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Theorem lnopeq0i 22587
Description: A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 22408 for arbitrary operators, when the operator is linear we need to consider only the values of the quadratic form  ( T `  x )  .ih  x
). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
lnopeq0.1  |-  T  e. 
LinOp
Assertion
Ref Expression
lnopeq0i  |-  ( A. x  e.  ~H  (
( T `  x
)  .ih  x )  =  0  <->  T  =  0hop )
Distinct variable group:    x, T

Proof of Theorem lnopeq0i
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnopeq0.1 . . . . . . 7  |-  T  e. 
LinOp
21lnopeq0lem2 22586 . . . . . 6  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( ( T `  y )  .ih  z
)  =  ( ( ( ( ( T `
 ( y  +h  z ) )  .ih  ( y  +h  z
) )  -  (
( T `  (
y  -h  z ) )  .ih  ( y  -h  z ) ) )  +  ( _i  x.  ( ( ( T `  ( y  +h  ( _i  .h  z ) ) ) 
.ih  ( y  +h  ( _i  .h  z
) ) )  -  ( ( T `  ( y  -h  (
_i  .h  z )
) )  .ih  (
y  -h  ( _i  .h  z ) ) ) ) ) )  /  4 ) )
32adantl 452 . . . . 5  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( ( T `  y )  .ih  z )  =  ( ( ( ( ( T `  ( y  +h  z ) ) 
.ih  ( y  +h  z ) )  -  ( ( T `  ( y  -h  z
) )  .ih  (
y  -h  z ) ) )  +  ( _i  x.  ( ( ( T `  (
y  +h  ( _i  .h  z ) ) )  .ih  ( y  +h  ( _i  .h  z ) ) )  -  ( ( T `
 ( y  -h  ( _i  .h  z
) ) )  .ih  ( y  -h  (
_i  .h  z )
) ) ) ) )  /  4 ) )
4 hvaddcl 21592 . . . . . . . . . . . 12  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( y  +h  z
)  e.  ~H )
5 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( x  =  ( y  +h  z )  ->  ( T `  x )  =  ( T `  ( y  +h  z
) ) )
6 id 19 . . . . . . . . . . . . . . 15  |-  ( x  =  ( y  +h  z )  ->  x  =  ( y  +h  z ) )
75, 6oveq12d 5876 . . . . . . . . . . . . . 14  |-  ( x  =  ( y  +h  z )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  +h  z ) )  .ih  ( y  +h  z
) ) )
87eqeq1d 2291 . . . . . . . . . . . . 13  |-  ( x  =  ( y  +h  z )  ->  (
( ( T `  x )  .ih  x
)  =  0  <->  (
( T `  (
y  +h  z ) )  .ih  ( y  +h  z ) )  =  0 ) )
98rspccva 2883 . . . . . . . . . . . 12  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  +h  z
)  e.  ~H )  ->  ( ( T `  ( y  +h  z
) )  .ih  (
y  +h  z ) )  =  0 )
104, 9sylan2 460 . . . . . . . . . . 11  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( ( T `  ( y  +h  z ) )  .ih  ( y  +h  z
) )  =  0 )
11 hvsubcl 21597 . . . . . . . . . . . 12  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( y  -h  z
)  e.  ~H )
12 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( x  =  ( y  -h  z )  ->  ( T `  x )  =  ( T `  ( y  -h  z
) ) )
13 id 19 . . . . . . . . . . . . . . 15  |-  ( x  =  ( y  -h  z )  ->  x  =  ( y  -h  z ) )
1412, 13oveq12d 5876 . . . . . . . . . . . . . 14  |-  ( x  =  ( y  -h  z )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  -h  z ) )  .ih  ( y  -h  z
