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Theorem lnopeq0i 27352
Description: A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 27173 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 27351 . . . . . 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 26356 . . . . . . . . . . . 12  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( y  +h  z
)  e.  ~H )
5 fveq2 5851 . . . . . . . . . . . . . . 15  |-  ( x  =  ( y  +h  z )  ->  ( T `  x )  =  ( T `  ( y  +h  z
) ) )
6 id 23 . . . . . . . . . . . . . . 15  |-  ( x  =  ( y  +h  z )  ->  x  =  ( y  +h  z ) )
75, 6oveq12d 6298 . . . . . . . . . . . . . 14  |-  ( x  =  ( y  +h  z )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  +h  z ) )  .ih  ( y  +h  z
) ) )
87eqeq1d 2406 . . . . . . . . . . . . 13  |-  ( x  =  ( y  +h  z )  ->  (
( ( T `  x )  .ih  x
)  =  0  <->  (
( T `  (
y  +h  z ) )  .ih  ( y  +h  z ) )  =  0 ) )
98rspccva 3161 . . . . . . . . . . . 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 26361 . . . . . . . . . . . 12  |-  ( ( y  e.  ~H  /\  z  e.  ~H )  ->  ( y  -h  z
)  e.  ~H )
12 fveq2 5851 . . . . . . . . . . . . . . 15  |-  ( x  =  ( y  -h  z )  ->  ( T `  x )  =  ( T `  ( y  -h  z
) ) )
13 id 23 . . . . . . . . . . . . . . 15  |-  ( x  =  ( y  -h  z )  ->  x  =  ( y  -h  z ) )
1412, 13oveq12d 6298 . . . . . . . . . . . . . 14  |-  ( x  =  ( y  -h  z )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  -h  z ) )  .ih  ( y  -h  z
) ) )
1514eqeq1d 2406 . . . . . . . . . . . . 13  |-  ( x  =  ( y  -h  z )  ->  (
( ( T `  x )  .ih  x
)  =  0  <->  (
( T `  (
y  -h  z ) )  .ih  ( y  -h  z ) )  =  0 ) )
1615rspccva 3161 . . . . . . . . . . . 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 6298 . . . . . . . . . 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 10688 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
2018, 19syl6eq 2461 . . . . . . . . 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 9583 . . . . . . . . . . . . . . . 16  |-  _i  e.  CC
22 hvmulcl 26357 . . . . . . . . . . . . . . . 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 26356 . . . . . . . . . . . . . . 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 5851 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  +h  ( _i  .h  z
) )  ->  ( T `  x )  =  ( T `  ( y  +h  (
_i  .h  z )
) ) )
27 id 23 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  +h  ( _i  .h  z
) )  ->  x  =  ( y  +h  ( _i  .h  z
) ) )
2826, 27oveq12d 6298 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( y  +h  ( _i  .h  z
) )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  +h  ( _i  .h  z
) ) )  .ih  ( y  +h  (
_i  .h  z )
) ) )
2928eqeq1d 2406 . . . . . . . . . . . . . . 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 3161 . . . . . . . . . . . . . 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 26361 . . . . . . . . . . . . . . 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 5851 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  -h  ( _i  .h  z
) )  ->  ( T `  x )  =  ( T `  ( y  -h  (
_i  .h  z )
) ) )
35 id 23 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( y  -h  ( _i  .h  z
) )  ->  x  =  ( y  -h  ( _i  .h  z
) ) )
3634, 35oveq12d 6298 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( y  -h  ( _i  .h  z
) )  ->  (
( T `  x
)  .ih  x )  =  ( ( T `
 ( y  -h  ( _i  .h  z
) ) )  .ih  ( y  -h  (
_i  .h  z )
) ) )
3736eqeq1d 2406 . . . . . . . . . . . . . . 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 3161 . . . . . . . . . . . . . 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 6298 . . . . . . . . . . . 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 2461 . . . . . . . . . . 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 6296 . . . . . . . . . 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 10804 . . . . . . . . . 10  |-  ( _i  x.  0 )  =  0
4442, 43syl6eq 2461 . . . . . . . . 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 6298 . . . . . . . 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 9791 . . . . . . . 8  |-  ( 0  +  0 )  =  0
4745, 46syl6eq 2461 . . . . . . 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 6295 . . . . . 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 10656 . . . . . . 7  |-  4  e.  CC
50 4ne0 10675 . . . . . . 7  |-  4  =/=  0
5149, 50div0i 10321 . . . . . 6  |-  ( 0  /  4 )  =  0
5248, 51syl6eq 2461 . . . . 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 2445 . . . 4  |-  ( ( A. x  e.  ~H  ( ( T `  x )  .ih  x
)  =  0  /\  ( y  e.  ~H  /\  z  e.  ~H )
)  ->  ( ( T `  y )  .ih  z )  =  0 )
5453ralrimivva 2827 . . 3  |-  ( A. x  e.  ~H  (
( T `  x
)  .ih  x )  =  0  ->  A. y  e.  ~H  A. z  e. 
