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Theorem dv11cn 22130
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 5722 . . . . 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 5722 . . . . 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 6300 . . . . . 6  |-  ( A ( ball `  ( abs  o.  -  ) ) R )  e.  _V
97, 8eqeltri 2544 . . . . 5  |-  X  e. 
_V
109a1i 11 . . . 4  |-  ( ph  ->  X  e.  _V )
11 inidm 3700 . . . 4  |-  ( X  i^i  X )  =  X
123, 6, 10, 10, 11offn 6526 . . 3  |-  ( ph  ->  ( F  oF  -  G )  Fn  X )
13 0cn 9577 . . . 4  |-  0  e.  CC
14 fnconstg 5764 . . . 4  |-  ( 0  e.  CC  ->  ( X  X.  { 0 } )  Fn  X )
1513, 14mp1i 12 . . 3  |-  ( ph  ->  ( X  X.  {
0 } )  Fn  X )
16 subcl 9808 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  -  y
)  e.  CC )
1716adantl 466 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  -  y
)  e.  CC )
1817, 1, 4, 10, 10, 11off 6529 . . . . . 6  |-  ( ph  ->  ( F  oF  -  G ) : X --> CC )
1918ffvelrnda 6012 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  x )  e.  CC )
20 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
21 dv11cn.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  X )
2221adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  C  e.  X )
2320, 22jca 532 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  X  /\  C  e.  X )
)
24 cnxmet 21008 . . . . . . . . . . . 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 20649 . . . . . . . . . . 11  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  A  e.  CC  /\  R  e.  RR* )  ->  ( A ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2925, 26, 27, 28syl3anc 1223 . . . . . . . . . 10  |-  ( ph  ->  ( A ( ball `  ( abs  o.  -  ) ) R ) 
C_  CC )
307, 29syl5eqss 3541 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
311ffvelrnda 6012 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
324ffvelrnda 6012 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
331feqmptd 5911 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
344feqmptd 5911 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
3510, 31, 32, 33, 34offval2 6531 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  oF  -  G )  =  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) )
3635oveq2d 6291 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  ( F  oF  -  G
) )  =  ( CC  _D  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) ) ) )
37 cnelprrecn 9574 . . . . . . . . . . . . . . 15  |-  CC  e.  { RR ,  CC }
3837a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  CC  e.  { RR ,  CC } )
39 fvex 5867 . . . . . . . . . . . . . . 15  |-  ( ( CC  _D  F ) `
 x )  e. 
_V
4039a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  _V )
4133oveq2d 6291 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  ( x  e.  X  |->  ( F `  x ) ) ) )
42 dvfcn 22040 . . . . . . . . . . . . . . . . 17  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
43 dv11cn.d . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  dom  ( CC  _D  F )  =  X )
4443feq2d 5709 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( CC  _D  F ) : dom  ( CC  _D  F
) --> CC  <->  ( CC  _D  F ) : X --> CC ) )
4542, 44mpbii 211 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( CC  _D  F
) : X --> CC )
4645feqmptd 5911 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( x  e.  X  |->  ( ( CC  _D  F ) `
 x ) ) )
4741, 46eqtr3d 2503 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( F `  x ) ) )  =  ( x  e.  X  |->  ( ( CC  _D  F
) `  x )
) )
48 dv11cn.e . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  G ) )
