Theorem negfcncf 7269

Description: The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.)

Hypotheses

Ref	Expression
negfcncf.1	|- A (_ CC
negfcncf.2	|- F e. (A-cn->CC)
negfcncf.3	|- G = {<.a, b>. | (a e. A /\ b = -u(F` a))}

Assertion

Ref	Expression
negfcncf	|- G e. (A-cn->CC)

Distinct variable groups:   A,a,b   F,a,b

Proof of Theorem negfcncf

Step	Hyp	Ref	Expression
1		negfcncf.1	. . 3 |- A (_ CC
2		ssid 2080	. . 3 |- CC (_ CC
3		elcncf 7265	. . 3 |- ((A (_ CC /\ CC (_ CC) -> (G e. (A-cn->CC) <-> (G:A-->CC /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((G` x) - (G` w))) < y))))
4	1, 2, 3	mp2an 697	. 2 |- (G e. (A-cn->CC) <-> (G:A-->CC /\ A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((G` x) - (G` w))) < y)))
5		negfcncf.2	. . . . . . . . 9 |- F e. (A-cn->CC)
6		elcncf 7265	. . . . . . . . . 10 |- ((A (_ CC /\ CC (_ CC) -> (F e. (A-cn->CC) <-> (F:A-->CC /\ 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))))
7	1, 2, 6	mp2an 697	. . . . . . . . 9 |- (F e. (A-cn->CC) <-> (F:A-->CC /\ 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)))
8	5, 7	mpbi 189	. . . . . . . 8 |- (F:A-->CC /\ 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))
9	8	pm3.26i 320	. . . . . . 7 |- F:A-->CC
10	9	ffvelrni 3815	. . . . . 6 |- (x e. A -> (F` x) e. CC)
11		negclt 5368	. . . . . 6 |- ((F` x) e. CC -> -u(F` x) e. CC)
12	10, 11	syl 10	. . . . 5 |- (x e. A -> -u(F` x) e. CC)
13	12	rgen 1698	. . . 4 |- A.x e. A -u(F` x) e. CC
14		fveq2 3724	. . . . . . 7 |- (x = a -> (F` x) = (F` a))
15	14	negeqd 5361	. . . . . 6 |- (x = a -> -u(F` x) = -u(F` a))
16	15	eleq1d 1540	. . . . 5 |- (x = a -> (-u(F` x) e. CC <-> -u(F` a) e. CC))
17	16	cbvralv 1800	. . . 4 |- (A.x e. A -u(F` x) e. CC <-> A.a e. A -u(F` a) e. CC)
18	13, 17	mpbi 189	. . 3 |- A.a e. A -u(F` a) e. CC
19		negfcncf.3	. . . 4 |- G = {<.a, b>. | (a e. A /\ b = -u(F` a))}
20	19	fopab2 3823	. . 3 |- (A.a e. A -u(F` a) e. CC <-> G:A-->CC)
21	18, 20	mpbi 189	. 2 |- G:A-->CC
22	8	pm3.27i 324	. . 3 |- 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)
23		fveq2 3724	. . . . . . . . . . . . . . 15 |- (a = x -> (F` a) = (F` x))
24	23	negeqd 5361	. . . . . . . . . . . . . 14 |- (a = x -> -u(F` a) = -u(F` x))
25		negex 5365	. . . . . . . . . . . . . 14 |- -u(F` x) e. V
26	24, 19, 25	fvopab4 3780	. . . . . . . . . . . . 13 |- (x e. A -> (G` x) = -u(F` x))
27		fveq2 3724	. . . . . . . . . . . . . . 15 |- (a = w -> (F` a) = (F` w))
28	27	negeqd 5361	. . . . . . . . . . . . . 14 |- (a = w -> -u(F` a) = -u(F` w))
29		negex 5365	. . . . . . . . . . . . . 14 |- -u(F` w) e. V
30	28, 19, 29	fvopab4 3780	. . . . . . . . . . . . 13 |- (w e. A -> (G` w) = -u(F` w))
31	26, 30	opreqan12d 3979	. . . . . . . . . . . 12 |- ((x e. A /\ w e. A) -> ((G` x) - (G` w)) = (-u(F` x) - -u(F` w)))
32		neg2subt 5459	. . . . . . . . . . . . 13 |- (((F` x) e. CC /\ (F` w) e. CC) -> (-u(F` x) - -u(F` w)) = ((F` w) - (F` x)))
33	9	ffvelrni 3815	. . . . . . . . . . . . 13 |- (w e. A -> (F` w) e. CC)
34	32, 10, 33	syl2an 454	. . . . . . . . . . . 12 |- ((x e. A /\ w e. A) -> (-u(F` x) - -u(F` w)) = ((F` w) - (F` x)))
35	31, 34	eqtrd 1507	. . . . . . . . . . 11 |- ((x e. A /\ w e. A) -> ((G` x) - (G` w)) = ((F` w) - (F` x)))
36	35	fveq2d 3728	. . . . . . . . . 10 |- ((x e. A /\ w e. A) -> (abs` ((G` x) - (G` w))) = (abs` ((F` w) - (F` x))))
37		abssubt 6894	. . . . . . . . . . 11 |- (((F` x) e. CC /\ (F` w) e. CC) -> (abs` ((F` x) - (F` w))) = (abs` ((F` w) - (F` x))))
38	37, 10, 33	syl2an 454	. . . . . . . . . 10 |- ((x e. A /\ w e. A) -> (abs` ((F` x) - (F` w))) = (abs` ((F` w) - (F` x))))
39	36, 38	eqtr4d 1510	. . . . . . . . 9 |- ((x e. A /\ w e. A) -> (abs` ((G` x) - (G` w))) = (abs` ((F` x) - (F` w))))
40	39	breq1d 2629	. . . . . . . 8 |- ((x e. A /\ w e. A) -> ((abs` ((G` x) - (G` w))) < y <-> (abs` ((F` x) - (F` w))) < y))
41	40	imbi2d 612	. . . . . . 7 |- ((x e. A /\ w e. A) -> (((abs` (x - w)) < z -> (abs` ((G` x) - (G` w))) < y) <-> ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y)))
42	41	ralbidva 1659	. . . . . 6 |- (x e. A -> (A.w e. A ((abs` (x - w)) < z -> (abs` ((G` x) - (G` w))) < y) <-> A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y)))
43	42	rexbidv 1664	. . . . 5 |- (x e. A -> (E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((G` x) - (G` w))) < y) <-> E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y)))
44	43	ralbidv 1663	. . . 4 |- (x e. A -> (A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((G` x) - (G` w))) < y) <-> A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((F` x) - (F` w))) < y)))
45	44	ralbiia 1673	. . 3 |- (A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((G` x) - (G` w))) < y) <-> 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))
46	22, 45	mpbir 190	. 2 |- A.x e. A A.y e. RR+ E.z e. RR+ A.w e. A ((abs` (x - w)) < z -> (abs` ((G` x) - (G` w))) < y)
47	4, 21, 46	mpbir2an 730	1 |- G e. (A-cn->CC)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646   (_ wss 2047   class class class wbr 2619  {copab 2666  -->wf 3178  ` cfv 3182  (class class class)co 3963  CCcc 5232   - cmin 5292  -ucneg 5293  RR+crp 5300   < clt 5486  abscabs 6750  -cn->ccncf 7262

This theorem is referenced by:  dsupivthlem 7291

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-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  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-int 2534  df-iun 2568  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-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 51