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Theorem efchtdvds 20397
Description: The exponentiated Chebyshev function forms a divisibility chain between any two points. (Contributed by Mario Carneiro, 22-Sep-2014.)
Assertion
Ref Expression
efchtdvds  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A
) )  ||  ( exp `  ( theta `  B
) ) )

Proof of Theorem efchtdvds
Dummy variables  p  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 chtcl 20347 . . . . . . 7  |-  ( B  e.  RR  ->  ( theta `  B )  e.  RR )
213ad2ant2 977 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  B )  e.  RR )
32recnd 8861 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  B )  e.  CC )
4 chtcl 20347 . . . . . . 7  |-  ( A  e.  RR  ->  ( theta `  A )  e.  RR )
543ad2ant1 976 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  A )  e.  RR )
65recnd 8861 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  A )  e.  CC )
7 efsub 12380 . . . . 5  |-  ( ( ( theta `  B )  e.  CC  /\  ( theta `  A )  e.  CC )  ->  ( exp `  (
( theta `  B )  -  ( theta `  A
) ) )  =  ( ( exp `  ( theta `  B ) )  /  ( exp `  ( theta `  A ) ) ) )
83, 6, 7syl2anc 642 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( ( theta `  B )  -  ( theta `  A ) ) )  =  ( ( exp `  ( theta `  B ) )  / 
( exp `  ( theta `  A ) ) ) )
9 chtfl 20387 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( theta `  ( |_ `  B ) )  =  ( theta `  B )
)
1093ad2ant2 977 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  ( |_ `  B ) )  =  ( theta `  B )
)
11 chtfl 20387 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( theta `  ( |_ `  A ) )  =  ( theta `  A )
)
12113ad2ant1 976 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  ( |_ `  A ) )  =  ( theta `  A )
)
1310, 12oveq12d 5876 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( theta `  ( |_ `  B ) )  -  ( theta `  ( |_ `  A ) ) )  =  ( ( theta `  B )  -  ( theta `  A ) ) )
14 flword2 10943 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( |_ `  B )  e.  ( ZZ>= `  ( |_ `  A ) ) )
15 chtdif 20396 . . . . . . . 8  |-  ( ( |_ `  B )  e.  ( ZZ>= `  ( |_ `  A ) )  ->  ( ( theta `  ( |_ `  B
) )  -  ( theta `  ( |_ `  A ) ) )  =  sum_ p  e.  ( ( ( ( |_
`  A )  +  1 ) ... ( |_ `  B ) )  i^i  Prime ) ( log `  p ) )
1614, 15syl 15 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( theta `  ( |_ `  B ) )  -  ( theta `  ( |_ `  A ) ) )  =  sum_ p  e.  ( ( ( ( |_
`  A )  +  1 ) ... ( |_ `  B ) )  i^i  Prime ) ( log `  p ) )
1713, 16eqtr3d 2317 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( theta `  B )  -  ( theta `  A
) )  =  sum_ p  e.  ( ( ( ( |_ `  A
)  +  1 ) ... ( |_ `  B ) )  i^i 
Prime ) ( log `  p
) )
18 ssrab2 3258 . . . . . . . . 9  |-  { x  e.  RR  |  ( exp `  x )  e.  NN }  C_  RR
19 ax-resscn 8794 . . . . . . . . 9  |-  RR  C_  CC
2018, 19sstri 3188 . . . . . . . 8  |-  { x  e.  RR  |  ( exp `  x )  e.  NN }  C_  CC
2120a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  { x  e.  RR  |  ( exp `  x )  e.  NN }  C_  CC )
22 fveq2 5525 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( exp `  x )  =  ( exp `  y
) )
2322eleq1d 2349 . . . . . . . . . 10  |-  ( x  =  y  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  y )  e.  NN ) )
2423elrab 2923 . . . . . . . . 9  |-  ( y  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( y  e.  RR  /\  ( exp `  y
)  e.  NN ) )
25 fveq2 5525 . . . . . . . . . . 11  |-  ( x  =  z  ->  ( exp `  x )  =  ( exp `  z
) )
2625eleq1d 2349 . . . . . . . . . 10  |-  ( x  =  z  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  z )  e.  NN ) )
2726elrab 2923 . . . . . . . . 9  |-  ( z  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( z  e.  RR  /\  ( exp `  z
)  e.  NN ) )
28 simpll 730 . . . . . . . . . . 11  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  y  e.  RR )
29 simprl 732 . . . . . . . . . . 11  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  z  e.  RR )
3028, 29readdcld 8862 . . . . . . . . . 10  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
y  +  z )  e.  RR )
