ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  limccl GIF version

Theorem limccl 13999
Description: Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
Assertion
Ref Expression
limccl (𝐹 lim 𝐵) ⊆ ℂ

Proof of Theorem limccl
Dummy variables 𝑑 𝑒 𝑓 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . 4 (𝑤 ∈ (𝐹 lim 𝐵) → 𝑤 ∈ (𝐹 lim 𝐵))
2 df-limced 13996 . . . . . 6 lim = (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‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒)))})
32elmpocl1 6067 . . . . 5 (𝑤 ∈ (𝐹 lim 𝐵) → 𝐹 ∈ (ℂ ↑pm ℂ))
4 limcrcl 13998 . . . . . 6 (𝑤 ∈ (𝐹 lim 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
54simp3d 1011 . . . . 5 (𝑤 ∈ (𝐹 lim 𝐵) → 𝐵 ∈ ℂ)
6 cnex 7932 . . . . . . 7 ℂ ∈ V
76rabex 4146 . . . . . 6 {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))} ∈ V
87a1i 9 . . . . 5 (𝑤 ∈ (𝐹 lim 𝐵) → {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))} ∈ V)
9 simpl 109 . . . . . . . . . 10 ((𝑓 = 𝐹𝑥 = 𝐵) → 𝑓 = 𝐹)
109dmeqd 4828 . . . . . . . . . 10 ((𝑓 = 𝐹𝑥 = 𝐵) → dom 𝑓 = dom 𝐹)
119, 10feq12d 5354 . . . . . . . . 9 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑓:dom 𝑓⟶ℂ ↔ 𝐹:dom 𝐹⟶ℂ))
1210sseq1d 3184 . . . . . . . . 9 ((𝑓 = 𝐹𝑥 = 𝐵) → (dom 𝑓 ⊆ ℂ ↔ dom 𝐹 ⊆ ℂ))
1311, 12anbi12d 473 . . . . . . . 8 ((𝑓 = 𝐹𝑥 = 𝐵) → ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ)))
14 simpr 110 . . . . . . . . . 10 ((𝑓 = 𝐹𝑥 = 𝐵) → 𝑥 = 𝐵)
1514eleq1d 2246 . . . . . . . . 9 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑥 ∈ ℂ ↔ 𝐵 ∈ ℂ))
1614breq2d 4014 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑧 # 𝑥𝑧 # 𝐵))
1714oveq2d 5888 . . . . . . . . . . . . . . . 16 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑧𝑥) = (𝑧𝐵))
1817fveq2d 5518 . . . . . . . . . . . . . . 15 ((𝑓 = 𝐹𝑥 = 𝐵) → (abs‘(𝑧𝑥)) = (abs‘(𝑧𝐵)))
1918breq1d 4012 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑥 = 𝐵) → ((abs‘(𝑧𝑥)) < 𝑑 ↔ (abs‘(𝑧𝐵)) < 𝑑))
2016, 19anbi12d 473 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑥 = 𝐵) → ((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) ↔ (𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑)))
219fveq1d 5516 . . . . . . . . . . . . . . 15 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑓𝑧) = (𝐹𝑧))
2221fvoveq1d 5894 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑥 = 𝐵) → (abs‘((𝑓𝑧) − 𝑦)) = (abs‘((𝐹𝑧) − 𝑦)))
2322breq1d 4012 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑥 = 𝐵) → ((abs‘((𝑓𝑧) − 𝑦)) < 𝑒 ↔ (abs‘((𝐹𝑧) − 𝑦)) < 𝑒))
2420, 23imbi12d 234 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑥 = 𝐵) → (((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒) ↔ ((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))
2510, 24raleqbidv 2684 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑥 = 𝐵) → (∀𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒) ↔ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))
2625rexbidv 2478 . . . . . . . . . 10 ((𝑓 = 𝐹𝑥 = 𝐵) → (∃𝑑 ∈ ℝ+𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒) ↔ ∃𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))
2726ralbidv 2477 . . . . . . . . 9 ((𝑓 = 𝐹𝑥 = 𝐵) → (∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒) ↔ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))
2815, 27anbi12d 473 . . . . . . . 8 ((𝑓 = 𝐹𝑥 = 𝐵) → ((𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒)) ↔ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒))))
2913, 28anbi12d 473 . . . . . . 7 ((𝑓 = 𝐹𝑥 = 𝐵) → (((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ∧ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒))) ↔ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))))
3029rabbidv 2726 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝐵) → {𝑦 ∈ ℂ ∣ ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ∧ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒)))} = {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))})
3130, 2ovmpoga 6001 . . . . 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‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))} ∈ V) → (𝐹 lim 𝐵) = {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))})
323, 5, 8, 31syl3anc 1238 . . . 4 (𝑤 ∈ (𝐹 lim 𝐵) → (𝐹 lim 𝐵) = {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))})
331, 32eleqtrd 2256 . . 3 (𝑤 ∈ (𝐹 lim 𝐵) → 𝑤 ∈ {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))})
34 elrabi 2890 . . 3 (𝑤 ∈ {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))} → 𝑤 ∈ ℂ)
3533, 34syl 14 . 2 (𝑤 ∈ (𝐹 lim 𝐵) → 𝑤 ∈ ℂ)
3635ssriv 3159 1 (𝐹 lim 𝐵) ⊆ ℂ
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  wrex 2456  {crab 2459  Vcvv 2737  wss 3129   class class class wbr 4002  dom cdm 4625  wf 5211  cfv 5215  (class class class)co 5872  pm cpm 6646  cc 7806   < clt 7988  cmin 8124   # cap 8534  +crp 9649  abscabs 10999   lim climc 13994
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 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899
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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pm 6648  df-limced 13996
This theorem is referenced by:  reldvg  14019  dvfvalap  14021  dvcl  14023
  Copyright terms: Public domain W3C validator