Theorem nvcni 9661

Description: Epsilon-delta property of a continuous operator. (Normed complex vector space version of metcni 9172.)

Hypotheses

Ref	Expression
nvcni.1	|- X = (BaseSet` U)
nvcni.m	|- M = (norm` U)
nvcni.n	|- N = (norm` W)
nvcni.r	|- R = (-v` U)
nvcni.s	|- S = (-v` W)
nvcni.8	|- C = (IndMet` U)
nvcni.d	|- D = (IndMet` W)
nvcni.j	|- J = (Open` C)
nvcni.k	|- K = (Open` D)

Assertion

Ref	Expression
nvcni	|- (((U e. NrmCVec /\ W e. NrmCVec /\ F e. (J Cn K)) /\ (P e. X /\ A e. RR /\ 0 < A)) -> E.x e. RR (0 < x /\ A.y e. X ((M` (PRy)) < x -> (N` ((F` P)S(F` y))) < A)))

Distinct variable groups:   x,y,A   x,C,y   x,D,y   x,F,y   x,J,y   x,K,y   x,P,y   x,U,y   x,W,y   x,X,y

Proof of Theorem nvcni

Step	Hyp	Ref	Expression
1		eqid 1884	. . . . . . 7 |- dom dom C = dom dom C
2		nvcni.j	. . . . . . 7 |- J = (Open` C)
3		eqid 1884	. . . . . . 7 |- dom dom D = dom dom D
4		nvcni.k	. . . . . . 7 |- K = (Open` D)
5	1, 2, 3, 4	metcni 9172	. . . . . 6 |- (((C e. Met /\ D e. Met /\ F e. (J Cn K)) /\ (P e. dom dom C /\ A e. RR /\ 0 < A)) -> E.x e. RR (0 < x /\ A.y e. dom dom C((PCy) < x -> ((F` P)D(F` y)) < A)))
6	5	ex 402	. . . . 5 |- ((C e. Met /\ D e. Met /\ F e. (J Cn K)) -> ((P e. dom dom C /\ A e. RR /\ 0 < A) -> E.x e. RR (0 < x /\ A.y e. dom dom C((PCy) < x -> ((F` P)D(F` y)) < A))))
7		nvcni.8	. . . . . 6 |- C = (IndMet` U)
8	7	imsmet 9656	. . . . 5 |- (U e. NrmCVec -> C e. Met)
9		nvcni.d	. . . . . 6 |- D = (IndMet` W)
10	9	imsmet 9656	. . . . 5 |- (W e. NrmCVec -> D e. Met)
11		id 73	. . . . 5 |- (F e. (J Cn K) -> F e. (J Cn K))
12	6, 8, 10, 11	syl3an 1139	. . . 4 |- ((U e. NrmCVec /\ W e. NrmCVec /\ F e. (J Cn K)) -> ((P e. dom dom C /\ A e. RR /\ 0 < A) -> E.x e. RR (0 < x /\ A.y e. dom dom C((PCy) < x -> ((F` P)D(F` y)) < A))))
13		nvcni.1	. . . . . . . . 9 |- X = (BaseSet` U)
14	13, 7	imsba 9653	. . . . . . . 8 |- (U e. NrmCVec -> X = dom dom C)
15	14	eleq2d 1964	. . . . . . 7 |- (U e. NrmCVec -> (P e. X <-> P e. dom dom C))
16	15	3anbi1d 1172	. . . . . 6 |- (U e. NrmCVec -> ((P e. X /\ A e. RR /\ 0 < A) <-> (P e. dom dom C /\ A e. RR /\ 0 < A)))
17	14	raleqdv 2269	. . . . . . . 8 |- (U e. NrmCVec -> (A.y e. X ((PCy) < x -> ((F` P)D(F` y)) < A) <-> A.y e. dom dom C((PCy) < x -> ((F` P)D(F` y)) < A)))
18	17	anbi2d 678	. . . . . . 7 |- (U e. NrmCVec -> ((0 < x /\ A.y e. X ((PCy) < x -> ((F` P)D(F` y)) < A)) <-> (0 < x /\ A.y e. dom dom C((PCy) < x -> ((F` P)D(F` y)) < A))))
19	18	rexbidv 2124	. . . . . 6 |- (U e. NrmCVec -> (E.x e. RR (0 < x /\ A.y e. X ((PCy) < x -> ((F` P)D(F` y)) < A)) <-> E.x e. RR (0 < x /\ A.y e. dom dom C((PCy) < x -> ((F` P)D(F` y)) < A))))
20	16, 19	imbi12d 688	. . . . 5 |- (U e. NrmCVec -> (((P e. X /\ A e. RR /\ 0 < A) -> E.x e. RR (0 < x /\ A.y e. X ((PCy) < x -> ((F` P)D(F` y)) < A))) <-> ((P e. dom dom C /\ A e. RR /\ 0 < A) -> E.x e. RR (0 < x /\ A.y e. dom dom C((PCy) < x -> ((F` P)D(F` y)) < A)))))
