Theorem elcnfnt 9766

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 1814	. 2 |- (T e. ConFn -> T e. V)
2		ax-hilex 8824	. . . 4 |- H~ e. V
3		fex 3647	. . . 4 |- ((T:H~-->CC /\ H~ e. V) -> T e. V)
4	2, 3	mpan2 695	. . 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 3616	. . . 4 |- (t = T -> (t:H~-->CC <-> T:H~-->CC))
7		fveq1 3718	. . . . . . . . . . . . 13 |- (t = T -> (t` w) = (T` w))
8		fveq1 3718	. . . . . . . . . . . . 13 |- (t = T -> (t` x) = (T` x))
9	7, 8	opreq12d 3973	. . . . . . . . . . . 12 |- (t = T -> ((t` w) - (t` x)) = ((T` w) - (T` x)))
10	9	fveq2d 3723	. . . . . . . . . . 11 |- (t = T -> (abs` ((t` w) - (t` x))) = (abs` ((T` w) - (T` x))))
11	10	breq1d 2625	. . . . . . . . . 10 |- (t = T -> ((abs` ((t` w) - (t` x))) < y <-> (abs` ((T` w) - (T` x))) < y))
12	11	imbi2d 611	. . . . . . . . 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 1661	. . . . . . . 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 615	. . . . . . 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 1662	. . . . . 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 611	. . . . 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 1678	. . . 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 627	. . 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 9730	. . 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 1897	. 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 678	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 955   e. wcel 957  A.wral 1643  E.wrex 1644  Vcvv 1808   class class class wbr 2615  -->wf 3174  ` cfv 3178  (class class class)co 3958  CCcc 5215  RRcr 5216  0cc0 5217   - cmin 5275   < clt 5469  abscabs 6696  H~chil 8743   -h cmv 8747  normhcno 8749  ConFnccnf 8777

This theorem is referenced by:  cnfnct 9811  0cnfn 9861  lnfncon 9946

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-hilex 8824

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194  df-opr 3960  df-cnfn 9730

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