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Theorem cnopc 23377
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 23321 . . . 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 451 . . 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 6056 . . . . . . . 8  |-  ( z  =  A  ->  (
y  -h  z )  =  ( y  -h  A ) )
43fveq2d 5699 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( y  -h  z ) )  =  ( normh `  ( y  -h  A ) ) )
54breq1d 4190 . . . . . 6  |-  ( z  =  A  ->  (
( normh `  ( y  -h  z ) )  < 
x  <->  ( normh `  (
y  -h  A ) )  <  x ) )
6 fveq2 5695 . . . . . . . . 9  |-  ( z  =  A  ->  ( T `  z )  =  ( T `  A ) )
76oveq2d 6064 . . . . . . . 8  |-  ( z  =  A  ->  (
( T `  y
)  -h  ( T `
 z ) )  =  ( ( T `
 y )  -h  ( T `  A
) ) )
87fveq2d 5699 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( ( T `
 y )  -h  ( T `  z
) ) )  =  ( normh `  ( ( T `  y )  -h  ( T `  A
) ) ) )
98breq1d 4190 . . . . . 6  |-  ( z  =  A  ->  (
( normh `  ( ( T `  y )  -h  ( T `  z
) ) )  < 
w  <->  ( normh `  (
( T `  y
)  -h  ( T `
 A ) ) )  <  w ) )
105, 9imbi12d 312 . . . . 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 2718 . . . 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 4184 . . . . . 6  |-  ( w  =  B  ->  (
( normh `  ( ( T `  y )  -h  ( T `  A
) ) )  < 
w  <->  ( normh `  (
( T `  y
)  -h  ( T `
 A ) ) )  <  B ) )
1312imbi2d 308 . . . . 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 2718 . . . 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 3026 . . 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 28 . 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 1151 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674   E.wrex 2675   class class class wbr 4180   -->wf 5417   ` cfv 5421  (class class class)co 6048    < clt 9084   RR+crp 10576   ~Hchil 22383   normhcno 22387    -h cmv 22389   ConOpccop 22410
This theorem is referenced by:  nmcopexi  23491
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-hilex 22463
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-cnop 23304
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