HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  nmlnop0iALT Unicode version

Theorem nmlnop0iALT 23009
Description: A linear operator with a zero norm is identically zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
nmlnop0.1  |-  T  e. 
LinOp
Assertion
Ref Expression
nmlnop0iALT  |-  ( (
normop `  T )  =  0  <->  T  =  0hop )

Proof of Theorem nmlnop0iALT
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 normcl 22138 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  RR )
21recnd 9008 . . . . . . . . . . . . 13  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  CC )
32adantr 451 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  x )  e.  CC )
4 norm-i 22142 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~H  ->  (
( normh `  x )  =  0  <->  x  =  0h ) )
5 fveq2 5632 . . . . . . . . . . . . . . . 16  |-  ( x  =  0h  ->  ( T `  x )  =  ( T `  0h ) )
6 nmlnop0.1 . . . . . . . . . . . . . . . . 17  |-  T  e. 
LinOp
76lnop0i 22984 . . . . . . . . . . . . . . . 16  |-  ( T `
 0h )  =  0h
85, 7syl6eq 2414 . . . . . . . . . . . . . . 15  |-  ( x  =  0h  ->  ( T `  x )  =  0h )
94, 8syl6bi 219 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  (
( normh `  x )  =  0  ->  ( T `  x )  =  0h ) )
109necon3d 2567 . . . . . . . . . . . . 13  |-  ( x  e.  ~H  ->  (
( T `  x
)  =/=  0h  ->  (
normh `  x )  =/=  0 ) )
1110imp 418 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  x )  =/=  0 )
123, 11recne0d 9677 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( 1  /  ( normh `  x ) )  =/=  0 )
13 simpr 447 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( T `  x
)  =/=  0h )
143, 11reccld 9676 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( 1  /  ( normh `  x ) )  e.  CC )
156lnopfi 22983 . . . . . . . . . . . . . . . 16  |-  T : ~H
--> ~H
1615ffvelrni 5771 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
1716adantr 451 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( T `  x
)  e.  ~H )
18 hvmul0or 22038 . . . . . . . . . . . . . 14  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  ( T `  x )  e.  ~H )  ->  (
( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) )  =  0h  <->  ( (
1  /  ( normh `  x ) )  =  0  \/  ( T `
 x )  =  0h ) ) )
1914, 17, 18syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
)  =  0h  <->  ( (
1  /  ( normh `  x ) )  =  0  \/  ( T `
 x )  =  0h ) ) )
2019necon3abid 2562 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
)  =/=  0h  <->  -.  (
( 1  /  ( normh `  x ) )  =  0  \/  ( T `  x )  =  0h ) ) )
21 neanior 2614 . . . . . . . . . . . 12  |-  ( ( ( 1  /  ( normh `  x ) )  =/=  0  /\  ( T `  x )  =/=  0h )  <->  -.  (
( 1  /  ( normh `  x ) )  =  0  \/  ( T `  x )  =  0h ) )
2220, 21syl6bbr 254 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
)  =/=  0h  <->  ( (
1  /  ( normh `  x ) )  =/=  0  /\  ( T `
 x )  =/= 
0h ) ) )
2312, 13, 22mpbir2and 888 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) )  =/=  0h )
24 hvmulcl 22027 . . . . . . . . . . . 12  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  ( T `  x )  e.  ~H )  ->  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) )  e. 
~H )
2514, 17, 24syl2anc 642 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) )  e.  ~H )
26 normgt0 22140 . . . . . . . . . . 11  |-  ( ( ( 1  /  ( normh `  x ) )  .h  ( T `  x ) )  e. 
~H  ->  ( ( ( 1  /  ( normh `  x ) )  .h  ( T `  x
) )  =/=  0h  <->  0  <  ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) ) ) )
2725, 26syl 15 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
)  =/=  0h  <->  0  <  (
normh `  ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
) ) ) )
2823, 27mpbid 201 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
0  <  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) ) )
2928ex 423 . . . . . . . 8  |-  ( x  e.  ~H  ->  (
( T `  x
)  =/=  0h  ->  0  <  ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) ) ) )
3029adantl 452 . . . . . . 7  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  (
( T `  x
)  =/=  0h  ->  0  <  ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) ) ) )
31 nmopsetretHIL 22878 . . . . . . . . . . . . . 14  |-  ( T : ~H --> ~H  ->  { y  |  E. z  e.  ~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) }  C_  RR )
3215, 31ax-mp 8 . . . . . . . . . . . . 13  |-  { y  |  E. z  e. 
~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) }  C_  RR
33 ressxr 9023 . . . . . . . . . . . . 13  |-  RR  C_  RR*
3432, 33sstri 3274 . . . . . . . . . . . 12  |-  { y  |  E. z  e. 
