Theorem rlimres 15531

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 4203	. . . . . . . 8 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹
2		ssralv 4018	. . . . . . . 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 3068	. . . . . 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 3067	. . . . 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 15474	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹:dom 𝐹⟶ℂ)
9		rlimss 15475	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
10		eqidd 2731	. . . 4 ⊢ ((𝐹 ⇝𝑟 𝐴 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) = (𝐹‘𝑧))
11	8, 9, 10	rlim 15468	. . 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 6729	. . . . . 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 5966	. . . . . . 7 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹 ∩ 𝐵))
15		ffn 6691	. . . . . . . . 9 ⊢ (𝐹:dom 𝐹⟶ℂ → 𝐹 Fn dom 𝐹)
16		fnresdm 6640	. . . . . . . . 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 5952	. . . . . . 7 ⊢ (𝐹 ⇝𝑟 𝐴 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ 𝐵))
19	14, 18	eqtr3id 2779	. . . . . 6 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
20	19	feq1d 6673	. . . . 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 3961	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → (dom 𝐹 ∩ 𝐵) ⊆ ℝ)
23		elinel2 4168	. . . . . 6 ⊢ (𝑧 ∈ (dom 𝐹 ∩ 𝐵) → 𝑧 ∈ 𝐵)
24	23	fvresd 6881	. . . . 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 15468	. . 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 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ∩ cin 3916   ⊆ wss 3917   class class class wbr 5110  dom cdm 5641   ↾ cres 5643   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  ℂcc 11073  ℝcr 11074   < clt 11215   ≤ cle 11216   − cmin 11412  ℝ⁺crp 12958  abscabs 15207   ⇝_𝑟 crli 15458

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-pm 8805  df-rlim 15462

This theorem is referenced by:  rlimres2  15534  pnt  27532