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Theorem limccl 13268

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

**Assertion**
Ref	Expression
limccl	⊢ (𝐹 lim_ℂ 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 lim_ℂ 𝐵) → 𝑤 ∈ (𝐹 lim_ℂ 𝐵))
2		df-limced 13265	. . . . . 6 ⊢ lim_ℂ = (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 6037	. . . . 5 ⊢ (𝑤 ∈ (𝐹 lim_ℂ 𝐵) → 𝐹 ∈ (ℂ ↑_pm ℂ))
4		limcrcl 13267	. . . . . 6 ⊢ (𝑤 ∈ (𝐹 lim_ℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
5	4	simp3d 1001	. . . . 5 ⊢ (𝑤 ∈ (𝐹 lim_ℂ 𝐵) → 𝐵 ∈ ℂ)
6		cnex 7877	. . . . . . 7 ⊢ ℂ ∈ V
7	6	rabex 4126	. . . . . 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 lim_ℂ 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 4806	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → dom 𝑓 = dom 𝐹)
11	9, 10	feq12d 5327	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑓:dom 𝑓⟶ℂ ↔ 𝐹:dom 𝐹⟶ℂ))
12	10	sseq1d 3171	. . . . . . . . 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 2235	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑥 ∈ ℂ ↔ 𝐵 ∈ ℂ))
16	14	breq2d 3994	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑧 # 𝑥 ↔ 𝑧 # 𝐵))
17	14	oveq2d 5858	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑧 − 𝑥) = (𝑧 − 𝐵))
18	17	fveq2d 5490	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (abs‘(𝑧 − 𝑥)) = (abs‘(𝑧 − 𝐵)))
19	18	breq1d 3992	. . . . . . . . . . . . . 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 5488	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑓‘𝑧) = (𝐹‘𝑧))
22	21	fvoveq1d 5864	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (abs‘((𝑓‘𝑧) − 𝑦)) = (abs‘((𝐹‘𝑧) − 𝑦)))
23	22	breq1d 3992	. . . . . . . . . . . . 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 2673	. . . . . . . . . . 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 2467	. . . . . . . . . 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 2466	. . . . . . . . 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 2715	. . . . . 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 5971	. . . . 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 lim_ℂ 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 1228	. . . 4 ⊢ (𝑤 ∈ (𝐹 lim_ℂ 𝐵) → (𝐹 lim_ℂ 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 2245	. . 3 ⊢ (𝑤 ∈ (𝐹 lim_ℂ 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 2879	. . 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 lim_ℂ 𝐵) → 𝑤 ∈ ℂ)
36	35	ssriv 3146	1 ⊢ (𝐹 lim_ℂ 𝐵) ⊆ ℂ

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 = wceq 1343 ∈ wcel 2136 ∀wral 2444 ∃wrex 2445 {crab 2448 Vcvv 2726 ⊆ wss 3116 class class class wbr 3982 dom cdm 4604 ⟶wf 5184 ‘cfv 5188 (class class class)co 5842 ↑_pm cpm 6615 ℂcc 7751 < clt 7933 − cmin 8069 # cap 8479 ℝ⁺crp 9589 abscabs 10939 lim_ℂ climc 13263

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pm 6617 df-limced 13265

This theorem is referenced by: reldvg 13288 dvfvalap 13290 dvcl 13292

W3C validator