Theorem clim 4877

Description: Express the predicate: The limit of complex number sequence F is A, or F converges to A. This means that for any real x, no matter how small, there always exists an integer y such that the absolute difference of any later complex number in the sequence and the limit is less than x.

Hypotheses

Ref	Expression
clim.1	⊢ F ∈ V
clim.2	⊢ A ∈ V

Assertion

Ref	Expression
clim	⊢ (F ⇝ A ↔ ((F:ℕ–→ℂ ∧ A ∈ ℂ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − A)) < x))))

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

Proof of Theorem clim

Step	Hyp	Ref	Expression
1		clim.1	. 2 ⊢ F ∈ V
2		clim.2	. 2 ⊢ A ∈ V
3		feq1 2748	. . . 4 ⊢ (f = F → (f:ℕ–→ℂ ↔ F:ℕ–→ℂ))
4	3	anbi1d 469	. . 3 ⊢ (f = F → ((f:ℕ–→ℂ ∧ w ∈ ℂ) ↔ (F:ℕ–→ℂ ∧ w ∈ ℂ)))
5		fveq1 2831	. . . . . . . . . . 11 ⊢ (f = F → (f ‘z) = (F ‘z))
6	5	opreq1d 3012	. . . . . . . . . 10 ⊢ (f = F → ((f ‘z) − w) = ((F ‘z) − w))
7	6	fveq2d 2836	. . . . . . . . 9 ⊢ (f = F → (abs ‘((f ‘z) − w)) = (abs ‘((F ‘z) − w)))
8	7	breq1d 2071	. . . . . . . 8 ⊢ (f = F → ((abs ‘((f ‘z) − w)) < x ↔ (abs ‘((F ‘z) − w)) < x))
9	8	imbi2d 464	. . . . . . 7 ⊢ (f = F → ((y ≤ z → (abs ‘((f ‘z) − w)) < x) ↔ (y ≤ z → (abs ‘((F ‘z) − w)) < x)))
10	9	biraldv 1219	. . . . . 6 ⊢ (f = F → (∀z ∈ ℕ (y ≤ z → (abs ‘((f ‘z) − w)) < x) ↔ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − w)) < x)))
11	10	birexdv 1220	. . . . 5 ⊢ (f = F → (∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((f ‘z) − w)) < x) ↔ ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − w)) < x)))
12	11	imbi2d 464	. . . 4 ⊢ (f = F → ((0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((f ‘z) − w)) < x)) ↔ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − w)) < x))))
13	12	biraldv 1219	. . 3 ⊢ (f = F → (∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((f ‘z) − w)) < x)) ↔ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − w)) < x))))
14	4, 13	anbi12d 476	. 2 ⊢ (f = F → (((f:ℕ–→ℂ ∧ w ∈ ℂ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((f ‘z) − w)) < x))) ↔ ((F:ℕ–→ℂ ∧ w ∈ ℂ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − w)) < x)))))
15		eleq1 1149	. . . 4 ⊢ (w = A → (w ∈ ℂ ↔ A ∈ ℂ))
16	15	anbi2d 468	. . 3 ⊢ (w = A → ((F:ℕ–→ℂ ∧ w ∈ ℂ) ↔ (F:ℕ–→ℂ ∧ A ∈ ℂ)))
17		opreq2 3007	. . . . . . . . . 10 ⊢ (w = A → ((F ‘z) − w) = ((F ‘z) − A))
18	17	fveq2d 2836	. . . . . . . . 9 ⊢ (w = A → (abs ‘((F ‘z) − w)) = (abs ‘((F ‘z) − A)))
19	18	breq1d 2071	. . . . . . . 8 ⊢ (w = A → ((abs ‘((F ‘z) − w)) < x ↔ (abs ‘((F ‘z) − A)) < x))
20	19	imbi2d 464	. . . . . . 7 ⊢ (w = A → ((y ≤ z → (abs ‘((F ‘z) − w)) < x) ↔ (y ≤ z → (abs ‘((F ‘z) − A)) < x)))
21	20	biraldv 1219	. . . . . 6 ⊢ (w = A → (∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − w)) < x) ↔ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − A)) < x)))
22	21	birexdv 1220	. . . . 5 ⊢ (w = A → (∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − w)) < x) ↔ ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − A)) < x)))
23	22	imbi2d 464	. . . 4 ⊢ (w = A → ((0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − w)) < x)) ↔ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − A)) < x))))
24	23	biraldv 1219	. . 3 ⊢ (w = A → (∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − w)) < x)) ↔ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − A)) < x))))
25	16, 24	anbi12d 476	. 2 ⊢ (w = A → (((F:ℕ–→ℂ ∧ w ∈ ℂ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − w)) < x))) ↔ ((F:ℕ–→ℂ ∧ A ∈ ℂ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − A)) < x)))))
26		df-clim 4876	. 2 ⊢ ⇝ = {⟨f, w⟩∣((f:ℕ–→ℂ ∧ w ∈ ℂ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((f ‘z) − w)) < x)))}
27	1, 2, 14, 25, 26	brab 2118	1 ⊢ (F ⇝ A ↔ ((F:ℕ–→ℂ ∧ A ∈ ℂ) ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (abs ‘((F ‘z) − 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  –→wf 2418   ‘cfv 2422  (class class class)co 3001  ℂcc 4026  ℝcr 4027  0cc0 4028   < clt 4033   − cmin 4089   ≤ cle 4092  ℕcn 4093  abscabs 4789   ⇝ cli 4875

This theorem is referenced by:  climseq 4878  climcn 4879  climconv 4880  clim0 4882  occllem6 5185  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-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-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  df-opr 3003  df-clim 4876

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