Theorem rescncf 25251

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 490	. . . . . 6 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐹 ∈ (𝐴–cn→𝐵))
2		cncfrss 25245	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ)
3	2	adantl 487	. . . . . . 7 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐴 ⊆ ℂ)
4		cncfrss2 25246	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ)
5	4	adantl 487	. . . . . . 7 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐵 ⊆ ℂ)
6		elcncf 25243	. . . . . . 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 596	. . . . . 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 235	. . . . 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 500	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐹:𝐴⟶𝐵)
10		simpl 488	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐶 ⊆ 𝐴)
11	9, 10	fssresd 6774	. . 3 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
12	8	simprd 501	. . . 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 3982	. . . . 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 3982	. . . . . . . . 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 6930	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
16		fvres 6930	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑤) = (𝐹‘𝑤))
17	15, 16	oveqan12d 7465	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤)) = ((𝐹‘𝑥) − (𝐹‘𝑤)))
18	17	fveq2d 6915	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) = (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))))
19	18	breq1d 5116	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → ((abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) < 𝑦 ↔ (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦))
20	19	imbi2d 343	. . . . . . . . . . 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 251	. . . . . . . . . 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 3178	. . . . . . . . 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 517	. . . . . . . 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 3181	. . . . . . 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 3180	. . . . . 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 3178	. . . . 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 48	. . . 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 66	. . 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 3923	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐶 ⊆ ℂ)
30		elcncf 25243	. . . 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 596	. . 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 726	. 2 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → (𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵))
33	32	ex 418	1 ⊢ (𝐶 ⊆ 𝐴 → (𝐹 ∈ (𝐴–cn→𝐵) → (𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   class class class wbr 5106   ↾ cres 5658  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446  ℂcc 11228   < clt 11373   − cmin 11571  ℝ⁺crp 13150  abscabs 15433  –cn→ccncf 25230

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 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286

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 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-map 8880  df-cncf 25232

This theorem is used by:  cpnres  26290  dvlip  26346  dvlip2  26348  c1liplem1  26349  c1lip2  26351  dvgt0lem1  26355  dvivthlem1  26361  dvne0  26364  lhop1lem  26366  dvcnvrelem1  26370  dvcnvrelem2  26371  dvcvx  26373  dvfsumle  26374  dvfsumabs  26376  dvfsumlem2  26380  ftc2ditglem  26398  itgparts  26400  itgsubstlem  26401  itgpowd  26403  psercn2  26783  abelth  26801  abelth2  26802  efcvx  26809  pige3ALT  26881  dvrelog  26998  logcn  27008  logccv  27024  loglesqrt  27122  rpsqrtcn  35305  cxpcncf1  35307  ftc2re  35310  fdvposlt  35311  fdvposle  35313  itgexpif  35318  ftc1cnnclem  38814  ftc2nc  38825  areacirc  38836  cncfres  38902  resopunitintvd  43279  resclunitintvd  43280  lcmineqlem2  43283  aks4d1p1p5  43328  areaquad  44421  lhe4.4ex1a  45516  cncfmptss  46814  resincncf  47100  dvbdfbdioolem1  47153  itgsbtaddcnst  47207  fourierdlem38  47370  fourierdlem46  47377  fourierdlem72  47403  fourierdlem90  47421  fourierdlem111  47442  fouriercn  47457