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

Theorem dv11cn 19452
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 5472 . . . . 5  |-  ( F : X --> CC  ->  F  Fn  X )
31, 2syl 15 . . . 4  |-  ( ph  ->  F  Fn  X )
4 dv11cn.g . . . . 5  |-  ( ph  ->  G : X --> CC )
5 ffn 5472 . . . . 5  |-  ( G : X --> CC  ->  G  Fn  X )
64, 5syl 15 . . . 4  |-  ( ph  ->  G  Fn  X )
7 dv11cn.x . . . . . 6  |-  X  =  ( A ( ball `  ( abs  o.  -  ) ) R )
8 ovex 5970 . . . . . 6  |-  ( A ( ball `  ( abs  o.  -  ) ) R )  e.  _V
97, 8eqeltri 2428 . . . . 5  |-  X  e. 
_V
109a1i 10 . . . 4  |-  ( ph  ->  X  e.  _V )
11 inidm 3454 . . . 4  |-  ( X  i^i  X )  =  X
123, 6, 10, 10, 11offn 6176 . . 3  |-  ( ph  ->  ( F  o F  -  G )  Fn  X )
13 0cn 8921 . . . 4  |-  0  e.  CC
14 fnconstg 5512 . . . 4  |-  ( 0  e.  CC  ->  ( X  X.  { 0 } )  Fn  X )
1513, 14mp1i 11 . . 3  |-  ( ph  ->  ( X  X.  {
0 } )  Fn  X )
16 subcl 9141 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  -  y
)  e.  CC )
1716adantl 452 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  -  y
)  e.  CC )
1817, 1, 4, 10, 10, 11off 6180 . . . . . 6  |-  ( ph  ->  ( F  o F  -  G ) : X --> CC )
19 ffvelrn 5746 . . . . . 6  |-  ( ( ( F  o F  -  G ) : X --> CC  /\  x  e.  X )  ->  (
( F  o F  -  G ) `  x )  e.  CC )
2018, 19sylan 457 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  o F  -  G ) `  x )  e.  CC )
21 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
22 dv11cn.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  X )
2322adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  C  e.  X )
2421, 23jca 518 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  X  /\  C  e.  X )
)
25 cnxmet 18384 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
2625a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  ( abs  o.  -  )  e.  ( * Met `  CC ) )
27 dv11cn.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  CC )
28 dv11cn.r . . . . . . . . . . 11  |-  ( ph  ->  R  e.  RR* )
29 blssm 18070 . . . . . . . . . . 11  |-  ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  A  e.  CC  /\  R  e.  RR* )  ->  ( A ( ball `  ( abs  o.  -  ) ) R )  C_  CC )
3026, 27, 28, 29syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  ( A ( ball `  ( abs  o.  -  ) ) R ) 
C_  CC )
317, 30syl5eqss 3298 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
32 ffvelrn 5746 . . . . . . . . . . . . . . . 16  |-  ( ( F : X --> CC  /\  x  e.  X )  ->  ( F `  x
)  e.  CC )
331, 32sylan 457 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
34 ffvelrn 5746 . . . . . . . . . . . . . . . 16  |-  ( ( G : X --> CC  /\  x  e.  X )  ->  ( G `  x
)  e.  CC )
354, 34sylan 457 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
361feqmptd 5658 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
374feqmptd 5658 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
3810, 33, 35, 36, 37offval2 6182 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  o F  -  G )  =  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) )
3938oveq2d 5961 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  ( F  o F  -  G
) )  =  ( CC  _D  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) ) ) )
40 cnex 8908 . . . . . . . . . . . . . . . 16  |-  CC  e.  _V
4140prid2 3811 . . . . . . . . . . . . . . 15  |-  CC  e.  { RR ,  CC }
4241a1i 10 . . . . . . . . . . . . . 14  |-  ( ph  ->  CC  e.  { RR ,  CC } )
43 fvex 5622 . . . . . . . . . . . . . . 15  |-  ( ( CC  _D  F ) `
 x )  e. 
