Theorem hlim2 5112

Description: The limit of a sequence on a Hilbert space.

Assertion

Ref	Expression
hlim2	⊢ ((F:ℕ–→ ℋ ∧ A ∈ ℋ ) → (F ⇝v A ↔ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x))))

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

Proof of Theorem hlim2

Step	Hyp	Ref	Expression
1		feq1 2748	. . . . . 6 ⊢ (f = F → (f:ℕ–→ ℋ ↔ F:ℕ–→ ℋ ))
2	1	anbi1d 469	. . . . 5 ⊢ (f = F → ((f:ℕ–→ ℋ ∧ w ∈ ℋ ) ↔ (F:ℕ–→ ℋ ∧ w ∈ ℋ )))
3		fveq1 2831	. . . . . . . . . . . . 13 ⊢ (f = F → (f ‘z) = (F ‘z))
4	3	opreq1d 3012	. . . . . . . . . . . 12 ⊢ (f = F → ((f ‘z) −v w) = ((F ‘z) −v w))
5	4	fveq2d 2836	. . . . . . . . . . 11 ⊢ (f = F → (norm ‘((f ‘z) −v w)) = (norm ‘((F ‘z) −v w)))
6	5	breq1d 2071	. . . . . . . . . 10 ⊢ (f = F → ((norm ‘((f ‘z) −v w)) < x ↔ (norm ‘((F ‘z) −v w)) < x))
7	6	imbi2d 464	. . . . . . . . 9 ⊢ (f = F → ((y ≤ z → (norm ‘((f ‘z) −v w)) < x) ↔ (y ≤ z → (norm ‘((F ‘z) −v w)) < x)))
8	7	biraldv 1219	. . . . . . . 8 ⊢ (f = F → (∀z ∈ ℕ (y ≤ z → (norm ‘((f ‘z) −v w)) < x) ↔ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v w)) < x)))
9	8	birexdv 1220	. . . . . . 7 ⊢ (f = F → (∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((f ‘z) −v w)) < x) ↔ ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v w)) < x)))
10	9	imbi2d 464	. . . . . 6 ⊢ (f = F → ((0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((f ‘z) −v w)) < x)) ↔ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v w)) < x))))
11	10	biraldv 1219	. . . . 5 ⊢ (f = F → (∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((f ‘z) −v w)) < x)) ↔ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v w)) < x))))
12	2, 11	anbi12d 476	. . . 4 ⊢ (f = F → (((f:ℕ–→ ℋ ∧ w ∈ ℋ ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((f ‘z) −v w)) < x))) ↔ ((F:ℕ–→ ℋ ∧ w ∈ ℋ ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v w)) < x)))))
13		eleq1 1149	. . . . . 6 ⊢ (w = A → (w ∈ ℋ ↔ A ∈ ℋ ))
14	13	anbi2d 468	. . . . 5 ⊢ (w = A → ((F:ℕ–→ ℋ ∧ w ∈ ℋ ) ↔ (F:ℕ–→ ℋ ∧ A ∈ ℋ )))
15		opreq2 3007	. . . . . . . . . . . 12 ⊢ (w = A → ((F ‘z) −v w) = ((F ‘z) −v A))
16	15	fveq2d 2836	. . . . . . . . . . 11 ⊢ (w = A → (norm ‘((F ‘z) −v w)) = (norm ‘((F ‘z) −v A)))
17	16	breq1d 2071	. . . . . . . . . 10 ⊢ (w = A → ((norm ‘((F ‘z) −v w)) < x ↔ (norm ‘((F ‘z) −v A)) < x))
18	17	imbi2d 464	. . . . . . . . 9 ⊢ (w = A → ((y ≤ z → (norm ‘((F ‘z) −v w)) < x) ↔ (y ≤ z → (norm ‘((F ‘z) −v A)) < x)))
19	18	biraldv 1219	. . . . . . . 8 ⊢ (w = A → (∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v w)) < x) ↔ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x)))
20	19	birexdv 1220	. . . . . . 7 ⊢ (w = A → (∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v w)) < x) ↔ ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x)))
21	20	imbi2d 464	. . . . . 6 ⊢ (w = A → ((0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v w)) < x)) ↔ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x))))
22	21	biraldv 1219	. . . . 5 ⊢ (w = A → (∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v w)) < x)) ↔ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x))))
23	14, 22	anbi12d 476	. . . 4 ⊢ (w = A → (((F:ℕ–→ ℋ ∧ w ∈ ℋ ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v w)) < x))) ↔ ((F:ℕ–→ ℋ ∧ A ∈ ℋ ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x)))))
24		df-hlim 5107	. . . 4 ⊢ ⇝v = {⟨f, w⟩∣((f:ℕ–→ ℋ ∧ w ∈ ℋ ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((f ‘z) −v w)) < x)))}
25	12, 23, 24	brabg 2116	. . 3 ⊢ ((F ∈ V ∧ A ∈ ℋ ) → (F ⇝v A ↔ ((F:ℕ–→ ℋ ∧ A ∈ ℋ ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x)))))
26		ffn 2752	. . . 4 ⊢ (F:ℕ–→ ℋ → F Fn ℕ)
27		nnex 4431	. . . . 5 ⊢ ℕ ∈ V
28		fnex 2740	. . . . 5 ⊢ (ℕ ∈ V → (F Fn ℕ → F ∈ V))
29	27, 28	ax-mp 6	. . . 4 ⊢ (F Fn ℕ → F ∈ V)
30	26, 29	syl 12	. . 3 ⊢ (F:ℕ–→ ℋ → F ∈ V)
31	25, 30	sylan 343	. 2 ⊢ ((F:ℕ–→ ℋ ∧ A ∈ ℋ ) → (F ⇝v A ↔ ((F:ℕ–→ ℋ ∧ A ∈ ℋ ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x)))))
32		ibar 487	. 2 ⊢ ((F:ℕ–→ ℋ ∧ A ∈ ℋ ) → (∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x)) ↔ ((F:ℕ–→ ℋ ∧ A ∈ ℋ ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x)))))
33	31, 32	bitr4d 409	1 ⊢ ((F:ℕ–→ ℋ ∧ A ∈ ℋ ) → (F ⇝v A ↔ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x))))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   class class class wbr 2054   Fn wfn 2417  –→wf 2418   ‘cfv 2422  (class class class)co 3001  ℝcr 4027  0cc0 4028   < clt 4033   ≤ cle 4092  ℕcn 4093   ℋ chil 4958   −_v cmv 4962  normcno 4964   ⇝_v chli 4966

This theorem is referenced by:  hlimcaui 5141

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

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-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-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  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-ltp 3884  df-plpr 3958  df-enr 3960  df-nr 3961  df-plr 3962  df-0r 3965  df-1r 3966  df-c 4034  df-1 4036  df-r 4038  df-plus 4039  df-n 4423  df-hlim 5107

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