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Theorem elaa2 37919
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 23258 . . . 4  |-  AA  C_  CC
2 eldifi 3587 . . . 4  |-  ( A  e.  ( AA  \  { 0 } )  ->  A  e.  AA )
31, 2sseldi 3462 . . 3  |-  ( A  e.  ( AA  \  { 0 } )  ->  A  e.  CC )
4 elaa 23256 . . . . . 6  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. g  e.  ( (Poly `  ZZ )  \  { 0p } ) ( g `
 A )  =  0 ) )
52, 4sylib 199 . . . . 5  |-  ( A  e.  ( AA  \  { 0 } )  ->  ( A  e.  CC  /\  E. g  e.  ( (Poly `  ZZ )  \  { 0p } ) ( g `
 A )  =  0 ) )
65simprd 464 . . . 4  |-  ( A  e.  ( AA  \  { 0 } )  ->  E. g  e.  ( (Poly `  ZZ )  \  { 0p }
) ( g `  A )  =  0 )
723ad2ant1 1026 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  ->  A  e.  AA )
8 eldifsni 4123 . . . . . . 7  |-  ( A  e.  ( AA  \  { 0 } )  ->  A  =/=  0
)
983ad2ant1 1026 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  ->  A  =/=  0 )
10 eldifi 3587 . . . . . . 7  |-  ( g  e.  ( (Poly `  ZZ )  \  { 0p } )  -> 
g  e.  (Poly `  ZZ ) )
11103ad2ant2 1027 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  -> 
g  e.  (Poly `  ZZ ) )
12 eldifsni 4123 . . . . . . 7  |-  ( g  e.  ( (Poly `  ZZ )  \  { 0p } )  -> 
g  =/=  0p )
13123ad2ant2 1027 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  -> 
g  =/=  0p )
14 simp3 1007 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  -> 
( g `  A
)  =  0 )
15 fveq2 5878 . . . . . . . . 9  |-  ( m  =  n  ->  (
(coeff `  g ) `  m )  =  ( (coeff `  g ) `  n ) )
1615neeq1d 2701 . . . . . . . 8  |-  ( m  =  n  ->  (
( (coeff `  g
) `  m )  =/=  0  <->  ( (coeff `  g ) `  n
)  =/=  0 ) )
1716cbvrabv 3080 . . . . . . 7  |-  { m  e.  NN0  |  ( (coeff `  g ) `  m
)  =/=  0 }  =  { n  e. 
NN0  |  ( (coeff `  g ) `  n
)  =/=  0 }
1817infeq1i 7997 . . . . . 6  |- inf ( { m  e.  NN0  | 
( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  )  = inf ( { n  e.  NN0  |  ( (coeff `  g
) `  n )  =/=  0 } ,  RR ,  <  )
19 oveq1 6309 . . . . . . . 8  |-  ( j  =  k  ->  (
j  + inf ( {
m  e.  NN0  | 
( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  ) )  =  ( k  + inf ( { m  e.  NN0  |  ( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  ) ) )
2019fveq2d 5882 . . . . . . 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 4513 . . . . . 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 2422 . . . . . 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 37917 . . . . 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 2919 . . . 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 534 . 2  |-  ( A  e.  ( AA  \  { 0 } )  ->  ( A  e.  CC  /\  E. f  e.  (Poly `  ZZ )
( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) ) )
27 simpl 458 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  ->  f  e.  (Poly `  ZZ )
)
28 fveq2 5878 . . . . . . . . . . . . . . 15  |-  ( f  =  0p  -> 
(coeff `  f )  =  (coeff `  0p
) )
29 coe0 23197 . . . . . . . . . . . . . . 15  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
3028, 29syl6eq 2479 . . . . . . . . . . . . . 14  |-  ( f  =  0p  -> 
(coeff `  f )  =  ( NN0  X.  { 0 } ) )
3130fveq1d 5880 . . . . . . . . . . . . 13  |-  ( f  =  0p  -> 
( (coeff `  f
) `  0 )  =  ( ( NN0 
X.  { 0 } ) `  0 ) )
32 0nn0 10885 . . . . . . . . . . . . . 14  |-  0  e.  NN0
33 fvconst2g 6130 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  NN0  /\  0  e.  NN0 )  -> 
( ( NN0  X.  { 0 } ) `
 0 )  =  0 )
3432, 32, 33mp2an 676 . . . . . . . . . . . . 13  |-  ( ( NN0  X.  { 0 } ) `  0
)  =  0
3531, 34syl6eq 2479 . . . . . . . . . . . 12  |-  ( f  =  0p  -> 
( (coeff `  f
) `  0 )  =  0 )
3635adantl 467 . . . . . . . . . . 11  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  f  =  0p )  ->  ( (coeff `  f ) `  0
)  =  0 )
37 neneq 2626 . . . . . . . . . . . 12  |-  ( ( (coeff `  f ) `  0 )  =/=  0  ->  -.  (
(coeff `  f ) `  0 )  =  0 )
3837ad2antlr 731 . . . . . . . . . . 11  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  f  =  0p )  ->  -.  ( (coeff `  f ) `  0
)  =  0 )
3936, 38pm2.65da 578 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  ->  -.  f  =  0p
)
40 elsn 4010 . . . . . . . . . 10  |-  ( f  e.  { 0p }  <->  f  =  0p )
4139, 40sylnibr 306 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  ->  -.  f  e.  { 0p } )
4227, 41eldifd 3447 . . . . . . . 8  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  ->  f  e.  ( (Poly `  ZZ )  \  { 0p } ) )