_V
4443a1i 10 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  _V )
4536oveq2d 5961 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  ( x  e.  X  |->  ( F `  x ) ) ) )
46 dvfcn 19362 . . . . . . . . . . . . . . . . 17  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
47 dv11cn.d . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  dom  ( CC  _D  F )  =  X )
4847feq2d 5462 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( CC  _D  F ) : dom  ( CC  _D  F
) --> CC  <->  ( CC  _D  F ) : X --> CC ) )
4946, 48mpbii 202 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( CC  _D  F
) : X --> CC )
5049feqmptd 5658 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( x  e.  X  |->  ( ( CC  _D  F ) `
 x ) ) )
5145, 50eqtr3d 2392 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( F `  x ) ) )  =  ( x  e.  X  |->  ( ( CC  _D  F
) `  x )
) )
52 dv11cn.e . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  G ) )
5337oveq2d 5961 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  G
)  =  ( CC 
_D  ( x  e.  X  |->  ( G `  x ) ) ) )
5452, 50, 533eqtr3rd 2399 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( CC  _D  F
) `  x )
) )
5542, 33, 44, 51, 35, 44, 54dvmptsub 19420 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( ( F `  x
)  -  ( G `
 x ) ) ) )  =  ( x  e.  X  |->  ( ( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) ) ) )
5642, 33, 44, 51dvmptcl 19412 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  CC )
5756subidd 9235 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) )  =  0 )
5857mpteq2dva 4187 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( x  e.  X  |->  0 ) )
59 fconstmpt 4814 . . . . . . . . . . . . . 14  |-  ( X  X.  { 0 } )  =  ( x  e.  X  |->  0 )
6058, 59syl6eqr 2408 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( X  X.  { 0 } ) )
6139, 55, 603eqtrd 2394 . . . . . . . . . . . 12  |-  ( ph  ->  ( CC  _D  ( F  o F  -  G
) )  =  ( X  X.  { 0 } ) )
6261dmeqd 4963 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( CC  _D  ( F  o F  -  G ) )  =  dom  ( X  X.  { 0 } ) )
63 snnzg 3819 . . . . . . . . . . . 12  |-  ( 0  e.  CC  ->  { 0 }  =/=  (/) )
64 dmxp 4979 . . . . . . . . . . . 12  |-  ( { 0 }  =/=  (/)  ->  dom  ( X  X.  { 0 } )  =  X )
6513, 63, 64mp2b 9 . . . . . . . . . . 11  |-  dom  ( X  X.  { 0 } )  =  X
6662, 65syl6eq 2406 . . . . . . . . . 10  |-  ( ph  ->  dom  ( CC  _D  ( F  o F  -  G ) )  =  X )
67 eqimss2 3307 . . . . . . . . . 10  |-  ( dom  ( CC  _D  ( F  o F  -  G
) )  =  X  ->  X  C_  dom  ( CC  _D  ( F  o F  -  G
) ) )
6866, 67syl 15 . . . . . . . . 9  |-  ( ph  ->  X  C_  dom  ( CC 
_D  ( F  o F  -  G )
) )
69 0re 8928 . . . . . . . . . 10  |-  0  e.  RR
7069a1i 10 . . . . . . . . 9  |-  ( ph  ->  0  e.  RR )
7161fveq1d 5610 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( CC  _D  ( F  o F  -  G ) ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
72 c0ex 8922 . . . . . . . . . . . . . 14  |-  0  e.  _V
7372fvconst2 5813 . . . . . . . . . . . . 13  |-  ( x  e.  X  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
7471, 73sylan9eq 2410 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  ( F  o F  -  G
) ) `  x
)  =  0 )
7574fveq2d 5612 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  o F  -  G )
) `  x )
)  =  ( abs `  0 ) )
76 abs0 11866 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
7775, 76syl6eq 2406 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  o F  -  G )
) `  x )
