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Theorem nmlnop0iALT 23503
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 22632 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  RR )
21recnd 9119 . . . . . . . . . . . . 13  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  CC )
32adantr 453 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  x )  e.  CC )
4 norm-i 22636 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~H  ->  (
( normh `  x )  =  0  <->  x  =  0h ) )
5 fveq2 5731 . . . . . . . . . . . . . . . 16  |-  ( x  =  0h  ->  ( T `  x )  =  ( T `  0h ) )
6 nmlnop0.1 . . . . . . . . . . . . . . . . 17  |-  T  e. 
LinOp
76lnop0i 23478 . . . . . . . . . . . . . . . 16  |-  ( T `
 0h )  =  0h
85, 7syl6eq 2486 . . . . . . . . . . . . . . 15  |-  ( x  =  0h  ->  ( T `  x )  =  0h )
94, 8syl6bi 221 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  (
( normh `  x )  =  0  ->  ( T `  x )  =  0h ) )
109necon3d 2641 . . . . . . . . . . . . 13  |-  ( x  e.  ~H  ->  (
( T `  x
)  =/=  0h  ->  (
normh `  x )  =/=  0 ) )
1110imp 420 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  x )  =/=  0 )
123, 11recne0d 9789 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( 1  /  ( normh `  x ) )  =/=  0 )
13 simpr 449 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( T `  x
)  =/=  0h )
143, 11reccld 9788 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( 1  /  ( normh `  x ) )  e.  CC )
156lnopfi 23477 . . . . . . . . . . . . . . . 16  |-  T : ~H
--> ~H
1615ffvelrni 5872 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
1716adantr 453 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( T `  x
)  e.  ~H )
18 hvmul0or 22532 . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
)  =  0h  <->  ( (
1  /  ( normh `  x ) )  =  0  \/  ( T `
 x )  =  0h ) ) )
2019necon3abid 2636 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
)  =/=  0h  <->  -.  (
( 1  /  ( normh `  x ) )  =  0  \/  ( T `  x )  =  0h ) ) )
21 neanior 2691 . . . . . . . . . . . 12  |-  ( ( ( 1  /  ( normh `  x ) )  =/=  0  /\  ( T `  x )  =/=  0h )  <->  -.  (
( 1  /  ( normh `  x ) )  =  0  \/  ( T `  x )  =  0h ) )
2220, 21syl6bbr 256 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
)  =/=  0h  <->  ( (
1  /  ( normh `  x ) )  =/=  0  /\  ( T `
 x )  =/= 
0h ) ) )
2312, 13, 22mpbir2and 890 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) )  =/=  0h )
24 hvmulcl 22521 . . . . . . . . . . . 12  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  ( T `  x )  e.  ~H )  ->  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) )  e. 
~H )
2514, 17, 24syl2anc 644 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) )  e.  ~H )
26 normgt0 22634 . . . . . . . . . . 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 203 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
0  <  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) ) )
2928ex 425 . . . . . . . 8  |-  ( x  e.  ~H  ->  (
( T `  x
)  =/=  0h  ->  0  <  ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) ) ) )
3029adantl 454 . . . . . . 7  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  (
( T `  x
)  =/=  0h  ->  0  <  ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) ) ) )
31 nmopsetretHIL 23372 . . . . . . . . . . . . . 14  |-  ( T : ~H --> ~H  ->  { y  |  E. z  e.  ~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) }  C_  RR )
3215, 31ax-mp 5 . . . . . . . . . . . . 13  |-  { y  |  E. z  e. 
~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) }  C_  RR
33 ressxr 9134 . . . . . . . . . . . . 13  |-  RR  C_  RR*
3432, 33sstri 3359 . . . . . . . . . . . 12  |-  { y  |  E. z  e. 
~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) }  C_  RR*
35 simpl 445 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  ->  x  e.  ~H )
36 hvmulcl 22521 . . . . . . . . . . . . . . 15  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  ~H )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e. 
