MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dv11cn Structured version   Unicode version

Theorem dv11cn 21476
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 5562 . . . . 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 5562 . . . . 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 6119 . . . . . 6  |-  ( A ( ball `  ( abs  o.  -  ) ) R )  e.  _V
97, 8eqeltri 2513 . . . . 5  |-  X  e. 
_V
109a1i 11 . . . 4  |-  ( ph  ->  X  e.  _V )
11 inidm 3562 . . . 4  |-  ( X  i^i  X )  =  X
123, 6, 10, 10, 11offn 6334 . . 3  |-  ( ph  ->  ( F  oF  -  G )  Fn  X )
13 0cn 9381 . . . 4  |-  0  e.  CC
14 fnconstg 5601 . . . 4  |-  ( 0  e.  CC  ->  ( X  X.  { 0 } )  Fn  X )
1513, 14mp1i 12 . . 3  |-  ( ph  ->  ( X  X.  {
0 } )  Fn  X )
16 subcl 9612 . . . . . . . 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 6337 . . . . . 6  |-  ( ph  ->  ( F  oF  -  G ) : X --> CC )
1918ffvelrnda 5846 . . . . 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 20355 . . . . . . . . . . . 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 19996 . . . . . . . . . . 11  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  A  e.  CC  /\  R  e.  RR* )  ->  ( A ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
2925, 26, 27, 28syl3anc 1218 . . . . . . . . . 10  |-  ( ph  ->  ( A ( ball `  ( abs  o.  -  ) ) R ) 
C_  CC )
307, 29syl5eqss 3403 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
311ffvelrnda 5846 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
324ffvelrnda 5846 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
331feqmptd 5747 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
344feqmptd 5747 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
3510, 31, 32, 33, 34offval2 6339 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  oF  -  G )  =  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) )
3635oveq2d 6110 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  ( F  oF  -  G
) )  =  ( CC  _D  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) ) ) )
37 cnelprrecn 9378 . . . . . . . . . . . . . . 15  |-  CC  e.  { RR ,  CC }
3837a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  CC  e.  { RR ,  CC } )
39 fvex 5704 . . . . . . . . . . . . . . 15  |-  ( ( CC  _D  F ) `
 x )  e. 
_V
4039a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  _V )
4133oveq2d 6110 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  ( x  e.  X  |->  ( F `  x ) ) ) )
42 dvfcn 21386 . . . . . . . . . . . . . . . . 17  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
43 dv11cn.d . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  dom  ( CC  _D  F )  =  X )
4443feq2d 5550 . . . . . . . . . . . . . . . . 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 5747 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( x  e.  X  |->  ( ( CC  _D  F ) `
 x ) ) )
4741, 46eqtr3d 2477 . . . . . . . . . . . . . 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 6110 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  G
)  =  ( CC 
_D  ( x  e.  X  |->  ( G `  x ) ) ) )
5048, 46, 493eqtr3rd 2484 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( CC  _D  F
) `  x )
) )
5138, 31, 40, 47, 32, 40, 50dvmptsub 21444 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( ( F `  x
)  -  ( G `
 x ) ) ) )  =  ( x  e.  X  |->  ( ( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) ) ) )
5245ffvelrnda 5846 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  CC )
5352subidd 9710 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) )  =  0 )
5453mpteq2dva 4381 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( x  e.  X  |->  0 ) )
55 fconstmpt 4885 . . . . . . . . . . . . . 14  |-  ( X  X.  { 0 } )  =  ( x  e.  X  |->  0 )
5654, 55syl6eqr 2493 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( X  X.  { 0 } ) )
5736, 51, 563eqtrd 2479 . . . . . . . . . . . 12  |-  ( ph  ->  ( CC  _D  ( F  oF  -  G
) )  =  ( X  X.  { 0 } ) )
5857dmeqd 5045 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( CC  _D  ( F  oF  -  G ) )  =  dom  ( X  X.  { 0 } ) )
59 snnzg 3995 . . . . . . . . . . . 12  |-  ( 0  e.  CC  ->  { 0 }  =/=  (/) )
60 dmxp 5061 . . . . . . . . . . . 12  |-  ( { 0 }  =/=  (/)  ->  dom  ( X  X.  { 0 } )  =  X )
6113, 59, 60mp2b 10 . . . . . . . . . . 11  |-  dom  ( X  X.  { 0 } )  =  X
6258, 61syl6eq 2491 . . . . . . . . . 10  |-  ( ph  ->  dom  ( CC  _D  ( F  oF  -  G ) )  =  X )
63 eqimss2 3412 . . . . . . . . . 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 9390 . . . . . . . . 9  |-  ( ph  ->  0  e.  RR )
6657fveq1d 5696 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( CC  _D  ( F  oF  -  G ) ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
67 c0ex 9383 . . . . . . . . . . . . 13  |-  0  e.  _V
6867fvconst2 5936 . . . . . . . . . . . 12  |-  ( x  e.  X  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
6966, 68sylan9eq 2495 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  ( F  oF  -  G
) ) `  x
)  =  0 )
7069abs00bd 12783 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  oF  -  G )
) `  x )
)  =  0 )
71 0le0 10414 . . . . . . . . . 10  |-  0  <_  0
7270, 71syl6eqbr 4332 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  oF  -  G )
) `  x )
)  <_  0 )
7330, 18, 26, 27, 7, 64, 65, 72dvlipcn 21469 . . . . . . . 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 5696 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F  oF  -  G ) `  C )  =  ( ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) `  C
) )
76 fveq2 5694 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
77 fveq2 5694 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( G `  x )  =  ( G `  C ) )
7876, 77oveq12d 6112 . . . . . . . . . . . . . 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 6119 . . . . . . . . . . . . . 14  |-  ( ( F `  C )  -  ( G `  C ) )  e. 
