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Theorem dv11cn 22275
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 5721 . . . . 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 5721 . . . . 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 6309 . . . . . 6  |-  ( A ( ball `  ( abs  o.  -  ) ) R )  e.  _V
97, 8eqeltri 2527 . . . . 5  |-  X  e. 
_V
109a1i 11 . . . 4  |-  ( ph  ->  X  e.  _V )
11 inidm 3692 . . . 4  |-  ( X  i^i  X )  =  X
123, 6, 10, 10, 11offn 6536 . . 3  |-  ( ph  ->  ( F  oF  -  G )  Fn  X )
13 0cn 9591 . . . 4  |-  0  e.  CC
14 fnconstg 5763 . . . 4  |-  ( 0  e.  CC  ->  ( X  X.  { 0 } )  Fn  X )
1513, 14mp1i 12 . . 3  |-  ( ph  ->  ( X  X.  {
0 } )  Fn  X )
16 subcl 9824 . . . . . . . 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 6539 . . . . . 6  |-  ( ph  ->  ( F  oF  -  G ) : X --> CC )
1918ffvelrnda 6016 . . . . 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 21153 . . . . . . . . . . . 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 20794 . . . . . . . . . . 11  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  A  e.  CC  /\  R  e.  RR* )  ->  ( A ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2925, 26, 27, 28syl3anc 1229 . . . . . . . . . 10  |-  ( ph  ->  ( A ( ball `  ( abs  o.  -  ) ) R ) 
C_  CC )
307, 29syl5eqss 3533 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
311ffvelrnda 6016 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
324ffvelrnda 6016 . . . . . . . . . . . . . . 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 6541 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  oF  -  G )  =  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) )
3635oveq2d 6297 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  ( F  oF  -  G
) )  =  ( CC  _D  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) ) ) )
37 cnelprrecn 9588 . . . . . . . . . . . . . . 15  |-  CC  e.  { RR ,  CC }
3837a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  CC  e.  { RR ,  CC } )
39 fvex 5866 . . . . . . . . . . . . . . 15  |-  ( ( CC  _D  F ) `
 x )  e. 
_V
4039a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  _V )
4133oveq2d 6297 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  ( x  e.  X  |->  ( F `  x ) ) ) )
42 dvfcn 22185 . . . . . . . . . . . . . . . . 17  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
43 dv11cn.d . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  dom  ( CC  _D  F )  =  X )
4443feq2d 5708 . . . . . . . . . . . . . . . . 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 2486 . . . . . . . . . . . . . 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 6297 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  G
)  =  ( CC 
_D  ( x  e.  X  |->  ( G `  x ) ) ) )
5048, 46, 493eqtr3rd 2493 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( CC  _D  F
) `  x )
) )
5138, 31, 40, 47, 32, 40, 50dvmptsub 22243 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( ( F `  x
)  -  ( G `
 x ) ) ) )  =  ( x  e.  X  |->  ( ( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) ) ) )
5245ffvelrnda 6016 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  CC )
5352subidd 9924 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) )  =  0 )
5453mpteq2dva 4523 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( x  e.  X  |->  0 ) )
55 fconstmpt 5033 . . . . . . . . . . . . . 14  |-  ( X  X.  { 0 } )  =  ( x  e.  X  |->  0 )
5654, 55syl6eqr 2502 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( X  X.  { 0 } ) )
5736, 51, 563eqtrd 2488 . . . . . . . . . . . 12  |-  ( ph  ->  ( CC  _D  ( F  oF  -  G
) )  =  ( X  X.  { 0 } ) )
5857dmeqd 5195 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( CC  _D  ( F  oF  -  G ) )  =  dom  ( X  X.  { 0 } ) )
59 snnzg 4132 . . . . . . . . . . . 12  |-  ( 0  e.  CC  ->  { 0 }  =/=  (/) )
60 dmxp 5211 . . . . . . . . . . . 12  |-  ( { 0 }  =/=  (/)  ->  dom  ( X  X.  { 0 } )  =  X )
6113, 59, 60mp2b 10 . . . . . . . . . . 11  |-  dom  ( X  X.  { 0 } )  =  X
6258, 61syl6eq 2500 . . . . . . . . . 10  |-  ( ph  ->  dom  ( CC  _D  ( F  oF  -  G ) )  =  X )
63 eqimss2 3542 . . . . . . . . . 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 9600 . . . . . . . . 9  |-  ( ph  ->  0  e.  RR )
6657fveq1d 5858 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  _D  ( F  oF  -  G ) ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
67 c0ex 9593 . . . . . . . . . . . . 13  |-  0  e.  _V
6867fvconst2 6111 . . . . . . . . . . . 12  |-  ( x  e.  X  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
6966, 68sylan9eq 2504 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  ( F  oF  -  G
) ) `  x
)  =  0 )
7069abs00bd 13103 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  oF  -  G )
) `  x )
)  =  0 )
