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Theorem dv11cn 21432
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 5556 . . . . 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 5556 . . . . 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 6115 . . . . . 6  |-  ( A ( ball `  ( abs  o.  -  ) ) R )  e.  _V
97, 8eqeltri 2511 . . . . 5  |-  X  e. 
_V
109a1i 11 . . . 4  |-  ( ph  ->  X  e.  _V )
11 inidm 3556 . . . 4  |-  ( X  i^i  X )  =  X
123, 6, 10, 10, 11offn 6330 . . 3  |-  ( ph  ->  ( F  oF  -  G )  Fn  X )
13 0cn 9374 . . . 4  |-  0  e.  CC
14 fnconstg 5595 . . . 4  |-  ( 0  e.  CC  ->  ( X  X.  { 0 } )  Fn  X )
1513, 14mp1i 12 . . 3  |-  ( ph  ->  ( X  X.  {
0 } )  Fn  X )
16 subcl 9605 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  -  y
)  e.  CC )
1716adantl 463 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  -  y
)  e.  CC )
1817, 1, 4, 10, 10, 11off 6333 . . . . . 6  |-  ( ph  ->  ( F  oF  -  G ) : X --> CC )
1918ffvelrnda 5840 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  x )  e.  CC )
20 simpr 458 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
21 dv11cn.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  X )
2221adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  C  e.  X )
2320, 22jca 529 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  X  /\  C  e.  X )
)
24 cnxmet 20311 . . . . . . . . . . . 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 19952 . . . . . . . . . . 11  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  A  e.  CC  /\  R  e.  RR* )  ->  ( A ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2925, 26, 27, 28syl3anc 1213 . . . . . . . . . 10  |-  ( ph  ->  ( A ( ball `  ( abs  o.  -  ) ) R ) 
C_  CC )
307, 29syl5eqss 3397 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
311ffvelrnda 5840 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
324ffvelrnda 5840 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
331feqmptd 5741 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
344feqmptd 5741 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
3510, 31, 32, 33, 34offval2 6335 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  oF  -  G )  =  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) )
3635oveq2d 6106 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  ( F  oF  -  G
) )  =  ( CC  _D  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) ) ) )
37 cnelprrecn 9371 . . . . . . . . . . . . . . 15  |-  CC  e.  { RR ,  CC }
3837a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  CC  e.  { RR ,  CC } )
39 fvex 5698 . . . . . . . . . . . . . . 15  |-  ( ( CC  _D  F ) `
 x )  e. 
_V
4039a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  _V )
4133oveq2d 6106 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  ( x  e.  X  |->  ( F `  x ) ) ) )
42 dvfcn 21342 . . . . . . . . . . . . . . . . 17  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
43 dv11cn.d . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  dom  ( CC  _D  F )  =  X )
4443feq2d 5544 . . . . . . . . . . . . . . . . 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 5741 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( x  e.  X  |->  ( ( CC  _D  F ) `
 x ) ) )
4741, 46eqtr3d 2475 . . . . . . . . . . . . . 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 6106 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  G
)  =  ( CC 
_D  ( x  e.  X  |->  ( G `  x ) ) ) )
5048, 46, 493eqtr3rd 2482 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( CC  _D  F
) `  x )
) )
5138, 31, 40, 47, 32, 40, 50dvmptsub 21400 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( ( F `  x
)  -  ( G `
 x ) ) ) )  =  ( x  e.  X  |->  ( ( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) ) ) )
5245ffvelrnda 5840 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  CC )
5352subidd 9703 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) )  =  0 )
5453mpteq2dva 4375 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( x  e.  X  |->  0 ) )
55 fconstmpt 4878 . . . . . . . . . . . . . 14  |-  ( X  X.  { 0 } )  =  ( x  e.  X  |->  0 )
5654, 55syl6eqr 2491 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( X  X.  { 0 } ) )
5736, 51, 563eqtrd 2477 . . . . . . . . . . . 12  |-  ( ph  ->  ( CC  _D  ( F  oF  -  G
) )  =  ( X  X.  { 0 } ) )
5857dmeqd 5038 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( CC  _D  ( F  oF  -  G ) )  =  dom  ( X  X.  { 0 } ) )
59 snnzg 3989 . . . . . . . . . . . 12  |-  ( 0  e.  CC  ->  { 0 }  =/=  (/) )
60 dmxp 5054 . . . . . . . . . . . 12  |-  ( { 0 }  =/=  (/)  ->  dom  ( X  X.  { 0 } )  =  X )
6113, 59, 60mp2b 10 . . . . . . . . . . 11  |-  dom  ( X  X.  { 0 } )  =  X
6258, 61syl6eq 2489 . . . . . . . . . 10  |-  ( ph  ->  dom  ( CC  _D  ( F  oF  -  G ) )  =  X )
63 eqimss2 3406 . . . . . . . . . 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 9383 . . . . . . . . 9  |-  ( ph  ->  0  e.  RR )
6657fveq1d 5690 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  _D  ( F  oF  -  G ) ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
