Theorem limccl 13028

Description: Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)

Assertion

Ref	Expression
limccl	⊢ (𝐹 limℂ 𝐵) ⊆ ℂ

Proof of Theorem limccl

Dummy variables 𝑑 𝑒 𝑓 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . 4 ⊢ (𝑤 ∈ (𝐹 limℂ 𝐵) → 𝑤 ∈ (𝐹 limℂ 𝐵))
2		df-limced 13025	. . . . . 6 ⊢ limℂ = (𝑓 ∈ (ℂ ↑pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦 ∈ ℂ ∣ ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ∧ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧 − 𝑥)) < 𝑑) → (abs‘((𝑓‘𝑧) − 𝑦)) < 𝑒)))})
3	2	elmpocl1 6019	. . . . 5 ⊢ (𝑤 ∈ (𝐹 limℂ 𝐵) → 𝐹 ∈ (ℂ ↑pm ℂ))
4		limcrcl 13027	. . . . . 6 ⊢ (𝑤 ∈ (𝐹 limℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
5	4	simp3d 996	. . . . 5 ⊢ (𝑤 ∈ (𝐹 limℂ 𝐵) → 𝐵 ∈ ℂ)
6		cnex 7856	. . . . . . 7 ⊢ ℂ ∈ V
7	6	rabex 4108	. . . . . 6 ⊢ {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))} ∈ V
8	7	a1i 9	. . . . 5 ⊢ (𝑤 ∈ (𝐹 limℂ 𝐵) → {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))} ∈ V)
9		simpl 108	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → 𝑓 = 𝐹)
10	9	dmeqd 4788	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → dom 𝑓 = dom 𝐹)
11	9, 10	feq12d 5309	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑓:dom 𝑓⟶ℂ ↔ 𝐹:dom 𝐹⟶ℂ))
12	10	sseq1d 3157	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (dom 𝑓 ⊆ ℂ ↔ dom 𝐹 ⊆ ℂ))
13	11, 12	anbi12d 465	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ)))
14		simpr 109	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
15	14	eleq1d 2226	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑥 ∈ ℂ ↔ 𝐵 ∈ ℂ))
16	14	breq2d 3977	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑧 # 𝑥 ↔ 𝑧 # 𝐵))
17	14	oveq2d 5840	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑧 − 𝑥) = (𝑧 − 𝐵))
18	17	fveq2d 5472	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (abs‘(𝑧 − 𝑥)) = (abs‘(𝑧 − 𝐵)))
19	18	breq1d 3975	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → ((abs‘(𝑧 − 𝑥)) < 𝑑 ↔ (abs‘(𝑧 − 𝐵)) < 𝑑))
20	16, 19	anbi12d 465	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → ((𝑧 # 𝑥 ∧ (abs‘(𝑧 − 𝑥)) < 𝑑) ↔ (𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑)))
21	9	fveq1d 5470	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑓‘𝑧) = (𝐹‘𝑧))
22	21	fvoveq1d 5846	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (abs‘((𝑓‘𝑧) − 𝑦)) = (abs‘((𝐹‘𝑧) − 𝑦)))
23	22	breq1d 3975	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → ((abs‘((𝑓‘𝑧) − 𝑦)) < 𝑒 ↔ (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒))
24	20, 23	imbi12d 233	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (((𝑧 # 𝑥 ∧ (abs‘(𝑧 − 𝑥)) < 𝑑) → (abs‘((𝑓‘𝑧) − 𝑦)) < 𝑒) ↔ ((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))
25	10, 24	raleqbidv 2664	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (∀𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧 − 𝑥)) < 𝑑) → (abs‘((𝑓‘𝑧) − 𝑦)) < 𝑒) ↔ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))
26	25	rexbidv 2458	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧 − 𝑥)) < 𝑑) → (abs‘((𝑓‘𝑧) − 𝑦)) < 𝑒) ↔ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))
27	26	ralbidv 2457	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧 − 𝑥)) < 𝑑) → (abs‘((𝑓‘𝑧) − 𝑦)) < 𝑒) ↔ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))
28	15, 27	anbi12d 465	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → ((𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧 − 𝑥)) < 𝑑) → (abs‘((𝑓‘𝑧) − 𝑦)) < 𝑒)) ↔ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒))))
29	13, 28	anbi12d 465	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ∧ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧 − 𝑥)) < 𝑑) → (abs‘((𝑓‘𝑧) − 𝑦)) < 𝑒))) ↔ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))))
30	29	rabbidv 2701	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → {𝑦 ∈ ℂ ∣ ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ∧ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧 − 𝑥)) < 𝑑) → (abs‘((𝑓‘𝑧) − 𝑦)) < 𝑒)))} = {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))})
31	30, 2	ovmpoga 5950	. . . . 5 ⊢ ((𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐵 ∈ ℂ ∧ {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))} ∈ V) → (𝐹 limℂ 𝐵) = {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))})
32	3, 5, 8, 31	syl3anc 1220	. . . 4 ⊢ (𝑤 ∈ (𝐹 limℂ 𝐵) → (𝐹 limℂ 𝐵) = {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))})
33	1, 32	eleqtrd 2236	. . 3 ⊢ (𝑤 ∈ (𝐹 limℂ 𝐵) → 𝑤 ∈ {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))})
34		elrabi 2865	. . 3 ⊢ (𝑤 ∈ {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒)))} → 𝑤 ∈ ℂ)
35	33, 34	syl 14	. 2 ⊢ (𝑤 ∈ (𝐹 limℂ 𝐵) → 𝑤 ∈ ℂ)
36	35	ssriv 3132	1 ⊢ (𝐹 limℂ 𝐵) ⊆ ℂ

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1335   ∈ wcel 2128  ∀wral 2435  ∃wrex 2436  {crab 2439  Vcvv 2712   ⊆ wss 3102   class class class wbr 3965  dom cdm 4586  ⟶wf 5166  ‘cfv 5170  (class class class)co 5824   ↑_pm cpm 6594  ℂcc 7730   < clt 7912   − cmin 8046   # cap 8456  ℝ⁺crp 9560  abscabs 10897   lim_ℂ climc 13023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pm 6596  df-limced 13025

This theorem is referenced by:  reldvg  13048  dvfvalap  13050  dvcl  13052