Theorem elcnop 11420

Description: Property defining a continuous Hilbert space operator.

Assertion

Ref	Expression
elcnop	|- (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 elcnop

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (T e. ConOp -> T e. _V)
2		ax-hilex 10501	. . . 4 |- ~H e. _V
3		fex 4595	. . . 4 |- ((T:~H-->~H /\ ~H e. _V) -> T e. _V)
4	2, 3	mpan2 760	. . 3 |- (T:~H-->~H -> T e. _V)
5	4	adantr 425	. 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 4551	. . . 4 |- (t = T -> (t:~H-->~H <-> T:~H-->~H))
7		fveq1 4680	. . . . . . . . . . . . 13 |- (t = T -> (t` w) = (T` w))
8		fveq1 4680	. . . . . . . . . . . . 13 |- (t = T -> (t` x) = (T` x))
9	7, 8	opreq12d 4900	. . . . . . . . . . . 12 |- (t = T -> ((t` w) -h (t` x)) = ((T` w) -h (T` x)))
10	9	fveq2d 4685	. . . . . . . . . . 11 |- (t = T -> (normh` ((t` w) -h (t` x))) = (normh` ((T` w) -h (T` x))))
11	10	breq1d 3348	. . . . . . . . . 10 |- (t = T -> ((normh` ((t` w) -h (t` x))) < y <-> (normh` ((T` w) -h (T` x))) < y))
12	11	imbi2d 674	. . . . . . . . 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 2123	. . . . . . . 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 678	. . . . . . 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 2124	. . . . . 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 674	. . . . 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 2140	. . . 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 690	. . 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 11403	. . 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 2406	. 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 743	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 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   class class class wbr 3338  -->wf 3994  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386   < clt 6653  ~Hchil 10420   -h cmv 10424  normhcno 10426  ConOpcco 10447

This theorem is referenced by:  cnopc 11474  0cnop 11540  idcnop 11542  lnopconi 11600

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-hilex 10501

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  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-fv 4014  df-opr 4886  df-cnop 11403

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