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Theorem nmlnop0iALT 22575
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 21704 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  RR )
21recnd 8861 . . . . . . . . . . . . 13  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  CC )
32adantr 451 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  x )  e.  CC )
4 norm-i 21708 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~H  ->  (
( normh `  x )  =  0  <->  x  =  0h ) )
5 fveq2 5525 . . . . . . . . . . . . . . . 16  |-  ( x  =  0h  ->  ( T `  x )  =  ( T `  0h ) )
6 nmlnop0.1 . . . . . . . . . . . . . . . . 17  |-  T  e. 
LinOp
76lnop0i 22550 . . . . . . . . . . . . . . . 16  |-  ( T `
 0h )  =  0h
85, 7syl6eq 2331 . . . . . . . . . . . . . . 15  |-  ( x  =  0h  ->  ( T `  x )  =  0h )
94, 8syl6bi 219 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  (
( normh `  x )  =  0  ->  ( T `  x )  =  0h ) )
109necon3d 2484 . . . . . . . . . . . . 13  |-  ( x  e.  ~H  ->  (
( T `  x
)  =/=  0h  ->  (
normh `  x )  =/=  0 ) )
1110imp 418 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  x )  =/=  0 )
123, 11recne0d 9530 . . . . . . . . . . 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 9529 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( 1  /  ( normh `  x ) )  e.  CC )
156lnopfi 22549 . . . . . . . . . . . . . . . 16  |-  T : ~H
--> ~H
1615ffvelrni 5664 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
1716adantr 451 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( T `  x
)  e.  ~H )
18 hvmul0or 21604 . . . . . . . . . . . . . 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 2479 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
)  =/=  0h  <->  -.  (
( 1  /  ( normh `  x ) )  =  0  \/  ( T `  x )  =  0h ) ) )
21 neanior 2531 . . . . . . . . . . . 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 21593 . . . . . . . . . . . 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 21706 . . . . . . . . . . 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 22444 . . . . . . . . . . . . . 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 8876 . . . . . . . . . . . . 13  |-  RR  C_  RR*
3432, 33sstri 3188 . . . . . . . . . . . 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 21593 . . . . . . . . . . . . . . 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 2485 . . . . . . . . . . . . . . . . 17  |-  ( ( T `  x )  =/=  0h  ->  x  =/=  0h )
39 norm1 21828 . . . . . . . . . . . . . . . . 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 8837 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
4240, 41syl6eqel 2371 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  e.  RR )
43 eqle 8923 . . . . . . . . . . . . . . 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 22552 . . . . . . . . . . . . . . . . 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 2288 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) )  =  ( T `  ( ( 1  / 
( normh `  x )
)  .h  x ) ) )
4847fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  =  ( normh `  ( T `  ( ( 1  / 
( normh `  x )
)  .h  x ) ) ) )
49 fveq2 5525 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( normh `  z )  =  (
normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) ) )
5049breq1d 4033 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( ( normh `  z )  <_ 
1  <->  ( normh `  (
( 1  /  ( normh `  x ) )  .h  x ) )  <_  1 ) )
51 fveq2 5525 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( T `  z )  =  ( T `  ( ( 1  /  ( normh `  x ) )  .h  x ) ) )
5251fveq2d 5529 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( normh `  ( T `  z
) )  =  (
normh `  ( T `  ( ( 1  / 
( normh `  x )
)  .h  x ) ) ) )
5352eqeq2d 2294 . . . . . . . . . . . . . . . 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 2884 . . . . . . . . . . . . . 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 1180 . . . . . . . . . . . . 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 5539 . . . . . . . . . . . . . 14  |-  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) )  e.  _V
58 eqeq1 2289 . . . . . . . . . . . . . . . 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 2564 . . . . . . . . . . . . . 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 2914 . . . . . . . . . . . . 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 10643 . . . . . . . . . . . 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 22436 . . . . . . . . . . . . . 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 2290 . . . . . . . . . . . 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 4047 . . . . . . . . 9  |-  ( ( ( ( normop `  T
)  =  0  /\  x  e.  ~H )  /\  ( T `  x
)  =/=  0h )  ->  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  <_ 
0 )
72 normcl 21704 . . . . . . . . . . . 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 8838 . . . . . . . . . . 11  |-  0  e.  RR
75 lenlt 8901 . . . . . . . . . . 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 2450 . . . . . 6  |-  ( -.  ( T `  x
)  =/=  0h  <->  ( T `  x )  =  0h )
8280, 81sylib 188 . . . . 5  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  ( T `  x )  =  0h )
83 ho0val 22330 . . . . . 6  |-  ( x  e.  ~H  ->  ( 0hop `  x )  =  0h )
8483adantl 452 . . . . 5  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  ( 0hop `  x )  =  0h )
8582, 84eqtr4d 2318 . . . 4  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  ( T `  x )  =  ( 0hop `  x
) )
8685ralrimiva 2626 . . 3  |-  ( (
normop `  T )  =  0  ->  A. x  e.  ~H  ( T `  x )  =  (
0hop `  x )
)
87 ffn 5389 . . . . 5  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
8815, 87ax-mp 8 . . . 4  |-  T  Fn  ~H
89 ho0f 22331 . . . . 5  |-  0hop : ~H --> ~H
90 ffn 5389 . . . . 5  |-  ( 0hop
: ~H --> ~H  ->  0hop 
Fn  ~H )
9189, 90ax-mp 8 . . . 4  |-  0hop  Fn  ~H
92 eqfnfv 5622 . . . 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 5525 . . 3  |-  ( T  =  0hop  ->  ( normop `  T )  =  (
normop `  0hop ) )
96 nmop0 22566 . . 3  |-  ( normop ` 
0hop )  =  0
9795, 96syl6eq 2331 . 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 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   RR*cxr 8866    < clt 8867    <_ cle 8868    / cdiv 9423   ~Hchil 21499    .h csm 21501   normhcno 21503   0hc0v 21504   0hopch0o 21523   normopcnop 21525   LinOpclo 21527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cc 8061  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664  ax-hcompl 21781
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-cn 16957  df-cnp 16958  df-lm 16959  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cfil 18681  df-cau 18682  df-cmet 18683  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-subgo 20969  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-dip 21274  df-ssp 21298  df-ph 21391  df-cbn 21442  df-hnorm 21548  df-hba 21549  df-hvsub 21551  df-hlim 21552  df-hcau 21553  df-sh 21786  df-ch 21801  df-oc 21831  df-ch0 21832  df-shs 21887  df-pjh 21974  df-h0op 22328  df-nmop 22419  df-lnop 22421
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