Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rlimres Structured version   Visualization version   GIF version

Theorem rlimres 14137
 Description: The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
rlimres (𝐹𝑟 𝐴 → (𝐹𝐵) ⇝𝑟 𝐴)

Proof of Theorem rlimres
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3795 . . . . . . . 8 (dom 𝐹𝐵) ⊆ dom 𝐹
2 ssralv 3629 . . . . . . . 8 ((dom 𝐹𝐵) ⊆ dom 𝐹 → (∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥) → ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥)))
31, 2ax-mp 5 . . . . . . 7 (∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥) → ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))
43reximi 2994 . . . . . 6 (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))
54ralimi 2936 . . . . 5 (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥) → ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))
65anim2i 591 . . . 4 ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥)) → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥)))
76a1i 11 . . 3 (𝐹𝑟 𝐴 → ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥)) → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))))
8 rlimf 14080 . . . 4 (𝐹𝑟 𝐴𝐹:dom 𝐹⟶ℂ)
9 rlimss 14081 . . . 4 (𝐹𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
10 eqidd 2611 . . . 4 ((𝐹𝑟 𝐴𝑧 ∈ dom 𝐹) → (𝐹𝑧) = (𝐹𝑧))
118, 9, 10rlim 14074 . . 3 (𝐹𝑟 𝐴 → (𝐹𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))))
12 fssres 5983 . . . . . 6 ((𝐹:dom 𝐹⟶ℂ ∧ (dom 𝐹𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶ℂ)
138, 1, 12sylancl 693 . . . . 5 (𝐹𝑟 𝐴 → (𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶ℂ)
14 resres 5329 . . . . . . 7 ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹𝐵))
15 ffn 5958 . . . . . . . . 9 (𝐹:dom 𝐹⟶ℂ → 𝐹 Fn dom 𝐹)
16 fnresdm 5914 . . . . . . . . 9 (𝐹 Fn dom 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
178, 15, 163syl 18 . . . . . . . 8 (𝐹𝑟 𝐴 → (𝐹 ↾ dom 𝐹) = 𝐹)
1817reseq1d 5316 . . . . . . 7 (𝐹𝑟 𝐴 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹𝐵))
1914, 18syl5eqr 2658 . . . . . 6 (𝐹𝑟 𝐴 → (𝐹 ↾ (dom 𝐹𝐵)) = (𝐹𝐵))
2019feq1d 5943 . . . . 5 (𝐹𝑟 𝐴 → ((𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶ℂ ↔ (𝐹𝐵):(dom 𝐹𝐵)⟶ℂ))
2113, 20mpbid 221 . . . 4 (𝐹𝑟 𝐴 → (𝐹𝐵):(dom 𝐹𝐵)⟶ℂ)
221, 9syl5ss 3579 . . . 4 (𝐹𝑟 𝐴 → (dom 𝐹𝐵) ⊆ ℝ)
23 inss2 3796 . . . . . . 7 (dom 𝐹𝐵) ⊆ 𝐵
2423sseli 3564 . . . . . 6 (𝑧 ∈ (dom 𝐹𝐵) → 𝑧𝐵)
25 fvres 6117 . . . . . 6 (𝑧𝐵 → ((𝐹𝐵)‘𝑧) = (𝐹𝑧))
2624, 25syl 17 . . . . 5 (𝑧 ∈ (dom 𝐹𝐵) → ((𝐹𝐵)‘𝑧) = (𝐹𝑧))
2726adantl 481 . . . 4 ((𝐹𝑟 𝐴𝑧 ∈ (dom 𝐹𝐵)) → ((𝐹𝐵)‘𝑧) = (𝐹𝑧))
2821, 22, 27rlim 14074 . . 3 (𝐹𝑟 𝐴 → ((𝐹𝐵) ⇝𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))))
297, 11, 283imtr4d 282 . 2 (𝐹𝑟 𝐴 → (𝐹𝑟 𝐴 → (𝐹𝐵) ⇝𝑟 𝐴))
3029pm2.43i 50 1 (𝐹𝑟 𝐴 → (𝐹𝐵) ⇝𝑟 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583  dom cdm 5038   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814   < clt 9953   ≤ cle 9954   − cmin 10145  ℝ+crp 11708  abscabs 13822   ⇝𝑟 crli 14064 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pm 7747  df-rlim 14068 This theorem is referenced by:  rlimres2  14140  pnt  25103
 Copyright terms: Public domain W3C validator