Theorem rescncf 7272

Description: A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.)

Assertion

Ref	Expression
rescncf	|- ((A (_ CC /\ B (_ CC /\ C (_ A) -> (F e. (A-cn->B) -> (F |` C) e. (C-cn->B)))

Proof of Theorem rescncf

Step	Hyp	Ref	Expression
1		fssres 3649	. . . . 5 |- ((F:A-->B /\ C (_ A) -> (F |` C):C-->B)
2	1	expcom 374	. . . 4 |- (C (_ A -> (F:A-->B -> (F |` C):C-->B))
3		ssralv 2117	. . . . 5 |- (C (_ A -> (A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y) -> A.x e. C A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y)))
4		ssralv 2117	. . . . . . . . 9 |- (C (_ A -> (A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y) -> A.w e. C ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y)))
5		fvres 3740	. . . . . . . . . . . . . . 15 |- (x e. C -> ((F |` C)` x) = (F` x))
6		fvres 3740	. . . . . . . . . . . . . . 15 |- (w e. C -> ((F |` C)` w) = (F` w))
7	5, 6	opreqan12d 3985	. . . . . . . . . . . . . 14 |- ((x e. C /\ w e. C) -> (((F |` C)` x) - ((F |` C)` w)) = ((F` x) - (F` w)))
8	7	fveq2d 3734	. . . . . . . . . . . . 13 |- ((x e. C /\ w e. C) -> (abs` (((F |` C)` x) - ((F |` C)` w))) = (abs` ((F` x) - (F` w))))
9	8	breq1d 2634	. . . . . . . . . . . 12 |- ((x e. C /\ w e. C) -> ((abs` (((F |` C)` x) - ((F |` C)` w))) < y <-> (abs` ((F` x) - (F` w))) < y))
10	9	imbi2d 614	. . . . . . . . . . 11 |- ((x e. C /\ w e. C) -> (((abs` (x - w)) < z -> (abs` (((F |` C)` x) - ((F |` C)` w))) < y) <-> ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y)))
11	10	biimprd 154	. . . . . . . . . 10 |- ((x e. C /\ w e. C) -> (((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y) -> ((abs` (x - w)) < z -> (abs` (((F |` C)` x) - ((F |` C)` w))) < y)))
12	11	r19.20dva 1712	. . . . . . . . 9 |- (x e. C -> (A.w e. C ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y) -> A.w e. C ((abs` (x - w)) < z -> (abs` (((F |` C)` x) - ((F |` C)` w))) < y)))
13	4, 12	sylan9 470	. . . . . . . 8 |- ((C (_ A /\ x e. C) -> (A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y) -> A.w e. C ((abs` (x - w)) < z -> (abs` (((F |` C)` x) - ((F |` C)` w))) < y)))
14	13	r19.22sdv 1741	. . . . . . 7 |- ((C (_ A /\ x e. C) -> (E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y) -> E.z e. RR+ A.w e. C ((abs` (x - w)) < z -> (abs` (((F |` C)` x) - ((F |` C)` w))) < y)))
15	14	r19.20sdv 1713	. . . . . 6 |- ((C (_ A /\ x e. C) -> (A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y) -> A.y e. RR+ E.z e. RR+ A.w e. C ((abs` (x - w)) < z -> (abs` (((F |` C)` x) - ((F |` C)` w))) < y)))
16	15	r19.20dva 1712	. . . . 5 |- (C (_ A -> (A.x e. C A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y) -> A.x e. C A.y e. RR+ E.z e. RR+ A.w e. C ((abs` (x - w)) < z -> (abs` (((F |` C)` x) - ((F |` C)` w))) < y)))
17	3, 16	syld 27	. . . 4 |- (C (_ A -> (A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y) -> A.x e. C A.y e. RR+ E.z e. RR+ A.w e. C ((abs` (x - w)) < z -> (abs` (((F |` C)` x) - ((F |` C)` w))) < y)))
18	2, 17	anim12d 560	. . 3 |- (C (_ A -> ((F:A-->B /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y)) -> ((F |` C):C-->B /\ A.x e. C A.y e. RR+ E.z e. RR+ A.w e. C ((abs` (x - w)) < z -> (abs` (((F |` C)` x) - ((F |` C)` w))) < y))))
19	18	3ad2ant3 804	. 2 |- ((A (_ CC /\ B (_ CC /\ C (_ A) -> ((F:A-->B /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y)) -> ((F |` C):C-->B /\ A.x e. C A.y e. RR+ E.z e. RR+ A.w e. C ((abs` (x - w)) < z -> (abs` (((F |` C)` x) - ((F |` C)` w))) < y))))
20		elcncf 7265	. . 3 |- ((A (_ CC /\ B (_ CC) -> (F e. (A-cn->B) <-> (F:A-->B /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y))))
21	20	3adant3 801	. 2 |- ((A (_ CC /\ B (_ CC /\ C (_ A) -> (F e. (A-cn->B) <-> (F:A-->B /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y))))
22		sstr 2075	. . . . . 6 |- ((C (_ A /\ A (_ CC) -> C (_ CC)
23	22	anim1i 334	. . . . 5 |- (((C (_ A /\ A (_ CC) /\ B (_ CC) -> (C (_ CC /\ B (_ CC))
24	23	3impa 830	. . . 4 |- ((C (_ A /\ A (_ CC /\ B (_ CC) -> (C (_ CC /\ B (_ CC))
25	24	3coml 842	. . 3 |- ((A (_ CC /\ B (_ CC /\ C (_ A) -> (C (_ CC /\ B (_ CC))
26		elcncf 7265	. . 3 |- ((C (_ CC /\ B (_ CC) -> ((F |` C) e. (C-cn->B) <-> ((F |` C):C-->B /\ A.x e. C A.y e. RR+ E.z e. RR+ A.w e. C ((abs` (x - w)) < z -> (abs` (((F |` C)` x) - ((F |` C)` w))) < y))))
27	25, 26	syl 10	. 2 |- ((A (_ CC /\ B (_ CC /\ C (_ A) -> ((F |` C) e. (C-cn->B) <-> ((F |` C):C-->B /\ A.x e. C A.y e. RR+ E.z e. RR+ A.w e. C ((abs` (x - w)) < z -> (abs` (((F |` C)` x) - ((F |` C)` w))) < y))))
28	19, 21, 27	3imtr4d 545	1 |- ((A (_ CC /\ B (_ CC /\ C (_ A) -> (F e. (A-cn->B) -> (F |` C) e. (C-cn->B)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   e. wcel 960  A.wral 1648  E.wrex 1649   (_ wss 2050   class class class wbr 2624   |` cres 3178  -->wf 3184  ` cfv 3188  (class class class)co 3969  CCcc 5244   - cmin 5304  RR+crp