Theorem elcnfnt 9794

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 1817	. 2 |- (T e. ConFn -> T e. V)
2		ax-hilex 8854	. . . 4 |- H~ e. V
3		fex 3652	. . . 4 |- ((T:H~-->CC /\ H~ e. V) -> T e. V)
4	2, 3	mpan2 696	. . 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 3620	. . . 4 |- (t = T -> (t:H~-->CC <-> T:H~-->CC))
7		fveq1 3723	. . . . . . . . . . . . 13 |- (t = T -> (t` w) = (T` w))
8		fveq1 3723	. . . . . . . . . . . . 13 |- (t = T -> (t` x) = (T` x))
9	7, 8	opreq12d 3978	. . . . . . . . . . . 12 |- (t = T -> ((t` w) - (t` x)) = ((T` w) - (T` x)))
10	9	fveq2d 3728	. . . . . . . . . . 11 |- (t = T -> (abs` ((t` w) - (t` x))) = (abs` ((T` w) - (T` x))))
11	10	breq1d 2629	. . . . . . . . . 10 |- (t = T -> ((abs` ((t` w) - (t` x))) < y <-> (abs` ((T` w) - (T` x))) < y))
12	11	imbi2d 612	. . . . . . . . 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 1663	. . . . . . . 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 616	. . . . . . 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 1664	. . . . . 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 612	. . . . 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 1680	. . . 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 628	. . 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 9758	. . 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 1900	. 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 679	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 956   e. wcel 958  A.wral 1645  E.wrex 1646  Vcvv 1811   class class class wbr 2619  -->wf 3178  ` cfv 3182  (class class class)co 3963  CCcc 5224  RRcr 5225  0cc0 5226   - cmin 5284   < clt 5478  abscabs 6736  H~chil 8773   -h cmv 8777  normhcno 8779  ConFnccnf 8807

This theorem is referenced by:  cnfnct 9839  0cnfn 9889  lnfncon 9975

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-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-pow 2742  ax-pr 2779  ax-un 2866  ax-hilex 8854

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  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-fv 3198  df-opr 3965  df-cnfn 9758

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