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Theorem dv11cn 22895
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 5689 . . . . 5  |-  ( F : X --> CC  ->  F  Fn  X )
31, 2syl 17 . . . 4  |-  ( ph  ->  F  Fn  X )
4 dv11cn.g . . . . 5  |-  ( ph  ->  G : X --> CC )
5 ffn 5689 . . . . 5  |-  ( G : X --> CC  ->  G  Fn  X )
64, 5syl 17 . . . 4  |-  ( ph  ->  G  Fn  X )
7 dv11cn.x . . . . . 6  |-  X  =  ( A ( ball `  ( abs  o.  -  ) ) R )
8 ovex 6277 . . . . . 6  |-  ( A ( ball `  ( abs  o.  -  ) ) R )  e.  _V
97, 8eqeltri 2502 . . . . 5  |-  X  e. 
_V
109a1i 11 . . . 4  |-  ( ph  ->  X  e.  _V )
11 inidm 3614 . . . 4  |-  ( X  i^i  X )  =  X
123, 6, 10, 10, 11offn 6500 . . 3  |-  ( ph  ->  ( F  oF  -  G )  Fn  X )
13 0cn 9586 . . . 4  |-  0  e.  CC
14 fnconstg 5731 . . . 4  |-  ( 0  e.  CC  ->  ( X  X.  { 0 } )  Fn  X )
1513, 14mp1i 13 . . 3  |-  ( ph  ->  ( X  X.  {
0 } )  Fn  X )
16 subcl 9825 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  -  y
)  e.  CC )
1716adantl 467 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  -  y
)  e.  CC )
1817, 1, 4, 10, 10, 11off 6504 . . . . . 6  |-  ( ph  ->  ( F  oF  -  G ) : X --> CC )
1918ffvelrnda 5981 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  x )  e.  CC )
20 simpr 462 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
21 dv11cn.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  X )
2221adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  C  e.  X )
2320, 22jca 534 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  X  /\  C  e.  X )
)
24 cnxmet 21735 . . . . . . . . . . . 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 21375 . . . . . . . . . . 11  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  A  e.  CC  /\  R  e.  RR* )  ->  ( A ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2925, 26, 27, 28syl3anc 1264 . . . . . . . . . 10  |-  ( ph  ->  ( A ( ball `  ( abs  o.  -  ) ) R ) 
C_  CC )
307, 29syl5eqss 3451 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
311ffvelrnda 5981 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
324ffvelrnda 5981 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
331feqmptd 5878 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
344feqmptd 5878 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
3510, 31, 32, 33, 34offval2 6506 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  oF  -  G )  =  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) )
3635oveq2d 6265 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  ( F  oF  -  G
) )  =  ( CC  _D  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) ) ) )
37 cnelprrecn 9583 . . . . . . . . . . . . . . 15  |-  CC  e.  { RR ,  CC }
3837a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  CC  e.  { RR ,  CC } )
39 fvex 5835 . . . . . . . . . . . . . . 15  |-  ( ( CC  _D  F ) `
 x )  e. 
_V
4039a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  _V )
4133oveq2d 6265 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  ( x  e.  X  |->  ( F `  x ) ) ) )
42 dvfcn 22805 . . . . . . . . . . . . . . . . 17  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
43 dv11cn.d . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  dom  ( CC  _D  F )  =  X )
4443feq2d 5676 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( CC  _D  F ) : dom  ( CC  _D  F
) --> CC  <->  ( CC  _D  F ) : X --> CC ) )
4542, 44mpbii 214 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( CC  _D  F
) : X --> CC )
4645feqmptd 5878 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( x  e.  X  |->  ( ( CC  _D  F ) `
 x ) ) )
4741, 46eqtr3d 2464 . . . . . . . . . . . . . 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 6265 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  G
)  =  ( CC 
_D  ( x  e.  X  |->  ( G `  x ) ) ) )
5048, 46, 493eqtr3rd 2471 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( CC  _D  F
) `  x )
) )
5138, 31, 40, 47, 32, 40, 50dvmptsub 22863 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( ( F `  x
)  -  ( G `
 x ) ) ) )  =  ( x  e.  X  |->  ( ( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) ) ) )
5245ffvelrnda 5981 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  CC )
5352subidd 9925 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) )  =  0 )
