Theorem rescncf 14817

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 14811	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ)
3	2	adantl 277	. . . . . . 7 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐴 ⊆ ℂ)
4		cncfrss2 14812	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ)
5	4	adantl 277	. . . . . . 7 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐵 ⊆ ℂ)
6		elcncf 14809	. . . . . . 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 5434	. . 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 3247	. . . . 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 3247	. . . . . . . . 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 5582	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
16		fvres 5582	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑤) = (𝐹‘𝑤))
17	15, 16	oveqan12d 5941	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤)) = ((𝐹‘𝑥) − (𝐹‘𝑤)))
18	17	fveq2d 5562	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑤 ∈ 𝐶) → (abs‘(((𝐹 ↾ 𝐶)‘𝑥) − ((𝐹 ↾ 𝐶)‘𝑤))) = (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))))
19	18	breq1d 4043	. . . . . . . . . . . 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 2564	. . . . . . . . 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 2598	. . . . . . 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 2565	. . . . . 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 2564	. . . . 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 3193	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐶 ⊆ ℂ)
30		elcncf 14809	. . . 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 946	. 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 2167  ∀wral 2475  ∃wrex 2476   ⊆ wss 3157   class class class wbr 4033   ↾ cres 4665  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922  ℂcc 7877   < clt 8061   − cmin 8197  ℝ⁺crp 9728  abscabs 11162  –cn→ccncf 14806

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-map 6709  df-cncf 14807

This theorem is referenced by:  hovercncf  14882