)  =  0 )
78 0le0 9917 . . . . . . . . . 10  |-  0  <_  0
7977, 78syl6eqbr 4141 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  o F  -  G )
) `  x )
)  <_  0 )
8031, 18, 27, 28, 7, 68, 70, 79dvlipcn 19445 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  X  /\  C  e.  X ) )  -> 
( abs `  (
( ( F  o F  -  G ) `  x )  -  (
( F  o F  -  G ) `  C ) ) )  <_  ( 0  x.  ( abs `  (
x  -  C ) ) ) )
8124, 80syldan 456 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( ( F  o F  -  G ) `  x
)  -  ( ( F  o F  -  G ) `  C
) ) )  <_ 
( 0  x.  ( abs `  ( x  -  C ) ) ) )
8238fveq1d 5610 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F  o F  -  G ) `  C )  =  ( ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) `  C
) )
83 fveq2 5608 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
84 fveq2 5608 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( G `  x )  =  ( G `  C ) )
8583, 84oveq12d 5963 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  (
( F `  x
)  -  ( G `
 x ) )  =  ( ( F `
 C )  -  ( G `  C ) ) )
86 eqid 2358 . . . . . . . . . . . . . 14  |-  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( F `
 x )  -  ( G `  x ) ) )
87 ovex 5970 . . . . . . . . . . . . . 14  |-  ( ( F `  C )  -  ( G `  C ) )  e. 
_V
8885, 86, 87fvmpt 5685 . . . . . . . . . . . . 13  |-  ( C  e.  X  ->  (
( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) `  C
)  =  ( ( F `  C )  -  ( G `  C ) ) )
8922, 88syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( x  e.  X  |->  ( ( F `
 x )  -  ( G `  x ) ) ) `  C
)  =  ( ( F `  C )  -  ( G `  C ) ) )
90 dv11cn.p . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F `  C
)  =  ( G `
 C ) )
9190oveq2d 5961 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( F `  C )  -  ( F `  C )
)  =  ( ( F `  C )  -  ( G `  C ) ) )
92 ffvelrn 5746 . . . . . . . . . . . . . . 15  |-  ( ( F : X --> CC  /\  C  e.  X )  ->  ( F `  C
)  e.  CC )
931, 22, 92syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F `  C
)  e.  CC )
9493subidd 9235 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( F `  C )  -  ( F `  C )
)  =  0 )
9591, 94eqtr3d 2392 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  C )  -  ( G `  C )
)  =  0 )
9682, 89, 953eqtrd 2394 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  o F  -  G ) `  C )  =  0 )
9796adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  o F  -  G ) `  C )  =  0 )
9897oveq2d 5961 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  o F  -  G ) `  x )  -  (
( F  o F  -  G ) `  C ) )  =  ( ( ( F  o F  -  G
) `  x )  -  0 ) )
9920subid1d 9236 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  o F  -  G ) `  x )  -  0 )  =  ( ( F  o F  -  G ) `  x
) )
10098, 99eqtrd 2390 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  o F  -  G ) `  x )  -  (
( F  o F  -  G ) `  C ) )  =  ( ( F  o F  -  G ) `  x ) )
101100fveq2d 5612 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( ( F  o F  -  G ) `  x
)  -  ( ( F  o F  -  G ) `  C
) ) )  =  ( abs `  (
( F  o F  -  G ) `  x ) ) )
10231sselda 3256 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  CC )
10331, 22sseldd 3257 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  CC )
104103adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  C  e.  CC )
105102, 104subcld 9247 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
x  -  C )  e.  CC )
106105abscld 12014 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  RR )
