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Theorem nmlnop0iALT 23451
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 22580 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  RR )
21recnd 9070 . . . . . . . . . . . . 13  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  CC )
32adantr 452 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  x )  e.  CC )
4 norm-i 22584 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~H  ->  (
( normh `  x )  =  0  <->  x  =  0h ) )
5 fveq2 5687 . . . . . . . . . . . . . . . 16  |-  ( x  =  0h  ->  ( T `  x )  =  ( T `  0h ) )
6 nmlnop0.1 . . . . . . . . . . . . . . . . 17  |-  T  e. 
LinOp
76lnop0i 23426 . . . . . . . . . . . . . . . 16  |-  ( T `
 0h )  =  0h
85, 7syl6eq 2452 . . . . . . . . . . . . . . 15  |-  ( x  =  0h  ->  ( T `  x )  =  0h )
94, 8syl6bi 220 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  (
( normh `  x )  =  0  ->  ( T `  x )  =  0h ) )
109necon3d 2605 . . . . . . . . . . . . 13  |-  ( x  e.  ~H  ->  (
( T `  x
)  =/=  0h  ->  (
normh `  x )  =/=  0 ) )
1110imp 419 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  x )  =/=  0 )
123, 11recne0d 9740 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( 1  /  ( normh `  x ) )  =/=  0 )
13 simpr 448 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( T `  x
)  =/=  0h )
143, 11reccld 9739 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( 1  /  ( normh `  x ) )  e.  CC )
156lnopfi 23425 . . . . . . . . . . . . . . . 16  |-  T : ~H
--> ~H
1615ffvelrni 5828 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
1716adantr 452 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( T `  x
)  e.  ~H )
18 hvmul0or 22480 . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
)  =  0h  <->  ( (
1  /  ( normh `  x ) )  =  0  \/  ( T `
 x )  =  0h ) ) )
2019necon3abid 2600 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
)  =/=  0h  <->  -.  (
( 1  /  ( normh `  x ) )  =  0  \/  ( T `  x )  =  0h ) ) )
21 neanior 2652 . . . . . . . . . . . 12  |-  ( ( ( 1  /  ( normh `  x ) )  =/=  0  /\  ( T `  x )  =/=  0h )  <->  -.  (
( 1  /  ( normh `  x ) )  =  0  \/  ( T `  x )  =  0h ) )
2220, 21syl6bbr 255 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
)  =/=  0h  <->  ( (
1  /  ( normh `  x ) )  =/=  0  /\  ( T `
 x )  =/= 
0h ) ) )
2312, 13, 22mpbir2and 889 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) )  =/=  0h )
24 hvmulcl 22469 . . . . . . . . . . . 12  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  ( T `  x )  e.  ~H )  ->  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) )  e. 
~H )
2514, 17, 24syl2anc 643 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) )  e.  ~H )
26 normgt0 22582 . . . . . . . . . . 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 16 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
)  =/=  0h  <->  0  <  (
normh `  ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
) ) ) )
2823, 27mpbid 202 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
0  <  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) ) )
2928ex 424 . . . . . . . 8  |-  ( x  e.  ~H  ->  (
( T `  x
)  =/=  0h  ->  0  <  ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) ) ) )
3029adantl 453 . . . . . . 7  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  (
( T `  x
)  =/=  0h  ->  0  <  ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) ) ) )
31 nmopsetretHIL 23320 . . . . . . . . . . . . . 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 9085 . . . . . . . . . . . . 13  |-  RR  C_  RR*
3432, 33sstri 3317 . . . . . . . . . . . 12  |-  { y  |  E. z  e. 
~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) }  C_  RR*
35 simpl 444 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  ->  x  e.  ~H )
36 hvmulcl 22469 . . . . . . . . . . . . . . 15  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  ~H )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e. 
~H )
3714, 35, 36syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( 1  / 
( normh `  x )
)  .h  x )  e.  ~H )
388necon3i 2606 . . . . . . . . . . . . . . . . 17  |-  ( ( T `  x )  =/=  0h  ->  x  =/=  0h )
39 norm1 22704 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  ~H  /\  x  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  =  1 )
4038, 39sylan2 461 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  =  1 )
41 1re 9046 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
4240, 41syl6eqel 2492 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  e.  RR )
43 eqle 9132 . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  <_ 
1 )
456lnopmuli 23428 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  ~H )  ->  ( T `  ( (
1  /  ( normh `  x ) )  .h  x ) )  =  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) )
4614, 35, 45syl2anc 643 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( T `  (
( 1  /  ( normh `  x ) )  .h  x ) )  =  ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
) )
4746eqcomd 2409 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) )  =  ( T `  ( ( 1  / 
( normh `  x )
)  .h  x ) ) )
4847fveq2d 5691 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  =  ( normh `  ( T `  ( ( 1  / 
( normh `  x )
)  .h  x ) ) ) )
49 fveq2 5687 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( normh `  z )  =  (
normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) ) )
5049breq1d 4182 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( ( normh `  z )  <_ 
1  <->  ( normh `  (
( 1  /  ( normh `  x ) )  .h  x ) )  <_  1 ) )
51 fveq2 5687 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( T `  z )  =  ( T `  ( ( 1  /  ( normh `  x ) )  .h  x ) ) )
5251fveq2d 5691 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( normh `  ( T `  z
) )  =  (
normh `  ( T `  ( ( 1  / 
( normh `  x )
)  .h  x ) ) ) )
5352eqeq2d 2415 . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . 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 3012 . . . . . . . . . . . . . 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 1182 . . . . . . . . . . . . 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 5701 . . . . . . . . . . . . . 14  |-  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) )  e.  _V
58 eqeq1 2410 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) )  ->  ( y  =  ( normh `  ( T `  z ) )  <->  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) )  =  ( normh `  ( T `  z
) ) ) )
5958anbi2d 685 . . . . . . . . . . . . . . 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 2687 . . . . . . . . . . . . . 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 3042 . . . . . . . . . . . . 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 204 . . . . . . . . . . . 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 10859 . . . . . . . . . . . 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 645 . . . . . . . . . . 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 695 . . . . . . . . . 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 23312 . . . . . . . . . . . . . 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 2411 . . . . . . . . . . . 12  |-  ( (
normop `  T )  =  0  <->  sup ( { y  |  E. z  e. 
