Theorem limccl 14131

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 14128	. . . . . 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 6070	. . . . 5 ⊢ (𝑤 ∈ (𝐹 limℂ 𝐵) → 𝐹 ∈ (ℂ ↑pm ℂ))
4		limcrcl 14130	. . . . . 6 ⊢ (𝑤 ∈ (𝐹 limℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
5	4	simp3d 1011	. . . . 5 ⊢ (𝑤 ∈ (𝐹 limℂ 𝐵) → 𝐵 ∈ ℂ)
6		cnex 7935	. . . . . . 7 ⊢ ℂ ∈ V
7	6	rabex 4148	. . . . . 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 109	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → 𝑓 = 𝐹)
10	9	dmeqd 4830	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → dom 𝑓 = dom 𝐹)
11	9, 10	feq12d 5356	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑓:dom 𝑓⟶ℂ ↔ 𝐹:dom 𝐹⟶ℂ))
12	10	sseq1d 3185	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (dom 𝑓 ⊆ ℂ ↔ dom 𝐹 ⊆ ℂ))
13	11, 12	anbi12d 473	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ)))
14		simpr 110	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
15	14	eleq1d 2246	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑥 ∈ ℂ ↔ 𝐵 ∈ ℂ))
16	14	breq2d 4016	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑧 # 𝑥 ↔ 𝑧 # 𝐵))
17	14	oveq2d 5891	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑧 − 𝑥) = (𝑧 − 𝐵))
18	17	fveq2d 5520	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (abs‘(𝑧 − 𝑥)) = (abs‘(𝑧 − 𝐵)))
19	18	breq1d 4014	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → ((abs‘(𝑧 − 𝑥)) < 𝑑 ↔ (abs‘(𝑧 − 𝐵)) < 𝑑))
20	16, 19	anbi12d 473	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → ((𝑧 # 𝑥 ∧ (abs‘(𝑧 − 𝑥)) < 𝑑) ↔ (𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑)))
21	9	fveq1d 5518	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑓‘𝑧) = (𝐹‘𝑧))
22	21	fvoveq1d 5897	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (abs‘((𝑓‘𝑧) − 𝑦)) = (abs‘((𝐹‘𝑧) − 𝑦)))
23	22	breq1d 4014	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → ((abs‘((𝑓‘𝑧) − 𝑦)) < 𝑒 ↔ (abs‘((𝐹‘𝑧) − 𝑦)) < 𝑒))
24	20, 23	imbi12d 234	. . . . . . . . . . . 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 2685	. . . . . . . . . . 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 2478	. . . . . . . . . 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 2477	. . . . . . . . 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 473	. . . . . . . 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 473	. . . . . . 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 2727	. . . . . 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 6004	. . . . 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 1238	. . . 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 2256	. . 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 2891	. . 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 3160	1 ⊢ (𝐹 limℂ 𝐵) ⊆ ℂ

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  {crab 2459  Vcvv 2738   ⊆ wss 3130   class class class wbr 4004  dom cdm 4627  ⟶wf 5213  ‘cfv 5217  (class class class)co 5875   ↑_pm cpm 6649  ℂcc 7809   < clt 7992   − cmin 8128   # cap 8538  ℝ⁺crp 9653  abscabs 11006   lim_ℂ climc 14126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pm 6651  df-limced 14128

This theorem is referenced by:  reldvg  14151  dvfvalap  14153  dvcl  14155