Theorem elcnfnt 9726

Description: Property defining a continuous functional.

Assertion

Ref	Expression
elcnfnt	|- (T e. ConFn <-> (T:H~-->CC /\ A.x e. H~ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((T` w) - (T` x))) < y)))))

Distinct variable group:   x,w,y,z,T

Proof of Theorem elcnfnt

Step	Hyp	Ref	Expression
1		elisset 1808	. 2 |- (T e. ConFn -> T e. V)
2		ax-hilex 8790	. . . 4 |- H~ e. V
3		fex 3637	. . . 4 |- ((T:H~-->CC /\ H~ e. V) -> T e. V)
4	2, 3	mpan2 694	. . 3 |- (T:H~-->CC -> T e. V)
5	4	adantr 389	. 2 |- ((T:H~-->CC /\ A.x e. H~ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((T` w) - (T` x))) < y)))) -> T e. V)
6		feq1 3606	. . . 4 |- (t = T -> (t:H~-->CC <-> T:H~-->CC))
7		fveq1 3708	. . . . . . . . . . . . 13 |- (t = T -> (t` w) = (T` w))
8		fveq1 3708	. . . . . . . . . . . . 13 |- (t = T -> (t` x) = (T` x))
9	7, 8	opreq12d 3963	. . . . . . . . . . . 12 |- (t = T -> ((t` w) - (t` x)) = ((T` w) - (T` x)))
10	9	fveq2d 3713	. . . . . . . . . . 11 |- (t = T -> (abs` ((t` w) - (t` x))) = (abs` ((T` w) - (T` x))))
11	10	breq1d 2619	. . . . . . . . . 10 |- (t = T -> ((abs` ((t` w) - (t` x))) < y <-> (abs` ((T` w) - (T` x))) < y))
12	11	imbi2d 610	. . . . . . . . 9 |- (t = T -> (((normh` (w -h x)) < z -> (abs` ((t` w) - (t` x))) < y) <-> ((normh` (w -h x)) < z -> (abs` ((T` w) - (T` x))) < y)))
13	12	ralbidv 1655	. . . . . . . 8 |- (t = T -> (A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((t` w) - (t` x))) < y) <-> A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((T` w) - (T` x))) < y)))
14	13	anbi2d 614	. . . . . . 7 |- (t = T -> ((0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((t` w) - (t` x))) < y)) <-> (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((T` w) - (T` x))) < y))))
15	14	rexbidv 1656	. . . . . 6 |- (t = T -> (E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((t` w) - (t` x))) < y)) <-> E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((T` w) - (T` x))) < y))))
16	15	imbi2d 610	. . . . 5 |- (t = T -> ((0 < y -> E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((t` w) - (t` x))) < y))) <-> (0 < y -> E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((T` w) - (T` x))) < y)))))
17	16	2ralbidv 1672	. . . 4 |- (t = T -> (A.x e. H~ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((t` w) - (t` x))) < y))) <-> A.x e. H~ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((T` w) - (T` x))) < y)))))
18	6, 17	anbi12d 626	. . 3 |- (t = T -> ((t:H~-->CC /\ A.x e. H~ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((t` w) - (t` x))) < y)))) <-> (T:H~-->CC /\ A.x e. H~ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((T` w) - (T` x))) < y))))))
19		df-cnfn 9690	. . 3 |- ConFn = {t | (t:H~-->CC /\ A.x e. H~ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((t` w) - (t` x))) < y))))}
20	18, 19	elab2g 1891	. 2 |- (T e. V -> (T e. ConFn <-> (T:H~-->CC /\ A.x e. H~ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((T` w) - (T` x))) < y))))))
21	1, 5, 20	pm5.21nii 677	1 |- (T e. ConFn <-> (T:H~-->CC /\ A.x e. H~ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (abs` ((T` w) - (T` x))) < y)))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  E.wrex 1638  Vcvv 1802   class class class wbr 2609  -->wf 3168  ` cfv 3172  (class class class)co 3948  CCcc 5204  RRcr 5205  0cc0 5206   - cmin 5264   < clt 5458  abscabs 6681  H~chil 8727   -h cmv 8731  normhcno 8733  ConFnccnf 8761

This theorem is referenced by:  cnfnct 9770  0cnfn 9820  lnfncon 9905

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-hilex 8790

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950  df-cnfn 9690

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