Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elaa2 Structured version   Visualization version   Unicode version

Theorem elaa2 38099
Description: Elementhood in the set of nonzero algebraic numbers: when  A is nonzero, the polynomial  f can be chosen with a nonzero constant term. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Proof shortened by AV, 1-Oct-2020.)
Assertion
Ref Expression
elaa2  |-  ( A  e.  ( AA  \  { 0 } )  <-> 
( A  e.  CC  /\ 
E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0 )  =/=  0  /\  ( f `
 A )  =  0 ) ) )
Distinct variable group:    A, f

Proof of Theorem elaa2
Dummy variables  g 
k  z  j  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aasscn 23271 . . . 4  |-  AA  C_  CC
2 eldifi 3555 . . . 4  |-  ( A  e.  ( AA  \  { 0 } )  ->  A  e.  AA )
31, 2sseldi 3430 . . 3  |-  ( A  e.  ( AA  \  { 0 } )  ->  A  e.  CC )
4 elaa 23269 . . . . . 6  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. g  e.  ( (Poly `  ZZ )  \  { 0p } ) ( g `
 A )  =  0 ) )
52, 4sylib 200 . . . . 5  |-  ( A  e.  ( AA  \  { 0 } )  ->  ( A  e.  CC  /\  E. g  e.  ( (Poly `  ZZ )  \  { 0p } ) ( g `
 A )  =  0 ) )
65simprd 465 . . . 4  |-  ( A  e.  ( AA  \  { 0 } )  ->  E. g  e.  ( (Poly `  ZZ )  \  { 0p }
) ( g `  A )  =  0 )
723ad2ant1 1029 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  ->  A  e.  AA )
8 eldifsni 4098 . . . . . . 7  |-  ( A  e.  ( AA  \  { 0 } )  ->  A  =/=  0
)
983ad2ant1 1029 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  ->  A  =/=  0 )
10 eldifi 3555 . . . . . . 7  |-  ( g  e.  ( (Poly `  ZZ )  \  { 0p } )  -> 
g  e.  (Poly `  ZZ ) )
11103ad2ant2 1030 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  -> 
g  e.  (Poly `  ZZ ) )
12 eldifsni 4098 . . . . . . 7  |-  ( g  e.  ( (Poly `  ZZ )  \  { 0p } )  -> 
g  =/=  0p )
13123ad2ant2 1030 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  -> 
g  =/=  0p )
14 simp3 1010 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  -> 
( g `  A
)  =  0 )
15 fveq2 5865 . . . . . . . . 9  |-  ( m  =  n  ->  (
(coeff `  g ) `  m )  =  ( (coeff `  g ) `  n ) )
1615neeq1d 2683 . . . . . . . 8  |-  ( m  =  n  ->  (
( (coeff `  g
) `  m )  =/=  0  <->  ( (coeff `  g ) `  n
)  =/=  0 ) )
1716cbvrabv 3044 . . . . . . 7  |-  { m  e.  NN0  |  ( (coeff `  g ) `  m
)  =/=  0 }  =  { n  e. 
NN0  |  ( (coeff `  g ) `  n
)  =/=  0 }
1817infeq1i 7994 . . . . . 6  |- inf ( { m  e.  NN0  | 
( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  )  = inf ( { n  e.  NN0  |  ( (coeff `  g
) `  n )  =/=  0 } ,  RR ,  <  )
19 oveq1 6297 . . . . . . . 8  |-  ( j  =  k  ->  (
j  + inf ( {
m  e.  NN0  | 
( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  ) )  =  ( k  + inf ( { m  e.  NN0  |  ( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  ) ) )
2019fveq2d 5869 . . . . . . 7  |-  ( j  =  k  ->  (
(coeff `  g ) `  ( j  + inf ( { m  e.  NN0  |  ( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  ) ) )  =  ( (coeff `  g ) `  (
k  + inf ( {
m  e.  NN0  | 
( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  ) ) ) )
2120cbvmptv 4495 . . . . . 6  |-  ( j  e.  NN0  |->  ( (coeff `  g ) `  (
j  + inf ( {
m  e.  NN0  | 
( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  ) ) ) )  =  ( k  e.  NN0  |->  ( (coeff `  g ) `  (
k  + inf ( {
m  e.  NN0  | 
( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  ) ) ) )
22 eqid 2451 . . . . . 6  |-  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (
(deg `  g )  - inf ( { m  e. 
