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Theorem efchtdvds 23189
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 23139 . . . . . . 7  |-  ( B  e.  RR  ->  ( theta `  B )  e.  RR )
213ad2ant2 1018 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  B )  e.  RR )
32recnd 9622 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  B )  e.  CC )
4 chtcl 23139 . . . . . . 7  |-  ( A  e.  RR  ->  ( theta `  A )  e.  RR )
543ad2ant1 1017 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  A )  e.  RR )
65recnd 9622 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  A )  e.  CC )
7 efsub 13696 . . . . 5  |-  ( ( ( theta `  B )  e.  CC  /\  ( theta `  A )  e.  CC )  ->  ( exp `  (
( theta `  B )  -  ( theta `  A
) ) )  =  ( ( exp `  ( theta `  B ) )  /  ( exp `  ( theta `  A ) ) ) )
83, 6, 7syl2anc 661 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( ( theta `  B )  -  ( theta `  A ) ) )  =  ( ( exp `  ( theta `  B ) )  / 
( exp `  ( theta `  A ) ) ) )
9 chtfl 23179 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( theta `  ( |_ `  B ) )  =  ( theta `  B )
)
1093ad2ant2 1018 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  ( |_ `  B ) )  =  ( theta `  B )
)
11 chtfl 23179 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( theta `  ( |_ `  A ) )  =  ( theta `  A )
)
12113ad2ant1 1017 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  ( |_ `  A ) )  =  ( theta `  A )
)
1310, 12oveq12d 6302 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( theta `  ( |_ `  B ) )  -  ( theta `  ( |_ `  A ) ) )  =  ( ( theta `  B )  -  ( theta `  A ) ) )
14 flword2 11917 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( |_ `  B )  e.  ( ZZ>= `  ( |_ `  A ) ) )
15 chtdif 23188 . . . . . . . 8  |-  ( ( |_ `  B )  e.  ( ZZ>= `  ( |_ `  A ) )  ->  ( ( theta `  ( |_ `  B
) )  -  ( theta `  ( |_ `  A ) ) )  =  sum_ p  e.  ( ( ( ( |_
`  A )  +  1 ) ... ( |_ `  B ) )  i^i  Prime ) ( log `  p ) )
1614, 15syl 16 . . . . . . 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 2510 . . . . . 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 3585 . . . . . . . . 9  |-  { x  e.  RR  |  ( exp `  x )  e.  NN }  C_  RR
19 ax-resscn 9549 . . . . . . . . 9  |-  RR  C_  CC
2018, 19sstri 3513 . . . . . . . 8  |-  { x  e.  RR  |  ( exp `  x )  e.  NN }  C_  CC
2120a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  { x  e.  RR  |  ( exp `  x )  e.  NN }  C_  CC )
22 fveq2 5866 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( exp `  x )  =  ( exp `  y
) )
2322eleq1d 2536 . . . . . . . . . 10  |-  ( x  =  y  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  y )  e.  NN ) )
2423elrab 3261 . . . . . . . . 9  |-  ( y  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( y  e.  RR  /\  ( exp `  y
)  e.  NN ) )
25 fveq2 5866 . . . . . . . . . . 11  |-  ( x  =  z  ->  ( exp `  x )  =  ( exp `  z
) )
2625eleq1d 2536 . . . . . . . . . 10  |-  ( x  =  z  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  z )  e.  NN ) )
2726elrab 3261 . . . . . . . . 9  |-  ( z  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( z  e.  RR  /\  ( exp `  z
)  e.  NN ) )
28 simpll 753 . . . . . . . . . . 11  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  y  e.  RR )
29 simprl 755 . . . . . . . . . . 11  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  z  e.  RR )
3028, 29readdcld 9623 . . . . . . . . . 10  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
y  +  z )  e.  RR )
3128recnd 9622 . . . . . . . . . . . 12  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  y  e.  CC )
3229recnd 9622 . . . . . . . . . . . 12  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  z  e.  CC )
33 efadd 13691 . . . . . . . . . . . 12  |-  ( ( y  e.  CC  /\  z  e.  CC )  ->  ( exp `  (
y  +  z ) )  =  ( ( exp `  y )  x.  ( exp `  z
) ) )
3431, 32, 33syl2anc 661 . . . . . . . . . . 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 10559 . . . . . . . . . . . 12  |-  ( ( ( exp `  y
)  e.  NN  /\  ( exp `  z )  e.  NN )  -> 
( ( exp `  y
)  x.  ( exp `  z ) )  e.  NN )
3635ad2ant2l 745 . . . . . . . . . . 11  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
( exp `  y
)  x.  ( exp `  z ) )  e.  NN )
3734, 36eqeltrd 2555 . . . . . . . . . 10  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  ( exp `  ( y  +  z ) )  e.  NN )
38 fveq2 5866 . . . . . . . . . . . 12  |-  ( x  =  ( y  +  z )  ->  ( exp `  x )  =  ( exp `  (
y  +  z ) ) )
3938eleq1d 2536 . . . . . . . . . . 11  |-  ( x  =  ( y  +  z )  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  ( y  +  z ) )  e.  NN ) )
4039elrab 3261 . . . . . . . . . 10  |-  ( ( y  +  z )  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( ( y  +  z )  e.  RR  /\  ( exp `  (
y  +  z ) )  e.  NN ) )
4130, 37, 40sylanbrc 664 . . . . . . . . 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 479 . . . . . . . 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 466 . . . . . . 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 12051 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  e.  Fin )
45 inss1 3718 . . . . . . . 8  |-  ( ( ( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime )  C_  (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )
46 ssfi 7740 . . . . . . . 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 662 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( ( ( |_
`  A )  +  1 ) ... ( |_ `  B ) )  i^i  Prime )  e.  Fin )
48 inss2 3719 . . . . . . . . . . . 12  |-  ( ( ( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime )  C_  Prime
49 simpr 461 . . . . . . . . . . . 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 3502 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  ->  p  e.  Prime )
51 prmnn 14079 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  p  e.  NN )
5250, 51syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  ->  p  e.  NN )
5352nnrpd 11255 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  ->  p  e.  RR+ )
5453relogcld 22764 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR )
5553reeflogd 22765 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  -> 
( exp `  ( log `  p ) )  =  p )
5655, 52eqeltrd 2555 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  -> 
( exp `  ( log `  p ) )  e.  NN )
57 fveq2 5866 . . . . . . . . . 10  |-  ( x  =  ( log `  p
)  ->  ( exp `  x )  =  ( exp `  ( log `  p ) ) )
5857eleq1d 2536 . . . . . . . . 9  |-  ( x  =  ( log `  p
)  ->  ( ( exp `  x )  e.  NN  <->  ( exp `  ( log `  p ) )  e.  NN ) )
5958elrab 3261 . . . . . . . 8  |-  ( ( log `  p )  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( ( log `  p
)  e.  RR  /\  ( exp `  ( log `  p ) )  e.  NN ) )
6054, 56, 59sylanbrc 664 . . . . . . 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 9596 . . . . . . . . 9  |-  0  e.  RR
62 1nn 10547 . . . . . . . . 9  |-  1  e.  NN
63 fveq2 5866 . . . . . . . . . . . 12  |-  ( x  =  0  ->  ( exp `  x )  =  ( exp `  0
) )
64 ef0 13688 . . . . . . . . . . . 12  |-  ( exp `  0 )  =  1
6563, 64syl6eq 2524 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( exp `  x )  =  1 )
6665eleq1d 2536 . . . . . . . . . 10  |-  ( x  =  0  ->  (
( exp `  x
)  e.  NN  <->  1  e.  NN ) )
6766elrab 3261 . . . . . . . . 9  |-  ( 0  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( 0  e.  RR  /\  1  e.  NN ) )
6861, 62, 67mpbir2an 918 . . . . . . . 8  |-  0  e.  { x  e.  RR  |  ( exp `  x
)  e.  NN }
6968a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  0  e.  { x  e.  RR  |  ( exp `  x
)  e.  NN }
)
7021, 43, 47, 60, 69fsumcllem 13517 . . . . . 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 2555 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( theta `  B )  -  ( theta `  A
) )  e.  {
x  e.  RR  | 
( exp `  x
)  e.  NN }
)
72 fveq2 5866 . . . . . . . 8  |-  ( x  =  ( ( theta `  B )  -  ( theta `  A ) )  ->  ( exp `  x
)  =  ( exp `  ( ( theta `  B
)  -  ( theta `  A ) ) ) )
7372eleq1d 2536 . . . . . . 7  |-  ( x  =  ( ( theta `  B )  -  ( theta `  A ) )  ->  ( ( exp `  x )  e.  NN  <->  ( exp `  ( (
theta `  B )  -  ( theta `  A )
) )  e.  NN ) )
7473elrab 3261 . . . . . 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 464 . . . . 5  |-  ( ( ( theta `  B )  -  ( theta `  A
) )  e.  {
x  e.  RR  | 
( exp `  x
)  e.  NN }  ->  ( exp `  (
( theta `  B )  -  ( theta `  A
) ) )  e.  NN )
7671, 75syl 16 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( ( theta `  B )  -  ( theta `  A ) ) )  e.  NN )
778, 76eqeltrrd 2556 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( exp `  ( theta `  B ) )  /  ( exp `  ( theta `  A ) ) )  e.  NN )
7877nnzd 10965 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( exp `  ( theta `  B ) )  /  ( exp `  ( theta `  A ) ) )  e.  ZZ )
79 efchtcl 23141 . . . . 5  |-  ( A  e.  RR  ->  ( exp `  ( theta `  A
) )  e.  NN )
80793ad2ant1 1017 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A
) )  e.  NN )
8180nnzd 10965 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A
) )  e.  ZZ )
8280nnne0d 10580 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A
) )  =/=  0
)
83 efchtcl 23141 . . . . 5  |-  ( B  e.  RR  ->  ( exp `  ( theta `  B
) )  e.  NN )
84833ad2ant2 1018 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  B
) )  e.  NN )
8584nnzd 10965 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  B
) )  e.  ZZ )
86 dvdsval2 13850 . . 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 1228 . 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 232 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A
) )  ||  ( exp `  ( theta `  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818    i^i cin 3475    C_ wss 3476   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Fincfn 7516   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    <_ cle 9629    - cmin 9805    / cdiv 10206   NNcn 10536   ZZcz 10864   ZZ>=cuz 11082   ...cfz 11672   |_cfl 11895   sum_csu 13471   expce 13659    || cdivides 13847   Primecprime 14076   logclog 22698   thetaccht 23120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-sum 13472  df-ef 13665  df-sin 13667  df-cos 13668  df-pi 13670  df-dvds 13848  df-prm 14077  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-haus 19610  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-xms 20586  df-ms 20587  df-tms 20588  df-cncf 21145  df-limc 22033  df-dv 22034  df-log 22700  df-cht 23126
This theorem is referenced by:  bposlem6  23320
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