Theorem negfcncfi 8531

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

Hypotheses

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

Assertion

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

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

Proof of Theorem negfcncfi

Step	Hyp	Ref	Expression
1		negfcncf.1	. . 3 |- A C_ CC
2		ssid 2634	. . 3 |- CC C_ CC
3		elcncf 8527	. . 3 |- ((A C_ CC /\ CC C_ 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 761	. 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 8527	. . . . . . . . . 10 |- ((A C_ CC /\ CC C_ 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 761	. . . . . . . . 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 206	. . . . . . . 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	simpli 347	. . . . . . 7 |- F:A-->CC
10	9	ffvelrni 4788	. . . . . 6 |- (x e. A -> (F` x) e. CC)
11		negcl 6525	. . . . . 6 |- ((F` x) e. CC -> -u(F` x) e. CC)
12	10, 11	syl 12	. . . . 5 |- (x e. A -> -u(F` x) e. CC)
13	12	rgen 2159	. . . 4 |- A.x e. A -u(F` x) e. CC
14		fveq2 4681	. . . . . . 7 |- (x = a -> (F` x) = (F` a))
15	14	negeqd 6516	. . . . . 6 |- (x = a -> -u(F` x) = -u(F` a))
16	15	eleq1d 1963	. . . . 5 |- (x = a -> (-u(F` x) e. CC <-> -u(F` a) e. CC))
17	16	cbvralv 2280	. . . 4 |- (A.x e. A -u(F` x) e. CC <-> A.a e. A -u(F` a) e. CC)
18	13, 17	mpbi 206	. . 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 4796	. . 3 |- (A.a e. A -u(F` a) e. CC <-> G:A-->CC)
21	18, 20	mpbi 206	. 2 |- G:A-->CC
22	8	simpri 351	. . 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 4681	. . . . . . . . . . . . . . 15 |- (a = x -> (F` a) = (F` x))
24	23	negeqd 6516	. . . . . . . . . . . . . 14 |- (a = x -> -u(F` a) = -u(F` x))
25		negex 6522	. . . . . . . . . . . . . 14 |- -u(F` x) e. _V
26	24, 19, 25	fvopab4 4743	. . . . . . . . . . . . 13 |- (x e. A -> (G` x) = -u(F` x))
27		fveq2 4681	. . . . . . . . . . . . . . 15 |- (a = w -> (F` a) = (F` w))
28	27	negeqd 6516	. . . . . . . . . . . . . 14 |- (a = w -> -u(F` a) = -u(F` w))
29		negex 6522	. . . . . . . . . . . . . 14 |- -u(F` w) e. _V
30	28, 19, 29	fvopab4 4743	. . . . . . . . . . . . 13 |- (w e. A -> (G` w) = -u(F` w))
31	26, 30	opreqan12d 4902	. . . . . . . . . . . 12 |- ((x e. A /\ w e. A) -> ((G` x) - (G` w)) = (-u(F` x) - -u(F` w)))
32		neg2sub 6624	. . . . . . . . . . . . 13 |- (((F` x) e. CC /\ (F` w) e. CC) -> (-u(F` x) - -u(F` w)) = ((F` w) - (F` x)))
33	9	ffvelrni 4788	. . . . . . . . . . . . 13 |- (w e. A -> (F` w) e. CC)
34	32, 10, 33	syl2an 503	. . . . . . . . . . . 12 |- ((x e. A /\ w e. A) -> (-u(F` x) - -u(F` w)) = ((F` w) - (F` x)))
35	31, 34	eqtrd 1925	. . . . . . . . . . 11 |- ((x e. A /\ w e. A) -> ((G` x) - (G` w)) = ((F` w) - (F` x)))
36	35	fveq2d 4685	. . . . . . . . . 10 |- ((x e. A /\ w e. A) -> (abs` ((G` x) - (G` w))) = (abs` ((F` w) - (F` x))))
37		abssub 8146	. . . . . . . . . . 11 |- (((F` x) e. CC /\ (F` w) e. CC) -> (abs` ((F` x) - (F` w))) = (abs` ((F` w) - (F` x))))
38	37, 10, 33	syl2an 503	. . . . . . . . . 10 |- ((x e. A /\ w e. A) -> (abs` ((F` x) - (F` w))) = (abs` ((F` w) - (F` x))))
39	36, 38	eqtr4d 1928	. . . . . . . . 9 |- ((x e. A /\ w e. A) -> (abs` ((G` x) - (G` w))) = (abs` ((F` x) - (F` w))))
40	39	breq1d 3348	. . . . . . . 8 |- ((x e. A /\ w e. A) -> ((abs` ((G` x) - (G` w))) < y <-> (abs` ((F` x) - (F` w))) < y))
41	40	imbi2d 674	. . . . . . 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 2119	. . . . . 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 2124	. . . . 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 2123	. . . 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 2133	. . 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 207	. 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 800	1 |- G e. (A-cn->CC)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   C_ wss 2593   class class class wbr 3338  {copab 3395  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384   - cmin 6445  -ucneg 6446  RR+crp 6453   < clt 6653  abscabs 8000  -cn->ccncf 8524

This theorem is referenced by:  dsupivthlem 8553

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-cncf 8525

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