~H )
3714, 35, 36syl2anc 644 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( 1  / 
( normh `  x )
)  .h  x )  e.  ~H )
388necon3i 2645 . . . . . . . . . . . . . . . . 17  |-  ( ( T `  x )  =/=  0h  ->  x  =/=  0h )
39 norm1 22756 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  ~H  /\  x  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  =  1 )
4038, 39sylan2 462 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  =  1 )
41 1re 9095 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
4240, 41syl6eqel 2526 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  e.  RR )
43 eqle 9181 . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  x ) )  <_ 
1 )
456lnopmuli 23480 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  ~H )  ->  ( T `  ( (
1  /  ( normh `  x ) )  .h  x ) )  =  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) )
4614, 35, 45syl2anc 644 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( T `  (
( 1  /  ( normh `  x ) )  .h  x ) )  =  ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
) )
4746eqcomd 2443 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) )  =  ( T `  ( ( 1  / 
( normh `  x )
)  .h  x ) ) )
4847fveq2d 5735 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  =  ( normh `  ( T `  ( ( 1  / 
( normh `  x )
)  .h  x ) ) ) )
49 fveq2 5731 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( normh `  z )  =  (
normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) ) )
5049breq1d 4225 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( ( normh `  z )  <_ 
1  <->  ( normh `  (
( 1  /  ( normh `  x ) )  .h  x ) )  <_  1 ) )
51 fveq2 5731 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( T `  z )  =  ( T `  ( ( 1  /  ( normh `  x ) )  .h  x ) ) )
5251fveq2d 5735 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( normh `  ( T `  z
) )  =  (
normh `  ( T `  ( ( 1  / 
( normh `  x )
)  .h  x ) ) ) )
5352eqeq2d 2449 . . . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . . . 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 3054 . . . . . . . . . . . . . 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 1183 . . . . . . . . . . . . 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 5745 . . . . . . . . . . . . . 14  |-  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) )  e.  _V
58 eqeq1 2444 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) )  ->  ( y  =  ( normh `  ( T `  z ) )  <->  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) )  =  ( normh `  ( T `  z
) ) ) )
5958anbi2d 686 . . . . . . . . . . . . . . 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 2728 . . . . . . . . . . . . . 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 3084 . . . . . . . . . . . . 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 205 . . . . . . . . . . . 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 10908 . . . . . . . . . . . 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 646 . . . . . . . . . . 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 696 . . . . . . . . . 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 23364 . . . . . . . . . . . . . 14  |-  ( T : ~H --> ~H  ->  (
normop `  T )  =  sup ( { y  |  E. z  e. 
~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) } ,  RR* ,  <  ) )
6715, 66ax-mp 5 . . . . . . . . . . . . 13  |-  ( normop `  T )  =  sup ( { y  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) } ,  RR* ,  <  )
6867eqeq1i 2445 . . . . . . . . . . . 12  |-  ( (
normop `  T )  =  0  <->  sup ( { y  |  E. z  e. 
~H  ( ( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) } ,  RR* ,  <  )  =  0 )
6968biimpi 188 . . . . . . . . . . 11  |-  ( (
normop `  T )  =  0  ->  sup ( { y  |  E. z  e.  ~H  (
( normh `  z )  <_  1  /\  y  =  ( normh `  ( T `  z ) ) ) } ,  RR* ,  <  )  =  0 )
7069ad2antrr 708 . . . . . . . . . 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 4239 . . . . . . . . 9  |-  ( ( ( ( normop `  T
)  =  0  /\  x  e.  ~H )  /\  ( T `  x
)  =/=  0h )  ->  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) )  <_ 
0 )