~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) }  C_  RR*
35 simpl 443 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  ->  x  e.  ~H )
36 hvmulcl 22027 . . . . . . . . . . . . . . 15  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  ~H )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e. 
~H )
3714, 35, 36syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( 1  / 
( normh `  x )
)  .h  x )  e.  ~H )
388necon3i 2568 . . . . . . . . . . . . . . . . 17  |-  ( ( T `  x )  =/=  0h  ->  x  =/=  0h )
39 norm1 22262 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  ~H  /\  x  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  =  1 )
4038, 39sylan2 460 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  =  1 )
41 1re 8984 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
4240, 41syl6eqel 2454 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  e.  RR )
43 eqle 9070 . . . . . . . . . . . . . . 15  |-  ( ( ( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  e.  RR  /\  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  x ) )  =  1 )  ->  ( normh `  (
( 1  /  ( normh `  x ) )  .h  x ) )  <_  1 )
4442, 40, 43syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  <_ 
1 )
456lnopmuli 22986 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  ~H )  ->  ( T `  ( (
1  /  ( normh `  x ) )  .h  x ) )  =  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) )
4614, 35, 45syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( T `  (
( 1  /  ( normh `  x ) )  .h  x ) )  =  ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
) )
4746eqcomd 2371 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) )  =  ( T `  ( ( 1  / 
( normh `  x )
)  .h  x ) ) )
4847fveq2d 5636 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  =  ( normh `  ( T `  ( ( 1  / 
( normh `  x )
)  .h  x ) ) ) )
49 fveq2 5632 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( normh `  z )  =  (
normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) ) )
5049breq1d 4135 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( ( normh `  z )  <_ 
1  <->  ( normh `  (
( 1  /  ( normh `  x ) )  .h  x ) )  <_  1 ) )
51 fveq2 5632 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( T `  z )  =  ( T `  ( ( 1  /  ( normh `  x ) )  .h  x ) ) )
5251fveq2d 5636 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( normh `  ( T `  z
) )  =  (
normh `  ( T `  ( ( 1  / 
( normh `  x )
)  .h  x ) ) ) )
5352eqeq2d 2377 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( ( normh `  ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
) )  =  (
normh `  ( T `  z ) )  <->  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) )  =  ( normh `  ( T `  (
( 1  /  ( normh `  x ) )  .h  x ) ) ) ) )
5450, 53anbi12d 691 . . . . . . . . . . . . . . 15  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( (
( normh `  z )  <_  1  /\  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) )  =  ( normh `  ( T `  z
) ) )  <->  ( ( normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) )  <_  1  /\  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  =  ( normh `  ( T `  ( ( 1  / 
( normh `  x )
)  .h  x ) ) ) ) ) )
5554rspcev 2969 . . . . . . . . . . . . . 14  |-  ( ( ( ( 1  / 
( normh `  x )
)  .h  x )  e.  ~H  /\  (
( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  <_ 
1  /\  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) )  =  ( normh `  ( T `  (
( 1  /  ( normh `  x ) )  .h  x ) ) ) ) )  ->  E. z  e.  ~H  ( ( normh `  z
)  <_  1  /\  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  =  ( normh `  ( T `  z ) ) ) )
5637, 44, 48, 55syl12anc 1181 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  ->  E. z  e.  ~H  ( ( normh `  z
)  <_  1  /\  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  =  ( normh `  ( T `  z ) ) ) )
57 fvex 5646 . . . . . . . . . . . . . 14  |-  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) )  e.  _V
58 eqeq1 2372 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) )  ->  ( y  =  ( normh `  ( T `  z ) )  <->  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) )  =  ( normh `  ( T `  z
) ) ) )
5958anbi2d 684 . . . . . . . . . . . . . . 15  |-  ( y  =  ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) )  ->  ( ( (
normh `  z )  <_ 
1  /\  y  =  ( normh `  ( T `  z ) ) )  <-> 
( ( normh `  z
)  <_  1  /\  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  =  ( normh `  ( T `  z ) ) ) ) )
6059rexbidv 2649 . . . . . . . . . . . . . 14  |-  ( y  =  ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) )  ->  ( E. z  e.  ~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) )  <->  E. z  e.  ~H  ( ( normh `  z
)  <_  1  /\  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  =  ( normh `  ( T `  z ) ) ) ) )
6157, 60elab 2999 . . . . . . . . . . . . 13  |-  ( (
normh `  ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
) )  e.  {
y  |  E. z  e.  ~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) }  <->  E. z  e.  ~H  ( ( normh `  z
)  <_  1  /\  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  =  ( normh `  ( T `  z ) ) ) )
6256, 61sylibr 203 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  e. 
