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 2623	. . . . . . . . . 10 |- (z = Z -> ((abs` (x - w)) < z <-> (abs` (x - w)) < Z))
4	3	imbi1d 613	. . . . . . . . 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 1663	. . . . . . . 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 1877	. . . . . . 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 485	. . . . . . . . . . 11 |- (((x e. A /\ w e. A) /\ y e. RR+) <-> ((x e. A /\ y e. RR+) /\ w e. A))
10	9	anbi2i 480	. . . . . . . . . 10 |- ((ph /\ ((x e. A /\ w e. A) /\ y e. RR+)) <-> (ph /\ ((x e. A /\ y e. RR+) /\ w e. A)))
11		anass 439	. . . . . . . . . 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 1714	. . . . . . 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 471	. . . . . 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 441	. . . . 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 1714	. . . 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 1714	. . 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 956   e. wcel 958  A.wral 1645  E.wrex 1646   (_ wss 2047   class class class wbr 2619  -->wf 3178  ` cfv 3182  (class class class)co 3963  CCcc 5232   - cmin 5292  RR+crp 5300   < clt 5486  abscabs 6750  -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-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-qs 4266  df-ni 5000  df-nq 5038  df-np 5086  df-nr 5167  df-c 5240  df-cncf 7263

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