Theorem rlimres 14917

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 4207	. . . . . . . 8 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹
2		ssralv 4035	. . . . . . . 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 3245	. . . . . 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 3162	. . . . 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 618	. . . 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 14860	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹:dom 𝐹⟶ℂ)
9		rlimss 14861	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
10		eqidd 2824	. . . 4 ⊢ ((𝐹 ⇝𝑟 𝐴 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) = (𝐹‘𝑧))
11	8, 9, 10	rlim 14854	. . 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 6546	. . . . . 6 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶ℂ)
13	8, 1, 12	sylancl 588	. . . . 5 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶ℂ)
14		resres 5868	. . . . . . 7 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹 ∩ 𝐵))
15		ffn 6516	. . . . . . . . 9 ⊢ (𝐹:dom 𝐹⟶ℂ → 𝐹 Fn dom 𝐹)
16		fnresdm 6468	. . . . . . . . 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 5854	. . . . . . 7 ⊢ (𝐹 ⇝𝑟 𝐴 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ 𝐵))
19	14, 18	syl5eqr 2872	. . . . . 6 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
20	19	feq1d 6501	. . . . 5 ⊢ (𝐹 ⇝𝑟 𝐴 → ((𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶ℂ ↔ (𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶ℂ))
21	13, 20	mpbid 234	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶ℂ)
22	1, 9	sstrid 3980	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → (dom 𝐹 ∩ 𝐵) ⊆ ℝ)
23		elinel2 4175	. . . . . 6 ⊢ (𝑧 ∈ (dom 𝐹 ∩ 𝐵) → 𝑧 ∈ 𝐵)
24	23	fvresd 6692	. . . . 5 ⊢ (𝑧 ∈ (dom 𝐹 ∩ 𝐵) → ((𝐹 ↾ 𝐵)‘𝑧) = (𝐹‘𝑧))
25	24	adantl 484	. . . 4 ⊢ ((𝐹 ⇝𝑟 𝐴 ∧ 𝑧 ∈ (dom 𝐹 ∩ 𝐵)) → ((𝐹 ↾ 𝐵)‘𝑧) = (𝐹‘𝑧))
26	21, 22, 25	rlim 14854	. . 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 296	. 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 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ∩ cin 3937   ⊆ wss 3938   class class class wbr 5068  dom cdm 5557   ↾ cres 5559   Fn wfn 6352  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  ℂcc 10537  ℝcr 10538   < clt 10677   ≤ cle 10678   − cmin 10872  ℝ⁺crp 12392  abscabs 14595   ⇝_𝑟 crli 14844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-pm 8411  df-rlim 14848

This theorem is referenced by:  rlimres2  14920  pnt  26192