Theorem rlimres 15591

Description: The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)

Assertion

Ref	Expression
rlimres	⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ 𝐵) ⇝𝑟 𝐴)

Proof of Theorem rlimres

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 4245	. . . . . . . 8 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹
2		ssralv 4064	. . . . . . . 8 ⊢ ((dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹 → (∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐴)) < 𝑥) → ∀𝑧 ∈ (dom 𝐹 ∩ 𝐵)(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐴)) < 𝑥)))
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ (∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐴)) < 𝑥) → ∀𝑧 ∈ (dom 𝐹 ∩ 𝐵)(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐴)) < 𝑥))
4	3	reximi 3082	. . . . . 6 ⊢ (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐴)) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹 ∩ 𝐵)(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐴)) < 𝑥))
5	4	ralimi 3081	. . . . 5 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐴)) < 𝑥) → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹 ∩ 𝐵)(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐴)) < 𝑥))
6	5	anim2i 617	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐴)) < 𝑥)) → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹 ∩ 𝐵)(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐴)) < 𝑥)))
7	6	a1i 11	. . 3 ⊢ (𝐹 ⇝𝑟 𝐴 → ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐴)) < 𝑥)) → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹 ∩ 𝐵)(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐴)) < 𝑥))))
8		rlimf 15534	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹:dom 𝐹⟶ℂ)
9		rlimss 15535	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
10		eqidd 2736	. . . 4 ⊢ ((𝐹 ⇝𝑟 𝐴 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) = (𝐹‘𝑧))
11	8, 9, 10	rlim 15528	. . 3 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ⇝𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐴)) < 𝑥))))
12		fssres 6775	. . . . . 6 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶ℂ)
13	8, 1, 12	sylancl 586	. . . . 5 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶ℂ)
14		resres 6013	. . . . . . 7 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹 ∩ 𝐵))
15		ffn 6737	. . . . . . . . 9 ⊢ (𝐹:dom 𝐹⟶ℂ → 𝐹 Fn dom 𝐹)
16		fnresdm 6688	. . . . . . . . 9 ⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
17	8, 15, 16	3syl 18	. . . . . . . 8 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ dom 𝐹) = 𝐹)
18	17	reseq1d 5999	. . . . . . 7 ⊢ (𝐹 ⇝𝑟 𝐴 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ 𝐵))
19	14, 18	eqtr3id 2789	. . . . . 6 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
20	19	feq1d 6721	. . . . 5 ⊢ (𝐹 ⇝𝑟 𝐴 → ((𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶ℂ ↔ (𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶ℂ))
21	13, 20	mpbid 232	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶ℂ)
22	1, 9	sstrid 4007	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → (dom 𝐹 ∩ 𝐵) ⊆ ℝ)
23		elinel2 4212	. . . . . 6 ⊢ (𝑧 ∈ (dom 𝐹 ∩ 𝐵) → 𝑧 ∈ 𝐵)
24	23	fvresd 6927	. . . . 5 ⊢ (𝑧 ∈ (dom 𝐹 ∩ 𝐵) → ((𝐹 ↾ 𝐵)‘𝑧) = (𝐹‘𝑧))
25	24	adantl 481	. . . 4 ⊢ ((𝐹 ⇝𝑟 𝐴 ∧ 𝑧 ∈ (dom 𝐹 ∩ 𝐵)) → ((𝐹 ↾ 𝐵)‘𝑧) = (𝐹‘𝑧))
26	21, 22, 25	rlim 15528	. . 3 ⊢ (𝐹 ⇝𝑟 𝐴 → ((𝐹 ↾ 𝐵) ⇝𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹 ∩ 𝐵)(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐴)) < 𝑥))))
27	7, 11, 26	3imtr4d 294	. 2 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ 𝐵) ⇝𝑟 𝐴))
28	27	pm2.43i 52	1 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ 𝐵) ⇝𝑟 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ∩ cin 3962   ⊆ wss 3963   class class class wbr 5148  dom cdm 5689   ↾ cres 5691   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  ℂcc 11151  ℝcr 11152   < clt 11293   ≤ cle 11294   − cmin 11490  ℝ⁺crp 13032  abscabs 15270   ⇝_𝑟 crli 15518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-pm 8868  df-rlim 15522

This theorem is referenced by:  rlimres2  15594  pnt  27673