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Theorem idcnop 23476
Description: The identity function (restricted to Hilbert space) is a continuous operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
idcnop  |-  (  _I  |`  ~H )  e.  ConOp

Proof of Theorem idcnop
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oi 5705 . . 3  |-  (  _I  |`  ~H ) : ~H -1-1-onto-> ~H
2 f1of 5666 . . 3  |-  ( (  _I  |`  ~H ) : ~H -1-1-onto-> ~H  ->  (  _I  |` 
~H ) : ~H --> ~H )
31, 2ax-mp 8 . 2  |-  (  _I  |`  ~H ) : ~H --> ~H
4 id 20 . . . 4  |-  ( y  e.  RR+  ->  y  e.  RR+ )
5 fvresi 5916 . . . . . . . . 9  |-  ( w  e.  ~H  ->  (
(  _I  |`  ~H ) `  w )  =  w )
6 fvresi 5916 . . . . . . . . 9  |-  ( x  e.  ~H  ->  (
(  _I  |`  ~H ) `  x )  =  x )
75, 6oveqan12rd 6093 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( ( (  _I  |`  ~H ) `  w
)  -h  ( (  _I  |`  ~H ) `  x ) )  =  ( w  -h  x
) )
87fveq2d 5724 . . . . . . 7  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( normh `  ( (
(  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  =  ( normh `  (
w  -h  x ) ) )
98breq1d 4214 . . . . . 6  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( ( normh `  (
( (  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  <  y  <->  ( normh `  ( w  -h  x
) )  <  y
) )
109biimprd 215 . . . . 5  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( ( normh `  (
w  -h  x ) )  <  y  -> 
( normh `  ( (
(  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  <  y ) )
1110ralrimiva 2781 . . . 4  |-  ( x  e.  ~H  ->  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  y  ->  ( normh `  ( (
(  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  <  y ) )
12 breq2 4208 . . . . . . 7  |-  ( z  =  y  ->  (
( normh `  ( w  -h  x ) )  < 
z  <->  ( normh `  (
w  -h  x ) )  <  y ) )
1312imbi1d 309 . . . . . 6  |-  ( z  =  y  ->  (
( ( normh `  (
w  -h  x ) )  <  z  -> 
( normh `  ( (
(  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  <  y )  <->  ( ( normh `  ( w  -h  x ) )  < 
y  ->  ( normh `  ( ( (  _I  |`  ~H ) `  w
)  -h  ( (  _I  |`  ~H ) `  x ) ) )  <  y ) ) )
1413ralbidv 2717 . . . . 5  |-  ( z  =  y  ->  ( A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( normh `  ( (
(  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  <  y )  <->  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  y  ->  ( normh `  ( (
(  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  <  y ) ) )
1514rspcev 3044 . . . 4  |-  ( ( y  e.  RR+  /\  A. w  e.  ~H  (
( normh `  ( w  -h  x ) )  < 
y  ->  ( normh `  ( ( (  _I  |`  ~H ) `  w
)  -h  ( (  _I  |`  ~H ) `  x ) ) )  <  y ) )  ->  E. z  e.  RR+  A. w  e.  ~H  (
( normh `  ( w  -h  x ) )  < 
z  ->  ( normh `  ( ( (  _I  |`  ~H ) `  w
)  -h  ( (  _I  |`  ~H ) `  x ) ) )  <  y ) )
164, 11, 15syl2anr 465 . . 3  |-  ( ( x  e.  ~H  /\  y  e.  RR+ )  ->  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( normh `  ( (
(  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  <  y ) )
1716rgen2 2794 . 2  |-  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
( normh `  ( w  -h  x ) )  < 
z  ->  ( normh `  ( ( (  _I  |`  ~H ) `  w
)  -h  ( (  _I  |`  ~H ) `  x ) ) )  <  y )
18 elcnop 23352 . 2  |-  ( (  _I  |`  ~H )  e.  ConOp 
<->  ( (  _I  |`  ~H ) : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
( normh `  ( w  -h  x ) )  < 
z  ->  ( normh `  ( ( (  _I  |`  ~H ) `  w
)  -h  ( (  _I  |`  ~H ) `  x ) ) )  <  y ) ) )
193, 17, 18mpbir2an 887 1  |-  (  _I  |`  ~H )  e.  ConOp
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   A.wral 2697   E.wrex 2698   class class class wbr 4204    _I cid 4485    |` cres 4872   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    < clt 9112   RR+crp 10604   ~Hchil 22414   normhcno 22418    -h cmv 22420   ConOpccop 22441
This theorem is referenced by:  nmcopex  23524  nmcoplb  23525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-hilex 22494
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-cnop 23335
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