72 normcl 22632 . . . . . . . . . . . 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 9096 . . . . . . . . . . 11  |-  0  e.  RR
75 lenlt 9159 . . . . . . . . . . 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 645 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  ( T `  x )  =/=  0h )  -> 
( ( normh `  (
( 1  /  ( normh `  x ) )  .h  ( T `  x ) ) )  <_  0  <->  -.  0  <  ( normh `  ( (
1  /  ( normh `  x ) )  .h  ( T `  x
) ) ) ) )
7776adantll 696 . . . . . . . . 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 203 . . . . . . . 8  |-  ( ( ( ( normop `  T
)  =  0  /\  x  e.  ~H )  /\  ( T `  x
)  =/=  0h )  ->  -.  0  <  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  ( T `  x )
) ) )
7978ex 425 . . . . . . 7  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  (
( T `  x
)  =/=  0h  ->  -.  0  <  ( normh `  ( ( 1  / 
( normh `  x )
)  .h  ( T `
 x ) ) ) ) )
8030, 79pm2.65d 169 . . . . . 6  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  -.  ( T `  x )  =/=  0h )
81 nne 2607 . . . . . 6  |-  ( -.  ( T `  x
)  =/=  0h  <->  ( T `  x )  =  0h )
8280, 81sylib 190 . . . . 5  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  ( T `  x )  =  0h )
83 ho0val 23258 . . . . . 6  |-  ( x  e.  ~H  ->  ( 0hop `  x )  =  0h )
8483adantl 454 . . . . 5  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  ( 0hop `  x )  =  0h )
8582, 84eqtr4d 2473 . . . 4  |-  ( ( ( normop `  T )  =  0  /\  x  e.  ~H )  ->  ( T `  x )  =  ( 0hop `  x
) )
8685ralrimiva 2791 . . 3  |-  ( (
normop `  T )  =  0  ->  A. x  e.  ~H  ( T `  x )  =  (
0hop `  x )
)
87 ffn 5594 . . . . 5  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
8815, 87ax-mp 5 . . . 4  |-  T  Fn  ~H
89 ho0f 23259 . . . . 5  |-  0hop : ~H --> ~H
90 ffn 5594 . . . . 5  |-  ( 0hop
: ~H --> ~H  ->  0hop 
Fn  ~H )
9189, 90ax-mp 5 . . . 4  |-  0hop  Fn  ~H
92 eqfnfv 5830 . . . 4  |-  ( ( T  Fn  ~H  /\  0hop 
Fn  ~H )  ->  ( T  =  0hop  <->  A. x  e.  ~H  ( T `  x )  =  (
0hop `  x )
) )
9388, 91, 92mp2an 655 . . 3  |-  ( T  =  0hop  <->  A. x  e.  ~H  ( T `  x )  =  ( 0hop `  x
) )
9486, 93sylibr 205 . 2  |-  ( (
normop `  T )  =  0  ->  T  =  0hop )
95 fveq2 5731 . . 3  |-  ( T  =  0hop  ->  ( normop `  T )  =  (
normop `  0hop ) )
96 nmop0 23494 . . 3  |-  ( normop ` 
0hop )  =  0
9795, 96syl6eq 2486 . 2  |-  ( T  =  0hop  ->  ( normop `  T )  =  0 )
9894, 97impbii 182 1  |-  ( (
normop `  T )  =  0  <->  T  =  0hop )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424    =/= wne 2601   A.wral 2707   E.wrex 2708    C_ wss 3322   class class class wbr 4215    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084   supcsup 7448   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996   RR*cxr 9124    < clt 9125    <_ cle 9126    / cdiv 9682   ~Hchil 22427    .h csm 22429   normhcno 22431   0hc0v 22432   0hopch0o 22451   normopcnop 22453   LinOpclo 22455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cc 8320  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075  ax-hilex 22507  ax-hfvadd 22508  ax-hvcom 22509  ax-hvass 22510  ax-hv0cl 22511  ax-hvaddid 22512  ax-hfvmul 22513  ax-hvmulid 22514  ax-hvmulass 22515  ax-hvdistr1 22516  ax-hvdistr2 22517  ax-hvmul0 22518  ax-hfi 22586  ax-his1 22589  ax-his2 22590  ax-his3 22591  ax-his4 22592  ax-hcompl 22709
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-omul 6732  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-acn 7834  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-cn 17296  df-cnp 17297  df-lm 17298  df-haus 17384  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cfil 19213  df-cau 19214  df-cmet 19215  df-grpo 21784  df-gid 21785  df-ginv 21786  df-gdiv 21787  df-ablo 21875  df-subgo 21895  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-vs 22083  df-nmcv 22084  df-ims 22085  df-dip 22202  df-ssp 22226  df-ph 22319  df-cbn 22370  df-hnorm 22476  df-hba 22477  df-hvsub 22479  df-hlim 22480  df-hcau 22481  df-sh 22714  df-ch 22729  df-oc 22759  df-ch0 22760  df-shs 22815  df-pjh 22902  df-h0op 23256  df-nmop 23347  df-lnop 23349
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