NN0  |  ( (coeff `  g ) `  m
)  =/=  0 } ,  RR ,  <  ) ) ) ( ( ( j  e.  NN0  |->  ( (coeff `  g ) `  ( j  + inf ( { m  e.  NN0  |  ( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  ) ) ) ) `  k )  x.  ( z ^
k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... ( (deg
`  g )  - inf ( { m  e.  NN0  |  ( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  ) ) ) ( ( ( j  e.  NN0  |->  ( (coeff `  g ) `  (
j  + inf ( {
m  e.  NN0  | 
( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  ) ) ) ) `  k )  x.  ( z ^
k ) ) )
237, 9, 11, 13, 14, 18, 21, 22elaa2lem 38097 . . . . 5  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  ->  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0 )  =/=  0  /\  ( f `
 A )  =  0 ) )
2423rexlimdv3a 2881 . . . 4  |-  ( A  e.  ( AA  \  { 0 } )  ->  ( E. g  e.  ( (Poly `  ZZ )  \  { 0p } ) ( g `
 A )  =  0  ->  E. f  e.  (Poly `  ZZ )
( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) ) )
256, 24mpd 15 . . 3  |-  ( A  e.  ( AA  \  { 0 } )  ->  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0 )  =/=  0  /\  ( f `
 A )  =  0 ) )
263, 25jca 535 . 2  |-  ( A  e.  ( AA  \  { 0 } )  ->  ( A  e.  CC  /\  E. f  e.  (Poly `  ZZ )
( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) ) )
27 simpl 459 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  ->  f  e.  (Poly `  ZZ )
)
28 fveq2 5865 . . . . . . . . . . . . . . 15  |-  ( f  =  0p  -> 
(coeff `  f )  =  (coeff `  0p
) )
29 coe0 23210 . . . . . . . . . . . . . . 15  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
3028, 29syl6eq 2501 . . . . . . . . . . . . . 14  |-  ( f  =  0p  -> 
(coeff `  f )  =  ( NN0  X.  { 0 } ) )
3130fveq1d 5867 . . . . . . . . . . . . 13  |-  ( f  =  0p  -> 
( (coeff `  f
) `  0 )  =  ( ( NN0 
X.  { 0 } ) `  0 ) )
32 0nn0 10884 . . . . . . . . . . . . . 14  |-  0  e.  NN0
33 fvconst2g 6118 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  NN0  /\  0  e.  NN0 )  -> 
( ( NN0  X.  { 0 } ) `
 0 )  =  0 )
3432, 32, 33mp2an 678 . . . . . . . . . . . . 13  |-  ( ( NN0  X.  { 0 } ) `  0
)  =  0
3531, 34syl6eq 2501 . . . . . . . . . . . 12  |-  ( f  =  0p  -> 
( (coeff `  f
) `  0 )  =  0 )
3635adantl 468 . . . . . . . . . . 11  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  f  =  0p )  ->  ( (coeff `  f ) `  0
)  =  0 )
37 neneq 2630 . . . . . . . . . . . 12  |-  ( ( (coeff `  f ) `  0 )  =/=  0  ->  -.  (
(coeff `  f ) `  0 )  =  0 )
3837ad2antlr 733 . . . . . . . . . . 11  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  f  =  0p )  ->  -.  ( (coeff `  f ) `  0
)  =  0 )
3936, 38pm2.65da 580 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  ->  -.  f  =  0p
)
40 elsn 3982 . . . . . . . . . 10  |-  ( f  e.  { 0p }  <->  f  =  0p )
4139, 40sylnibr 307 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  ->  -.  f  e.  { 0p } )
4227, 41eldifd 3415 . . . . . . . 8  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  ->  f  e.  ( (Poly `  ZZ )  \  { 0p } ) )
4342adantrr 723 . . . . . . 7  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  ->  f  e.  ( (Poly `  ZZ )  \  { 0p }
) )
44 simprr 766 . . . . . . 7  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  ->  ( f `  A )  =  0 )
4543, 44jca 535 . . . . . 6  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  ->  ( f  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
f `  A )  =  0 ) )
4645reximi2 2854 . . . . 5  |-  ( E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 )  ->  E. f  e.  ( (Poly `  ZZ )  \  { 0p }
) ( f `  A )  =  0 )
4746anim2i 573 . . . 4  |-  ( ( A  e.  CC  /\  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  { 0p } ) ( f `
 A )  =  0 ) )
48 elaa 23269 . . . 4  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  { 0p } ) ( f `
 A )  =  0 ) )
4947, 48sylibr 216 . . 3  |-  ( ( A  e.  CC  /\  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  A  e.  AA )
50 simpr 463 . . . 4  |-  ( ( A  e.  CC  /\  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  E. f  e.  (Poly `  ZZ )
( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )
51 nfv 1761 . . . . . 6  |-  F/ f  A  e.  CC
52 nfre1 2848 . . . . . 6  |-  F/ f E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0 )  =/=  0  /\  ( f `
 A )  =  0 )
5351, 52nfan 2011 . . . . 5  |-  F/ f ( A  e.  CC  /\ 
E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0 )  =/=  0  /\  ( f `
 A )  =  0 ) )
54 nfv 1761 . . . . 5  |-  F/ f  -.  A  e.  {
0 }
55 simpl3r 1064 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  /\  A  =  0 )  ->  ( f `  A )  =  0 )
56 fveq2 5865 . . . . . . . . . . . . . . 15  |-  ( A  =  0  ->  (
f `  A )  =  ( f ` 
0 ) )
57 eqid 2451 . . . . . . . . . . . . . . . 16  |-  (coeff `  f )  =  (coeff `  f )
5857coefv0 23202 . . . . . . . . . . . . . . 15  |-  ( f  e.  (Poly `  ZZ )  ->  ( f ` 
0 )  =  ( (coeff `  f ) `  0 ) )
5956, 58sylan9eqr 2507 . . . . . . . . . . . . . 14  |-  ( ( f  e.  (Poly `  ZZ )  /\  A  =  0 )  ->  (
f `  A )  =  ( (coeff `  f ) `  0
) )
6059adantlr 721 . . . . . . . . . . . . 13  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  A  =  0 )  -> 
( f `  A
)  =  ( (coeff `  f ) `  0
) )
61 simplr 762 . . . . . . . . . . . . 13  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  A  =  0 )  -> 
( (coeff `  f
) `  0 )  =/=  0 )
6260, 61eqnetrd 2691 . . . . . . . . . . . 12  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  A  =  0 )  -> 
( f `  A
)  =/=  0 )
6362neneqd 2629 . . . . . . . . . . 11  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  A  =  0 )  ->  -.  ( f `  A
)  =  0 )
6463adantlrr 727 . . . . . . . . . 10  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  /\  A  =  0 )  ->  -.  (
f `  A )  =  0 )
65643adantl1 1164 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  /\  A  =  0 )  ->  -.  (
f `  A )  =  0 )
6655, 65pm2.65da 580 . . . . . . . 8  |-  ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  -.  A  =  0 )
67 elsncg 3991 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  e.  { 0 } 
<->  A  =  0 ) )
6867biimpa 487 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  e.  { 0 } )  ->  A  =  0 )
69683ad2antl1 1170 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  /\  A  e.  {
0 } )  ->  A  =  0 )
7066, 69mtand 665 . . . . . . 7  |-  ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  -.  A  e.  { 0 } )
71703exp 1207 . . . . . 6  |-  ( A  e.  CC  ->  (
f  e.  (Poly `  ZZ )  ->  ( ( ( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 )  ->  -.  A  e.  { 0 } ) ) )
7271adantr 467 . . . . 5  |-  ( ( A  e.  CC  /\  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  ( f  e.  (Poly `  ZZ )  ->  ( ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 )  ->  -.  A  e.  { 0 } ) ) )
7353, 54, 72rexlimd 2871 . . . 4  |-  ( ( A  e.  CC  /\  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  ( E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 )  ->  -.  A  e.  { 0 } ) )
7450, 73mpd 15 . . 3  |-  ( ( A  e.  CC  /\  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  -.  A  e.  { 0 } )
7549, 74eldifd 3415 . 2  |-  ( ( A  e.  CC  /\  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  A  e.  ( AA  \  { 0 } ) )
7626, 75impbii 191 1  |-  ( A  e.  ( AA  \  { 0 } )  <-> 
( A  e.  CC  /\ 
E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0 )  =/=  0  /\  ( f `
 A )  =  0 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   E.wrex 2738   {crab 2741    \ cdif 3401   {csn 3968    |-> cmpt 4461    X. cxp 4832   ` cfv 5582  (class class class)co 6290  infcinf 7955   CCcc 9537   RRcr 9538   0cc0 9539    + caddc 9542    x. cmul 9544    < clt 9675    - cmin 9860   NN0cn0 10869   ZZcz 10937   ...cfz 11784   ^cexp 12272   sum_csu 13752   0pc0p 22627  Polycply 23138  coeffccoe 23140  degcdgr 23141   AAcaa 23267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-rlim 13553  df-sum 13753  df-0p 22628  df-ply 23142  df-coe 23144  df-dgr 23145  df-aa 23268
This theorem is referenced by:  etransc  38149
  Copyright terms: Public domain W3C validator