4934oveq2d 6291 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  G
)  =  ( CC 
_D  ( x  e.  X  |->  ( G `  x ) ) ) )
5048, 46, 493eqtr3rd 2510 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( CC  _D  F
) `  x )
) )
5138, 31, 40, 47, 32, 40, 50dvmptsub 22098 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( ( F `  x
)  -  ( G `
 x ) ) ) )  =  ( x  e.  X  |->  ( ( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) ) ) )
5245ffvelrnda 6012 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  CC )
5352subidd 9907 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) )  =  0 )
5453mpteq2dva 4526 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( x  e.  X  |->  0 ) )
55 fconstmpt 5035 . . . . . . . . . . . . . 14  |-  ( X  X.  { 0 } )  =  ( x  e.  X  |->  0 )
5654, 55syl6eqr 2519 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( X  X.  { 0 } ) )
5736, 51, 563eqtrd 2505 . . . . . . . . . . . 12  |-  ( ph  ->  ( CC  _D  ( F  oF  -  G
) )  =  ( X  X.  { 0 } ) )
5857dmeqd 5196 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( CC  _D  ( F  oF  -  G ) )  =  dom  ( X  X.  { 0 } ) )
59 snnzg 4137 . . . . . . . . . . . 12  |-  ( 0  e.  CC  ->  { 0 }  =/=  (/) )
60 dmxp 5212 . . . . . . . . . . . 12  |-  ( { 0 }  =/=  (/)  ->  dom  ( X  X.  { 0 } )  =  X )
6113, 59, 60mp2b 10 . . . . . . . . . . 11  |-  dom  ( X  X.  { 0 } )  =  X
6258, 61syl6eq 2517 . . . . . . . . . 10  |-  ( ph  ->  dom  ( CC  _D  ( F  oF  -  G ) )  =  X )
63 eqimss2 3550 . . . . . . . . . 10  |-  ( dom  ( CC  _D  ( F  oF  -  G
) )  =  X  ->  X  C_  dom  ( CC  _D  ( F  oF  -  G
) ) )
6462, 63syl 16 . . . . . . . . 9  |-  ( ph  ->  X  C_  dom  ( CC 
_D  ( F  oF  -  G )
) )
65 0red 9586 . . . . . . . . 9  |-  ( ph  ->  0  e.  RR )
6657fveq1d 5859 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  _D  ( F  oF  -  G ) ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
67 c0ex 9579 . . . . . . . . . . . . 13  |-  0  e.  _V
6867fvconst2 6107 . . . . . . . . . . . 12  |-  ( x  e.  X  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
6966, 68sylan9eq 2521 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  ( F  oF  -  G
) ) `  x
)  =  0 )
7069abs00bd 13074 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  oF  -  G )
) `  x )
)  =  0 )
71 0le0 10614 . . . . . . . . . 10  |-  0  <_  0
7270, 71syl6eqbr 4477 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  oF  -  G )
) `  x )
)  <_  0 )
7330, 18, 26, 27, 7, 64, 65, 72dvlipcn 22123 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  X  /\  C  e.  X ) )  -> 
( abs `  (
( ( F  oF  -  G ) `  x )  -  (
( F  oF  -  G ) `  C ) ) )  <_  ( 0  x.  ( abs `  (
x  -  C ) ) ) )
7423, 73syldan 470 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( ( F  oF  -  G ) `  x
)  -  ( ( F  oF  -  G ) `  C
) ) )  <_ 
( 0  x.  ( abs `  ( x  -  C ) ) ) )
7535fveq1d 5859 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F  oF  -  G ) `  C )  =  ( ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) `  C
) )
76 fveq2 5857 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
77 fveq2 5857 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( G `  x )  =  ( G `  C ) )
7876, 77oveq12d 6293 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  (
( F `  x
)  -  ( G `
 x ) )  =  ( ( F `
 C )  -  ( G `  C ) ) )
79 eqid 2460 . . . . . . . . . . . . . 14  |-  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( F `
 x )  -  ( G `  x ) ) )
80 ovex 6300 . . . . . . . . . . . . . 14  |-  ( ( F `  C )  -  ( G `  C ) )  e. 