67 c0ex 9376 . . . . . . . . . . . . 13  |-  0  e.  _V
6867fvconst2 5930 . . . . . . . . . . . 12  |-  ( x  e.  X  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
6966, 68sylan9eq 2493 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  ( F  oF  -  G
) ) `  x
)  =  0 )
7069abs00bd 12776 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  oF  -  G )
) `  x )
)  =  0 )
71 0le0 10407 . . . . . . . . . 10  |-  0  <_  0
7270, 71syl6eqbr 4326 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  oF  -  G )
) `  x )
)  <_  0 )
7330, 18, 26, 27, 7, 64, 65, 72dvlipcn 21425 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  X  /\  C  e.  X ) )  -> 
( abs `  (
( ( F  oF  -  G ) `  x )  -  (
( F  oF  -  G ) `  C ) ) )  <_  ( 0  x.  ( abs `  (
x  -  C ) ) ) )
7423, 73syldan 467 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( ( F  oF  -  G ) `  x
)  -  ( ( F  oF  -  G ) `  C
) ) )  <_ 
( 0  x.  ( abs `  ( x  -  C ) ) ) )
7535fveq1d 5690 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F  oF  -  G ) `  C )  =  ( ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) `  C
) )
76 fveq2 5688 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
77 fveq2 5688 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( G `  x )  =  ( G `  C ) )
7876, 77oveq12d 6108 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  (
( F `  x
)  -  ( G `
 x ) )  =  ( ( F `
 C )  -  ( G `  C ) ) )
79 eqid 2441 . . . . . . . . . . . . . 14  |-  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( F `
 x )  -  ( G `  x ) ) )
80 ovex 6115 . . . . . . . . . . . . . 14  |-  ( ( F `  C )  -  ( G `  C ) )  e. 
_V
8178, 79, 80fvmpt 5771 . . . . . . . . . . . . 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 5841 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F `  C
)  e.  CC )
84 dv11cn.p . . . . . . . . . . . . 13  |-  ( ph  ->  ( F `  C
)  =  ( G `
 C ) )
8583, 84subeq0bd 9770 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  C )  -  ( G `  C )
)  =  0 )
8675, 82, 853eqtrd 2477 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  oF  -  G ) `  C )  =  0 )
8786adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  C )  =  0 )
8887oveq2d 6106 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  (
( F  oF  -  G ) `  C ) )  =  ( ( ( F  oF  -  G
) `  x )  -  0 ) )
8919subid1d 9704 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  0 )  =  ( ( F  oF  -  G ) `  x
) )
9088, 89eqtrd 2473 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  (
( F  oF  -  G ) `  C ) )  =  ( ( F  oF  -  G ) `  x ) )
9190fveq2d 5692 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( ( F  oF  -  G ) `  x
)  -  ( ( F  oF  -  G ) `  C
) ) )  =  ( abs `  (
( F  oF  -  G ) `  x ) ) )
9230sselda 3353 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  CC )
9330, 21sseldd 3354 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  CC )
9493adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  C  e.  CC )
9592, 94subcld 9715 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
x  -  C )  e.  CC )
9695abscld 12918 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  RR )
9796recnd 9408 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  CC )
9897mul02d 9563 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
0  x.  ( abs `  ( x  -  C
) ) )  =  0 )
9974, 91, 983brtr3d 4318 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  <_  0 )
10019absge0d 12926 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  0  <_  ( abs `  (
( F  oF  -  G ) `  x ) ) )
10119abscld 12918 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  e.  RR )
102 0re 9382 . . . . . . 7  |-  0  e.  RR
103 letri3 9456 . . . . . . 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 657 . . . . . 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 908 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  =  0 )
10619, 105abs00d 12928 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  x )  =  0 )
10768adantl 463 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
108106, 107eqtr4d 2476 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
10912, 15, 108eqfnfvd 5797 . 2  |-  ( ph  ->  ( F  oF  -  G )  =  ( X  X.  {
0 } ) )
110 ofsubeq0 10315 . . 3  |-  ( ( X  e.  _V  /\  F : X --> CC  /\  G : X --> CC )  ->  ( ( F  oF  -  G
)  =  ( X  X.  { 0 } )  <->  F  =  G
) )
11110, 1, 4, 110syl3anc 1213 . 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 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970    C_ wss 3325   (/)c0 3634   {csn 3874   {cpr 3876   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   dom cdm 4836    o. ccom 4840    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   CCcc 9276   RRcr 9277   0cc0 9278    x. cmul 9283   RR*cxr 9413    <_ cle 9415    - cmin 9591   abscabs 12719   *Metcxmt 17760   ballcbl 17762    _D cdv 21297
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-cmp 18949  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301
This theorem is referenced by:  logtayl  22064
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