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Theorem lnopeq0i 26602
Description: A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 26423 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 26601 . . . . . 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 25605 . . . . . . . . . . . 12  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( y  +h  z
)  e.  ~H )
5 fveq2 5864 . . . . . . . . . . . . . . 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 6300 . . . . . . . . . . . . . 14  |-  ( x  =  ( y  +h  z )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  +h  z ) )  .ih  ( y  +h  z
) ) )
87eqeq1d 2469 . . . . . . . . . . . . 13  |-  ( x  =  ( y  +h  z )  ->  (
( ( T `  x )  .ih  x
)  =  0  <->  (
( T `  (
y  +h  z ) )  .ih  ( y  +h  z ) )  =  0 ) )
98rspccva 3213 . . . . . . . . . . . 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 25610 . . . . . . . . . . . 12  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( y  -h  z
)  e.  ~H )
12 fveq2 5864 . . . . . . . . . . . . . . 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 6300 . . . . . . . . . . . . . 14  |-  ( x  =  ( y  -h  z )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  -h  z ) )  .ih  ( y  -h  z
) ) )
1514eqeq1d 2469 . . . . . . . . . . . . 13  |-  ( x  =  ( y  -h  z )  ->  (
( ( T `  x )  .ih  x
)  =  0  <->  (
( T `  (
y  -h  z ) )  .ih  ( y  -h  z ) )  =  0 ) )
1615rspccva 3213 . . . . . . . . . . . 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 6300 . . . . . . . . . 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 10641 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
2018, 19syl6eq 2524 . . . . . . . . 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 9547 . . . . . . . . . . . . . . . 16  |-  _i  e.  CC
22 hvmulcl 25606 . . . . . . . . . . . . . . . 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 25605 . . . . . . . . . . . . . . 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 5864 . . . . . . . . . . . . . . . . 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 6300 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( y  +h  ( _i  .h  z
) )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  +h  ( _i  .h  z
) ) )  .ih  ( y  +h  (
_i  .h  z )
) ) )
2928eqeq1d 2469 . . . . . . . . . . . . . . 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 3213 . . . . . . . . . . . . . 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 25610 . . . . . . . . . . . . . . 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 5864 . . . . . . . . . . . . . . . . 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 6300 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( y  -h  ( _i  .h  z
) )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  -h  ( _i  .h  z
) ) )  .ih  ( y  -h  (
_i  .h  z )
) ) )
3736eqeq1d 2469 . . . . . . . . . . . . . . 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 3213 . . . . . . . . . . . . . 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 6300 . . . . . . . . . . . 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 2524 . . . . . . . . . . 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 6298 . . . . . . . . . 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 10757 . . . . . . . . . 10  |-  ( _i  x.  0 )  =  0
4442, 43syl6eq 2524 . . . . . . . . 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 6300 . . . . . . . 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 9750 . . . . . . . 8  |-  ( 0  +  0 )  =  0
4745, 46syl6eq 2524 . . . . . . 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 6297 . . . . . 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 10609 . . . . . . 7  |-  4  e.  CC
50 4ne0 10628 . . . . . . 7  |-  4  =/=  0
5149, 50div0i 10274 . . . . . 6  |-  ( 0  /  4 )  =  0
5248, 51syl6eq 2524 . . . . 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 2508 . . . 4  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( ( T `  y )  .ih  z )  =  0 )
5453ralrimivva 2885 . . 3  |-  ( A. x  e.  ~H  (
( T `  x
)  .ih  x )  =  0  ->  A. y  e.  ~H  A. z  e. 
~H  ( ( T `
 y )  .ih  z )  =  0 )
551lnopfi 26564 . . . 4  |-  T : ~H
--> ~H
5655ho01i 26423 . . 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 5863 . . . . . 6  |-  ( T  =  0hop  ->  ( T `
 x )  =  ( 0hop `  x
) )
59 ho0val 26345 . . . . . 6  |-  ( x  e.  ~H  ->  ( 0hop `  x )  =  0h )
6058, 59sylan9eq 2528 . . . . 5  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  ( T `  x )  =  0h )
6160oveq1d 6297 . . . 4  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  =  ( 0h  .ih  x ) )
62 hi01 25689 . . . . 5  |-  ( x  e.  ~H  ->  ( 0h  .ih  x )  =  0 )
6362adantl 466 . . . 4  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  ( 0h  .ih  x )  =  0 )
6461, 63eqtrd 2508 . . 3  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  =  0 )
6564ralrimiva 2878 . 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 1379    e. wcel 1767   A.wral 2814   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   _ici 9490    + caddc 9491    x. cmul 9493    - cmin 9801    / cdiv 10202   4c4 10583   ~Hchil 25512    +h cva 25513    .h csm 25514    .ih csp 25515   0hc0v 25517    -h cmv 25518   0hopch0o 25536   LinOpclo 25540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cc 8811  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568  ax-hilex 25592  ax-hfvadd 25593  ax-hvcom 25594  ax-hvass 25595  ax-hv0cl 25596  ax-hvaddid 25597  ax-hfvmul 25598  ax-hvmulid 25599  ax-hvmulass 25600  ax-hvdistr1 25601  ax-hvdistr2 25602  ax-hvmul0 25603  ax-hfi 25672  ax-his1 25675  ax-his2 25676  ax-his3 25677  ax-his4 25678  ax-hcompl 25795
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-acn 8319  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-cn 19494  df-cnp 19495  df-lm 19496  df-haus 19582  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cfil 21429  df-cau 21430  df-cmet 21431  df-grpo 24869  df-gid 24870  df-ginv 24871  df-gdiv 24872  df-ablo 24960  df-subgo 24980  df-vc 25115  df-nv 25161  df-va 25164  df-ba 25165  df-sm 25166  df-0v 25167  df-vs 25168  df-nmcv 25169  df-ims 25170  df-dip 25287  df-ssp 25311  df-ph 25404  df-cbn 25455  df-hnorm 25561  df-hba 25562  df-hvsub 25564  df-hlim 25565  df-hcau 25566  df-sh 25800  df-ch 25815  df-oc 25846  df-ch0 25847  df-shs 25902  df-pjh 25989  df-h0op 26343  df-lnop 26436
This theorem is referenced by:  lnopeqi  26603
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