_V
8178, 79, 80fvmpt 5941 . . . . . . . . . . . . 13  |-  ( C  e.  X  ->  (
( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) `  C
)  =  ( ( F `  C )  -  ( G `  C ) ) )
8221, 81syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( x  e.  X  |->  ( ( F `
 x )  -  ( G `  x ) ) ) `  C
)  =  ( ( F `  C )  -  ( G `  C ) ) )
831, 21ffvelrnd 6013 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F `  C
)  e.  CC )
84 dv11cn.p . . . . . . . . . . . . 13  |-  ( ph  ->  ( F `  C
)  =  ( G `
 C ) )
8583, 84subeq0bd 9974 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  C )  -  ( G `  C )
)  =  0 )
8675, 82, 853eqtrd 2505 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  oF  -  G ) `  C )  =  0 )
8786adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  C )  =  0 )
8887oveq2d 6291 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  (
( F  oF  -  G ) `  C ) )  =  ( ( ( F  oF  -  G
) `  x )  -  0 ) )
8919subid1d 9908 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  0 )  =  ( ( F  oF  -  G ) `  x
) )
9088, 89eqtrd 2501 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  (
( F  oF  -  G ) `  C ) )  =  ( ( F  oF  -  G ) `  x ) )
9190fveq2d 5861 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( ( F  oF  -  G ) `  x
)  -  ( ( F  oF  -  G ) `  C
) ) )  =  ( abs `  (
( F  oF  -  G ) `  x ) ) )
9230sselda 3497 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  CC )
9330, 21sseldd 3498 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  CC )
9493adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  C  e.  CC )
9592, 94subcld 9919 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
x  -  C )  e.  CC )
9695abscld 13216 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  RR )
9796recnd 9611 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  CC )
9897mul02d 9766 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
0  x.  ( abs `  ( x  -  C
) ) )  =  0 )
9974, 91, 983brtr3d 4469 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  <_  0 )
10019absge0d 13224 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  0  <_  ( abs `  (
( F  oF  -  G ) `  x ) ) )
10119abscld 13216 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  e.  RR )
102 0re 9585 . . . . . . 7  |-  0  e.  RR
103 letri3 9659 . . . . . . 7  |-  ( ( ( abs `  (
( F  oF  -  G ) `  x ) )  e.  RR  /\  0  e.  RR )  ->  (
( abs `  (
( F  oF  -  G ) `  x ) )  =  0  <->  ( ( abs `  ( ( F  oF  -  G ) `  x ) )  <_ 
0  /\  0  <_  ( abs `  ( ( F  oF  -  G ) `  x
) ) ) ) )
104101, 102, 103sylancl 662 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( abs `  (
( F  oF  -  G ) `  x ) )  =  0  <->  ( ( abs `  ( ( F  oF  -  G ) `  x ) )  <_ 
0  /\  0  <_  ( abs `  ( ( F  oF  -  G ) `  x
) ) ) ) )
10599, 100, 104mpbir2and 915 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  =  0 )
10619, 105abs00d 13226 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  x )  =  0 )
10768adantl 466 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
108106, 107eqtr4d 2504 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
10912, 15, 108eqfnfvd 5969 . 2  |-  ( ph  ->  ( F  oF  -  G )  =  ( X  X.  {
0 } ) )
110 ofsubeq0 10522 . . 3  |-  ( ( X  e.  _V  /\  F : X --> CC  /\  G : X --> CC )  ->  ( ( F  oF  -  G
)  =  ( X  X.  { 0 } )  <->  F  =  G
) )
11110, 1, 4, 110syl3anc 1223 . 2  |-  ( ph  ->  ( ( F  oF  -  G )  =  ( X  X.  { 0 } )  <-> 
F  =  G ) )
112109, 111mpbid 210 1  |-  ( ph  ->  F  =  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   _Vcvv 3106    C_ wss 3469   (/)c0 3778   {csn 4020   {cpr 4022   class class class wbr 4440    |-> cmpt 4498    X. cxp 4990   dom cdm 4992    o. ccom 4996    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275    oFcof 6513   CCcc 9479   RRcr 9480   0cc0 9481    x. cmul 9486   RR*cxr 9616    <_ cle 9618    - cmin 9794   abscabs 13017   *Metcxmt 18167   ballcbl 18169    _D cdv 21995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-cn 19487  df-cnp 19488  df-haus 19575  df-cmp 19646  df-tx 19791  df-hmeo 19984  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-xms 20551  df-ms 20552  df-tms 20553  df-cncf 21110  df-limc 21998  df-dv 21999
This theorem is referenced by:  logtayl  22762
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