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Theorem idcnop 23332
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 5653 . . 3  |-  (  _I  |`  ~H ) : ~H -1-1-onto-> ~H
2 f1of 5614 . . 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 5863 . . . . . . . . 9  |-  ( w  e.  ~H  ->  (
(  _I  |`  ~H ) `  w )  =  w )
6 fvresi 5863 . . . . . . . . 9  |-  ( x  e.  ~H  ->  (
(  _I  |`  ~H ) `  x )  =  x )
75, 6oveqan12rd 6040 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( ( (  _I  |`  ~H ) `  w
)  -h  ( (  _I  |`  ~H ) `  x ) )  =  ( w  -h  x
) )
87fveq2d 5672 . . . . . . 7  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( normh `  ( (
(  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  =  ( normh `  (
w  -h  x ) ) )
98breq1d 4163 . . . . . 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 2732 . . . 4  |-  ( x  e.  ~H  ->  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  y  ->  ( normh `  ( (
(  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  <  y ) )
12 breq2 4157 . . . . . . 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 2669 . . . . 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 2995 . . . 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 2745 . 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 23208 . 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 1717   A.wral 2649   E.wrex 2650   class class class wbr 4153    _I cid 4434    |` cres 4820   -->wf 5390   -1-1-onto->wf1o 5393   ` cfv 5394  (class class class)co 6020    < clt 9053   RR+crp 10544   ~Hchil 22270   normhcno 22274    -h cmv 22276   ConOpccop 22297
This theorem is referenced by:  nmcopex  23380  nmcoplb  23381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-hilex 22350
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-map 6956  df-cnop 23191
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