Theorem rescncf 15372

Description: A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)

Assertion

Ref	Expression
rescncf	⊢ (𝐶 ⊆ 𝐴 → (𝐹 ∈ (𝐴–cn→𝐵) → (𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵)))

Proof of Theorem rescncf

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

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . 6 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐹 ∈ (𝐴–cn→𝐵))
2		cncfrss 15366	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ)
3	2	adantl 277	. . . . . . 7 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐴 ⊆ ℂ)
4		cncfrss2 15367	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ)
5	4	adantl 277	. . . . . . 7 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐵 ⊆ ℂ)
6		elcncf 15364	. . . . . . 7 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴–cn→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦))))
7	3, 5, 6	syl2anc 411	. . . . . 6 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → (𝐹 ∈ (𝐴–cn→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦))))
8	1, 7	mpbid 147	. . . . 5 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦)))
9	8	simpld 112	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐹:𝐴⟶𝐵)
10		simpl 109	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐶 ⊆ 𝐴)
11	9, 10	fssresd 5521	. . 3 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
12	8	simprd 114	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦))
13		ssralv 3292	. . . . 5 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦)))
14		ssralv 3292	. . . . . . . . 9 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦) → ∀𝑤 ∈ 𝐶 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦)))
15		fvres 5672	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
16		fvres 5672	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑤) = (𝐹‘𝑤))
17	15, 16	oveqan12d 6047	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤)) = ((𝐹‘𝑥) − (𝐹‘𝑤)))
18	17	fveq2d 5652	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) = (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))))
19	18	breq1d 4103	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → ((abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) < 𝑦 ↔ (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦))
20	19	imbi2d 230	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) < 𝑦) ↔ ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦)))
21	20	biimprd 158	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦) → ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) < 𝑦)))
22	21	ralimdva 2600	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐶 → (∀𝑤 ∈ 𝐶 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦) → ∀𝑤 ∈ 𝐶 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) < 𝑦)))
23	14, 22	sylan9 409	. . . . . . . 8 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐶) → (∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦) → ∀𝑤 ∈ 𝐶 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) < 𝑦)))
24	23	reximdv 2634	. . . . . . 7 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐶) → (∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐶 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) < 𝑦)))
25	24	ralimdv 2601	. . . . . 6 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐶) → (∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦) → ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐶 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) < 𝑦)))
26	25	ralimdva 2600	. . . . 5 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐶 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) < 𝑦)))
27	13, 26	syld 45	. . . 4 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐶 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) < 𝑦)))
28	10, 12, 27	sylc 62	. . 3 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐶 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) < 𝑦))
29	10, 3	sstrd 3238	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐶 ⊆ ℂ)
30		elcncf 15364	. . . 4 ⊢ ((𝐶 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → ((𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵) ↔ ((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐶 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) < 𝑦))))
31	29, 5, 30	syl2anc 411	. . 3 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → ((𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵) ↔ ((𝐹 ↾ 𝐶):𝐶⟶𝐵 ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐶 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) < 𝑦))))
32	11, 28, 31	mpbir2and 953	. 2 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → (𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵))
33	32	ex 115	1 ⊢ (𝐶 ⊆ 𝐴 → (𝐹 ∈ (𝐴–cn→𝐵) → (𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512   ⊆ wss 3201   class class class wbr 4093   ↾ cres 4733  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028  ℂcc 8073   < clt 8257   − cmin 8393  ℝ⁺crp 9931  abscabs 11618  –cn→ccncf 15361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-map 6862  df-cncf 15362

This theorem is referenced by:  hovercncf  15437