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Theorem cnopc 25322
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 25266 . . . 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 464 . . 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 6104 . . . . . . . 8  |-  ( z  =  A  ->  (
y  -h  z )  =  ( y  -h  A ) )
43fveq2d 5700 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( y  -h  z ) )  =  ( normh `  ( y  -h  A ) ) )
54breq1d 4307 . . . . . 6  |-  ( z  =  A  ->  (
( normh `  ( y  -h  z ) )  < 
x  <->  ( normh `  (
y  -h  A ) )  <  x ) )
6 fveq2 5696 . . . . . . . . 9  |-  ( z  =  A  ->  ( T `  z )  =  ( T `  A ) )
76oveq2d 6112 . . . . . . . 8  |-  ( z  =  A  ->  (
( T `  y
)  -h  ( T `
 z ) )  =  ( ( T `
 y )  -h  ( T `  A
) ) )
87fveq2d 5700 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( ( T `
 y )  -h  ( T `  z
) ) )  =  ( normh `  ( ( T `  y )  -h  ( T `  A
) ) ) )
98breq1d 4307 . . . . . 6  |-  ( z  =  A  ->  (
( normh `  ( ( T `  y )  -h  ( T `  z
) ) )  < 
w  <->  ( normh `  (
( T `  y
)  -h  ( T `
 A ) ) )  <  w ) )
105, 9imbi12d 320 . . . . 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 2764 . . . 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 4301 . . . . . 6  |-  ( w  =  B  ->  (
( normh `  ( ( T `  y )  -h  ( T `  A
) ) )  < 
w  <->  ( normh `  (
( T `  y
)  -h  ( T `
 A ) ) )  <  B ) )
1312imbi2d 316 . . . . 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 2764 . . . 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 3084 . . 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 30 . 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 1185 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   class class class wbr 4297   -->wf 5419   ` cfv 5423  (class class class)co 6096    < clt 9423   RR+crp 10996   ~Hchil 24326   normhcno 24330    -h cmv 24332   ConOpccop 24353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-hilex 24406
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-map 7221  df-cnop 25249
This theorem is referenced by:  nmcopexi  25436
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