Theorem rlimres 15605

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 4189	. . . . . . . 8 ⊢ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹
2		ssralv 4006	. . . . . . . 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 3103	. . . . . 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 3102	. . . . 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 628	. . . 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 15548	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹:dom 𝐹⟶ℂ)
9		rlimss 15549	. . . 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 15542	. . 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 6744	. . . . . 6 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ (dom 𝐹 ∩ 𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶ℂ)
13	8, 1, 12	sylancl 597	. . . . 5 ⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ (dom 𝐹 ∩ 𝐵)):(dom 𝐹 ∩ 𝐵)⟶ℂ)
14		resres 5991	. . . . . . 7 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹 ∩ 𝐵))
15		ffn 6705	. . . . . . . . 9 ⊢ (𝐹:dom 𝐹⟶ℂ → 𝐹 Fn dom 𝐹)
16		fnresdm 6654	. . . . . . . . 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 5977	. . . . . . 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 6687	. . . . 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 3948	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐴 → (dom 𝐹 ∩ 𝐵) ⊆ ℝ)
23		elinel2 4155	. . . . . 6 ⊢ (𝑧 ∈ (dom 𝐹 ∩ 𝐵) → 𝑧 ∈ 𝐵)
24	23	fvresd 6901	. . . . 5 ⊢ (𝑧 ∈ (dom 𝐹 ∩ 𝐵) → ((𝐹 ↾ 𝐵)‘𝑧) = (𝐹‘𝑧))
25	24	adantl 486	. . . 4 ⊢ ((𝐹 ⇝𝑟 𝐴 ∧ 𝑧 ∈ (dom 𝐹 ∩ 𝐵)) → ((𝐹 ↾ 𝐵)‘𝑧) = (𝐹‘𝑧))
26	21, 22, 25	rlim 15542	. . 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)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ∩ cin 3904   ⊆ wss 3905   class class class wbr 5109  dom cdm 5661   ↾ cres 5663   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ℂcc 11093  ℝcr 11094   < clt 11238   ≤ cle 11239   − cmin 11436  ℝ⁺crp 13011  abscabs 15281   ⇝_𝑟 crli 15532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-pm 8823  df-rlim 15536

This theorem is referenced by:  rlimres2  15608  pnt  27778