~H  ( ( T `
 y )  .ih  z )  =  0 )
551lnopfi 27314 . . . 4  |-  T : ~H
--> ~H
5655ho01i 27173 . . 3  |-  ( A. y  e.  ~H  A. z  e.  ~H  ( ( T `
 y )  .ih  z )  =  0  <-> 
T  =  0hop )
5754, 56sylib 198 . 2  |-  ( A. x  e.  ~H  (
( T `  x
)  .ih  x )  =  0  ->  T  =  0hop )
58 fveq1 5850 . . . . . 6  |-  ( T  =  0hop  ->  ( T `
 x )  =  ( 0hop `  x
) )
59 ho0val 27095 . . . . . 6  |-  ( x  e.  ~H  ->  ( 0hop `  x )  =  0h )
6058, 59sylan9eq 2465 . . . . 5  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  ( T `  x )  =  0h )
6160oveq1d 6295 . . . 4  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  =  ( 0h  .ih  x ) )
62 hi01 26440 . . . . 5  |-  ( x  e.  ~H  ->  ( 0h  .ih  x )  =  0 )
6362adantl 466 . . . 4  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  ( 0h  .ih  x )  =  0 )
6461, 63eqtrd 2445 . . 3  |-  ( ( T  =  0hop  /\  x  e.  ~H )  ->  (
( T `  x
)  .ih  x )  =  0 )
6564ralrimiva 2820 . 2  |-  ( T  =  0hop  ->  A. x  e.  ~H  ( ( T `
 x )  .ih  x )  =  0 )
6657, 65impbii 189 1  |-  ( A. x  e.  ~H  (
( T `  x
)  .ih  x )  =  0  <->  T  =  0hop )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   A.wral 2756   ` cfv 5571  (class class class)co 6280   CCcc 9522   0cc0 9524   _ici 9526    + caddc 9527    x. cmul 9529    - cmin 9843    / cdiv 10249   4c4 10630   ~Hchil 26263    +h cva 26264    .h csm 26265    .ih csp 26266   0hc0v 26268    -h cmv 26269   0hopch0o 26287   LinOpclo 26291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cc 8849  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602  ax-addf 9603  ax-mulf 9604  ax-hilex 26343  ax-hfvadd 26344  ax-hvcom 26345  ax-hvass 26346  ax-hv0cl 26347  ax-hvaddid 26348  ax-hfvmul 26349  ax-hvmulid 26350  ax-hvmulass 26351  ax-hvdistr1 26352  ax-hvdistr2 26353  ax-hvmul0 26354  ax-hfi 26423  ax-his1 26426  ax-his2 26427  ax-his3 26428  ax-his4 26429  ax-hcompl 26546
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523  df-om 6686  df-1st 6786  df-2nd 6787  df-supp 6905  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-2o 7170  df-oadd 7173  df-omul 7174  df-er 7350  df-map 7461  df-pm 7462  df-ixp 7510  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-fsupp 7866  df-fi 7907  df-sup 7937  df-oi 7971  df-card 8354  df-acn 8357  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-q 11230  df-rp 11268  df-xneg 11373  df-xadd 11374  df-xmul 11375  df-ioo 11588  df-ico 11590  df-icc 11591  df-fz 11729  df-fzo 11857  df-fl 11968  df-seq 12154  df-exp 12213  df-hash 12455  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-clim 13462  df-rlim 13463  df-sum 13660  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-starv 14926  df-sca 14927  df-vsca 14928  df-ip 14929  df-tset 14930  df-ple 14931  df-ds 14933  df-unif 14934  df-hom 14935  df-cco 14936  df-rest 15039  df-topn 15040  df-0g 15058  df-gsum 15059  df-topgen 15060  df-pt 15061  df-prds 15064  df-xrs 15118  df-qtop 15123  df-imas 15124  df-xps 15126  df-mre 15202  df-mrc 15203  df-acs 15205  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-submnd 16293  df-mulg 16386  df-cntz 16681  df-cmn 17126  df-psmet 18733  df-xmet 18734  df-met 18735  df-bl 18736  df-mopn 18737  df-fbas 18738  df-fg 18739  df-cnfld 18743  df-top 19693  df-bases 19695  df-topon 19696  df-topsp 19697  df-cld 19814  df-ntr 19815  df-cls 19816  df-nei 19894  df-cn 20023  df-cnp 20024  df-lm 20025  df-haus 20111  df-tx 20357  df-hmeo 20550  df-fil 20641  df-fm 20733  df-flim 20734  df-flf 20735  df-xms 21117  df-ms 21118  df-tms 21119  df-cfil 21988  df-cau 21989  df-cmet 21990  df-grpo 25620  df-gid 25621  df-ginv 25622  df-gdiv 25623  df-ablo 25711  df-subgo 25731  df-vc 25866  df-nv 25912  df-va 25915  df-ba 25916  df-sm 25917  df-0v 25918  df-vs 25919  df-nmcv 25920  df-ims 25921  df-dip 26038  df-ssp 26062  df-ph 26155  df-cbn 26206  df-hnorm 26312  df-hba 26313  df-hvsub 26315  df-hlim 26316  df-hcau 26317  df-sh 26551  df-ch 26566  df-oc 26597  df-ch0 26598  df-shs 26653  df-pjh 26740  df-h0op 27093  df-lnop 27186
This theorem is referenced by:  lnopeqi  27353
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