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Theorem lnopeq0i 25362
Description: A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 25183 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 25361 . . . . . 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 466 . . . . 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 24365 . . . . . . . . . . . 12  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( y  +h  z
)  e.  ~H )
5 fveq2 5686 . . . . . . . . . . . . . . 15  |-  ( x  =  ( y  +h  z )  ->  ( T `  x )  =  ( T `  ( y  +h  z
) ) )
6 id 22 . . . . . . . . . . . . . . 15  |-  ( x  =  ( y  +h  z )  ->  x  =  ( y  +h  z ) )
75, 6oveq12d 6104 . . . . . . . . . . . . . 14  |-  ( x  =  ( y  +h  z )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  +h  z ) )  .ih  ( y  +h  z
) ) )
87eqeq1d 2446 . . . . . . . . . . . . 13  |-  ( x  =  ( y  +h  z )  ->  (
( ( T `  x )  .ih  x
)  =  0  <->  (
( T `  (
y  +h  z ) )  .ih  ( y  +h  z ) )  =  0 ) )
98rspccva 3067 . . . . . . . . . . . 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 474 . . . . . . . . . . 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 24370 . . . . . . . . . . . 12  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( y  -h  z
)  e.  ~H )
12 fveq2 5686 . . . . . . . . . . . . . . 15  |-  ( x  =  ( y  -h  z )  ->  ( T `  x )  =  ( T `  ( y  -h  z
) ) )
13 id 22 . . . . . . . . . . . . . . 15  |-  ( x  =  ( y  -h  z )  ->  x  =  ( y  -h  z ) )
1412, 13oveq12d 6104 . . . . . . . . . . . . . 14  |-  ( x  =  ( y  -h  z )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  -h  z ) )  .ih  ( y  -h  z
) ) )
1514eqeq1d 2446 . . . . . . . . . . . . 13  |-  ( x  =  ( y  -h  z )  ->  (
( ( T `  x )  .ih  x
)  =  0  <->  (
( T `  (
y  -h  z ) )  .ih  ( y  -h  z ) )  =  0 ) )
1615rspccva 3067 . . . . . . . . . . . 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 474 . . . . . . . . . . 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 6104 . . . . . . . . . 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 0m0e0 10423 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
2018, 19syl6eq 2486 . . . . . . . . 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 )
21 ax-icn 9333 . . . . . . . . . . . . . . . 16  |-  _i  e.  CC
22 hvmulcl 24366 . . . . . . . . . . . . . . . 16  |-  ( ( _i  e.  CC  /\  z  e.  ~H )  ->  ( _i  .h  z
)  e.  ~H )
2321, 22mpan 670 . . . . . . . . . . . . . . 15  |-  ( z  e.  ~H  ->  (
_i  .h  z )  e.  ~H )
24 hvaddcl 24365 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  ~H  /\  ( _i  .h  z
)  e.  ~H )  ->  ( y  +h  (
_i  .h  z )
)  e.  ~H )
2523, 24sylan2 474 . . . . . . . . . . . . . 14  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( y  +h  (
_i  .h  z )
)  e.  ~H )
26 fveq2 5686 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  +h  ( _i  .h  z
) )  ->  ( T `  x )  =  ( T `  ( y  +h  (
_i  .h  z )
) ) )
27 id 22 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  +h  ( _i  .h  z
) )  ->  x  =  ( y  +h  ( _i  .h  z
) ) )
2826, 27oveq12d 6104 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( y  +h  ( _i  .h  z
) )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  +h  ( _i  .h  z
) ) )  .ih  ( y  +h  (
_i  .h  z )
) ) )
2928eqeq1d 2446 . . . . . . . . . . . . . . 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 ) )
3029rspccva 3067 . . . . . . . . . . . . . 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 )
3125, 30sylan2 474 . . . . . . . . . . . . 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 )
32 hvsubcl 24370 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  ~H  /\  ( _i  .h  z
)  e.  ~H )  ->  ( y  -h  (
_i  .h  z )
)  e.  ~H )
3323, 32sylan2 474 . . . . . . . . . . . . . 14  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( y  -h  (
_i  .h  z )
)  e.  ~H )
34 fveq2 5686 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  -h  ( _i  .h  z
) )  ->  ( T `  x )  =  ( T `  ( y  -h  (
_i  .h  z )
) ) )
35 id 22 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  -h  ( _i  .h  z
) )  ->  x  =  ( y  -h  ( _i  .h  z
) ) )
3634, 35oveq12d 6104 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( y  -h  ( _i  .h  z
) )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  -h  ( _i  .h  z
) ) )  .ih  ( y  -h  (
_i  .h  z )
) ) )
3736eqeq1d 2446 . . . . . . . . . . . . . . 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 ) )
3837rspccva 3067 . . . . . . . . . . . . . 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 )
3933, 38sylan2 474 . . . . . . . . . . . . 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 )
4031, 39oveq12d 6104 . . . . . . . . . . . 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 ) )
4140, 19syl6eq 2486 . . . . . . . . . . 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 )
4241oveq2d 6102 . . . . . . . . . 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 ) )
43 it0e0 10539 . . . . . . . . . 10  |-  ( _i  x.  0 )  =  0
4442, 43syl6eq 2486 . . . . . . . . 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 )
4520, 44oveq12d 6104 . . . . . . . 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 ) )
46 00id 9536 . . . . . . . 8  |-  ( 0  +  0 )  =  0
4745, 46syl6eq 2486 . . . . . . 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 )
4847oveq1d 6101 . . . . . 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
) )
49 4cn 10391 . . . . . . 7  |-  4  e.  CC
50 4ne0 10410 . . . . . . 7  |-  4  =/=  0
5149, 50div0i 10057 . . . . . 6  |-  ( 0  /  4 )  =  0
5248, 51syl6eq 2486 . . . . 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 )
533, 52eqtrd 2470 . . . 4  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( ( T `  y )  .ih  z )  =  0 )
5453ralrimivva 2803 . . 3  |-  ( A. x  e.  ~H  (
( T `  x
)  .ih  x )  =  0  ->  A. y  e.  ~H  A. z  e. 
