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Theorem dv11cn 19348
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 5389 . . . . 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 5389 . . . . 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 5883 . . . . . 6  |-  ( A ( ball `  ( abs  o.  -  ) ) R )  e.  _V
97, 8eqeltri 2353 . . . . 5  |-  X  e. 
_V
109a1i 10 . . . 4  |-  ( ph  ->  X  e.  _V )
11 inidm 3378 . . . 4  |-  ( X  i^i  X )  =  X
123, 6, 10, 10, 11offn 6089 . . 3  |-  ( ph  ->  ( F  o F  -  G )  Fn  X )
13 0cn 8831 . . . 4  |-  0  e.  CC
14 fnconstg 5429 . . . 4  |-  ( 0  e.  CC  ->  ( X  X.  { 0 } )  Fn  X )
1513, 14mp1i 11 . . 3  |-  ( ph  ->  ( X  X.  {
0 } )  Fn  X )
16 subcl 9051 . . . . . . . 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 6093 . . . . . 6  |-  ( ph  ->  ( F  o F  -  G ) : X --> CC )
19 ffvelrn 5663 . . . . . 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 18282 . . . . . . . . . . . 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 17968 . . . . . . . . . . 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 3222 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
32 ffvelrn 5663 . . . . . . . . . . . . . . . 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 5663 . . . . . . . . . . . . . . . 16  |-  ( ( G : X --> CC  /\  x  e.  X )  ->  ( G `  x
)  e.  CC )
354, 34sylan 457 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
361feqmptd 5575 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
374feqmptd 5575 . . . . . . . . . . . . . . 15  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
3810, 33, 35, 36, 37offval2 6095 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  o F  -  G )  =  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) )
3938oveq2d 5874 . . . . . . . . . . . . 13  |-  ( ph  ->  ( CC  _D  ( F  o F  -  G
) )  =  ( CC  _D  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) ) ) )
40 cnex 8818 . . . . . . . . . . . . . . . 16  |-  CC  e.  _V
4140prid2 3735 . . . . . . . . . . . . . . 15  |-  CC  e.  { RR ,  CC }
4241a1i 10 . . . . . . . . . . . . . 14  |-  ( ph  ->  CC  e.  { RR ,  CC } )
43 fvex 5539 . . . . . . . . . . . . . . 15  |-  ( ( CC  _D  F ) `
 x )  e. 
_V
4443a1i 10 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  _V )
4536oveq2d 5874 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( CC 
_D  ( x  e.  X  |->  ( F `  x ) ) ) )
46 dvfcn 19258 . . . . . . . . . . . . . . . . 17  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
47 dv11cn.d . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  dom  ( CC  _D  F )  =  X )
4847feq2d 5380 . . . . . . . . . . . . . . . . 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 5575 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  F
)  =  ( x  e.  X  |->  ( ( CC  _D  F ) `
 x ) ) )
5145, 50eqtr3d 2317 . . . . . . . . . . . . . 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 5874 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( CC  _D  G
)  =  ( CC 
_D  ( x  e.  X  |->  ( G `  x ) ) ) )
5452, 50, 533eqtr3rd 2324 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( CC  _D  (
x  e.  X  |->  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( CC  _D  F
) `  x )
) )
5542, 33, 44, 51, 35, 44, 54dvmptsub 19316 . . . . . . . . . . . . 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 19308 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  F
) `  x )  e.  CC )
5756subidd 9145 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( CC  _D  F ) `  x
)  -  ( ( CC  _D  F ) `
 x ) )  =  0 )
5857mpteq2dva 4106 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( x  e.  X  |->  0 ) )
59 fconstmpt 4732 . . . . . . . . . . . . . 14  |-  ( X  X.  { 0 } )  =  ( x  e.  X  |->  0 )
6058, 59syl6eqr 2333 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  X  |->  ( ( ( CC 
_D  F ) `  x )  -  (
( CC  _D  F
) `  x )
) )  =  ( X  X.  { 0 } ) )
6139, 55, 603eqtrd 2319 . . . . . . . . . . . 12  |-  ( ph  ->  ( CC  _D  ( F  o F  -  G
) )  =  ( X  X.  { 0 } ) )
6261dmeqd 4881 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( CC  _D  ( F  o F  -  G ) )  =  dom  ( X  X.  { 0 } ) )
63 snnzg 3743 . . . . . . . . . . . 12  |-  ( 0  e.  CC  ->  { 0 }  =/=  (/) )
64 dmxp 4897 . . . . . . . . . . . 12  |-  ( { 0 }  =/=  (/)  ->  dom  ( X  X.  { 0 } )  =  X )
6513, 63, 64mp2b 9 . . . . . . . . . . 11  |-  dom  ( X  X.  { 0 } )  =  X
6662, 65syl6eq 2331 . . . . . . . . . 10  |-  ( ph  ->  dom  ( CC  _D  ( F  o F  -  G ) )  =  X )
67 eqimss2 3231 . . . . . . . . . 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 8838 . . . . . . . . . 10  |-  0  e.  RR
7069a1i 10 . . . . . . . . 9  |-  ( ph  ->  0  e.  RR )
7161fveq1d 5527 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( CC  _D  ( F  o F  -  G ) ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
72 c0ex 8832 . . . . . . . . . . . . . 14  |-  0  e.  _V