) ) )
1514eqeq1d 2291 . . . . . . . . . . . . 13  |-  ( x  =  ( y  -h  z )  ->  (
( ( T `  x )  .ih  x
)  =  0  <->  (
( T `  (
y  -h  z ) )  .ih  ( y  -h  z ) )  =  0 ) )
1615rspccva 2883 . . . . . . . . . . . 12  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  -h  z
)  e.  ~H )  ->  ( ( T `  ( y  -h  z
) )  .ih  (
y  -h  z ) )  =  0 )
1711, 16sylan2 460 . . . . . . . . . . 11  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( ( T `  ( y  -h  z ) )  .ih  ( y  -h  z
) )  =  0 )
1810, 17oveq12d 5876 . . . . . . . . . 10  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( (
( T `  (
y  +h  z ) )  .ih  ( y  +h  z ) )  -  ( ( T `
 ( y  -h  z ) )  .ih  ( y  -h  z
) ) )  =  ( 0  -  0 ) )
19 0cn 8831 . . . . . . . . . . 11  |-  0  e.  CC
2019subidi 9117 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
2118, 20syl6eq 2331 . . . . . . . . 9  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( (
( T `  (
y  +h  z ) )  .ih  ( y  +h  z ) )  -  ( ( T `
 ( y  -h  z ) )  .ih  ( y  -h  z
) ) )  =  0 )
22 ax-icn 8796 . . . . . . . . . . . . . . . 16  |-  _i  e.  CC
23 hvmulcl 21593 . . . . . . . . . . . . . . . 16  |-  ( ( _i  e.  CC  /\  z  e.  ~H )  ->  ( _i  .h  z
)  e.  ~H )
2422, 23mpan 651 . . . . . . . . . . . . . . 15  |-  ( z  e.  ~H  ->  (
_i  .h  z )  e.  ~H )
25 hvaddcl 21592 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  ~H  /\  ( _i  .h  z
)  e.  ~H )  ->  ( y  +h  (
_i  .h  z )
)  e.  ~H )
2624, 25sylan2 460 . . . . . . . . . . . . . 14  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( y  +h  (
_i  .h  z )
)  e.  ~H )
27 fveq2 5525 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  +h  ( _i  .h  z
) )  ->  ( T `  x )  =  ( T `  ( y  +h  (
_i  .h  z )
) ) )
28 id 19 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  +h  ( _i  .h  z
) )  ->  x  =  ( y  +h  ( _i  .h  z
) ) )
2927, 28oveq12d 5876 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( y  +h  ( _i  .h  z
) )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  +h  ( _i  .h  z
) ) )  .ih  ( y  +h  (
_i  .h  z )
) ) )
3029eqeq1d 2291 . . . . . . . . . . . . . . 15  |-  ( x  =  ( y  +h  ( _i  .h  z
) )  ->  (
( ( T `  x )  .ih  x
)  =  0  <->  (
( T `  (
y  +h  ( _i  .h  z ) ) )  .ih  ( y  +h  ( _i  .h  z ) ) )  =  0 ) )
3130rspccva 2883 . . . . . . . . . . . . . 14  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  +h  (
_i  .h  z )
)  e.  ~H )  ->  ( ( T `  ( y  +h  (
_i  .h  z )
) )  .ih  (
y  +h  ( _i  .h  z ) ) )  =  0 )
3226, 31sylan2 460 . . . . . . . . . . . . 13  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( ( T `  ( y  +h  ( _i  .h  z
) ) )  .ih  ( y  +h  (
_i  .h  z )
) )  =  0 )
33 hvsubcl 21597 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  ~H  /\  ( _i  .h  z
)  e.  ~H )  ->  ( y  -h  (
_i  .h  z )
)  e.  ~H )
