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 2083	. . 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 699	. 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 699	. . . . . . . . 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 3821	. . . . . 6 |- (x e. A -> (F` x) e. CC)
11		negclt 5380	. . . . . 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 1701	. . . 4 |- A.x e. A -u(F` x) e. CC
14		fveq2 3730	. . . . . . 7 |- (x = a -> (F` x) = (F` a))
15	14	negeqd 5373	. . . . . 6 |- (x = a -> -u(F` x) = -u(F` a))
16	15	eleq1d 1543	. . . . 5 |- (x = a -> (-u(F` x) e. CC <-> -u(F` a) e. CC))
17	16	cbvralv 1803	. . . 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 3829	. . 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 3730	. . . . . . . . . . . . . . 15 |- (a = x -> (F` a) = (F` x))
24	23	negeqd 5373	. . . . . . . . . . . . . 14 |- (a = x -> -u(F` a) = -u(F` x))
25		negex 5377	. . . . . . . . . . . . . 14 |- -u(F` x) e. V
26	24, 19, 25	fvopab4 3786	. . . . . . . . . . . . 13 |- (x e. A -> (G` x) = -u(F` x))
27		fveq2 3730	. . . . . . . . . . . . . . 15 |- (a = w -> (F` a) = (F` w))
28	27	negeqd 5373	. . . . . . . . . . . . . 14 |- (a = w -> -u(F` a) = -u(F` w))
29		negex 5377	. . . . . . . . . . . . . 14 |- -u(F` w) e. V
30	28, 19, 29	fvopab4 3786	. . . . . . . . . . . . 13 |- (w e. A -> (G` w) = -u(F` w))
31	26, 30	opreqan12d 3985	. . . . . . . . . . . 12 |- ((x e. A /\ w e. A) -> ((G` x) - (G` w)) = (-u(F` x) - -u(F` w)))
32		neg2subt 5471	. . . . . . . . . . . . 13 |- (((F` x) e. CC /\ (F` w) e. CC) -> (-u(F` x) - -u(F` w)) = ((F` w) - (F` x)))
33	9	ffvelrni 3821	. . . . . . . . . . . . 13 |- (w e. A -> (F` w) e. CC)
34	32, 10, 33	syl2an 456	. . . . . . . . . . . 12 |- ((x e. A /\ w e. A) -> (-u(F` x) - -u(F` w)) = ((F` w) - (F` x)))
35	31, 34	eqtrd 1510	. . . . . . . . . . 11 |- ((x e. A /\ w e. A) -> ((G` x) - (G` w)) = ((F` w) - (F` x)))
36	35	fveq2d 3734	. . . . . . . . . 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 456	. . . . . . . . . 10 |- ((x e. A /\ w e. A) -> (abs` ((F` x) - (F` w))) = (abs` ((F` w) - (F` x))))
39	36, 38	eqtr4d 1513	. . . . . . . . 9 |- ((x e. A /\ w e. A) -> (abs` ((G` x) - (G` w))) = (abs` ((F` x) - (F` w))))
40	39	breq1d 2634	. . . . . . . 8 |- ((x e. A /\ w e. A) -> ((abs` ((G` x) - (G` w))) < y <-> (abs` ((F` x) - (F` w))) < y))
41	40	imbi2d 614	. . . . . . 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 1662	. . . . . 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 1667	. . . . 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 1666	. . . 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 1676	. . 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 732	1 |- G e. (A-cn->CC)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649   (_ wss 2050   class class class wbr 2624  {copab 2671  -->wf 3184  ` cfv 3188  (class class class)co 3969  CCcc 5244   - cmin 5304  -ucneg 5305  RR+crp 5312   < clt 5498  abscabs 6751  -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-7 964  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-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  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-int 2538  df-iun 2572  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-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 51