{ y  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) } )
63 supxrub 10796 . . . . . . . . . . . 12  |-  ( ( { y  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) }  C_  RR*  /\  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
) )  e.  {
y  |  E. z  e.  ~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) } )  ->  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
) )  <_  sup ( { y  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) } ,  RR* ,  <  ) )
6434, 62, 63sylancr 644 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  <_  sup ( { y  |  E. z  e.  ~H  ( ( normh `  z
)  <_  1  /\  y  =  ( normh `  ( T `  z
) ) ) } ,  RR* ,  <  )
)
6564adantll 694 . . . . . . . . . 10  |-  ( ( ( ( normop `  T
)  =  0  /\  x  e.  ~H )  /\  ( T `  x
)  =/=  0h )  ->  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  <_  sup ( { y  |  E. z  e.  ~H  ( ( normh `  z
)  <_  1  /\  y  =  ( normh `  ( T `  z
) ) ) } ,  RR* ,  <  )
)
66 nmopval 22870 . . . . . . . . . . . . . 14  |-  ( T : ~H --> ~H  ->  (
normop `  T )  =  sup ( { y  |  E. z  e. 
~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) } ,  RR* ,  <  ) )
6715, 66ax-mp 8 . . . . . . . . . . . . 13  |-  ( normop `  T )  =  sup ( { y  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) } ,  RR* ,  <  )
6867eqeq1i 2373 . . . . . . . . . . . 12  |-  ( (
normop `  T )  =  0  <->  sup ( { y  |  E. z  e. 
~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) } ,  RR* ,  <  )  =  0 )
6968biimpi 186 . . . . . . . . . . 11  |-  ( (
normop `  T )  =  0  ->  sup ( { y  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) } ,  RR* ,  <  )  =  0 )
7069ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ( normop `  T
)  =  0  /\  x  e.  ~H )  /\  ( T `  x
)  =/=  0h )  ->  sup ( { y  |  E. z  e. 
~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) } ,  RR* ,  <  )  =  0 )
7165, 70breqtrd 4149 . . . . . . . . 9  |-  ( ( ( ( normop `  T
)  =  0  /\  x  e.  ~H )  /\  ( T `  x
)  =/=  0h )  ->  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  <_ 
0 )
72 normcl 22138 . . . . . . . . . . . 12  |-  ( ( ( 1  /  ( normh `  x ) )  .h  ( T `  x ) )  e. 
~H  ->  ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) )  e.  RR )
7325, 72syl 15 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  e.  RR )
74 0re 8985 . . . . . . . . . . 11  |-  0  e.  RR
75 lenlt 9048 . . . . . . . . . . 11  |-  ( ( ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  e.  RR  /\  0  e.  RR )  ->  (
( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  <_ 
0  <->  -.  0  <  (
normh `  ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
) ) ) )
7673, 74, 75sylancl 643 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) )  <_  0  <->  -.  0  <  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) ) ) )
7776adantll 694 . . . . . . . . 9  |-  ( ( ( ( normop `  T
)  =  0  /\  x  e.  ~H )  /\  ( T `  x
)  =/=  0h )  ->  ( ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) )  <_  0  <->  -.  0  <  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) ) ) )
7871, 77mpbid 201 . . . . . . . 8  |-  ( ( ( ( normop `  T
)  =  0  /\  x  e.  ~H )  /\  ( T `  x
)  =/=  0h )  ->  -.  0  <  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
) ) )
7978ex 423 . . . . . . 7  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  (
( T `  x
)  =/=  0h  ->  -.  0  <  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) ) ) )
8030, 79pm2.65d 166 . . . . . 6  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  -.  ( T `  x )  =/=  0h )
81 nne 2533 . . . . . 6  |-  ( -.  ( T `  x
)  =/=  0h  <->  ( T `  x )  =  0h )
8280, 81sylib 188 . . . . 5  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  ( T `  x )  =  0h )
83 ho0val 22764 . . . . . 6  |-  ( x  e.  ~H  ->  ( 0hop `  x )  =  0h )
8483adantl 452 . . . . 5  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  ( 0hop `  x )  =  0h )