7372fvconst2 5729 . . . . . . . . . . . . 13  |-  ( x  e.  X  ->  (
( X  X.  {
0 } ) `  x )  =  0 )
7471, 73sylan9eq 2335 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  (
( CC  _D  ( F  o F  -  G
) ) `  x
)  =  0 )
7574fveq2d 5529 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  o F  -  G )
) `  x )
)  =  ( abs `  0 ) )
76 abs0 11770 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
7775, 76syl6eq 2331 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  o F  -  G )
) `  x )
)  =  0 )
78 0le0 9827 . . . . . . . . . 10  |-  0  <_  0
7977, 78syl6eqbr 4060 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( CC 
_D  ( F  o F  -  G )
) `  x )
)  <_  0 )
8031, 18, 27, 28, 7, 68, 70, 79dvlipcn 19341 . . . . . . . 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 5527 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F  o F  -  G ) `  C )  =  ( ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x )
) ) `  C
) )
83 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( F `  x )  =  ( F `  C ) )
84 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( x  =  C  ->  ( G `  x )  =  ( G `  C ) )
8583, 84oveq12d 5876 . . . . . . . . . . . . . 14  |-  ( x  =  C  ->  (
( F `  x
)  -  ( G `
 x ) )  =  ( ( F `
 C )  -  ( G `  C ) ) )
86 eqid 2283 . . . . . . . . . . . . . 14  |-  ( x  e.  X  |->  ( ( F `  x )  -  ( G `  x ) ) )  =  ( x  e.  X  |->  ( ( F `
 x )  -  ( G `  x ) ) )
87 ovex 5883 . . . . . . . . . . . . . 14  |-  ( ( F `  C )  -  ( G `  C ) )  e. 
_V
8885, 86, 87fvmpt 5602 . . . . . . . . . . . . 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 5874 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( F `  C )  -  ( F `  C )
)  =  ( ( F `  C )  -  ( G `  C ) ) )
92 ffvelrn 5663 . . . . . . . . . . . . . . 15  |-  ( ( F : X --> CC  /\  C  e.  X )  ->  ( F `  C
)  e.  CC )
931, 22, 92syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F `  C
)  e.  CC )
9493subidd 9145 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( F `  C )  -  ( F `  C )
)  =  0 )
9591, 94eqtr3d 2317 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  C )  -  ( G `  C )
)  =  0 )
9682, 89, 953eqtrd 2319 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  o F  -  G ) `  C )  =  0 )
9796adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  o F  -  G ) `  C )  =  0 )
9897oveq2d 5874 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  o F  -  G ) `  x )  -  (
( F  o F  -  G ) `  C ) )  =  ( ( ( F  o F  -  G
) `  x )  -  0 ) )
9920subid1d 9146 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  o F  -  G ) `  x )  -  0 )  =  ( ( F  o F  -  G ) `  x
) )
10098, 99eqtrd 2315 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( F  o F  -  G ) `  x )  -  (
( F  o F  -  G ) `  C ) )  =  ( ( F  o F  -  G ) `  x ) )
101100fveq2d 5529 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( ( F  o F  -  G ) `  x
)  -  ( ( F  o F  -  G ) `  C
) ) )  =  ( abs `  (
( F  o F  -  G ) `  x ) ) )
10231sselda 3180 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  CC )
10331, 22sseldd 3181 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  CC )
104103adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  C  e.  CC )
105102, 104subcld 9157 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
x  -  C )  e.  CC )
106105abscld 11918 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  RR )
107106recnd 8861 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( x  -  C ) )  e.  CC )
108107mul02d 9010 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
0  x.  ( abs `  ( x  -  C
) ) )  =  0 )
10981, 101, 1083brtr3d 4052 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  o F  -  G
) `  x )
)  <_  0 )
11020absge0d 11926 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  0  <_  ( abs `  (
( F  o F  -  G ) `  x ) ) )
11120abscld 11918 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( abs `  ( ( F  o F  -  G
) `  x )
)  e.  RR )
112 letri3 8907 . . . . . . 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 11928 . . . 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 2318 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( F  o F  -  G ) `  x )  =  ( ( X  X.  {
0 } ) `  x ) )
11812, 15, 117eqfnfvd 5625 . 2  |-  ( ph  ->  ( F  o F  -  G )  =  ( X  X.  {
0 } ) )
119 ofsubeq0 9743 . . 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    C_ wss 3152   (/)c0 3455   {csn 3640   {cpr 3641   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   dom cdm 4689    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   RRcr 8736   0cc0 8737    x. cmul 8742   RR*cxr 8866    <_ cle 8868    - cmin 9037   abscabs 11719   * Metcxmt 16369   ballcbl 16371    _D cdv 19213
This theorem is referenced by:  logtayl  20007
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
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