Theorem elcnopt 9723

Description: Property defining a continuous Hilbert space operator.

Assertion

Ref	Expression
elcnopt	|- (T e. ConOp <-> (T:H~-->H~ /\ 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 -> (normh` ((T` w) -h (T` x))) < y)))))

Distinct variable group:   x,w,y,z,T

Proof of Theorem elcnopt

Step	Hyp	Ref	Expression
1		elisset 1813	. 2 |- (T e. ConOp -> T e. V)
2		ax-hilex 8808	. . . 4 |- H~ e. V
3		fex 3643	. . . 4 |- ((T:H~-->H~ /\ H~ e. V) -> T e. V)
4	2, 3	mpan2 695	. . 3 |- (T:H~-->H~ -> T e. V)
5	4	adantr 389	. 2 |- ((T:H~-->H~ /\ 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 -> (normh` ((T` w) -h (T` x))) < y)))) -> T e. V)
6		feq1 3612	. . . 4 |- (t = T -> (t:H~-->H~ <-> T:H~-->H~))
7		fveq1 3714	. . . . . . . . . . . . 13 |- (t = T -> (t` w) = (T` w))
8		fveq1 3714	. . . . . . . . . . . . 13 |- (t = T -> (t` x) = (T` x))
9	7, 8	opreq12d 3969	. . . . . . . . . . . 12 |- (t = T -> ((t` w) -h (t` x)) = ((T` w) -h (T` x)))
10	9	fveq2d 3719	. . . . . . . . . . 11 |- (t = T -> (normh` ((t` w) -h (t` x))) = (normh` ((T` w) -h (T` x))))
11	10	breq1d 2624	. . . . . . . . . 10 |- (t = T -> ((normh` ((t` w) -h (t` x))) < y <-> (normh` ((T` w) -h (T` x))) < y))
12	11	imbi2d 611	. . . . . . . . 9 |- (t = T -> (((normh` (w -h x)) < z -> (normh` ((t` w) -h (t` x))) < y) <-> ((normh` (w -h x)) < z -> (normh` ((T` w) -h (T` x))) < y)))
13	12	ralbidv 1660	. . . . . . . 8 |- (t = T -> (A.w e. H~ ((normh` (w -h x)) < z -> (normh` ((t` w) -h (t` x))) < y) <-> A.w e. H~ ((normh` (w -h x)) < z -> (normh` ((T` w) -h (T` x))) < y)))
14	13	anbi2d 615	. . . . . . 7 |- (t = T -> ((0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (normh` ((t` w) -h (t` x))) < y)) <-> (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (normh` ((T` w) -h (T` x))) < y))))
15	14	rexbidv 1661	. . . . . 6 |- (t = T -> (E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (normh` ((t` w) -h (t` x))) < y)) <-> E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (normh` ((T` w) -h (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 -> (normh` ((t` w) -h (t` x))) < y))) <-> (0 < y -> E.z e. RR (0 < z /\ A.w e. H~ ((normh` (w -h x)) < z -> (normh` ((T` w) -h (T` x))) < y)))))
17	16	2ralbidv 1677	. . . 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 -> (normh` ((t` w) -h (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 -> (normh` ((T` w) -h (T` x))) < y)))))
18	6, 17	anbi12d 627	. . 3 |- (t = T -> ((t:H~-->H~ /\ 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 -> (normh` ((t` w) -h (t` x))) < y)))) <-> (T:H~-->H~ /\ 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 -> (normh` ((T` w) -h (T` x))) < y))))))
19		df-cnop 9706	. . 3 |- ConOp = {t | (t:H~-->H~ /\ 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 -> (normh` ((t` w) -h (t` x))) < y))))}
20	18, 19	elab2g 1896	. 2 |- (T e. V -> (T e. ConOp <-> (T:H~-->H~ /\ 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 -> (normh` ((T` w) -h (T` x))) < y))))))
21	1, 5, 20	pm5.21nii 678	1 |- (T e. ConOp <-> (T:H~-->H~ /\ 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 -> (normh` ((T` w) -h (T` x))) < y)))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  A.wral 1642  E.wrex 1643  Vcvv 1807   class class class wbr 2614  -->wf 3173  ` cfv 3177  (class class class)co 3954  RRcr 5213  0cc0 5214   < clt 5466  H~chil 8727   -h cmv 8731  normhcno 8733  ConOpcco 8754

This theorem is referenced by:  cnopct 9776  0cnop 9842  idcnop 9844  lnopcon 9901

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-hilex 8808

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-opr 3956  df-cnop 9706

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