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Theorem efchtdvds 23816
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 23766 . . . . . . 7  |-  ( B  e.  RR  ->  ( theta `  B )  e.  RR )
213ad2ant2 1021 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  B )  e.  RR )
32recnd 9654 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  B )  e.  CC )
4 chtcl 23766 . . . . . . 7  |-  ( A  e.  RR  ->  ( theta `  A )  e.  RR )
543ad2ant1 1020 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  A )  e.  RR )
65recnd 9654 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  A )  e.  CC )
7 efsub 14046 . . . . 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 23806 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( theta `  ( |_ `  B ) )  =  ( theta `  B )
)
1093ad2ant2 1021 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  ( |_ `  B ) )  =  ( theta `  B )
)
11 chtfl 23806 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( theta `  ( |_ `  A ) )  =  ( theta `  A )
)
12113ad2ant1 1020 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( theta `  ( |_ `  A ) )  =  ( theta `  A )
)
1310, 12oveq12d 6298 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( theta `  ( |_ `  B ) )  -  ( theta `  ( |_ `  A ) ) )  =  ( ( theta `  B )  -  ( theta `  A ) ) )
14 flword2 11988 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( |_ `  B )  e.  ( ZZ>= `  ( |_ `  A ) ) )
15 chtdif 23815 . . . . . . . 8  |-  ( ( |_ `  B )  e.  ( ZZ>= `  ( |_ `  A ) )  ->  ( ( theta `  ( |_ `  B
) )  -  ( theta `  ( |_ `  A ) ) )  =  sum_ p  e.  ( ( ( ( |_
`  A )  +  1 ) ... ( |_ `  B ) )  i^i  Prime ) ( log `  p ) )
1614, 15syl 17 . . . . . . 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 2447 . . . . . 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 3526 . . . . . . . . 9  |-  { x  e.  RR  |  ( exp `  x )  e.  NN }  C_  RR
19 ax-resscn 9581 . . . . . . . . 9  |-  RR  C_  CC
2018, 19sstri 3453 . . . . . . . 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 5851 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( exp `  x )  =  ( exp `  y
) )
2322eleq1d 2473 . . . . . . . . . 10  |-  ( x  =  y  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  y )  e.  NN ) )
2423elrab 3209 . . . . . . . . 9  |-  ( y  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( y  e.  RR  /\  ( exp `  y
)  e.  NN ) )
25 fveq2 5851 . . . . . . . . . . 11  |-  ( x  =  z  ->  ( exp `  x )  =  ( exp `  z
) )
2625eleq1d 2473 . . . . . . . . . 10  |-  ( x  =  z  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  z )  e.  NN ) )
2726elrab 3209 . . . . . . . . 9  |-  ( z  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( z  e.  RR  /\  ( exp `  z
)  e.  NN ) )
28 simpll 754 . . . . . . . . . . 11  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  y  e.  RR )
29 simprl 758 . . . . . . . . . . 11  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  z  e.  RR )
3028, 29readdcld 9655 . . . . . . . . . 10  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
y  +  z )  e.  RR )
3128recnd 9654 . . . . . . . . . . . 12  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  y  e.  CC )
3229recnd 9654 . . . . . . . . . . . 12  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  z  e.  CC )
33 efadd 14040 . . . . . . . . . . . 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 10601 . . . . . . . . . . . 12  |-  ( ( ( exp `  y
)  e.  NN  /\  ( exp `  z )  e.  NN )  -> 
( ( exp `  y
)  x.  ( exp `  z ) )  e.  NN )
3635ad2ant2l 746 . . . . . . . . . . 11  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
( exp `  y
)  x.  ( exp `  z ) )  e.  NN )
3734, 36eqeltrd 2492 . . . . . . . . . 10  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  ( exp `  ( y  +  z ) )  e.  NN )
38 fveq2 5851 . . . . . . . . . . . 12  |-  ( x  =  ( y  +  z )  ->  ( exp `  x )  =  ( exp `  (
y  +  z ) ) )
3938eleq1d 2473 . . . . . . . . . . 11  |-  ( x  =  ( y  +  z )  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  ( y  +  z ) )  e.  NN ) )
4039elrab 3209 . . . . . . . . . 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 12126 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  e.  Fin )
45 inss1 3661 . . . . . . . 8  |-  ( ( ( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime )  C_  (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )
46 ssfi 7777 . . . . . . . 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 3662 . . . . . . . . . . . 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 3442 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  ->  p  e.  Prime )
51 prmnn 14431 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  p  e.  NN )
5250, 51syl 17 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  ->  p  e.  NN )
5352nnrpd 11304 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  ->  p  e.  RR+ )
5453relogcld 23304 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR )
5553reeflogd 23305 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  -> 
( exp `  ( log `  p ) )  =  p )
5655, 52eqeltrd 2492 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  p  e.  ( (
( ( |_ `  A )  +  1 ) ... ( |_
`  B ) )  i^i  Prime ) )  -> 
( exp `  ( log `  p ) )  e.  NN )
57 fveq2 5851 . . . . . . . . . 10  |-  ( x  =  ( log `  p