5453mpteq2dva 4453 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( x  e.  X  |->  0 ) )
55 fconstmpt 4840 . . . . . . . . . . . . . 14  |-  ( X  X.  { 0 } )  =  ( x  e.  X  |->  0 )
5654, 55syl6eqr 2480 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( X  X.  { 0 } ) )
5736, 51, 563eqtrd 2466 . . . . . . . . . . . 12  |-  ( ph  ->  ( CC  _D  ( F  oF  -  G
) )  =  ( X  X.  { 0 } ) )
5857dmeqd 4999 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( CC  _D  ( F  oF  -  G ) )  =  dom  ( X  X.  { 0 } ) )
59 snnzg 4060 . . . . . . . . . . . 12  |-  ( 0  e.  CC  ->  { 0 }  =/=  (/) )
60 dmxp 5015 . . . . . . . . . . . 12  |-  ( { 0 }  =/=  (/)  ->  dom  ( X  X.  { 0 } )  =  X )
6113, 59, 60mp2b 10 . . . . . . . . . . 11  |-  dom  ( X  X.  { 0 } )  =  X
6258, 61syl6eq 2478 . . . . . . . . . 10  |-  ( ph  ->  dom  ( CC  _D  ( F  oF  -  G ) )  =  X )
63 eqimss2 3460 . . . . . . . . . 10  |-  ( dom  ( CC  _D  ( F  oF  -  G
) )  =  X  ->  X  C_  dom  ( CC  _D  ( F  oF  -  G
) ) )
6462, 63syl 17 . . . . . . . . 9  |-  ( ph  ->  X  C_  dom  ( CC 
_D  ( F  oF  -  G )
) )
65 0red 9595 . . . . . . . . 9  |-  ( ph  ->  0  e.  RR )
6657fveq1d 5827 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  _D  ( F  oF  -  G ) ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
67 c0ex 9588 . . . . . . . . . . . . 13  |-  0  e.  _V
6867fvconst2 6079 . . . . . . . . . . . 12  |-  ( x  e.  X  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
6966, 68sylan9eq 2482 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  ( F  oF  -  G
) ) `  x
)  =  0 )
7069abs00bd 13298 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  oF  -  G )
) `  x )
)  =  0 )
71 0le0 10650 . . . . . . . . . 10  |-  0  <_  0
7270, 71syl6eqbr 4404 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  oF  -  G )
) `  x )
)  <_  0 )
7330, 18, 26, 27, 7, 64, 65, 72dvlipcn 22888 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  X  /\  C  e.  X ) )  -> 
( abs `  (
( ( F  oF  -  G ) `  x )  -  (
( F  oF  -  G ) `  C ) ) )  <_  ( 0  x.  ( abs `  (
x  -  C ) ) ) )
7423, 73syldan 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( ( F  oF  -  G ) `  x
)  -  ( ( F  oF  -  G ) `  C
) ) )  <_ 
( 0  x.  ( abs `  ( x  -  C ) ) ) )
7535fveq1d 5827 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F  oF  -  G ) `  C )  =  ( ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) `  C
) )
76 fveq2 5825 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
77 fveq2 5825 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( G `  x )  =  ( G `  C ) )
7876, 77oveq12d 6267 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  (
( F `  x
)  -  ( G `
 x ) )  =  ( ( F `
 C )  -  ( G `  C ) ) )
79 eqid 2428 . . . . . . . . . . . . . 14  |-  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( F `
 x )  -  ( G `  x ) ) )
80 ovex 6277 . . . . . . . . . . . . . 14  |-  ( ( F `  C )  -  ( G `  C ) )  e. 
_V
8178, 79, 80fvmpt 5908 . . . . . . . . . . . . 13  |-  ( C  e.  X  ->  (
( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) `  C
)  =  ( ( F `  C )  -  ( G `  C ) ) )
8221, 81syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( x  e.  X  |->  ( ( F `
 x )  -  ( G `  x ) ) ) `  C
)  =  ( ( F `  C )  -  ( G `  C ) ) )
831, 21ffvelrnd 5982 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F `  C
)  e.  CC )
84 dv11cn.p . . . . . . . . . . . . 13  |-  ( ph  ->  ( F `  C
)  =  ( G `
 C ) )
8583, 84subeq0bd 9996 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  C )  -  ( G `  C )
)  =  0 )
8675, 82, 853eqtrd 2466 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  oF  -  G ) `  C )  =  0 )
8786adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  C )  =  0 )
8887oveq2d 6265 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  (
( F  oF  -  G ) `  C ) )  =  ( ( ( F  oF  -  G
) `  x )  -  0 ) )
8919subid1d 9926 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  0 )  =  ( ( F  oF  -  G ) `  x
) )
9088, 89eqtrd 2462 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  (
( F  oF  -  G ) `  C ) )  =  ( ( F  oF  -  G ) `  x ) )
9190fveq2d 5829 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( ( F  oF  -  G ) `  x
)  -  ( ( F  oF  -  G ) `  C
) ) )  =  ( abs `  (
( F  oF  -  G ) `  x ) ) )
9230sselda 3407 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  CC )
