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Theorem limccl 14813

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 14810	. . . . . 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 6114	. . . . 5 ⊢ (𝑤 ∈ (𝐹 lim_ℂ 𝐵) → 𝐹 ∈ (ℂ ↑_pm ℂ))
4		limcrcl 14812	. . . . . 6 ⊢ (𝑤 ∈ (𝐹 lim_ℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
5	4	simp3d 1013	. . . . 5 ⊢ (𝑤 ∈ (𝐹 lim_ℂ 𝐵) → 𝐵 ∈ ℂ)
6		cnex 7996	. . . . . . 7 ⊢ ℂ ∈ V
7	6	rabex 4173	. . . . . 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 109	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → 𝑓 = 𝐹)
10	9	dmeqd 4864	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → dom 𝑓 = dom 𝐹)
11	9, 10	feq12d 5393	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑓:dom 𝑓⟶ℂ ↔ 𝐹:dom 𝐹⟶ℂ))
12	10	sseq1d 3208	. . . . . . . . 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 2262	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑥 ∈ ℂ ↔ 𝐵 ∈ ℂ))
16	14	breq2d 4041	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑧 # 𝑥 ↔ 𝑧 # 𝐵))
17	14	oveq2d 5934	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑧 − 𝑥) = (𝑧 − 𝐵))
18	17	fveq2d 5558	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (abs‘(𝑧 − 𝑥)) = (abs‘(𝑧 − 𝐵)))
19	18	breq1d 4039	. . . . . . . . . . . . . 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 5556	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑓‘𝑧) = (𝐹‘𝑧))
22	21	fvoveq1d 5940	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (abs‘((𝑓‘𝑧) − 𝑦)) = (abs‘((𝐹‘𝑧) − 𝑦)))
23	22	breq1d 4039	. . . . . . . . . . . . 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 2706	. . . . . . . . . . 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 2495	. . . . . . . . . 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 2494	. . . . . . . . 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 2749	. . . . . 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 6048	. . . . 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 1249	. . . 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 2272	. . 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 2913	. . 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 3183	1 ⊢ (𝐹 lim_ℂ 𝐵) ⊆ ℂ

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 ∀wral 2472 ∃wrex 2473 {crab 2476 Vcvv 2760 ⊆ wss 3153 class class class wbr 4029 dom cdm 4659 ⟶wf 5250 ‘cfv 5254 (class class class)co 5918 ↑_pm cpm 6703 ℂcc 7870 < clt 8054 − cmin 8190 # cap 8600 ℝ⁺crp 9719 abscabs 11141 lim_ℂ climc 14808

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fv 5262 df-ov 5921 df-oprab 5922 df-mpo 5923 df-pm 6705 df-limced 14810

This theorem is referenced by: reldvg 14833 dvfvalap 14835 dvcl 14837

W3C validator