8582, 84eqtr4d 2401 . . . 4  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  ( T `  x )  =  ( 0hop `  x
) )
8685ralrimiva 2711 . . 3  |-  ( (
normop `  T )  =  0  ->  A. x  e.  ~H  ( T `  x )  =  (
0hop `  x )
)
87 ffn 5495 . . . . 5  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
8815, 87ax-mp 8 . . . 4  |-  T  Fn  ~H
89 ho0f 22765 . . . . 5  |-  0hop : ~H --> ~H
90 ffn 5495 . . . . 5  |-  ( 0hop
: ~H --> ~H  ->  0hop 
Fn  ~H )
9189, 90ax-mp 8 . . . 4  |-  0hop  Fn  ~H
92 eqfnfv 5729 . . . 4  |-  ( ( T  Fn  ~H  /\  0hop 
Fn  ~H )  ->  ( T  =  0hop  <->  A. x  e.  ~H  ( T `  x )  =  (
0hop `  x )
) )
9388, 91, 92mp2an 653 . . 3  |-  ( T  =  0hop  <->  A. x  e.  ~H  ( T `  x )  =  ( 0hop `  x
) )
9486, 93sylibr 203 . 2  |-  ( (
normop `  T )  =  0  ->  T  =  0hop )
95 fveq2 5632 . . 3  |-  ( T  =  0hop  ->  ( normop `  T )  =  (
normop `  0hop ) )
96 nmop0 23000 . . 3  |-  ( normop ` 
0hop )  =  0
9795, 96syl6eq 2414 . 2  |-  ( T  =  0hop  ->  ( normop `  T )  =  0 )
9894, 97impbii 180 1  |-  ( (
normop `  T )  =  0  <->  T  =  0hop )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1647    e. wcel 1715   {cab 2352    =/= wne 2529   A.wral 2628   E.wrex 2629    C_ wss 3238   class class class wbr 4125    Fn wfn 5353   -->wf 5354   ` cfv 5358  (class class class)co 5981   supcsup 7340   CCcc 8882   RRcr 8883   0cc0 8884   1c1 8885   RR*cxr 9013    < clt 9014    <_ cle 9015    / cdiv 9570   ~Hchil 21933    .h csm 21935   normhcno 21937   0hc0v 21938   0hopch0o 21957   normopcnop 21959   LinOpclo 21961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cc 8208  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-addf 8963  ax-mulf 8964  ax-hilex 22013  ax-hfvadd 22014  ax-hvcom 22015  ax-hvass 22016  ax-hv0cl 22017  ax-hvaddid 22018  ax-hfvmul 22019  ax-hvmulid 22020  ax-hvmulass 22021  ax-hvdistr1 22022  ax-hvdistr2 22023  ax-hvmul0 22024  ax-hfi 22092  ax-his1 22095  ax-his2 22096  ax-his3 22097  ax-his4 22098  ax-hcompl 22215
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-omul 6626  df-er 6802  df-map 6917  df-pm 6918  df-ixp 6961  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-fi 7312  df-sup 7341  df-oi 7372  df-card 7719  df-acn 7722  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-q 10468  df-rp 10506  df-xneg 10603  df-xadd 10604  df-xmul 10605  df-ioo 10813  df-ico 10815  df-icc 10816  df-fz 10936  df-fzo 11026  df-fl 11089  df-seq 11211  df-exp 11270  df-hash 11506  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-clim 12169  df-rlim 12170  df-sum 12367  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-starv 13431  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-unif 13439  df-hom 13440  df-cco 13441  df-rest 13537  df-topn 13538  df-topgen 13554  df-pt 13555  df-prds 13558  df-xrs 13613  df-0g 13614  df-gsum 13615  df-qtop 13620  df-imas 13621  df-xps 13623  df-mre 13698  df-mrc 13699  df-acs 13701  df-mnd 14577  df-submnd 14626  df-mulg 14702  df-cntz 15003  df-cmn 15301  df-xmet 16586  df-met 16587  df-bl 16588  df-mopn 16589  df-fbas 16590  df-fg 16591  df-cnfld 16594  df-top 16853  df-bases 16855  df-topon 16856  df-topsp 16857  df-cld 16973  df-ntr 16974  df-cls 16975  df-nei 17052  df-cn 17174  df-cnp 17175  df-lm 17176  df-haus 17260  df-tx 17474  df-hmeo 17663  df-fil 17754  df-fm 17846  df-flim 17847  df-flf 17848  df-xms 18098  df-ms 18099  df-tms 18100  df-cfil 18896  df-cau 18897  df-cmet 18898  df-grpo 21290  df-gid 21291  df-ginv 21292  df-gdiv 21293  df-ablo 21381  df-subgo 21401  df-vc 21536  df-nv 21582  df-va 21585  df-ba 21586  df-sm 21587  df-0v 21588  df-vs 21589  df-nmcv 21590  df-ims 21591  df-dip 21708  df-ssp 21732  df-ph 21825  df-cbn 21876  df-hnorm 21982  df-hba 21983  df-hvsub 21985  df-hlim 21986  df-hcau 21987  df-sh 22220  df-ch 22235  df-oc 22265  df-ch0 22266  df-shs 22321  df-pjh 22408  df-h0op 22762  df-nmop 22853  df-lnop 22855
  Copyright terms: Public domain W3C validator