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Theorem dv11cn 19890
Description: Two functions defined on a ball whose derivatives are the same and which are equal at any given point 
C in the ball must be equal everywhere. (Contributed by Mario Carneiro, 31-Mar-2015.)
Hypotheses
Ref Expression
dv11cn.x  |-  X  =  ( A ( ball `  ( abs  o.  -  ) ) R )
dv11cn.a  |-  ( ph  ->  A  e.  CC )
dv11cn.r  |-  ( ph  ->  R  e.  RR* )
dv11cn.f  |-  ( ph  ->  F : X --> CC )
dv11cn.g  |-  ( ph  ->  G : X --> CC )
dv11cn.d  |-  ( ph  ->  dom  ( CC  _D  F )  =  X )
dv11cn.e  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  G ) )
dv11cn.c  |-  ( ph  ->  C  e.  X )
dv11cn.p  |-  ( ph  ->  ( F `  C
)  =  ( G `
 C ) )
Assertion
Ref Expression
dv11cn  |-  ( ph  ->  F  =  G )

Proof of Theorem dv11cn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dv11cn.f . . . . 5  |-  ( ph  ->  F : X --> CC )
2 ffn 5594 . . . . 5  |-  ( F : X --> CC  ->  F  Fn  X )
31, 2syl 16 . . . 4  |-  ( ph  ->  F  Fn  X )
4 dv11cn.g . . . . 5  |-  ( ph  ->  G : X --> CC )
5 ffn 5594 . . . . 5  |-  ( G : X --> CC  ->  G  Fn  X )
64, 5syl 16 . . . 4  |-  ( ph  ->  G  Fn  X )
7 dv11cn.x . . . . . 6  |-  X  =  ( A ( ball `  ( abs  o.  -  ) ) R )
8 ovex 6109 . . . . . 6  |-  ( A ( ball `  ( abs  o.  -  ) ) R )  e.  _V
97, 8eqeltri 2508 . . . . 5  |-  X  e. 
_V
109a1i 11 . . . 4  |-  ( ph  ->  X  e.  _V )
11 inidm 3552 . . . 4  |-  ( X  i^i  X )  =  X
123, 6, 10, 10, 11offn 6319 . . 3  |-  ( ph  ->  ( F  o F  -  G )  Fn  X )
13 0cn 9089 . . . 4  |-  0  e.  CC
14 fnconstg 5634 . . . 4  |-  ( 0  e.  CC  ->  ( X  X.  { 0 } )  Fn  X )
1513, 14mp1i 12 . . 3  |-  ( ph  ->  ( X  X.  {
0 } )  Fn  X )
16 subcl 9310 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  -  y
)  e.  CC )
1716adantl 454 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  -  y
)  e.  CC )
1817, 1, 4, 10, 10, 11off 6323 . . . . . 6  |-  ( ph  ->  ( F  o F  -  G ) : X --> CC )
1918ffvelrnda 5873 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  o F  -  G ) `  x )  e.  CC )
20 simpr 449 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
21 dv11cn.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  X )
2221adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  C  e.  X )
2320, 22jca 520 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  X  /\  C  e.  X )
)
24 cnxmet 18812 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
2524a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( abs  o.  -  )  e.  ( * Met `  CC ) )
26 dv11cn.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  CC )
27 dv11cn.r . . . . . . . . . . 11  |-  ( ph  ->  R  e.  RR* )
28 blssm 18453 . . . . . . . . . . 11  |-  ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  A  e.  CC  /\  R  e.  RR* )  ->  ( A ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2925, 26, 27, 28syl3anc 1185 . . . . . . . . . 10  |-  ( ph  ->  ( A ( ball `  ( abs  o.  -  ) ) R ) 
C_  CC )
307, 29syl5eqss 3394 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
311ffvelrnda 5873 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
324ffvelrnda 5873 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
331feqmptd 5782 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
344feqmptd 5782 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
3510, 31, 32, 33, 34offval2 6325 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  o F  -  G )  =  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) )
3635oveq2d 6100 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  ( F  o F  -  G
) )  =  ( CC  _D  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) ) ) )
37 cnex 9076 . . . . . . . . . . . . . . . 16  |-  CC  e.  _V
3837prid2 3915 . . . . . . . . . . . . . . 15  |-  CC  e.  { RR ,  CC }
3938a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  CC  e.  { RR ,  CC } )
40 fvex 5745 . . . . . . . . . . . . . . 15  |-  ( ( CC  _D  F ) `
 x )  e. 