_V
8178, 79, 80fvmpt 5777 . . . . . . . . . . . . 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 5847 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F `  C
)  e.  CC )
84 dv11cn.p . . . . . . . . . . . . 13  |-  ( ph  ->  ( F `  C
)  =  ( G `
 C ) )
8583, 84subeq0bd 9777 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  C )  -  ( G `  C )
)  =  0 )
8675, 82, 853eqtrd 2479 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  oF  -  G ) `  C )  =  0 )
8786adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  C )  =  0 )
8887oveq2d 6110 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  (
( F  oF  -  G ) `  C ) )  =  ( ( ( F  oF  -  G
) `  x )  -  0 ) )
8919subid1d 9711 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  0 )  =  ( ( F  oF  -  G ) `  x
) )
9088, 89eqtrd 2475 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  oF  -  G ) `  x )  -  (
( F  oF  -  G ) `  C ) )  =  ( ( F  oF  -  G ) `  x ) )
9190fveq2d 5698 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( ( F  oF  -  G ) `  x
)  -  ( ( F  oF  -  G ) `  C
) ) )  =  ( abs `  (
( F  oF  -  G ) `  x ) ) )
9230sselda 3359 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  CC )
9330, 21sseldd 3360 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  CC )
9493adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  C  e.  CC )
9592, 94subcld 9722 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
x  -  C )  e.  CC )
9695abscld 12925 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  RR )
9796recnd 9415 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  CC )
9897mul02d 9570 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
0  x.  ( abs `  ( x  -  C
) ) )  =  0 )
9974, 91, 983brtr3d 4324 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  <_  0 )
10019absge0d 12933 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  0  <_  ( abs `  (
( F  oF  -  G ) `  x ) ) )
10119abscld 12925 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  e.  RR )
102 0re 9389 . . . . . . 7  |-  0  e.  RR
103 letri3 9463 . . . . . . 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 913 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  oF  -  G
) `  x )
)  =  0 )
10619, 105abs00d 12935 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  x )  =  0 )
10768adantl 466 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
108106, 107eqtr4d 2478 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  oF  -  G ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
10912, 15, 108eqfnfvd 5803 . 2  |-  ( ph  ->  ( F  oF  -  G )  =  ( X  X.  {
0 } ) )
110 ofsubeq0 10322 . . 3  |-  ( ( X  e.  _V  /\  F : X --> CC  /\  G : X --> CC )  ->  ( ( F  oF  -  G
)  =  ( X  X.  { 0 } )  <->  F  =  G
) )
11110, 1, 4, 110syl3anc 1218 . 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 1369    e. wcel 1756    =/= wne 2609   _Vcvv 2975    C_ wss 3331   (/)c0 3640   {csn 3880   {cpr 3882   class class class wbr 4295    e. cmpt 4353    X. cxp 4841   dom cdm 4843    o. ccom 4847    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6094    oFcof 6321   CCcc 9283   RRcr 9284   0cc0 9285    x. cmul 9290   RR*cxr 9420    <_ cle 9422    - cmin 9598   abscabs 12726   *Metcxmt 17804   ballcbl 17806    _D cdv 21341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363  ax-addf 9364  ax-mulf 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-iin 4177  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-om 6480  df-1st 6580  df-2nd 6581  df-supp 6694  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-ixp 7267  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-fsupp 7624  df-fi 7664  df-sup 7694  df-oi 7727  df-card 8112  df-cda 8340  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-9 10390  df-10 10391  df-n0 10583  df-z 10650  df-dec 10759  df-uz 10865  df-q 10957  df-rp 10995  df-xneg 11092  df-xadd 11093  df-xmul 11094  df-ioo 11307  df-ico 11309  df-icc 11310  df-fz 11441  df-fzo 11552  df-seq 11810  df-exp 11869  df-hash 12107  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-mulr 14255  df-starv 14256  df-sca 14257  df-vsca 14258  df-ip 14259  df-tset 14260  df-ple 14261  df-ds 14263  df-unif 14264  df-hom 14265  df-cco 14266  df-rest 14364  df-topn 14365  df-0g 14383  df-gsum 14384  df-topgen 14385  df-pt 14386  df-prds 14389  df-xrs 14443  df-qtop 14448  df-imas 14449  df-xps 14451  df-mre 14527  df-mrc 14528  df-acs 14530  df-mnd 15418  df-submnd 15468  df-mulg 15551  df-cntz 15838  df-cmn 16282  df-psmet 17812  df-xmet 17813  df-met 17814  df-bl 17815  df-mopn 17816  df-fbas 17817  df-fg 17818  df-cnfld 17822  df-top 18506  df-bases 18508  df-topon 18509  df-topsp 18510  df-cld 18626  df-ntr 18627  df-cls 18628  df-nei 18705  df-lp 18743  df-perf 18744  df-cn 18834  df-cnp 18835  df-haus 18922  df-cmp 18993  df-tx 19138  df-hmeo 19331  df-fil 19422  df-fm 19514  df-flim 19515  df-flf 19516  df-xms 19898  df-ms 19899  df-tms 19900  df-cncf 20457  df-limc 21344  df-dv 21345
This theorem is referenced by:  logtayl  22108
  Copyright terms: Public domain W3C validator