4342adantrr 721 . . . . . . 7  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  ->  f  e.  ( (Poly `  ZZ )  \  { 0p }
) )
44 simprr 764 . . . . . . 7  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  ->  ( f `  A )  =  0 )
4543, 44jca 534 . . . . . 6  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  ->  ( f  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
f `  A )  =  0 ) )
4645reximi2 2892 . . . . 5  |-  ( E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 )  ->  E. f  e.  ( (Poly `  ZZ )  \  { 0p }
) ( f `  A )  =  0 )
4746anim2i 571 . . . 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 23256 . . . 4  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  { 0p } ) ( f `
 A )  =  0 ) )
4947, 48sylibr 215 . . 3  |-  ( ( A  e.  CC  /\  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  A  e.  AA )
50 simpr 462 . . . 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 1751 . . . . . 6  |-  F/ f  A  e.  CC
52 nfre1 2886 . . . . . 6  |-  F/ f E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0 )  =/=  0  /\  ( f `
 A )  =  0 )
5351, 52nfan 1984 . . . . 5  |-  F/ f ( A  e.  CC  /\ 
E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0 )  =/=  0  /\  ( f `
 A )  =  0 ) )
54 nfv 1751 . . . . 5  |-  F/ f  -.  A  e.  {
0 }
55 simpl3r 1061 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  /\  A  =  0 )  ->  ( f `  A )  =  0 )
56 fveq2 5878 . . . . . . . . . . . . . . 15  |-  ( A  =  0  ->  (
f `  A )  =  ( f ` 
0 ) )
57 eqid 2422 . . . . . . . . . . . . . . . 16  |-  (coeff `  f )  =  (coeff `  f )
5857coefv0 23189 . . . . . . . . . . . . . . 15  |-  ( f  e.  (Poly `  ZZ )  ->  ( f ` 
0 )  =  ( (coeff `  f ) `  0 ) )
5956, 58sylan9eqr 2485 . . . . . . . . . . . . . 14  |-  ( ( f  e.  (Poly `  ZZ )  /\  A  =  0 )  ->  (
f `  A )  =  ( (coeff `  f ) `  0
) )
6059adantlr 719 . . . . . . . . . . . . 13  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  A  =  0 )  -> 
( f `  A
)  =  ( (coeff `  f ) `  0
) )
61 simplr 760 . . . . . . . . . . . . 13  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  A  =  0 )  -> 
( (coeff `  f
) `  0 )  =/=  0 )
6260, 61eqnetrd 2717 . . . . . . . . . . . 12  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  A  =  0 )  -> 
( f `  A
)  =/=  0 )
6362neneqd 2625 . . . . . . . . . . 11  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  A  =  0 )  ->  -.  ( f `  A
)  =  0 )
6463adantlrr 725 . . . . . . . . . 10  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  /\  A  =  0 )  ->  -.  (
f `  A )  =  0 )
65643adantl1 1161 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  /\  A  =  0 )  ->  -.  (
f `  A )  =  0 )
6655, 65pm2.65da 578 . . . . . . . 8  |-  ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  -.  A  =  0 )
67 elsncg 4019 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  e.  { 0 } 
<->  A  =  0 ) )
6867biimpa 486 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  e.  { 0 } )  ->  A  =  0 )
69683ad2antl1 1167 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  /\  A  e.  {
0 } )  ->  A  =  0 )
7066, 69mtand 663 . . . . . . 7  |-  ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  -.  A  e.  { 0 } )
71703exp 1204 . . . . . 6  |-  ( A  e.  CC  ->  (
f  e.  (Poly `  ZZ )  ->  ( ( ( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 )  ->  -.  A  e.  { 0 } ) ) )
7271adantr 466 . . . . 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 2909 . . . 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 3447 . 2  |-  ( ( A  e.  CC  /\  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  A  e.  ( AA  \  { 0 } ) )
7626, 75impbii 190 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   E.wrex 2776   {crab 2779    \ cdif 3433   {csn 3996    |-> cmpt 4479    X. cxp 4848   ` cfv 5598  (class class class)co 6302  infcinf 7958   CCcc 9538   RRcr 9539   0cc0 9540    + caddc 9543    x. cmul 9545    < clt 9676    - cmin 9861   NN0cn0 10870   ZZcz 10938   ...cfz 11785   ^cexp 12272   sum_csu 13740   0pc0p 22614  Polycply 23125  coeffccoe 23127  degcdgr 23128   AAcaa 23254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-rp 11304  df-fz 11786  df-fzo 11917  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-rlim 13541  df-sum 13741  df-0p 22615  df-ply 23129  df-coe 23131  df-dgr 23132  df-aa 23255
This theorem is referenced by:  etransc  37969
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