_V
4140a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  _V )
4233oveq2d 6100 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  ( x  e.  X  |->  ( F `  x ) ) ) )
43 dvfcn 19800 . . . . . . . . . . . . . . . . 17  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
44 dv11cn.d . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  dom  ( CC  _D  F )  =  X )
4544feq2d 5584 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( CC  _D  F ) : dom  ( CC  _D  F
) --> CC  <->  ( CC  _D  F ) : X --> CC ) )
4643, 45mpbii 204 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( CC  _D  F
) : X --> CC )
4746feqmptd 5782 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( x  e.  X  |->  ( ( CC  _D  F ) `
 x ) ) )
4842, 47eqtr3d 2472 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( F `  x ) ) )  =  ( x  e.  X  |->  ( ( CC  _D  F
) `  x )
) )
49 dv11cn.e . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  G ) )
5034oveq2d 6100 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  G
)  =  ( CC 
_D  ( x  e.  X  |->  ( G `  x ) ) ) )
5149, 47, 503eqtr3rd 2479 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( CC  _D  F
) `  x )
) )
5239, 31, 41, 48, 32, 41, 51dvmptsub 19858 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( ( F `  x
)  -  ( G `
 x ) ) ) )  =  ( x  e.  X  |->  ( ( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) ) ) )
5346ffvelrnda 5873 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  CC )
5453subidd 9404 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) )  =  0 )
5554mpteq2dva 4298 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( x  e.  X  |->  0 ) )
56 fconstmpt 4924 . . . . . . . . . . . . . 14  |-  ( X  X.  { 0 } )  =  ( x  e.  X  |->  0 )
5755, 56syl6eqr 2488 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( X  X.  { 0 } ) )
5836, 52, 573eqtrd 2474 . . . . . . . . . . . 12  |-  ( ph  ->  ( CC  _D  ( F  o F  -  G
) )  =  ( X  X.  { 0 } ) )
5958dmeqd 5075 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( CC  _D  ( F  o F  -  G ) )  =  dom  ( X  X.  { 0 } ) )
60 snnzg 3923 . . . . . . . . . . . 12  |-  ( 0  e.  CC  ->  { 0 }  =/=  (/) )
61 dmxp 5091 . . . . . . . . . . . 12  |-  ( { 0 }  =/=  (/)  ->  dom  ( X  X.  { 0 } )  =  X )
6213, 60, 61mp2b 10 . . . . . . . . . . 11  |-  dom  ( X  X.  { 0 } )  =  X
6359, 62syl6eq 2486 . . . . . . . . . 10  |-  ( ph  ->  dom  ( CC  _D  ( F  o F  -  G ) )  =  X )
64 eqimss2 3403 . . . . . . . . . 10  |-  ( dom  ( CC  _D  ( F  o F  -  G
) )  =  X  ->  X  C_  dom  ( CC  _D  ( F  o F  -  G
) ) )
6563, 64syl 16 . . . . . . . . 9  |-  ( ph  ->  X  C_  dom  ( CC 
_D  ( F  o F  -  G )
) )
66 0re 9096 . . . . . . . . . 10  |-  0  e.  RR
6766a1i 11 . . . . . . . . 9  |-  ( ph  ->  0  e.  RR )
6858fveq1d 5733 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  _D  ( F  o F  -  G ) ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
69 c0ex 9090 . . . . . . . . . . . . 13  |-  0  e.  _V
7069fvconst2 5950 . . . . . . . . . . . 12  |-  ( x  e.  X  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
7168, 70sylan9eq 2490 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  ( F  o F  -  G
) ) `  x
)  =  0 )
7271abs00bd 12101 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  o F  -  G )
) `  x )
)  =  0 )
73 0le0 10086 . . . . . . . . . 10  |-  0  <_  0
7472, 73syl6eqbr 4252 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  o F  -  G )
) `  x )
)  <_  0 )
7530, 18, 26, 27, 7, 65, 67, 74dvlipcn 19883 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  X  /\  C  e.  X ) )  -> 
( abs `  (
( ( F  o F  -  G ) `  x )  -  (
( F  o F  -  G ) `  C ) ) )  <_  ( 0  x.  ( abs `  (
x  -  C ) ) ) )
7623, 75syldan 458 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( ( F  o F  -  G ) `  x
)  -  ( ( F  o F  -  G ) `  C
) ) )  <_ 
( 0  x.  ( abs `  ( x  -  C ) ) ) )
7735fveq1d 5733 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F  o F  -  G ) `  C )  =  ( ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) `  C
) )
78 fveq2 5731 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
79 fveq2 5731 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( G `  x )  =  ( G `  C ) )
8078, 79oveq12d 6102 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  (
( F `  x
)  -  ( G `
 x ) )  =  ( ( F `
 C )  -  ( G `  C ) ) )
81 eqid 2438 . . . . . . . . . . . . . 14  |-  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( F `
 x )  -  ( G `  x ) ) )
82 ovex 6109 . . . . . . . . . . . . . 14  |-  ( ( F `  C )  -  ( G `  C ) )  e. 