9330, 21sseldd 3408 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  CC )
9493adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  C  e.  CC )
9592, 94subcld 9937 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
x  -  C )  e.  CC )
9695abscld 13441 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  RR )
9796recnd 9620 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  CC )
9897mul02d 9782 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
0  x.  ( abs `  ( x  -  C
) ) )  =  0 )
9974, 91, 983brtr3d 4396 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  <_  0 )
10019absge0d 13449 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  0  <_  ( abs `  (
( F  oF  -  G ) `  x ) ) )
10119abscld 13441 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  e.  RR )
102 0re 9594 . . . . . . 7  |-  0  e.  RR
103 letri3 9670 . . . . . . 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 666 . . . . . 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 930 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  =  0 )
10619, 105abs00d 13451 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  x )  =  0 )
10768adantl 467 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
108106, 107eqtr4d 2465 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
10912, 15, 108eqfnfvd 5938 . 2  |-  ( ph  ->  ( F  oF  -  G )  =  ( X  X.  {
0 } ) )
110 ofsubeq0 10557 . . 3  |-  ( ( X  e.  _V  /\  F : X --> CC  /\  G : X --> CC )  ->  ( ( F  oF  -  G
)  =  ( X  X.  { 0 } )  <->  F  =  G
) )
11110, 1, 4, 110syl3anc 1264 . 2  |-  ( ph  ->  ( ( F  oF  -  G )  =  ( X  X.  { 0 } )  <-> 
F  =  G ) )
112109, 111mpbid 213 1  |-  ( ph  ->  F  =  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599   _Vcvv 3022    C_ wss 3379   (/)c0 3704   {csn 3941   {cpr 3943   class class class wbr 4366    |-> cmpt 4425    X. cxp 4794   dom cdm 4796    o. ccom 4800    Fn wfn 5539   -->wf 5540   ` cfv 5544  (class class class)co 6249    oFcof 6487   CCcc 9488   RRcr 9489   0cc0 9490    x. cmul 9495   RR*cxr 9625    <_ cle 9627    - cmin 9811   abscabs 13241   *Metcxmt 18898   ballcbl 18900    _D cdv 22760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-inf2 8099  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-iin 4245  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-of 6489  df-om 6651  df-1st 6751  df-2nd 6752  df-supp 6870  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-2o 7138  df-oadd 7141  df-er 7318  df-map 7429  df-pm 7430  df-ixp 7478  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-fsupp 7837  df-fi 7878  df-sup 7909  df-inf 7910  df-oi 7978  df-card 8325  df-cda 8549  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-3 10620  df-4 10621  df-5 10622  df-6 10623  df-7 10624  df-8 10625  df-9 10626  df-10 10627  df-n0 10821  df-z 10889  df-dec 11003  df-uz 11111  df-q 11216  df-rp 11254  df-xneg 11360  df-xadd 11361  df-xmul 11362  df-ioo 11590  df-ico 11592  df-icc 11593  df-fz 11736  df-fzo 11867  df-seq 12164  df-exp 12223  df-hash 12466  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-struct 15066  df-ndx 15067  df-slot 15068  df-base 15069  df-sets 15070  df-ress 15071  df-plusg 15146  df-mulr 15147  df-starv 15148  df-sca 15149  df-vsca 15150  df-ip 15151  df-tset 15152  df-ple 15153  df-ds 15155  df-unif 15156  df-hom 15157  df-cco 15158  df-rest 15264  df-topn 15265  df-0g 15283  df-gsum 15284  df-topgen 15285  df-pt 15286  df-prds 15289  df-xrs 15343  df-qtop 15349  df-imas 15350  df-xps 15353  df-mre 15435  df-mrc 15436  df-acs 15438  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-submnd 16526  df-mulg 16619  df-cntz 16914  df-cmn 17375  df-psmet 18905  df-xmet 18906  df-met 18907  df-bl 18908  df-mopn 18909  df-fbas 18910  df-fg 18911  df-cnfld 18914  df-top 19863  df-bases 19864  df-topon 19865  df-topsp 19866  df-cld 19976  df-ntr 19977  df-cls 19978  df-nei 20056  df-lp 20094  df-perf 20095  df-cn 20185  df-cnp 20186  df-haus 20273  df-cmp 20344  df-tx 20519  df-hmeo 20712  df-fil 20803  df-fm 20895  df-flim 20896  df-flf 20897  df-xms 21277  df-ms 21278  df-tms 21279  df-cncf 21852  df-limc 22763  df-dv 22764
This theorem is referenced by:  logtayl  23547  binomcxplemnotnn0  36618
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