21	20	3ad2ant1 897	. . . 4 |- ((U e. NrmCVec /\ W e. NrmCVec /\ F e. (J Cn K)) -> (((P e. X /\ A e. RR /\ 0 < A) -> E.x e. RR (0 < x /\ A.y e. X ((PCy) < x -> ((F` P)D(F` y)) < A))) <-> ((P e. dom dom C /\ A e. RR /\ 0 < A) -> E.x e. RR (0 < x /\ A.y e. dom dom C((PCy) < x -> ((F` P)D(F` y)) < A)))))
22	12, 21	mpbird 213	. . 3 |- ((U e. NrmCVec /\ W e. NrmCVec /\ F e. (J Cn K)) -> ((P e. X /\ A e. RR /\ 0 < A) -> E.x e. RR (0 < x /\ A.y e. X ((PCy) < x -> ((F` P)D(F` y)) < A))))
23	22	imp 377	. 2 |- (((U e. NrmCVec /\ W e. NrmCVec /\ F e. (J Cn K)) /\ (P e. X /\ A e. RR /\ 0 < A)) -> E.x e. RR (0 < x /\ A.y e. X ((PCy) < x -> ((F` P)D(F` y)) < A)))
24		nvcni.r	. . . . . . . . . . . . . . 15 |- R = (-v` U)
25		nvcni.m	. . . . . . . . . . . . . . 15 |- M = (norm` U)
26	13, 24, 25, 7	imsdval 9649	. . . . . . . . . . . . . 14 |- ((U e. NrmCVec /\ P e. X /\ y e. X) -> (PCy) = (M` (PRy)))
27	26	3expb 1068	. . . . . . . . . . . . 13 |- ((U e. NrmCVec /\ (P e. X /\ y e. X)) -> (PCy) = (M` (PRy)))
28	27	breq1d 3348	. . . . . . . . . . . 12 |- ((U e. NrmCVec /\ (P e. X /\ y e. X)) -> ((PCy) < x <-> (M` (PRy)) < x))
29	28	3ad2antl1 1038	. . . . . . . . . . 11 |- (((U e. NrmCVec /\ W e. NrmCVec /\ F:X-->(BaseSet` W)) /\ (P e. X /\ y e. X)) -> ((PCy) < x <-> (M` (PRy)) < x))
30		simp2 877	. . . . . . . . . . . . . 14 |- ((U e. NrmCVec /\ W e. NrmCVec /\ F:X-->(BaseSet` W)) -> W e. NrmCVec)
31	30	adantr 425	. . . . . . . . . . . . 13 |- (((U e. NrmCVec /\ W e. NrmCVec /\ F:X-->(BaseSet` W)) /\ (P e. X /\ y e. X)) -> W e. NrmCVec)
32		ffvelrn 4787	. . . . . . . . . . . . . . 15 |- ((F:X-->(BaseSet` W) /\ P e. X) -> (F` P) e. (BaseSet` W))
33	32	ad2ant2lr 446	. . . . . . . . . . . . . 14 |- (((W e. NrmCVec /\ F:X-->(BaseSet` W)) /\ (P e. X /\ y e. X)) -> (F` P) e. (BaseSet` W))
34	33	3adantl1 1032	. . . . . . . . . . . . 13 |- (((U e. NrmCVec /\ W e. NrmCVec /\ F:X-->(BaseSet` W)) /\ (P e. X /\ y e. X)) -> (F` P) e. (BaseSet` W))
35		ffvelrn 4787	. . . . . . . . . . . . . . 15 |- ((F:X-->(BaseSet` W) /\ y e. X) -> (F` y) e. (BaseSet` W))
36	35	ad2ant2l 444	. . . . . . . . . . . . . 14 |- (((W e. NrmCVec /\ F:X-->(BaseSet` W)) /\ (P e. X /\ y e. X)) -> (F` y) e. (BaseSet` W))
37	36	3adantl1 1032	. . . . . . . . . . . . 13 |- (((U e. NrmCVec /\ W e. NrmCVec /\ F:X-->(BaseSet` W)) /\ (P e. X /\ y e. X)) -> (F` y) e. (BaseSet` W))
38		eqid 1884	. . . . . . . . . . . . . 14 |- (BaseSet` W) = (BaseSet` W)
39		nvcni.s	. . . . . . . . . . . . . 14 |- S = (-v` W)
40		nvcni.n	. . . . . . . . . . . . . 14 |- N = (norm` W)
41	38, 39, 40, 9	imsdval 9649	. . . . . . . . . . . . 13 |- ((W e. NrmCVec /\ (F` P) e. (BaseSet` W) /\ (F` y) e. (BaseSet` W)) -> ((F` P)D(F` y)) = (N` ((F` P)S(F` y))))
42	31, 34, 37, 41	syl111anc 1100	. . . . . . . . . . . 12 |- (((U e. NrmCVec /\ W e. NrmCVec /\ F:X-->(BaseSet` W)) /\ (P e. X /\ y e. X)) -> ((F` P)D(F` y)) = (N` ((F` P)S(F` y))))
43	42	breq1d 3348	. . . . . . . . . . 11 |- (((U e. NrmCVec /\ W e. NrmCVec /\ F:X-->(BaseSet` W)) /\ (P e. X /\ y e. X)) -> (((F` P)D(F` y)) < A <-> (N` ((F` P)S(F` y))) < A))