_V
8380, 81, 82fvmpt 5809 . . . . . . . . . . . . 13  |-  ( C  e.  X  ->  (
( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) `  C
)  =  ( ( F `  C )  -  ( G `  C ) ) )
8421, 83syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( x  e.  X  |->  ( ( F `
 x )  -  ( G `  x ) ) ) `  C
)  =  ( ( F `  C )  -  ( G `  C ) ) )
851, 21ffvelrnd 5874 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F `  C
)  e.  CC )
86 dv11cn.p . . . . . . . . . . . . 13  |-  ( ph  ->  ( F `  C
)  =  ( G `
 C ) )
8785, 86subeq0bd 9468 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  C )  -  ( G `  C )
)  =  0 )
8877, 84, 873eqtrd 2474 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  o F  -  G ) `  C )  =  0 )
8988adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  o F  -  G ) `  C )  =  0 )
9089oveq2d 6100 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  o F  -  G ) `  x )  -  (
( F  o F  -  G ) `  C ) )  =  ( ( ( F  o F  -  G
) `  x )  -  0 ) )
9119subid1d 9405 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  o F  -  G ) `  x )  -  0 )  =  ( ( F  o F  -  G ) `  x
) )
9290, 91eqtrd 2470 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  o F  -  G ) `  x )  -  (
( F  o F  -  G ) `  C ) )  =  ( ( F  o F  -  G ) `  x ) )
9392fveq2d 5735 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( ( F  o F  -  G ) `  x
)  -  ( ( F  o F  -  G ) `  C
) ) )  =  ( abs `  (
( F  o F  -  G ) `  x ) ) )
9430sselda 3350 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  CC )
9530, 21sseldd 3351 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  CC )
9695adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  C  e.  CC )
9794, 96subcld 9416 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
x  -  C )  e.  CC )
9897abscld 12243 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  RR )
9998recnd 9119 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  CC )
10099mul02d 9269 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
0  x.  ( abs `  ( x  -  C
) ) )  =  0 )
10176, 93, 1003brtr3d 4244 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  o F  -  G
) `  x )
)  <_  0 )
10219absge0d 12251 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  0  <_  ( abs `  (
( F  o F  -  G ) `  x ) ) )
10319abscld 12243 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  o F  -  G
) `  x )
)  e.  RR )
104 letri3 9165 . . . . . . 7  |-  ( ( ( abs `  (
( F  o F  -  G ) `  x ) )  e.  RR  /\  0  e.  RR )  ->  (
( abs `  (
( F  o F  -  G ) `  x ) )  =  0  <->  ( ( abs `  ( ( F  o F  -  G ) `  x ) )  <_ 
0  /\  0  <_  ( abs `  ( ( F  o F  -  G ) `  x
) ) ) ) )
105103, 66, 104sylancl 645 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( abs `  (
( F  o F  -  G ) `  x ) )  =  0  <->  ( ( abs `  ( ( F  o F  -  G ) `  x ) )  <_ 
0  /\  0  <_  ( abs `  ( ( F  o F  -  G ) `  x
) ) ) ) )
106101, 102, 105mpbir2and 890 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  o F  -  G
) `  x )
)  =  0 )
10719, 106abs00d 12253 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  o F  -  G ) `  x )  =  0 )
10870adantl 454 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
109107, 108eqtr4d 2473 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  o F  -  G ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
11012, 15, 109eqfnfvd 5833 . 2  |-  ( ph  ->  ( F  o F  -  G )  =  ( X  X.  {
0 } ) )
111 ofsubeq0 10002 . . 3  |-  ( ( X  e.  _V  /\  F : X --> CC  /\  G : X --> CC )  ->  ( ( F  o F  -  G
)  =  ( X  X.  { 0 } )  <->  F  =  G
) )
11210, 1, 4, 111syl3anc 1185 . 2  |-  ( ph  ->  ( ( F  o F  -  G )  =  ( X  X.  { 0 } )  <-> 
F  =  G ) )
113110, 112mpbid 203 1  |-  ( ph  ->  F  =  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958    C_ wss 3322   (/)c0 3630   {csn 3816   {cpr 3817   class class class wbr 4215    e. cmpt 4269    X. cxp 4879   dom cdm 4881    o. ccom 4885    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084    o Fcof 6306   CCcc 8993   RRcr 8994   0cc0 8995    x. cmul 9000   RR*cxr 9124    <_ cle 9126    - cmin 9296   abscabs 12044   * Metcxmt 16691   ballcbl 16693    _D cdv 19755
This theorem is referenced by:  logtayl  20556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lp 17205  df-perf 17206  df-cn 17296  df-cnp 17297  df-haus 17384  df-cmp 17455  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cncf 18913  df-limc 19758  df-dv 19759
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