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Theorem cnopc 26958
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 26902 . . . 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 6304 . . . . . . . 8  |-  ( z  =  A  ->  (
y  -h  z )  =  ( y  -h  A ) )
43fveq2d 5876 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( y  -h  z ) )  =  ( normh `  ( y  -h  A ) ) )
54breq1d 4466 . . . . . 6  |-  ( z  =  A  ->  (
( normh `  ( y  -h  z ) )  < 
x  <->  ( normh `  (
y  -h  A ) )  <  x ) )
6 fveq2 5872 . . . . . . . . 9  |-  ( z  =  A  ->  ( T `  z )  =  ( T `  A ) )
76oveq2d 6312 . . . . . . . 8  |-  ( z  =  A  ->  (
( T `  y
)  -h  ( T `
 z ) )  =  ( ( T `
 y )  -h  ( T `  A
) ) )
87fveq2d 5876 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( ( T `
 y )  -h  ( T `  z
) ) )  =  ( normh `  ( ( T `  y )  -h  ( T `  A
) ) ) )
98breq1d 4466 . . . . . 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 2976 . . . 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 4460 . . . . . 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 2976 . . . 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 3219 . . 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 1194 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   class class class wbr 4456   -->wf 5590   ` cfv 5594  (class class class)co 6296    < clt 9645   RR+crp 11245   ~Hchil 25962   normhcno 25966    -h cmv 25968   ConOpccop 25989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-hilex 26042
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-cnop 26885
This theorem is referenced by:  nmcopexi  27072
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