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Theorem efchtdvds 20413
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 20363 . . . . . . 7  |-  ( B  e.  RR  ->  ( theta `  B )  e.  RR )
213ad2ant2 977 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  B )  e.  RR )
32recnd 8877 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  B )  e.  CC )
4 chtcl 20363 . . . . . . 7  |-  ( A  e.  RR  ->  ( theta `  A )  e.  RR )
543ad2ant1 976 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  A )  e.  RR )
65recnd 8877 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  A )  e.  CC )
7 efsub 12396 . . . . 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 20403 . . . . . . . . 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 20403 . . . . . . . . 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 5892 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( theta `  ( |_ `  B ) )  -  ( theta `  ( |_ `  A ) ) )  =  ( ( theta `  B )  -  ( theta `  A ) ) )
14 flword2 10959 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( |_ `  B )  e.  ( ZZ>= `  ( |_ `  A ) ) )
15 chtdif 20412 . . . . . . . 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 2330 . . . . . 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 3271 . . . . . . . . 9  |-  { x  e.  RR  |  ( exp `  x )  e.  NN }  C_  RR
19 ax-resscn 8810 . . . . . . . . 9  |-  RR  C_  CC
2018, 19sstri 3201 . . . . . . . 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 5541 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( exp `  x )  =  ( exp `  y
) )
2322eleq1d 2362 . . . . . . . . . 10  |-  ( x  =  y  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  y )  e.  NN ) )
2423elrab 2936 . . . . . . . . 9  |-  ( y  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( y  e.  RR  /\  ( exp `  y
)  e.  NN ) )
25 fveq2 5541 . . . . . . . . . . 11  |-  ( x  =  z  ->  ( exp `  x )  =  ( exp `  z
) )
2625eleq1d 2362 . . . . . . . . . 10  |-  ( x  =  z  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  z )  e.  NN ) )
2726elrab 2936 . . . . . . . . 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 8878 . . . . . . . . . 10  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
y  +  z )  e.  RR )
3128recnd 8877 . . . . . . . . . . . 12  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  y  e.  CC )
3229recnd 8877 . . . . . . . . . . . 12  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  z  e.  CC )
33 efadd 12391 . . . . . . . . . . . 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 9785 . . . . . . . . . . . 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 2370 . . . . . . . . . 10  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  ( exp `  ( y  +  z ) )  e.  NN )
38 fveq2 5541 . . . . . . . . . . . 12  |-  ( x  =  ( y  +  z )  ->  ( exp `  x )  =  ( exp `  (
y  +  z ) ) )
3938eleq1d 2362 . . . . . . . . . . 11  |-  ( x  =  ( y  +  z )  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  ( y  +  z ) )  e.  NN ) )
4039elrab 2936 . . . . . . . . . 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 11051 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  e.  Fin )
45 inss1 3402 . . . . . . . 8  |-  ( ( ( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime )  C_  (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )
46 ssfi 7099 . . . . . . . 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 3403 . . . . . . . . . . . 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 3191 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  ->  p  e.  Prime )
51 prmnn 12777 . . . . . . . . . . 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 10405 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  ->  p  e.  RR+ )
5453relogcld 19990 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR )
5553reeflogd 19991 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  -> 
( exp `  ( log `  p ) )  =  p )
5655, 52eqeltrd 2370 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  -> 
( exp `  ( log `  p ) )  e.  NN )
57 fveq2 5541 . . . . . . . . . 10  |-  ( x  =  ( log `  p
)  ->  ( exp `  x )  =  ( exp `  ( log `  p ) ) )
5857eleq1d 2362 . . . . . . . . 9  |-  ( x  =  ( log `  p
)  ->  ( ( exp `  x )  e.  NN  <->  ( exp `  ( log `  p ) )  e.  NN ) )
5958elrab 2936 . . . . . . . 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 8854 . . . . . . . . 9  |-  0  e.  RR
62 1nn 9773 . . . . . . . . 9  |-  1  e.  NN
63 fveq2 5541 . . . . . . . . . . . 12  |-  ( x  =  0  ->  ( exp `  x )  =  ( exp `  0
) )
64 ef0 12388 . . . . . . . . . . . 12  |-  ( exp `  0 )  =  1
6563, 64syl6eq 2344 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( exp `  x )  =  1 )
6665eleq1d 2362 . . . . . . . . . 10  |-  ( x  =  0  ->  (
( exp `  x
)  e.  NN  <->  1  e.  NN ) )
6766elrab 2936 . . . . . . . . 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 12221 . . . . . 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 2370 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( theta `  B )  -  ( theta `  A
) )  e.  {
x  e.  RR  | 
( exp `  x
)  e.  NN }
)
72 fveq2 5541 . . . . . . . 8  |-  ( x  =  ( ( theta `  B )  -  ( theta `  A ) )  ->  ( exp `  x
)  =  ( exp `  ( ( theta `  B
)  -  ( theta `  A ) ) ) )
7372eleq1d 2362 . . . . . . 7  |-  ( x  =  ( ( theta `  B )  -  ( theta `  A ) )  ->  ( ( exp `  x )  e.  NN  <->  ( exp `  ( (
theta `  B )  -  ( theta `  A )
) )  e.  NN ) )
7473elrab 2936 . . . . . 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 2371 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( exp `  ( theta `  B ) )  /  ( exp `  ( theta `  A ) ) )  e.  NN )
7877nnzd 10132 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( exp `  ( theta `  B ) )  /  ( exp `  ( theta `  A ) ) )  e.  ZZ )
79 efchtcl 20365 . . . . 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 10132 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A
) )  e.  ZZ )
8280nnne0d 9806 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A
) )  =/=  0
)
83 efchtcl 20365 . . . . 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 10132 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  B
) )  e.  ZZ )
86 dvdsval2 12550 . . 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 1632    e. wcel 1696    =/= wne 2459   {crab 2560    i^i cin 3164    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   |_cfl 10940   sum_csu 12174   expce 12359    || cdivides 12547   Primecprime 12774   logclog 19928   thetaccht 20344
This theorem is referenced by:  bposlem6  20544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-dvds 12548  df-prm 12775  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cht 20350
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