)  ->  ( exp `  x )  =  ( exp `  ( log `  p ) ) )
5857eleq1d 2473 . . . . . . . . 9  |-  ( x  =  ( log `  p
)  ->  ( ( exp `  x )  e.  NN  <->  ( exp `  ( log `  p ) )  e.  NN ) )
5958elrab 3209 . . . . . . . 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 9628 . . . . . . . . 9  |-  0  e.  RR
62 1nn 10589 . . . . . . . . 9  |-  1  e.  NN
63 fveq2 5851 . . . . . . . . . . . 12  |-  ( x  =  0  ->  ( exp `  x )  =  ( exp `  0
) )
64 ef0 14037 . . . . . . . . . . . 12  |-  ( exp `  0 )  =  1
6563, 64syl6eq 2461 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( exp `  x )  =  1 )
6665eleq1d 2473 . . . . . . . . . 10  |-  ( x  =  0  ->  (
( exp `  x
)  e.  NN  <->  1  e.  NN ) )
6766elrab 3209 . . . . . . . . 9  |-  ( 0  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( 0  e.  RR  /\  1  e.  NN ) )
6861, 62, 67mpbir2an 923 . . . . . . . 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 13705 . . . . . 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 2492 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( theta `  B )  -  ( theta `  A
) )  e.  {
x  e.  RR  | 
( exp `  x
)  e.  NN }
)
72 fveq2 5851 . . . . . . . 8  |-  ( x  =  ( ( theta `  B )  -  ( theta `  A ) )  ->  ( exp `  x
)  =  ( exp `  ( ( theta `  B
)  -  ( theta `  A ) ) ) )
7372eleq1d 2473 . . . . . . 7  |-  ( x  =  ( ( theta `  B )  -  ( theta `  A ) )  ->  ( ( exp `  x )  e.  NN  <->  ( exp `  ( (
theta `  B )  -  ( theta `  A )
) )  e.  NN ) )
7473elrab 3209 . . . . . 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 17 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( ( theta `  B )  -  ( theta `  A ) ) )  e.  NN )
778, 76eqeltrrd 2493 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( exp `  ( theta `  B ) )  /  ( exp `  ( theta `  A ) ) )  e.  NN )
7877nnzd 11009 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( exp `  ( theta `  B ) )  /  ( exp `  ( theta `  A ) ) )  e.  ZZ )
79 efchtcl 23768 . . . . 5  |-  ( A  e.  RR  ->  ( exp `  ( theta `  A
) )  e.  NN )
80793ad2ant1 1020 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A
) )  e.  NN )
8180nnzd 11009 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A
) )  e.  ZZ )
8280nnne0d 10623 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  A
) )  =/=  0
)
83 efchtcl 23768 . . . . 5  |-  ( B  e.  RR  ->  ( exp `  ( theta `  B
) )  e.  NN )
84833ad2ant2 1021 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  B
) )  e.  NN )
8584nnzd 11009 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( exp `  ( theta `  B
) )  e.  ZZ )
86 dvdsval2 14200 . . 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 1232 . 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 234 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 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    =/= wne 2600   {crab 2760    i^i cin 3415    C_ wss 3416   class class class wbr 4397   ` cfv 5571  (class class class)co 6280   Fincfn 7556   CCcc 9522   RRcr 9523   0cc0 9524   1c1 9525    + caddc 9527    x. cmul 9529    <_ cle 9661    - cmin 9843    / cdiv 10249   NNcn 10578   ZZcz 10907   ZZ>=cuz 11129   ...cfz 11728   |_cfl 11966   sum_csu 13659   expce 14008    || cdvds 14197   Primecprime 14428   logclog 23236   thetaccht 23747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602  ax-addf 9603  ax-mulf 9604
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523  df-om 6686  df-1st 6786  df-2nd 6787  df-supp 6905  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-2o 7170  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-ixp 7510  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-fsupp 7866  df-fi 7907  df-sup 7937  df-oi 7971  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-q 11230  df-rp 11268  df-xneg 11373  df-xadd 11374  df-xmul 11375  df-ioo 11588  df-ioc 11589  df-ico 11590  df-icc 11591  df-fz 11729  df-fzo 11857  df-fl 11968  df-mod 12037  df-seq 12154  df-exp 12213  df-fac 12400  df-bc 12427  df-hash 12455  df-shft 13051  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-limsup 13445  df-clim 13462  df-rlim 13463  df-sum 13660  df-ef 14014  df-sin 14016  df-cos 14017  df-pi 14019  df-dvds 14198  df-prm 14429  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-starv 14926  df-sca 14927  df-vsca 14928  df-ip 14929  df-tset 14930  df-ple 14931  df-ds 14933  df-unif 14934  df-hom 14935  df-cco 14936  df-rest 15039  df-topn 15040  df-0g 15058  df-gsum 15059  df-topgen 15060  df-pt 15061  df-prds 15064  df-xrs 15118  df-qtop 15123  df-imas 15124  df-xps 15126  df-mre 15202  df-mrc 15203  df-acs 15205  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-submnd 16293  df-mulg 16386  df-cntz 16681  df-cmn 17126  df-psmet 18733  df-xmet 18734  df-met 18735  df-bl 18736  df-mopn 18737  df-fbas 18738  df-fg 18739  df-cnfld 18743  df-top 19693  df-bases 19695  df-topon 19696  df-topsp 19697  df-cld 19814  df-ntr 19815  df-cls 19816  df-nei 19894  df-lp 19932  df-perf 19933  df-cn 20023  df-cnp 20024  df-haus 20111  df-tx 20357  df-hmeo 20550  df-fil 20641  df-fm 20733  df-flim 20734  df-flf 20735  df-xms 21117  df-ms 21118  df-tms 21119  df-cncf 21676  df-limc 22564  df-dv 22565  df-log 23238  df-cht 23753
This theorem is referenced by:  bposlem6  23947
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