3424, 33sylan2 460 . . . . . . . . . . . . . 14  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( y  -h  (
_i  .h  z )
)  e.  ~H )
35 fveq2 5525 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  -h  ( _i  .h  z
) )  ->  ( T `  x )  =  ( T `  ( y  -h  (
_i  .h  z )
) ) )
36 id 19 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  -h  ( _i  .h  z
) )  ->  x  =  ( y  -h  ( _i  .h  z
) ) )
3735, 36oveq12d 5876 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( y  -h  ( _i  .h  z
) )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  -h  ( _i  .h  z
) ) )  .ih  ( y  -h  (
_i  .h  z )
) ) )
3837eqeq1d 2291 . . . . . . . . . . . . . . 15  |-  ( x  =  ( y  -h  ( _i  .h  z
) )  ->  (
( ( T `  x )  .ih  x
)  =  0  <->  (
( T `  (
y  -h  ( _i  .h  z ) ) )  .ih  ( y  -h  ( _i  .h  z ) ) )  =  0 ) )
3938rspccva 2883 . . . . . . . . . . . . . 14  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  -h  (
_i  .h  z )
)  e.  ~H )  ->  ( ( T `  ( y  -h  (
_i  .h  z )
) )  .ih  (
y  -h  ( _i  .h  z ) ) )  =  0 )
4034, 39sylan2 460 . . . . . . . . . . . . 13  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( ( T `  ( y  -h  ( _i  .h  z
) ) )  .ih  ( y  -h  (
_i  .h  z )
) )  =  0 )
4132, 40oveq12d 5876 . . . . . . . . . . . 12  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( (
( T `  (
y  +h  ( _i  .h  z ) ) )  .ih  ( y  +h  ( _i  .h  z ) ) )  -  ( ( T `
 ( y  -h  ( _i  .h  z
) ) )  .ih  ( y  -h  (
_i  .h  z )
) ) )  =  ( 0  -  0 ) )
4241, 20syl6eq 2331 . . . . . . . . . . 11  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( (
( T `  (
y  +h  ( _i  .h  z ) ) )  .ih  ( y  +h  ( _i  .h  z ) ) )  -  ( ( T `
 ( y  -h  ( _i  .h  z
) ) )  .ih  ( y  -h  (
_i  .h  z )
) ) )  =  0 )
4342oveq2d 5874 . . . . . . . . . 10  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( _i  x.  ( ( ( T `
 ( y  +h  ( _i  .h  z
) ) )  .ih  ( y  +h  (
_i  .h  z )
) )  -  (
( T `  (
y  -h  ( _i  .h  z ) ) )  .ih  ( y  -h  ( _i  .h  z ) ) ) ) )  =  ( _i  x.  0 ) )
4422mul01i 9002 . . . . . . . . . 10  |-  ( _i  x.  0 )  =  0
4543, 44syl6eq 2331 . . . . . . . . 9  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( _i  x.  ( ( ( T `
 ( y  +h  ( _i  .h  z
) ) )  .ih  ( y  +h  (
_i  .h  z )
) )  -  (
( T `  (
y  -h  ( _i  .h  z ) ) )  .ih  ( y  -h  ( _i  .h  z ) ) ) ) )  =  0 )
4621, 45oveq12d 5876 . . . . . . . 8  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( (
( ( T `  ( y  +h  z
) )  .ih  (
y  +h  z ) )  -  ( ( T `  ( y  -h  z ) ) 
.ih  ( y  -h  z ) ) )  +  ( _i  x.  ( ( ( T `
 ( y  +h  ( _i  .h  z
) ) )  .ih  ( y  +h  (
_i  .h  z )
) )  -  (
( T `  (
y  -h  ( _i  .h  z ) ) )  .ih  ( y  -h  ( _i  .h  z ) ) ) ) ) )  =  ( 0  +  0 ) )
47 00id 8987 . . . . . . . 8  |-  ( 0  +  0 )  =  0
4846, 47syl6eq 2331 . . . . . . 7  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( (
( ( T `  ( y  +h  z
) )  .ih  (
y  +h  z ) )  -  ( ( T `  ( y  -h  z ) ) 