44	29, 43	imbi12d 688	. . . . . . . . . 10 |- (((U e. NrmCVec /\ W e. NrmCVec /\ F:X-->(BaseSet` W)) /\ (P e. X /\ y e. X)) -> (((PCy) < x -> ((F` P)D(F` y)) < A) <-> ((M` (PRy)) < x -> (N` ((F` P)S(F` y))) < A)))
45	44	anassrs 489	. . . . . . . . 9 |- ((((U e. NrmCVec /\ W e. NrmCVec /\ F:X-->(BaseSet` W)) /\ P e. X) /\ y e. X) -> (((PCy) < x -> ((F` P)D(F` y)) < A) <-> ((M` (PRy)) < x -> (N` ((F` P)S(F` y))) < A)))
46	45	ralbidva 2119	. . . . . . . 8 |- (((U e. NrmCVec /\ W e. NrmCVec /\ F:X-->(BaseSet` W)) /\ P e. X) -> (A.y e. X ((PCy) < x -> ((F` P)D(F` y)) < A) <-> A.y e. X ((M` (PRy)) < x -> (N` ((F` P)S(F` y))) < A)))
47	46	anbi2d 678	. . . . . . 7 |- (((U e. NrmCVec /\ W e. NrmCVec /\ F:X-->(BaseSet` W)) /\ P e. X) -> ((0 < x /\ A.y e. X ((PCy) < x -> ((F` P)D(F` y)) < A)) <-> (0 < x /\ A.y e. X ((M` (PRy)) < x -> (N` ((F` P)S(F` y))) < A))))
48	47	rexbidv 2124	. . . . . 6 |- (((U e. NrmCVec /\ W e. NrmCVec /\ F:X-->(BaseSet` W)) /\ P e. X) -> (E.x e. RR (0 < x /\ A.y e. X ((PCy) < x -> ((F` P)D(F` y)) < A)) <-> E.x e. RR (0 < x /\ A.y e. X ((M` (PRy)) < x -> (N` ((F` P)S(F` y))) < A))))
49	48	ex 402	. . . . 5 |- ((U e. NrmCVec /\ W e. NrmCVec /\ F:X-->(BaseSet` W)) -> (P e. X -> (E.x e. RR (0 < x /\ A.y e. X ((PCy) < x -> ((F` P)D(F` y)) < A)) <-> E.x e. RR (0 < x /\ A.y e. X ((M` (PRy)) < x -> (N` ((F` P)S(F` y))) < A)))))
50	13, 38, 7, 9, 2, 4	nvcnf 9659	. . . . 5 |- ((U e. NrmCVec /\ W e. NrmCVec /\ F e. (J Cn K)) -> F:X-->(BaseSet` W))
51	49, 50	syld3an3 1142	. . . 4 |- ((U e. NrmCVec /\ W e. NrmCVec /\ F e. (J Cn K)) -> (P e. X -> (E.x e. RR (0 < x /\ A.y e. X ((PCy) < x -> ((F` P)D(F` y)) < A)) <-> E.x e. RR (0 < x /\ A.y e. X ((M` (PRy)) < x -> (N` ((F` P)S(F` y))) < A)))))
52	51	imp 377	. . 3 |- (((U e. NrmCVec /\ W e. NrmCVec /\ F e. (J Cn K)) /\ P e. X) -> (E.x e. RR (0 < x /\ A.y e. X ((PCy) < x -> ((F` P)D(F` y)) < A)) <-> E.x e. RR (0 < x /\ A.y e. X ((M` (PRy)) < x -> (N` ((F` P)S(F` y))) < A))))
53	52	3ad2antr1 1041	. 2 |- (((U e. NrmCVec /\ W e. NrmCVec /\ F e. (J Cn K)) /\ (P e. X /\ A e. RR /\ 0 < A)) -> (E.x e. RR (0 < x /\ A.y e. X ((PCy) < x -> ((F` P)D(F` y)) < A)) <-> E.x e. RR (0 < x /\ A.y e. X ((M` (PRy)) < x -> (N` ((F` P)S(F` y))) < A))))
54	23, 53	mpbid 212	1 |- (((U e. NrmCVec /\ W e. NrmCVec /\ F e. (J Cn K)) /\ (P e. X /\ A e. RR /\ 0 < A)) -> E.x e. RR (0 < x /\ A.y e. X ((M` (PRy)) < x -> (N` ((F` P)S(F` y))) < A)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   class class class wbr 3338  dom cdm 3986  -->wf 3994  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386   < clt 6653   Cn ccn 9028  Metcme 9066  Opencopn 9069  NrmCVeccnv 9535  BaseSetcba 9537  -vcnsb 9540  normcnm 9541  IndMetcims 9542

This theorem is referenced by:  nvcni3 9663

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-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  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-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-top 8861  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-vs 9550  df-nm 9551  df-ims 9552

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