71 0le0 10631 . . . . . . . . . 10  |-  0  <_  0
7270, 71syl6eqbr 4474 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  oF  -  G )
) `  x )
)  <_  0 )
7330, 18, 26, 27, 7, 64, 65, 72dvlipcn 22268 . . . . . . . 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 5858 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F  oF  -  G ) `  C )  =  ( ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) `  C
) )
76 fveq2 5856 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
77 fveq2 5856 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( G `  x )  =  ( G `  C ) )
7876, 77oveq12d 6299 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  (
( F `  x
)  -  ( G `
 x ) )  =  ( ( F `
 C )  -  ( G `  C ) ) )
79 eqid 2443 . . . . . . . . . . . . . 14  |-  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( F `
 x )  -  ( G `  x ) ) )
80 ovex 6309 . . . . . . . . . . . . . 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 6017 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F `  C
)  e.  CC )
84 dv11cn.p . . . . . . . . . . . . 13  |-  ( ph  ->  ( F `  C
)  =  ( G `
 C ) )
8583, 84subeq0bd 9991 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  C )  -  ( G `  C )
)  =  0 )
8675, 82, 853eqtrd 2488 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  oF  -  G ) `  C )  =  0 )
8786adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  C )  =  0 )
8887oveq2d 6297 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  (
( F  oF  -  G ) `  C ) )  =  ( ( ( F  oF  -  G
) `  x )  -  0 ) )
8919subid1d 9925 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  0 )  =  ( ( F  oF  -  G ) `  x
) )
9088, 89eqtrd 2484 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  (
( F  oF  -  G ) `  C ) )  =  ( ( F  oF  -  G ) `  x ) )
9190fveq2d 5860 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( ( F  oF  -  G ) `  x
)  -  ( ( F  oF  -  G ) `  C
) ) )  =  ( abs `  (
( F  oF  -  G ) `  x ) ) )
9230sselda 3489 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  CC )
9330, 21sseldd 3490 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  CC )
9493adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  C  e.  CC )
9592, 94subcld 9936 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
x  -  C )  e.  CC )
9695abscld 13246 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  RR )
9796recnd 9625 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  CC )
9897mul02d 9781 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
0  x.  ( abs `  ( x  -  C
) ) )  =  0 )
9974, 91, 983brtr3d 4466 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  <_  0 )
10019absge0d 13254 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  0  <_  ( abs `  (
( F  oF  -  G ) `  x ) ) )
10119abscld 13246 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  e.  RR )
102 0re 9599 . . . . . . 7  |-  0  e.  RR
103 letri3 9673 . . . . . . 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 922 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  =  0 )
10619, 105abs00d 13256 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  x )  =  0 )
10768adantl 466 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
108106, 107eqtr4d 2487 . . 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 10539 . . 3  |-  ( ( X  e.  _V  /\  F : X --> CC  /\  G : X --> CC )  ->  ( ( F  oF  -  G
)  =  ( X  X.  { 0 } )  <->  F  =  G
) )
11110, 1, 4, 110syl3anc 1229 . 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 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095    C_ wss 3461   (/)c0 3770   {csn 4014   {cpr 4016   class class class wbr 4437    |-> cmpt 4495    X. cxp 4987   dom cdm 4989    o. ccom 4993    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    oFcof 6523   CCcc 9493   RRcr 9494   0cc0 9495    x. cmul 9500   RR*cxr 9630    <_ cle 9632    - cmin 9810   abscabs 13046   *Metcxmt 18277   ballcbl 18279    _D cdv 22140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-seq 12087  df-exp 12146  df-hash 12385  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-pt 14719  df-prds 14722  df-xrs 14776  df-qtop 14781  df-imas 14782  df-xps 14784  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-mulg 15934  df-cntz 16229  df-cmn 16674  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-fbas 18290  df-fg 18291  df-cnfld 18295  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-ntr 19394  df-cls 19395  df-nei 19472  df-lp 19510  df-perf 19511  df-cn 19601  df-cnp 19602  df-haus 19689  df-cmp 19760  df-tx 19936  df-hmeo 20129  df-fil 20220  df-fm 20312  df-flim 20313  df-flf 20314  df-xms 20696  df-ms 20697  df-tms 20698  df-cncf 21255  df-limc 22143  df-dv 22144
This theorem is referenced by:  logtayl  22913
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