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Theorem rlimres 14907
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 4203 . . . . . . . 8 (dom 𝐹𝐵) ⊆ dom 𝐹
2 ssralv 4031 . . . . . . . 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 3241 . . . . . 6 (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))
54ralimi 3158 . . . . 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 618 . . . 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 14850 . . . 4 (𝐹𝑟 𝐴𝐹:dom 𝐹⟶ℂ)
9 rlimss 14851 . . . 4 (𝐹𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
10 eqidd 2820 . . . 4 ((𝐹𝑟 𝐴𝑧 ∈ dom 𝐹) → (𝐹𝑧) = (𝐹𝑧))
118, 9, 10rlim 14844 . . 3 (𝐹𝑟 𝐴 → (𝐹𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))))
12 fssres 6537 . . . . . 6 ((𝐹:dom 𝐹⟶ℂ ∧ (dom 𝐹𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶ℂ)
138, 1, 12sylancl 588 . . . . 5 (𝐹𝑟 𝐴 → (𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶ℂ)
14 resres 5859 . . . . . . 7 ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹𝐵))
15 ffn 6507 . . . . . . . . 9 (𝐹:dom 𝐹⟶ℂ → 𝐹 Fn dom 𝐹)
16 fnresdm 6459 . . . . . . . . 9 (𝐹 Fn dom 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
178, 15, 163syl 18 . . . . . . . 8 (𝐹𝑟 𝐴 → (𝐹 ↾ dom 𝐹) = 𝐹)
1817reseq1d 5845 . . . . . . 7 (𝐹𝑟 𝐴 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹𝐵))
1914, 18syl5eqr 2868 . . . . . 6 (𝐹𝑟 𝐴 → (𝐹 ↾ (dom 𝐹𝐵)) = (𝐹𝐵))
2019feq1d 6492 . . . . 5 (𝐹𝑟 𝐴 → ((𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶ℂ ↔ (𝐹𝐵):(dom 𝐹𝐵)⟶ℂ))
2113, 20mpbid 234 . . . 4 (𝐹𝑟 𝐴 → (𝐹𝐵):(dom 𝐹𝐵)⟶ℂ)
221, 9sstrid 3976 . . . 4 (𝐹𝑟 𝐴 → (dom 𝐹𝐵) ⊆ ℝ)
23 elinel2 4171 . . . . . 6 (𝑧 ∈ (dom 𝐹𝐵) → 𝑧𝐵)
2423fvresd 6683 . . . . 5 (𝑧 ∈ (dom 𝐹𝐵) → ((𝐹𝐵)‘𝑧) = (𝐹𝑧))
2524adantl 484 . . . 4 ((𝐹𝑟 𝐴𝑧 ∈ (dom 𝐹𝐵)) → ((𝐹𝐵)‘𝑧) = (𝐹𝑧))
2621, 22, 25rlim 14844 . . 3 (𝐹𝑟 𝐴 → ((𝐹𝐵) ⇝𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))))
277, 11, 263imtr4d 296 . 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 398   = wceq 1530  wcel 2107  wral 3136  wrex 3137  cin 3933  wss 3934   class class class wbr 5057  dom cdm 5548  cres 5550   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7148  cc 10527  cr 10528   < clt 10667  cle 10668  cmin 10862  +crp 12381  abscabs 14585  𝑟 crli 14834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-pm 8401  df-rlim 14838
This theorem is referenced by:  rlimres2  14910  pnt  26182
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