107106recnd 8951 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  CC )
108107mul02d 9100 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
0  x.  ( abs `  ( x  -  C
) ) )  =  0 )
10981, 101, 1083brtr3d 4133 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  o F  -  G
) `  x )
)  <_  0 )
11020absge0d 12022 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  0  <_  ( abs `  (
( F  o F  -  G ) `  x ) ) )
11120abscld 12014 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  o F  -  G
) `  x )
)  e.  RR )
112 letri3 8997 . . . . . . 7  |-  ( ( ( abs `  (
( F  o F  -  G ) `  x ) )  e.  RR  /\  0  e.  RR )  ->  (
( abs `  (
( F  o F  -  G ) `  x ) )  =  0  <->  ( ( abs `  ( ( F  o F  -  G ) `  x ) )  <_ 
0  /\  0  <_  ( abs `  ( ( F  o F  -  G ) `  x
) ) ) ) )
113111, 69, 112sylancl 643 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( abs `  (
( F  o F  -  G ) `  x ) )  =  0  <->  ( ( abs `  ( ( F  o F  -  G ) `  x ) )  <_ 
0  /\  0  <_  ( abs `  ( ( F  o F  -  G ) `  x
) ) ) ) )
114109, 110, 113mpbir2and 888 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  o F  -  G
) `  x )
)  =  0 )
11520, 114abs00d 12024 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  o F  -  G ) `  x )  =  0 )
11673adantl 452 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
117115, 116eqtr4d 2393 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  o F  -  G ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
11812, 15, 117eqfnfvd 5708 . 2  |-  ( ph  ->  ( F  o F  -  G )  =  ( X  X.  {
0 } ) )
119 ofsubeq0 9833 . . 3  |-  ( ( X  e.  _V  /\  F : X --> CC  /\  G : X --> CC )  ->  ( ( F  o F  -  G
)  =  ( X  X.  { 0 } )  <->  F  =  G
) )
12010, 1, 4, 119syl3anc 1182 . 2  |-  ( ph  ->  ( ( F  o F  -  G )  =  ( X  X.  { 0 } )  <-> 
F  =  G ) )
121118, 120mpbid 201 1  |-  ( ph  ->  F  =  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864    C_ wss 3228   (/)c0 3531   {csn 3716   {cpr 3717   class class class wbr 4104    e. cmpt 4158    X. cxp 4769   dom cdm 4771    o. ccom 4775    Fn wfn 5332   -->wf 5333   ` cfv 5337  (class class class)co 5945    o Fcof 6163   CCcc 8825   RRcr 8826   0cc0 8827    x. cmul 8832   RR*cxr 8956    <_ cle 8958    - cmin 9127   abscabs 11815   * Metcxmt 16468   ballcbl 16470    _D cdv 19317
This theorem is referenced by:  logtayl  20118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-fi 7255  df-sup 7284  df-oi 7315  df-card 7662  df-cda 7884  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-q 10409  df-rp 10447  df-xneg 10544  df-xadd 10545  df-xmul 10546  df-ioo 10752  df-ico 10754  df-icc 10755  df-fz 10875  df-fzo 10963  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-hom 13329  df-cco 13330  df-rest 13426  df-topn 13427  df-topgen 13443  df-pt 13444  df-prds 13447  df-xrs 13502  df-0g 13503  df-gsum 13504  df-qtop 13509  df-imas 13510  df-xps 13512  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-submnd 14515  df-mulg 14591  df-cntz 14892  df-cmn 15190  df-xmet 16475  df-met 16476  df-bl 16477  df-mopn 16478  df-fbas 16479  df-fg 16480  df-cnfld 16483  df-top 16742  df-bases 16744  df-topon 16745  df-topsp 16746  df-cld 16862  df-ntr 16863  df-cls 16864  df-nei 16941  df-lp 16974  df-perf 16975  df-cn 17063  df-cnp 17064  df-haus 17149  df-cmp 17220  df-tx 17363  df-hmeo 17552  df-fil 17643  df-fm 17735  df-flim 17736  df-flf 17737  df-xms 17987  df-ms 17988  df-tms 17989  df-cncf 18485  df-limc 19320  df-dv 19321
  Copyright terms: Public domain W3C validator