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Theorem rlimres 15594
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 4237 . . . . . . . 8 (dom 𝐹𝐵) ⊆ dom 𝐹
2 ssralv 4052 . . . . . . . 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 3084 . . . . . 6 (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))
54ralimi 3083 . . . . 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 617 . . . 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 15537 . . . 4 (𝐹𝑟 𝐴𝐹:dom 𝐹⟶ℂ)
9 rlimss 15538 . . . 4 (𝐹𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
10 eqidd 2738 . . . 4 ((𝐹𝑟 𝐴𝑧 ∈ dom 𝐹) → (𝐹𝑧) = (𝐹𝑧))
118, 9, 10rlim 15531 . . 3 (𝐹𝑟 𝐴 → (𝐹𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))))
12 fssres 6774 . . . . . 6 ((𝐹:dom 𝐹⟶ℂ ∧ (dom 𝐹𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶ℂ)
138, 1, 12sylancl 586 . . . . 5 (𝐹𝑟 𝐴 → (𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶ℂ)
14 resres 6010 . . . . . . 7 ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹𝐵))
15 ffn 6736 . . . . . . . . 9 (𝐹:dom 𝐹⟶ℂ → 𝐹 Fn dom 𝐹)
16 fnresdm 6687 . . . . . . . . 9 (𝐹 Fn dom 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
178, 15, 163syl 18 . . . . . . . 8 (𝐹𝑟 𝐴 → (𝐹 ↾ dom 𝐹) = 𝐹)
1817reseq1d 5996 . . . . . . 7 (𝐹𝑟 𝐴 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹𝐵))
1914, 18eqtr3id 2791 . . . . . 6 (𝐹𝑟 𝐴 → (𝐹 ↾ (dom 𝐹𝐵)) = (𝐹𝐵))
2019feq1d 6720 . . . . 5 (𝐹𝑟 𝐴 → ((𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶ℂ ↔ (𝐹𝐵):(dom 𝐹𝐵)⟶ℂ))
2113, 20mpbid 232 . . . 4 (𝐹𝑟 𝐴 → (𝐹𝐵):(dom 𝐹𝐵)⟶ℂ)
221, 9sstrid 3995 . . . 4 (𝐹𝑟 𝐴 → (dom 𝐹𝐵) ⊆ ℝ)
23 elinel2 4202 . . . . . 6 (𝑧 ∈ (dom 𝐹𝐵) → 𝑧𝐵)
2423fvresd 6926 . . . . 5 (𝑧 ∈ (dom 𝐹𝐵) → ((𝐹𝐵)‘𝑧) = (𝐹𝑧))
2524adantl 481 . . . 4 ((𝐹𝑟 𝐴𝑧 ∈ (dom 𝐹𝐵)) → ((𝐹𝐵)‘𝑧) = (𝐹𝑧))
2621, 22, 25rlim 15531 . . 3 (𝐹𝑟 𝐴 → ((𝐹𝐵) ⇝𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))))
277, 11, 263imtr4d 294 . 2 (𝐹𝑟 𝐴 → (𝐹𝑟 𝐴 → (𝐹𝐵) ⇝𝑟 𝐴))
2827pm2.43i 52 1 (𝐹𝑟 𝐴 → (𝐹𝐵) ⇝𝑟 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cin 3950  wss 3951   class class class wbr 5143  dom cdm 5685  cres 5687   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  cr 11154   < clt 11295  cle 11296  cmin 11492  +crp 13034  abscabs 15273  𝑟 crli 15521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-pm 8869  df-rlim 15525
This theorem is referenced by:  rlimres2  15597  pnt  27658
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