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Theorem idcnop 22561
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 5511 . . 3  |-  (  _I  |`  ~H ) : ~H -1-1-onto-> ~H
2 f1of 5472 . . 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 5711 . . . . . . . . 9  |-  ( w  e.  ~H  ->  (
(  _I  |`  ~H ) `  w )  =  w )
6 fvresi 5711 . . . . . . . . 9  |-  ( x  e.  ~H  ->  (
(  _I  |`  ~H ) `  x )  =  x )
75, 6oveqan12rd 5878 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( ( (  _I  |`  ~H ) `  w
)  -h  ( (  _I  |`  ~H ) `  x ) )  =  ( w  -h  x
) )
87fveq2d 5529 . . . . . . 7  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( normh `  ( (
(  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  =  ( normh `  (
w  -h  x ) ) )
98breq1d 4033 . . . . . 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 2626 . . . 4  |-  ( x  e.  ~H  ->  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  y  ->  ( normh `  ( (
(  _I  |`  ~H ) `  w )  -h  (
(  _I  |`  ~H ) `  x ) ) )  <  y ) )
12 breq2 4027 . . . . . . 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 2563 . . . . 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 2884 . . . 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 2639 . 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 22437 . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023    _I cid 4304    |` cres 4691   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    < clt 8867   RR+crp 10354   ~Hchil 21499   normhcno 21503    -h cmv 21505   ConOpccop 21526
This theorem is referenced by:  nmcopex  22609  nmcoplb  22610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-cnop 22420
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