Theorem elcncf1d 7270

Description: Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 26-Nov-2007.)

Hypotheses

Ref	Expression
elcncf1d.1	|- (ph -> F:A-->B)
elcncf1d.2	|- (ph -> ((x e. A /\ y e. RR+) -> Z e. RR+))
elcncf1d.3	|- (ph -> (((x e. A /\ w e. A) /\ y e. RR+) -> ((abs` (x - w)) < Z -> (abs` ((F` x) - (F` w))) < y)))

Assertion

Ref	Expression
elcncf1d	|- (ph -> ((A (_ CC /\ B (_ CC) -> F e. (A-cn->B)))

Distinct variable groups:   w,A,x,y   w,F,x,y   w,Z   ph,w,x,y

Proof of Theorem elcncf1d

Step	Hyp	Ref	Expression
1		elcncf 7265	. 2 |- ((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))))
2		elcncf1d.1	. . 3 |- (ph -> F:A-->B)
3		breq2 2628	. . . . . . . . . 10 |- (z = Z -> ((abs` (x - w)) < z <-> (abs` (x - w)) < Z))
4	3	imbi1d 615	. . . . . . . . 9 |- (z = Z -> (((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y) <-> ((abs` (x - w)) < Z -> (abs` ((F` x) - (F` w))) < y)))
5	4	ralbidv 1666	. . . . . . . 8 |- (z = Z -> (A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y) <-> A.w e. A ((abs` (x - w)) < Z -> (abs` ((F` x) - (F` w))) < y)))
6	5	rcla4ev 1880	. . . . . . 7 |- ((Z e. RR+ /\ A.w e. A ((abs` (x - w)) < Z -> (abs` ((F` x) - (F` w))) < y)) -> E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y))
7		elcncf1d.2	. . . . . . . 8 |- (ph -> ((x e. A /\ y e. RR+) -> Z e. RR+))
8	7	imp 350	. . . . . . 7 |- ((ph /\ (x e. A /\ y e. RR+)) -> Z e. RR+)
9		an23 487	. . . . . . . . . . 11 |- (((x e. A /\ w e. A) /\ y e. RR+) <-> ((x e. A /\ y e. RR+) /\ w e. A))
10	9	anbi2i 482	. . . . . . . . . 10 |- ((ph /\ ((x e. A /\ w e. A) /\ y e. RR+)) <-> (ph /\ ((x e. A /\ y e. RR+) /\ w e. A)))
11		anass 441	. . . . . . . . . 10 |- (((ph /\ (x e. A /\ y e. RR+)) /\ w e. A) <-> (ph /\ ((x e. A /\ y e. RR+) /\ w e. A)))
12	10, 11	bitr4 176	. . . . . . . . 9 |- ((ph /\ ((x e. A /\ w e. A) /\ y e. RR+)) <-> ((ph /\ (x e. A /\ y e. RR+)) /\ w e. A))
13		elcncf1d.3	. . . . . . . . . 10 |- (ph -> (((x e. A /\ w e. A) /\ y e. RR+) -> ((abs` (x - w)) < Z -> (abs` ((F` x) - (F` w))) < y)))
14	13	imp 350	. . . . . . . . 9 |- ((ph /\ ((x e. A /\ w e. A) /\ y e. RR+)) -> ((abs` (x - w)) < Z -> (abs` ((F` x) - (F` w))) < y))
15	12, 14	sylbir 201	. . . . . . . 8 |- (((ph /\ (x e. A /\ y e. RR+)) /\ w e. A) -> ((abs` (x - w)) < Z -> (abs` ((F` x) - (F` w))) < y))
16	15	r19.21aiva 1717	. . . . . . 7 |- ((ph /\ (x e. A /\ y e. RR+)) -> A.w e. A ((abs` (x - w)) < Z -> (abs` ((F` x) - (F` w))) < y))
17	6, 8, 16	sylanc 473	. . . . . 6 |- ((ph /\ (x e. A /\ y e. RR+)) -> E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y))
18	17	anassrs 443	. . . . 5 |- (((ph /\ x e. A) /\ y e. RR+) -> E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y))
19	18	r19.21aiva 1717	. . . 4 |- ((ph /\ 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))
20	19	r19.21aiva 1717	. . 3 |- (ph -> 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	2, 20	jca 288	. 2 |- (ph -> (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	1, 21	syl5cbir 211	1 |- (ph -> ((A (_ CC /\ B (_ CC) -> F e. (A-cn->B)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649   (_ wss 2050   class class class wbr 2624  -->wf 3184  ` cfv 3188  (class class class)co 3969  CCcc 5244   - cmin 5304  RR+crp 5312   < clt 5498  abscabs 6751  -cn->ccncf 7262

This theorem is referenced by:  elcncf1i 7271  mulc1cncf 7279

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-oprab 3972  df-qs 4272  df-ni 5012  df-nq 5050  df-np 5098  df-nr 5179  df-c 5252  df-cncf 7263

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