Theorem rescncf 24815

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 484	. . . . . 6 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐹 ∈ (𝐴–cn→𝐵))
2		cncfrss 24809	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ)
3	2	adantl 481	. . . . . . 7 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐴 ⊆ ℂ)
4		cncfrss2 24810	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ)
5	4	adantl 481	. . . . . . 7 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐵 ⊆ ℂ)
6		elcncf 24807	. . . . . . 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 584	. . . . . 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 232	. . . . 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 494	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐹:𝐴⟶𝐵)
10		simpl 482	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐶 ⊆ 𝐴)
11	9, 10	fssresd 6690	. . 3 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
12	8	simprd 495	. . . 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 4003	. . . . 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 4003	. . . . . . . . 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 6841	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
16		fvres 6841	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑤) = (𝐹‘𝑤))
17	15, 16	oveqan12d 7365	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤)) = ((𝐹‘𝑥) − (𝐹‘𝑤)))
18	17	fveq2d 6826	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) = (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))))
19	18	breq1d 5101	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → ((abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) < 𝑦 ↔ (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦))
20	19	imbi2d 340	. . . . . . . . . . 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 248	. . . . . . . . . 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 3144	. . . . . . . . 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 507	. . . . . . . 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 3147	. . . . . . 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 3146	. . . . . 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 3144	. . . . 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 47	. . . 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 65	. . 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 3945	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐶 ⊆ ℂ)
30		elcncf 24807	. . . 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 584	. . 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 713	. 2 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → (𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵))
33	32	ex 412	1 ⊢ (𝐶 ⊆ 𝐴 → (𝐹 ∈ (𝐴–cn→𝐵) → (𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ⊆ wss 3902   class class class wbr 5091   ↾ cres 5618  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  ℂcc 11001   < clt 11143   − cmin 11341  ℝ⁺crp 12887  abscabs 15138  –cn→ccncf 24794

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-map 8752  df-cncf 24796

This theorem is referenced by:  cpnres  25864  dvlip  25923  dvlip2  25925  c1liplem1  25926  c1lip2  25928  dvgt0lem1  25932  dvivthlem1  25938  dvne0  25941  lhop1lem  25943  dvcnvrelem1  25947  dvcnvrelem2  25948  dvcvx  25950  dvfsumle  25951  dvfsumleOLD  25952  dvfsumabs  25954  dvfsumlem2  25958  dvfsumlem2OLD  25959  ftc2ditglem  25977  itgparts  25979  itgsubstlem  25980  itgpowd  25982  psercn2  26357  psercn2OLD  26358  abelth  26376  abelth2  26377  efcvx  26384  pige3ALT  26454  dvrelog  26571  logcn  26581  logccv  26597  loglesqrt  26696  rpsqrtcn  34601  cxpcncf1  34603  ftc2re  34606  fdvposlt  34607  fdvposle  34609  itgexpif  34614  ftc1cnnclem  37730  ftc2nc  37741  areacirc  37752  cncfres  37804  resopunitintvd  42058  resclunitintvd  42059  lcmineqlem2  42062  aks4d1p1p5  42107  areaquad  43248  lhe4.4ex1a  44361  cncfmptss  45626  resincncf  45912  dvbdfbdioolem1  45965  itgsbtaddcnst  46019  fourierdlem38  46182  fourierdlem46  46189  fourierdlem72  46215  fourierdlem90  46233  fourierdlem111  46254  fouriercn  46269