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Theorem cnopc 23421
Description: Basic continuity property of a continuous Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
cnopc  |-  ( ( T  e.  ConOp  /\  A  e.  ~H  /\  B  e.  RR+ )  ->  E. x  e.  RR+  A. y  e. 
~H  ( ( normh `  ( y  -h  A
) )  <  x  ->  ( normh `  ( ( T `  y )  -h  ( T `  A
) ) )  < 
B ) )
Distinct variable groups:    x, y, A    x, B, y    x, T, y

Proof of Theorem cnopc
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elcnop 23365 . . . 4  |-  ( T  e.  ConOp 
<->  ( T : ~H --> ~H  /\  A. z  e. 
~H  A. w  e.  RR+  E. x  e.  RR+  A. y  e.  ~H  ( ( normh `  ( y  -h  z
) )  <  x  ->  ( normh `  ( ( T `  y )  -h  ( T `  z
) ) )  < 
w ) ) )
21simprbi 452 . . 3  |-  ( T  e.  ConOp  ->  A. z  e.  ~H  A. w  e.  RR+  E. x  e.  RR+  A. y  e.  ~H  (
( normh `  ( y  -h  z ) )  < 
x  ->  ( normh `  ( ( T `  y )  -h  ( T `  z )
) )  <  w
) )
3 oveq2 6092 . . . . . . . 8  |-  ( z  =  A  ->  (
y  -h  z )  =  ( y  -h  A ) )
43fveq2d 5735 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( y  -h  z ) )  =  ( normh `  ( y  -h  A ) ) )
54breq1d 4225 . . . . . 6  |-  ( z  =  A  ->  (
( normh `  ( y  -h  z ) )  < 
x  <->  ( normh `  (
y  -h  A ) )  <  x ) )
6 fveq2 5731 . . . . . . . . 9  |-  ( z  =  A  ->  ( T `  z )  =  ( T `  A ) )
76oveq2d 6100 . . . . . . . 8  |-  ( z  =  A  ->  (
( T `  y
)  -h  ( T `
 z ) )  =  ( ( T `
 y )  -h  ( T `  A
) ) )
87fveq2d 5735 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( ( T `
 y )  -h  ( T `  z
) ) )  =  ( normh `  ( ( T `  y )  -h  ( T `  A
) ) ) )
98breq1d 4225 . . . . . 6  |-  ( z  =  A  ->  (
( normh `  ( ( T `  y )  -h  ( T `  z
) ) )  < 
w  <->  ( normh `  (
( T `  y
)  -h  ( T `
 A ) ) )  <  w ) )
105, 9imbi12d 313 . . . . 5  |-  ( z  =  A  ->  (
( ( normh `  (
y  -h  z ) )  <  x  -> 
( normh `  ( ( T `  y )  -h  ( T `  z
) ) )  < 
w )  <->  ( ( normh `  ( y  -h  A ) )  < 
x  ->  ( normh `  ( ( T `  y )  -h  ( T `  A )
) )  <  w
) ) )
1110rexralbidv 2751 . . . 4  |-  ( z  =  A  ->  ( E. x  e.  RR+  A. y  e.  ~H  ( ( normh `  ( y  -h  z
) )  <  x  ->  ( normh `  ( ( T `  y )  -h  ( T `  z
) ) )  < 
w )  <->  E. x  e.  RR+  A. y  e. 
~H  ( ( normh `  ( y  -h  A
) )  <  x  ->  ( normh `  ( ( T `  y )  -h  ( T `  A
) ) )  < 
w ) ) )
12 breq2 4219 . . . . . 6  |-  ( w  =  B  ->  (
( normh `  ( ( T `  y )  -h  ( T `  A
) ) )  < 
w  <->  ( normh `  (
( T `  y
)  -h  ( T `
 A ) ) )  <  B ) )
1312imbi2d 309 . . . . 5  |-  ( w  =  B  ->  (
( ( normh `  (
y  -h  A ) )  <  x  -> 
( normh `  ( ( T `  y )  -h  ( T `  A
) ) )  < 
w )  <->  ( ( normh `  ( y  -h  A ) )  < 
x  ->  ( normh `  ( ( T `  y )  -h  ( T `  A )
) )  <  B
) ) )
1413rexralbidv 2751 . . . 4  |-  ( w  =  B  ->  ( E. x  e.  RR+  A. y  e.  ~H  ( ( normh `  ( y  -h  A
) )  <  x  ->  ( normh `  ( ( T `  y )  -h  ( T `  A
) ) )  < 
w )  <->  E. x  e.  RR+  A. y  e. 
~H  ( ( normh `  ( y  -h  A
) )  <  x  ->  ( normh `  ( ( T `  y )  -h  ( T `  A
) ) )  < 
B ) ) )
1511, 14rspc2v 3060 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  RR+ )  -> 
( A. z  e. 
~H  A. w  e.  RR+  E. x  e.  RR+  A. y  e.  ~H  ( ( normh `  ( y  -h  z
) )  <  x  ->  ( normh `  ( ( T `  y )  -h  ( T `  z
) ) )  < 
w )  ->  E. x  e.  RR+  A. y  e. 
~H  ( ( normh `  ( y  -h  A
) )  <  x  ->  ( normh `  ( ( T `  y )  -h  ( T `  A
) ) )  < 
B ) ) )
162, 15syl5com 29 . 2  |-  ( T  e.  ConOp  ->  ( ( A  e.  ~H  /\  B  e.  RR+ )  ->  E. x  e.  RR+  A. y  e. 
~H  ( ( normh `  ( y  -h  A
) )  <  x  ->  ( normh `  ( ( T `  y )  -h  ( T `  A
) ) )  < 
B ) ) )
17163impib 1152 1  |-  ( ( T  e.  ConOp  /\  A  e.  ~H  /\  B  e.  RR+ )  ->  E. x  e.  RR+  A. y  e. 
~H  ( ( normh `  ( y  -h  A
) )  <  x  ->  ( normh `  ( ( T `  y )  -h  ( T `  A
) ) )  < 
B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   class class class wbr 4215   -->wf 5453   ` cfv 5457  (class class class)co 6084    < clt 9125   RR+crp 10617   ~Hchil 22427   normhcno 22431    -h cmv 22433   ConOpccop 22454
This theorem is referenced by:  nmcopexi  23535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-hilex 22507
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-cnop 23348
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