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Theorem idcnop 22577
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 5527 . . 3  |-  (  _I  |`  ~H ) : ~H -1-1-onto-> ~H
2 f1of 5488 . . 3  |-  ( (  _I  |`  ~H ) : ~H -1-1-onto-> ~H  ->  (  _I  |` 
~H ) : ~H --> ~H )
31, 2ax-mp 8 . 2  |-  (  _I  |`  ~H ) : ~H --> ~H
4 id 19 . . . 4  |-  ( y  e.  RR+  ->  y  e.  RR+ )
5 fvresi 5727 . . . . . . . . 9  |-  ( w  e.  ~H  ->  (
(  _I  |`  ~H ) `  w )  =  w )
6 fvresi 5727 . . . . . . . . 9  |-  ( x  e.  ~H  ->  (
(  _I  |`  ~H ) `  x )  =  x )
75, 6oveqan12rd 5894 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( ( (  _I  |`  ~H ) `  w
)  -h  ( (  _I  |`  ~H ) `  x ) )  =  ( w  -h  x
) )
87fveq2d 5545 . . . . . . 7  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( normh `  ( (
(  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  =  ( normh `  (
w  -h  x ) ) )
98breq1d 4049 . . . . . 6  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( ( normh `  (
( (  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  <  y  <->  ( normh `  ( w  -h  x
) )  <  y
) )
109biimprd 214 . . . . 5  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( ( normh `  (
w  -h  x ) )  <  y  -> 
( normh `  ( (
(  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  <  y ) )
1110ralrimiva 2639 . . . 4  |-  ( x  e.  ~H  ->  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  y  ->  ( normh `  ( (
(  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  <  y ) )
12 breq2 4043 . . . . . . 7  |-  ( z  =  y  ->  (
( normh `  ( w  -h  x ) )  < 
z  <->  ( normh `  (
w  -h  x ) )  <  y ) )
1312imbi1d 308 . . . . . 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 2576 . . . . 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 2897 . . . 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 464 . . 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 2652 . 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 22453 . 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 886 1  |-  (  _I  |`  ~H )  e.  ConOp
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   class class class wbr 4039    _I cid 4320    |` cres 4707   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    < clt 8883   RR+crp 10370   ~Hchil 21515   normhcno 21519    -h cmv 21521   ConOpccop 21542
This theorem is referenced by:  nmcopex  22625  nmcoplb  22626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-cnop 22436
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