Theorem rlimres 15769

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 4136	. . . . . . . 8 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹
2		ssralv 3982	. . . . . . . 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 3104	. . . . . 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 3103	. . . . 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 629	. . . 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 15712	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹:dom 𝐹⟶ℂ)
9		rlimss 15713	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
10		eqidd 2764	. . . 4 ⊢ ((𝐹 ⇝𝑟 𝐴 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) = (𝐹‘𝑧))
11	8, 9, 10	rlim 15706	. . 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 6778	. . . . . 6 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶ℂ)
13	8, 1, 12	sylancl 598	. . . . 5 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶ℂ)
14		resres 5999	. . . . . . 7 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹 ∩ 𝐵))
15		ffn 6739	. . . . . . . . 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 19	. . . . . . . 8 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ dom 𝐹) = 𝐹)
18	17	reseq1d 5985	. . . . . . 7 ⊢ (𝐹 ⇝𝑟 𝐴 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ 𝐵))
19	14, 18	eqtr3id 2812	. . . . . 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 235	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ 𝐵):(dom 𝐹 ∩ 𝐵)⟶ℂ)
22	1, 9	sstrid 3924	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → (dom 𝐹 ∩ 𝐵) ⊆ ℝ)
23		elinel2 4102	. . . . . 6 ⊢ (𝑧 ∈ (dom 𝐹 ∩ 𝐵) → 𝑧 ∈ 𝐵)
24	23	fvresd 6936	. . . . 5 ⊢ (𝑧 ∈ (dom 𝐹 ∩ 𝐵) → ((𝐹 ↾ 𝐵)‘𝑧) = (𝐹‘𝑧))
25	24	adantl 487	. . . 4 ⊢ ((𝐹 ⇝𝑟 𝐴 ∧ 𝑧 ∈ (dom 𝐹 ∩ 𝐵)) → ((𝐹 ↾ 𝐵)‘𝑧) = (𝐹‘𝑧))
26	21, 22, 25	rlim 15706	. . 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 297	. 2 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ 𝐵) ⇝𝑟 𝐴))
28	27	pm2.43i 53	1 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ 𝐵) ⇝𝑟 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092   class class class wbr 5108  dom cdm 5658   ↾ cres 5660   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  ℂcc 11237  ℝcr 11238   < clt 11382   ≤ cle 11383   − cmin 11580  ℝ⁺crp 13160  abscabs 15444   ⇝_𝑟 crli 15696

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-pm 8888  df-rlim 15700

This theorem is used by:  rlimres2  15772  pnt  27998