~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) } ,  RR* ,  <  )  =  0 )
6968biimpi 187 . . . . . . . . . . 11  |-  ( (
normop `  T )  =  0  ->  sup ( { y  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) } ,  RR* ,  <  )  =  0 )
7069ad2antrr 707 . . . . . . . . . 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 4196 . . . . . . . . 9  |-  ( ( ( ( normop `  T
)  =  0  /\  x  e.  ~H )  /\  ( T `  x
)  =/=  0h )  ->  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  <_ 
0 )
72 normcl 22580 . . . . . . . . . . . 12  |-  ( ( ( 1  /  ( normh `  x ) )  .h  ( T `  x ) )  e. 
~H  ->  ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) )  e.  RR )
7325, 72syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  e.  RR )
74 0re 9047 . . . . . . . . . . 11  |-  0  e.  RR
75 lenlt 9110 . . . . . . . . . . 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 644 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) )  <_  0  <->  -.  0  <  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) ) ) )
7776adantll 695 . . . . . . . . 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 202 . . . . . . . 8  |-  ( ( ( ( normop `  T
)  =  0  /\  x  e.  ~H )  /\  ( T `  x
)  =/=  0h )  ->  -.  0  <  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
) ) )
7978ex 424 . . . . . . 7  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  (
( T `  x
)  =/=  0h  ->  -.  0  <  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) ) ) )
8030, 79pm2.65d 168 . . . . . 6  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  -.  ( T `  x )  =/=  0h )
81 nne 2571 . . . . . 6  |-  ( -.  ( T `  x
)  =/=  0h  <->  ( T `  x )  =  0h )
8280, 81sylib 189 . . . . 5  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  ( T `  x )  =  0h )
83 ho0val 23206 . . . . . 6  |-  ( x  e.  ~H  ->  ( 0hop `  x )  =  0h )
8483adantl 453 . . . . 5  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  ( 0hop `  x )  =  0h )
8582, 84eqtr4d 2439 . . . 4  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  ( T `  x )  =  ( 0hop `  x
) )
8685ralrimiva 2749 . . 3  |-  ( (
normop `  T )  =  0  ->  A. x  e.  ~H  ( T `  x )  =  (
0hop `  x )
)
87 ffn 5550 . . . . 5  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
8815, 87ax-mp 8 . . . 4  |-  T  Fn  ~H
89 ho0f 23207 . . . . 5  |-  0hop : ~H --> ~H
90 ffn 5550 . . . . 5  |-  ( 0hop
: ~H --> ~H  ->  0hop 
Fn  ~H )
9189, 90ax-mp 8 . . . 4  |-  0hop  Fn  ~H
92 eqfnfv 5786 . . . 4  |-  ( ( T  Fn  ~H  /\  0hop 
Fn  ~H )  ->  ( T  =  0hop  <->  A. x  e.  ~H  ( T `  x )  =  (
0hop `  x )
) )
9388, 91, 92mp2an 654 . . 3  |-  ( T  =  0hop  <->  A. x  e.  ~H  ( T `  x )  =  ( 0hop `  x
) )
9486, 93sylibr 204 . 2  |-  ( (
normop `  T )  =  0  ->  T  =  0hop )
95 fveq2 5687 . . 3  |-  ( T  =  0hop  ->  ( normop `  T )  =  (
normop `  0hop ) )
96 nmop0 23442 . . 3  |-  ( normop ` 
0hop )  =  0
9795, 96syl6eq 2452 . 2  |-  ( T  =  0hop  ->  ( normop `  T )  =  0 )
9894, 97impbii 181 1  |-  ( (
normop `  T )  =  0  <->  T  =  0hop )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   A.wral 2666   E.wrex 2667    C_ wss 3280   class class class wbr 4172    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   supcsup 7403   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   RR*cxr 9075    < clt 9076    <_ cle 9077    / cdiv 9633   ~Hchil 22375    .h csm 22377   normhcno 22379   0hc0v 22380   0hopch0o 22399   normopcnop 22401   LinOpclo 22403
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cc 8271  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026  ax-hilex 22455  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvmulass 22463  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540  ax-hcompl 22657
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-cn 17245  df-cnp 17246  df-lm 17247  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cfil 19161  df-cau 19162  df-cmet 19163  df-grpo 21732  df-gid 21733  df-ginv 21734  df-gdiv 21735  df-ablo 21823  df-subgo 21843  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-vs 22031  df-nmcv 22032  df-ims 22033  df-dip 22150  df-ssp 22174  df-ph 22267  df-cbn 22318  df-hnorm 22424  df-hba 22425  df-hvsub 22427  df-hlim 22428  df-hcau 22429  df-sh 22662  df-ch 22677  df-oc 22707  df-ch0 22708  df-shs 22763  df-pjh 22850  df-h0op 23204  df-nmop 23295  df-lnop 23297
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