~H  ( ( T `
 y )  .ih  z )  =  0 )
551lnopfi 25324 . . . 4  |-  T : ~H
--> ~H
5655ho01i 25183 . . 3  |-  ( A. y  e.  ~H  A. z  e.  ~H  ( ( T `
 y )  .ih  z )  =  0  <-> 
T  =  0hop )
5754, 56sylib 196 . 2  |-  ( A. x  e.  ~H  (
( T `  x
)  .ih  x )  =  0  ->  T  =  0hop )
58 fveq1 5685 . . . . . 6  |-  ( T  =  0hop  ->  ( T `
 x )  =  ( 0hop `  x
) )
59 ho0val 25105 . . . . . 6  |-  ( x  e.  ~H  ->  ( 0hop `  x )  =  0h )
6058, 59sylan9eq 2490 . . . . 5  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  ( T `  x )  =  0h )
6160oveq1d 6101 . . . 4  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  =  ( 0h  .ih  x ) )
62 hi01 24449 . . . . 5  |-  ( x  e.  ~H  ->  ( 0h  .ih  x )  =  0 )
6362adantl 466 . . . 4  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  ( 0h  .ih  x )  =  0 )
6461, 63eqtrd 2470 . . 3  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  =  0 )
6564ralrimiva 2794 . 2  |-  ( T  =  0hop  ->  A. x  e.  ~H  ( ( T `
 x )  .ih  x )  =  0 )
6657, 65impbii 188 1  |-  ( A. x  e.  ~H  (
( T `  x
)  .ih  x )  =  0  <->  T  =  0hop )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274   _ici 9276    + caddc 9277    x. cmul 9279    - cmin 9587    / cdiv 9985   4c4 10365   ~Hchil 24272    +h cva 24273    .h csm 24274    .ih csp 24275   0hc0v 24277    -h cmv 24278   0hopch0o 24296   LinOpclo 24300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cc 8596  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354  ax-hilex 24352  ax-hfvadd 24353  ax-hvcom 24354  ax-hvass 24355  ax-hv0cl 24356  ax-hvaddid 24357  ax-hfvmul 24358  ax-hvmulid 24359  ax-hvmulass 24360  ax-hvdistr1 24361  ax-hvdistr2 24362  ax-hvmul0 24363  ax-hfi 24432  ax-his1 24435  ax-his2 24436  ax-his3 24437  ax-his4 24438  ax-hcompl 24555
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-omul 6917  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-acn 8104  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-cn 18806  df-cnp 18807  df-lm 18808  df-haus 18894  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cfil 20741  df-cau 20742  df-cmet 20743  df-grpo 23629  df-gid 23630  df-ginv 23631  df-gdiv 23632  df-ablo 23720  df-subgo 23740  df-vc 23875  df-nv 23921  df-va 23924  df-ba 23925  df-sm 23926  df-0v 23927  df-vs 23928  df-nmcv 23929  df-ims 23930  df-dip 24047  df-ssp 24071  df-ph 24164  df-cbn 24215  df-hnorm 24321  df-hba 24322  df-hvsub 24324  df-hlim 24325  df-hcau 24326  df-sh 24560  df-ch 24575  df-oc 24606  df-ch0 24607  df-shs 24662  df-pjh 24749  df-h0op 25103  df-lnop 25196
This theorem is referenced by:  lnopeqi  25363
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