3128recnd 8861 . . . . . . . . . . . 12  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  y  e.  CC )
3229recnd 8861 . . . . . . . . . . . 12  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  z  e.  CC )
33 efadd 12375 . . . . . . . . . . . 12  |-  ( ( y  e.  CC  /\  z  e.  CC )  ->  ( exp `  (
y  +  z ) )  =  ( ( exp `  y )  x.  ( exp `  z
) ) )
3431, 32, 33syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  ( exp `  ( y  +  z ) )  =  ( ( exp `  y
)  x.  ( exp `  z ) ) )
35 nnmulcl 9769 . . . . . . . . . . . 12  |-  ( ( ( exp `  y
)  e.  NN  /\  ( exp `  z )  e.  NN )  -> 
( ( exp `  y
)  x.  ( exp `  z ) )  e.  NN )
3635ad2ant2l 726 . . . . . . . . . . 11  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
( exp `  y
)  x.  ( exp `  z ) )  e.  NN )
3734, 36eqeltrd 2357 . . . . . . . . . 10  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  ( exp `  ( y  +  z ) )  e.  NN )
38 fveq2 5525 . . . . . . . . . . . 12  |-  ( x  =  ( y  +  z )  ->  ( exp `  x )  =  ( exp `  (
y  +  z ) ) )
3938eleq1d 2349 . . . . . . . . . . 11  |-  ( x  =  ( y  +  z )  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  ( y  +  z ) )  e.  NN ) )
4039elrab 2923 . . . . . . . . . 10  |-  ( ( y  +  z )  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( ( y  +  z )  e.  RR  /\  ( exp `  (
y  +  z ) )  e.  NN ) )
4130, 37, 40sylanbrc 645 . . . . . . . . 9  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
y  +  z )  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } )
4224, 27, 41syl2anb 465 . . . . . . . 8  |-  ( ( y  e.  { x  e.  RR  |  ( exp `  x )  e.  NN }  /\  z  e.  {
x  e.  RR  | 
( exp `  x
)  e.  NN }
)  ->  ( y  +  z )  e. 
{ x  e.  RR  |  ( exp `  x
)  e.  NN }
)
4342adantl 452 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  ( y  e.  {
x  e.  RR  | 
( exp `  x
)  e.  NN }  /\  z  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } ) )  -> 
( y  +  z )  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } )
44 fzfid 11035 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  e.  Fin )
45 inss1 3389 . . . . . . . 8  |-  ( ( ( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime )  C_  (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )
46 ssfi 7083 . . . . . . . 8  |-  ( ( ( ( ( |_
`  A )  +  1 ) ... ( |_ `  B ) )  e.  Fin  /\  (
( ( ( |_
`  A )  +  1 ) ... ( |_ `  B ) )  i^i  Prime )  C_  (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) ) )  ->  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime )  e.  Fin )
4744, 45, 46sylancl 643 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( ( ( |_
`  A )  +  1 ) ... ( |_ `  B ) )  i^i  Prime )  e.  Fin )
48 inss2 3390 . . . . . . . . . . . 12  |-  ( ( ( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime )  C_  Prime
49 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  ->  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )
5048, 49sseldi 3178 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  ->  p  e.  Prime )
51 prmnn 12761 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  p  e.  NN )
5250, 51syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  ->  p  e.  NN )
5352nnrpd 10389 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  ->  p  e.  RR+ )
5453relogcld 19974 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR )
5553reeflogd 19975 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  -> 
( exp `  ( log `  p ) )  =  p )
5655, 52eqeltrd 2357 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  -> 
( exp `  ( log `  p ) )  e.  NN )
57 fveq2 5525 . . . . . . . . . 10  |-  ( x  =  ( log `  p
)  ->  ( exp `  x )  =  ( exp `  ( log `  p ) ) )
5857eleq1d 2349 . . . . . . . . 9  |-  ( x  =  ( log `  p
)  ->  ( ( exp `  x )  e.  NN  <->  ( exp `  ( log `  p ) )  e.  NN ) )
5958elrab 2923 . . . . . . . 8  |-  ( ( log `  p )  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( ( log `  p
)  e.  RR  /\  ( exp `  ( log `  p ) )  e.  NN ) )
6054, 56, 59sylanbrc 645 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  -> 
( log `  p
)  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } )
61 0re 8838 . . . . . . . . 9  |-  0  e.  RR
62 1nn 9757 . . . . . . . . 9  |-  1  e.  NN
63 fveq2 5525 . . . . . . . . . . . 12  |-  ( x  =  0  ->  ( exp `  x )  =  ( exp `  0
) )
64 ef0 12372 . . . . . . . . . . . 12  |-  ( exp `  0 )  =  1
6563, 64syl6eq 2331 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( exp `  x )  =  1 )
6665eleq1d 2349 . . . . . . . . . 10  |-  ( x  =  0  ->  (
( exp `  x
)  e.  NN  <->  1  e.  NN ) )
6766elrab 2923 . . . . . . . . 9  |-  ( 0  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( 0  e.  RR  /\  1  e.  NN ) )
6861, 62, 67mpbir2an 886 . . . . . . . 8  |-  0  e.  { x  e.  RR  |  ( exp `  x
)  e.  NN }
6968a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  0  e.  { x  e.  RR  |  ( exp `  x
)  e.  NN }
)
7021, 43, 47, 60, 69fsumcllem 12205 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  sum_ p  e.  ( ( ( ( |_ `  A )  +  1 ) ... ( |_ `  B
) )  i^i  Prime ) ( log `  p
)  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } )
7117, 70eqeltrd 2357 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( theta `  B )  -  ( theta `  A
) )  e.  {
x  e.  RR  | 
( exp `  x
)  e.  NN }
)
72 fveq2 5525 . . . . . . . 8  |-  ( x  =  ( ( theta `  B )  -  ( theta `  A ) )  ->  ( exp `  x
)  =  ( exp `  ( ( theta `  B
)  -  ( theta `  A ) ) ) )
7372eleq1d 2349 . . . . . . 7  |-  ( x  =  ( ( theta `  B )  -  ( theta `  A ) )  ->  ( ( exp `  x )  e.  NN  <->  ( exp `  ( (
theta `  B )  -  ( theta `  A )
) )  e.  NN ) )
7473elrab 2923 . . . . . 6  |-  ( ( ( theta `  B )  -  ( theta `  A
) )  e.  {
x  e.  RR  | 
( exp `  x
)  e.  NN }  <->  ( ( ( theta `  B
)  -  ( theta `  A ) )  e.  RR  /\  ( exp `  ( ( theta `  B
)  -  ( theta `  A ) ) )  e.  NN ) )
7574simprbi 450 . . . . 5  |-  ( ( ( theta `  B )  -  ( theta `  A
) )  e.  {
x  e.  RR  | 
( exp `  x
)  e.  NN }  ->  ( exp `  (
( theta `  B )  -  ( theta `  A
) ) )  e.  NN )
7671, 75syl 15 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( ( theta `  B )  -  ( theta `  A ) ) )  e.  NN )
778, 76eqeltrrd 2358 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( exp `  ( theta `  B ) )  /  ( exp `  ( theta `  A ) ) )  e.  NN )
7877nnzd 10116 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( exp `  ( theta `  B ) )  /  ( exp `  ( theta `  A ) ) )  e.  ZZ )
79 efchtcl 20349 . . . . 5  |-  ( A  e.  RR  ->  ( exp `  ( theta `  A
) )  e.  NN )
80793ad2ant1 976 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A
) )  e.  NN )
8180nnzd 10116 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A
) )  e.  ZZ )
8280nnne0d 9790 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A
) )  =/=  0
)
83 efchtcl 20349 . . . . 5  |-  ( B  e.  RR  ->  ( exp `  ( theta `  B
) )  e.  NN )
84833ad2ant2 977 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  B
) )  e.  NN )
8584nnzd 10116 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  B
) )  e.  ZZ )
86 dvdsval2 12534 . . 3  |-  ( ( ( exp `  ( theta `  A ) )  e.  ZZ  /\  ( exp `  ( theta `  A
) )  =/=  0  /\  ( exp `  ( theta `  B ) )  e.  ZZ )  -> 
( ( exp `  ( theta `  A ) ) 
||  ( exp `  ( theta `  B ) )  <-> 
( ( exp `  ( theta `  B ) )  /  ( exp `  ( theta `  A ) ) )  e.  ZZ ) )
8781, 82, 85, 86syl3anc 1182 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( exp `  ( theta `  A ) ) 
||  ( exp `  ( theta `  B ) )  <-> 
( ( exp `  ( theta `  B ) )  /  ( exp `  ( theta `  A ) ) )  e.  ZZ ) )
8878, 87mpbird 223 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A
) )  ||  ( exp `  ( theta `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547    i^i cin 3151    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   |_cfl 10924   sum_csu 12158   expce 12343    || cdivides 12531   Primecprime 12758   logclog 19912   thetaccht 20328
This theorem is referenced by:  bposlem6  20528
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-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-prm 12759  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-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  df-log 19914  df-cht 20334
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