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Theorem lnopeq0i 25556
Description: A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 25377 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 25555 . . . . . 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 24559 . . . . . . . . . . . 12  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( y  +h  z
)  e.  ~H )
5 fveq2 5792 . . . . . . . . . . . . . . 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 6211 . . . . . . . . . . . . . 14  |-  ( x  =  ( y  +h  z )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  +h  z ) )  .ih  ( y  +h  z
) ) )
87eqeq1d 2453 . . . . . . . . . . . . 13  |-  ( x  =  ( y  +h  z )  ->  (
( ( T `  x )  .ih  x
)  =  0  <->  (
( T `  (
y  +h  z ) )  .ih  ( y  +h  z ) )  =  0 ) )
98rspccva 3171 . . . . . . . . . . . 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 24564 . . . . . . . . . . . 12  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( y  -h  z
)  e.  ~H )
12 fveq2 5792 . . . . . . . . . . . . . . 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 6211 . . . . . . . . . . . . . 14  |-  ( x  =  ( y  -h  z )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  -h  z ) )  .ih  ( y  -h  z
) ) )
1514eqeq1d 2453 . . . . . . . . . . . . 13  |-  ( x  =  ( y  -h  z )  ->  (
( ( T `  x )  .ih  x
)  =  0  <->  (
( T `  (
y  -h  z ) )  .ih  ( y  -h  z ) )  =  0 ) )
1615rspccva 3171 . . . . . . . . . . . 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 6211 . . . . . . . . . 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 10535 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
2018, 19syl6eq 2508 . . . . . . . . 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 9445 . . . . . . . . . . . . . . . 16  |-  _i  e.  CC
22 hvmulcl 24560 . . . . . . . . . . . . . . . 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 24559 . . . . . . . . . . . . . . 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 5792 . . . . . . . . . . . . . . . . 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 6211 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( y  +h  ( _i  .h  z
) )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  +h  ( _i  .h  z
) ) )  .ih  ( y  +h  (
_i  .h  z )
) ) )
2928eqeq1d 2453 . . . . . . . . . . . . . . 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 3171 . . . . . . . . . . . . . 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 24564 . . . . . . . . . . . . . . 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 5792 . . . . . . . . . . . . . . . . 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 6211 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( y  -h  ( _i  .h  z
) )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  -h  ( _i  .h  z
) ) )  .ih  ( y  -h  (
_i  .h  z )
) ) )
3736eqeq1d 2453 . . . . . . . . . . . . . . 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 3171 . . . . . . . . . . . . . 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 6211 . . . . . . . . . . . 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 2508 . . . . . . . . . . 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 6209 . . . . . . . . . 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 10651 . . . . . . . . . 10  |-  ( _i  x.  0 )  =  0
4442, 43syl6eq 2508 . . . . . . . . 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 6211 . . . . . . . 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 9648 . . . . . . . 8  |-  ( 0  +  0 )  =  0
4745, 46syl6eq 2508 . . . . . . 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 6208 . . . . . 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 10503 . . . . . . 7  |-  4  e.  CC
50 4ne0 10522 . . . . . . 7  |-  4  =/=  0
5149, 50div0i 10169 . . . . . 6  |-  ( 0  /  4 )  =  0
5248, 51syl6eq 2508 . . . . 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 2492 . . . 4  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( ( T `  y )  .ih  z )  =  0 )
5453ralrimivva 2907 . . 3  |-  ( A. x  e.  ~H  (
( T `  x
)  .ih  x )  =  0  ->  A. y  e.  ~H  A. z  e. 
~H  ( ( T `
 y )  .ih  z )  =  0 )
551lnopfi 25518 . . . 4  |-  T : ~H
--> ~H
5655ho01i 25377 . . 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 5791 . . . . . 6  |-  ( T  =  0hop  ->  ( T `
 x )  =  ( 0hop `  x
) )
59 ho0val 25299 . . . . . 6  |-  ( x  e.  ~H  ->  ( 0hop `  x )  =  0h )
6058, 59sylan9eq 2512 . . . . 5  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  ( T `  x )  =  0h )
6160oveq1d 6208 . . . 4  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  =  ( 0h  .ih  x ) )
62 hi01 24643 . . . . 5  |-  ( x  e.  ~H  ->  ( 0h  .ih  x )  =  0 )
6362adantl 466 . . . 4  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  ( 0h  .ih  x )  =  0 )
6461, 63eqtrd 2492 . . 3  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  =  0 )
6564ralrimiva 2825 . 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 1370    e. wcel 1758   A.wral 2795   ` cfv 5519  (class class class)co 6193   CCcc 9384   0cc0 9386   _ici 9388    + caddc 9389    x. cmul 9391    - cmin 9699    / cdiv 10097   4c4 10477   ~Hchil 24466    +h cva 24467    .h csm 24468    .ih csp 24469   0hc0v 24471    -h cmv 24472   0hopch0o 24490   LinOpclo 24494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cc 8708  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466  ax-hilex 24546  ax-hfvadd 24547  ax-hvcom 24548  ax-hvass 24549  ax-hv0cl 24550  ax-hvaddid 24551  ax-hfvmul 24552  ax-hvmulid 24553  ax-hvmulass 24554  ax-hvdistr1 24555  ax-hvdistr2 24556  ax-hvmul0 24557  ax-hfi 24626  ax-his1 24629  ax-his2 24630  ax-his3 24631  ax-his4 24632  ax-hcompl 24749
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-omul 7028  df-er 7204  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-fi 7765  df-sup 7795  df-oi 7828  df-card 8213  df-acn 8216  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-fl 11752  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-rlim 13078  df-sum 13275  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-rest 14472  df-topn 14473  df-0g 14491  df-gsum 14492  df-topgen 14493  df-pt 14494  df-prds 14497  df-xrs 14551  df-qtop 14556  df-imas 14557  df-xps 14559  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-submnd 15576  df-mulg 15659  df-cntz 15946  df-cmn 16392  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-fbas 17932  df-fg 17933  df-cnfld 17937  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-cld 18748  df-ntr 18749  df-cls 18750  df-nei 18827  df-cn 18956  df-cnp 18957  df-lm 18958  df-haus 19044  df-tx 19260  df-hmeo 19453  df-fil 19544  df-fm 19636  df-flim 19637  df-flf 19638  df-xms 20020  df-ms 20021  df-tms 20022  df-cfil 20891  df-cau 20892  df-cmet 20893  df-grpo 23823  df-gid 23824  df-ginv 23825  df-gdiv 23826  df-ablo 23914  df-subgo 23934  df-vc 24069  df-nv 24115  df-va 24118  df-ba 24119  df-sm 24120  df-0v 24121  df-vs 24122  df-nmcv 24123  df-ims 24124  df-dip 24241  df-ssp 24265  df-ph 24358  df-cbn 24409  df-hnorm 24515  df-hba 24516  df-hvsub 24518  df-hlim 24519  df-hcau 24520  df-sh 24754  df-ch 24769  df-oc 24800  df-ch0 24801  df-shs 24856  df-pjh 24943  df-h0op 25297  df-lnop 25390
This theorem is referenced by:  lnopeqi  25557
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