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Theorem cnopc 27565
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 27509 . . . 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 465 . . 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 6314 . . . . . . . 8  |-  ( z  =  A  ->  (
y  -h  z )  =  ( y  -h  A ) )
43fveq2d 5886 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( y  -h  z ) )  =  ( normh `  ( y  -h  A ) ) )
54breq1d 4433 . . . . . 6  |-  ( z  =  A  ->  (
( normh `  ( y  -h  z ) )  < 
x  <->  ( normh `  (
y  -h  A ) )  <  x ) )
6 fveq2 5882 . . . . . . . . 9  |-  ( z  =  A  ->  ( T `  z )  =  ( T `  A ) )
76oveq2d 6322 . . . . . . . 8  |-  ( z  =  A  ->  (
( T `  y
)  -h  ( T `
 z ) )  =  ( ( T `
 y )  -h  ( T `  A
) ) )
87fveq2d 5886 . . . . . . 7  |-  ( z  =  A  ->  ( normh `  ( ( T `
 y )  -h  ( T `  z
) ) )  =  ( normh `  ( ( T `  y )  -h  ( T `  A
) ) ) )
98breq1d 4433 . . . . . 6  |-  ( z  =  A  ->  (
( normh `  ( ( T `  y )  -h  ( T `  z
) ) )  < 
w  <->  ( normh `  (
( T `  y
)  -h  ( T `
 A ) ) )  <  w ) )
105, 9imbi12d 321 . . . . 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 2944 . . . 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 4427 . . . . . 6  |-  ( w  =  B  ->  (
( normh `  ( ( T `  y )  -h  ( T `  A
) ) )  < 
w  <->  ( normh `  (
( T `  y
)  -h  ( T `
 A ) ) )  <  B ) )
1312imbi2d 317 . . . . 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 2944 . . . 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 3191 . . 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 31 . 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 1203 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 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2771   E.wrex 2772   class class class wbr 4423   -->wf 5597   ` cfv 5601  (class class class)co 6306    < clt 9683   RR+crp 11310   ~Hchil 26571   normhcno 26575    -h cmv 26577   ConOpccop 26598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-hilex 26651
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-map 7486  df-cnop 27492
This theorem is referenced by:  nmcopexi  27679
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