.ih  ( y  -h  z ) ) )  +  ( _i  x.  ( ( ( T `
 ( y  +h  ( _i  .h  z
) ) )  .ih  ( y  +h  (
_i  .h  z )
) )  -  (
( T `  (
y  -h  ( _i  .h  z ) ) )  .ih  ( y  -h  ( _i  .h  z ) ) ) ) ) )  =  0 )
4948oveq1d 5873 . . . . . 6  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( (
( ( ( T `
 ( y  +h  z ) )  .ih  ( y  +h  z
) )  -  (
( T `  (
y  -h  z ) )  .ih  ( y  -h  z ) ) )  +  ( _i  x.  ( ( ( T `  ( y  +h  ( _i  .h  z ) ) ) 
.ih  ( y  +h  ( _i  .h  z
) ) )  -  ( ( T `  ( y  -h  (
_i  .h  z )
) )  .ih  (
y  -h  ( _i  .h  z ) ) ) ) ) )  /  4 )  =  ( 0  /  4
) )
50 4cn 9820 . . . . . . 7  |-  4  e.  CC
51 4re 9819 . . . . . . . 8  |-  4  e.  RR
52 4pos 9832 . . . . . . . 8  |-  0  <  4
5351, 52gt0ne0ii 9309 . . . . . . 7  |-  4  =/=  0
5450, 53div0i 9494 . . . . . 6  |-  ( 0  /  4 )  =  0
5549, 54syl6eq 2331 . . . . 5  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( (
( ( ( T `
 ( y  +h  z ) )  .ih  ( y  +h  z
) )  -  (
( T `  (
y  -h  z ) )  .ih  ( y  -h  z ) ) )  +  ( _i  x.  ( ( ( T `  ( y  +h  ( _i  .h  z ) ) ) 
.ih  ( y  +h  ( _i  .h  z
) ) )  -  ( ( T `  ( y  -h  (
_i  .h  z )
) )  .ih  (
y  -h  ( _i  .h  z ) ) ) ) ) )  /  4 )  =  0 )
563, 55eqtrd 2315 . . . 4  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( ( T `  y )  .ih  z )  =  0 )
5756ralrimivva 2635 . . 3  |-  ( A. x  e.  ~H  (
( T `  x
)  .ih  x )  =  0  ->  A. y  e.  ~H  A. z  e. 
~H  ( ( T `
 y )  .ih  z )  =  0 )
581lnopfi 22549 . . . 4  |-  T : ~H
--> ~H
5958ho01i 22408 . . 3  |-  ( A. y  e.  ~H  A. z  e.  ~H  ( ( T `
 y )  .ih  z )  =  0  <-> 
T  =  0hop )
6057, 59sylib 188 . 2  |-  ( A. x  e.  ~H  (
( T `  x
)  .ih  x )  =  0  ->  T  =  0hop )
61 fveq1 5524 . . . . . 6  |-  ( T  =  0hop  ->  ( T `
 x )  =  ( 0hop `  x
) )
62 ho0val 22330 . . . . . 6  |-  ( x  e.  ~H  ->  ( 0hop `  x )  =  0h )
6361, 62sylan9eq 2335 . . . . 5  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  ( T `  x )  =  0h )
6463oveq1d 5873 . . . 4  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  =  ( 0h  .ih  x ) )
65 hi01 21675 . . . . 5  |-  ( x  e.  ~H  ->  ( 0h  .ih  x )  =  0 )
6665adantl 452 . . . 4  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  ( 0h  .ih  x )  =  0 )
6764, 66eqtrd 2315 . . 3  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  =  0 )
6867ralrimiva 2626 . 2  |-  ( T  =  0hop  ->  A. x  e.  ~H  ( ( T `
 x )  .ih  x )  =  0 )
6960, 68impbii 180 1  |-  ( A. x  e.  ~H  (
( T `  x
)  .ih  x )  =  0  <->  T  =  0hop )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037    / cdiv 9423   4c4 9797   ~Hchil 21499    +h cva 21500    .h csm 21501    .ih csp 21502   0hc0v 21504    -h cmv 21505   0hopch0o 21523   LinOpclo 21527
This theorem is referenced by:  lnopeqi  22588
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-lnop 22421
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