ
4		vf	. . . . . 6 set f
5	4	cv 953	. . . . 5 class f
6	2, 3, 5	wf 3173	. . . 4 wff f:ℕ–→ ℋ
7		cc0 5214	. . . . . . 7 class 0
8		vx	. . . . . . . 8 set x
9	8	cv 953	. . . . . . 7 class x
10		clt 5466	. . . . . . 7 class <
11	7, 9, 10	wbr 2614	. . . . . 6 wff 0 < x
12		vy	. . . . . . . . . . . . 13 set y
13	12	cv 953	. . . . . . . . . . . 12 class y
14		vz	. . . . . . . . . . . . 13 set z
15	14	cv 953	. . . . . . . . . . . 12 class z
16		cle 5275	. . . . . . . . . . . 12 class ≤
17	13, 15, 16	wbr 2614	. . . . . . . . . . 11 wff y ≤ z
18		vw	. . . . . . . . . . . . 13 set w
19	18	cv 953	. . . . . . . . . . . 12 class w
20	13, 19, 16	wbr 2614	. . . . . . . . . . 11 wff y ≤ w
21	17, 20	wa 223	. . . . . . . . . 10 wff (y ≤ z ⋀ y ≤ w)
22	15, 5	cfv 3177	. . . . . . . . . . . . 13 class (f ‘z)
23	19, 5	cfv 3177	. . . . . . . . . . . . 13 class (f ‘w)
24		cmv 8731	. . . . . . . . . . . . 13 class −h
25	22, 23, 24	co 3954	. . . . . . . . . . . 12 class ((f ‘z) −h (f ‘w))
26		cno 8733	. . . . . . . . . . . 12 class normh
27	25, 26	cfv 3177	. . . . . . . . . . 11 class (normh ‘((f ‘z) −h (f ‘w)))
28	27, 9, 10	wbr 2614	. . . . . . . . . 10 wff (normh ‘((f ‘z) −h (f ‘w))) < x
29	21, 28	wi 3	. . . . . . . . 9 wff ((y ≤ z ⋀ y ≤ w) → (normh ‘((f ‘z) −h (f ‘w))) < x)
30	29, 18, 2	wral 1642	. . . . . . . 8 wff ∀w ∈ ℕ ((y ≤ z ⋀ y ≤ w) → (normh ‘((f ‘z) −h (f ‘w))) < x)
31	30, 14, 2	wral 1642	. . . . . . 7 wff ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ⋀ y ≤ w) → (normh ‘((f ‘z) −h (f ‘w))) < x)
32	31, 12, 2	wrex 1643	. . . . . 6 wff ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ⋀ y ≤ w) → (normh ‘((f ‘z) −h (f ‘w))) < x)
33	11, 32	wi 3	. . . . 5 wff (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ⋀ y ≤ w) → (normh ‘((f ‘z) −h (f ‘w))) < x))
34		cr 5213	. . . . 5 class ℝ
35	33, 8, 34	wral 1642	. . . 4 wff ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ⋀ y ≤ w) → (normh ‘((f ‘z) −h (f ‘w))) < x))
36	6, 35	wa 223	. . 3 wff (f:ℕ–→ ℋ ⋀ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ⋀ y ≤ w) → (normh ‘((f ‘z) −h (f ‘w))) < x)))
37	36, 4	cab 1461	. 2 class {f∣(f:ℕ–→ ℋ ⋀ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ⋀ y ≤ w) → (normh ‘((f ‘z) −h (f ‘w))) < x)))}
38	1, 37	wceq 954	1 wff Cauchy = {f∣(f:ℕ–→ ℋ ⋀ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ⋀ y ≤ w) → (normh ‘((f ‘z) −h (f ‘w))) < x)))}

This definition is referenced by:  h2hcau 8788  hcau 8990

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