Theorem limccl 12800

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 12797	. . . . . 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 5969	. . . . 5 ⊢ (𝑤 ∈ (𝐹 limℂ 𝐵) → 𝐹 ∈ (ℂ ↑pm ℂ))
4		limcrcl 12799	. . . . . 6 ⊢ (𝑤 ∈ (𝐹 limℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
5	4	simp3d 995	. . . . 5 ⊢ (𝑤 ∈ (𝐹 limℂ 𝐵) → 𝐵 ∈ ℂ)
6		cnex 7747	. . . . . . 7 ⊢ ℂ ∈ V
7	6	rabex 4072	. . . . . 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 4741	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → dom 𝑓 = dom 𝐹)
11	9, 10	feq12d 5262	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑓:dom 𝑓⟶ℂ ↔ 𝐹:dom 𝐹⟶ℂ))
12	10	sseq1d 3126	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (dom 𝑓 ⊆ ℂ ↔ dom 𝐹 ⊆ ℂ))
13	11, 12	anbi12d 464	. . . . . . . 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 2208	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑥 ∈ ℂ ↔ 𝐵 ∈ ℂ))
16	14	breq2d 3941	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑧 # 𝑥 ↔ 𝑧 # 𝐵))
17	14	oveq2d 5790	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑧 − 𝑥) = (𝑧 − 𝐵))
18	17	fveq2d 5425	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (abs‘(𝑧 − 𝑥)) = (abs‘(𝑧 − 𝐵)))
19	18	breq1d 3939	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → ((abs‘(𝑧 − 𝑥)) < 𝑑 ↔ (abs‘(𝑧 − 𝐵)) < 𝑑))
20	16, 19	anbi12d 464	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → ((𝑧 # 𝑥 ∧ (abs‘(𝑧 − 𝑥)) < 𝑑) ↔ (𝑧 # 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑑)))
21	9	fveq1d 5423	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (𝑓‘𝑧) = (𝐹‘𝑧))
22	21	fvoveq1d 5796	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝐵) → (abs‘((𝑓‘𝑧) − 𝑦)) = (abs‘((𝐹‘𝑧) − 𝑦)))
23	22	breq1d 3939	. . . . . . . . . . . . 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 2638	. . . . . . . . . . 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 2438	. . . . . . . . . 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 2437	. . . . . . . . 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 464	. . . . . . . 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 464	. . . . . . 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 2675	. . . . . 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 5900	. . . . 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 1216	. . . 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 2218	. . 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 2837	. . 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 3101	1 ⊢ (𝐹 limℂ 𝐵) ⊆ ℂ

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420  Vcvv 2686   ⊆ wss 3071   class class class wbr 3929  dom cdm 4539  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774   ↑_pm cpm 6543  ℂcc 7621   < clt 7803   − cmin 7936   # cap 8346  ℝ⁺crp 9444  abscabs 10772   lim_ℂ climc 12795

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7714

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pm 6545  df-limced 12797

This theorem is referenced by:  reldvg  12820  dvfvalap  12822  dvcl  12824