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Theorem elaa2 38211
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 23350 . . . 4  |-  AA  C_  CC
2 eldifi 3544 . . . 4  |-  ( A  e.  ( AA  \  { 0 } )  ->  A  e.  AA )
31, 2sseldi 3416 . . 3  |-  ( A  e.  ( AA  \  { 0 } )  ->  A  e.  CC )
4 elaa 23348 . . . . . 6  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. g  e.  ( (Poly `  ZZ )  \  { 0p } ) ( g `
 A )  =  0 ) )
52, 4sylib 201 . . . . 5  |-  ( A  e.  ( AA  \  { 0 } )  ->  ( A  e.  CC  /\  E. g  e.  ( (Poly `  ZZ )  \  { 0p } ) ( g `
 A )  =  0 ) )
65simprd 470 . . . 4  |-  ( A  e.  ( AA  \  { 0 } )  ->  E. g  e.  ( (Poly `  ZZ )  \  { 0p }
) ( g `  A )  =  0 )
723ad2ant1 1051 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  ->  A  e.  AA )
8 eldifsni 4089 . . . . . . 7  |-  ( A  e.  ( AA  \  { 0 } )  ->  A  =/=  0
)
983ad2ant1 1051 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  ->  A  =/=  0 )
10 eldifi 3544 . . . . . . 7  |-  ( g  e.  ( (Poly `  ZZ )  \  { 0p } )  -> 
g  e.  (Poly `  ZZ ) )
11103ad2ant2 1052 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  -> 
g  e.  (Poly `  ZZ ) )
12 eldifsni 4089 . . . . . . 7  |-  ( g  e.  ( (Poly `  ZZ )  \  { 0p } )  -> 
g  =/=  0p )
13123ad2ant2 1052 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  -> 
g  =/=  0p )
14 simp3 1032 . . . . . 6  |-  ( ( A  e.  ( AA 
\  { 0 } )  /\  g  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
g `  A )  =  0 )  -> 
( g `  A
)  =  0 )
15 fveq2 5879 . . . . . . . . 9  |-  ( m  =  n  ->  (
(coeff `  g ) `  m )  =  ( (coeff `  g ) `  n ) )
1615neeq1d 2702 . . . . . . . 8  |-  ( m  =  n  ->  (
( (coeff `  g
) `  m )  =/=  0  <->  ( (coeff `  g ) `  n
)  =/=  0 ) )
1716cbvrabv 3030 . . . . . . 7  |-  { m  e.  NN0  |  ( (coeff `  g ) `  m
)  =/=  0 }  =  { n  e. 
NN0  |  ( (coeff `  g ) `  n
)  =/=  0 }
1817infeq1i 8012 . . . . . 6  |- inf ( { m  e.  NN0  | 
( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  )  = inf ( { n  e.  NN0  |  ( (coeff `  g
) `  n )  =/=  0 } ,  RR ,  <  )
19 oveq1 6315 . . . . . . . 8  |-  ( j  =  k  ->  (
j  + inf ( {
m  e.  NN0  | 
( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  ) )  =  ( k  + inf ( { m  e.  NN0  |  ( (coeff `  g
) `  m )  =/=  0 } ,  RR ,  <  ) ) )
2019fveq2d 5883 . . . . . . 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 4488 . . . . . 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 2471 . . . . . 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 38209 . . . . 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 2873 . . . 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 541 . 2  |-  ( A  e.  ( AA  \  { 0 } )  ->  ( A  e.  CC  /\  E. f  e.  (Poly `  ZZ )
( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) ) )
27 simpl 464 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  ->  f  e.  (Poly `  ZZ )
)
28 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( f  =  0p  -> 
(coeff `  f )  =  (coeff `  0p
) )
29 coe0 23289 . . . . . . . . . . . . . . 15  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
3028, 29syl6eq 2521 . . . . . . . . . . . . . 14  |-  ( f  =  0p  -> 
(coeff `  f )  =  ( NN0  X.  { 0 } ) )
3130fveq1d 5881 . . . . . . . . . . . . 13  |-  ( f  =  0p  -> 
( (coeff `  f
) `  0 )  =  ( ( NN0 
X.  { 0 } ) `  0 ) )
32 0nn0 10908 . . . . . . . . . . . . . 14  |-  0  e.  NN0
33 fvconst2g 6134 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  NN0  /\  0  e.  NN0 )  -> 
( ( NN0  X.  { 0 } ) `
 0 )  =  0 )
3432, 32, 33mp2an 686 . . . . . . . . . . . . 13  |-  ( ( NN0  X.  { 0 } ) `  0
)  =  0
3531, 34syl6eq 2521 . . . . . . . . . . . 12  |-  ( f  =  0p  -> 
( (coeff `  f
) `  0 )  =  0 )
3635adantl 473 . . . . . . . . . . 11  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  f  =  0p )  ->  ( (coeff `  f ) `  0
)  =  0 )
37 neneq 2649 . . . . . . . . . . . 12  |-  ( ( (coeff `  f ) `  0 )  =/=  0  ->  -.  (
(coeff `  f ) `  0 )  =  0 )
3837ad2antlr 741 . . . . . . . . . . 11  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  f  =  0p )  ->  -.  ( (coeff `  f ) `  0
)  =  0 )
3936, 38pm2.65da 586 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  ->  -.  f  =  0p
)
40 elsn 3973 . . . . . . . . . 10  |-  ( f  e.  { 0p }  <->  f  =  0p )
4139, 40sylnibr 312 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  ->  -.  f  e.  { 0p } )
4227, 41eldifd 3401 . . . . . . . 8  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  ->  f  e.  ( (Poly `  ZZ )  \  { 0p } ) )
4342adantrr 731 . . . . . . 7  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  ->  f  e.  ( (Poly `  ZZ )  \  { 0p }
) )
44 simprr 774 . . . . . . 7  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  ->  ( f `  A )  =  0 )
4543, 44jca 541 . . . . . 6  |-  ( ( f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  ->  ( f  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  (
f `  A )  =  0 ) )
4645reximi2 2851 . . . . 5  |-  ( E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 )  ->  E. f  e.  ( (Poly `  ZZ )  \  { 0p }
) ( f `  A )  =  0 )
4746anim2i 579 . . . 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 23348 . . . 4  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  { 0p } ) ( f `
 A )  =  0 ) )
4947, 48sylibr 217 . . 3  |-  ( ( A  e.  CC  /\  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  A  e.  AA )
50 simpr 468 . . . 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 1769 . . . . . 6  |-  F/ f  A  e.  CC
52 nfre1 2846 . . . . . 6  |-  F/ f E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0 )  =/=  0  /\  ( f `
 A )  =  0 )
5351, 52nfan 2031 . . . . 5  |-  F/ f ( A  e.  CC  /\ 
E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0 )  =/=  0  /\  ( f `
 A )  =  0 ) )
54 nfv 1769 . . . . 5  |-  F/ f  -.  A  e.  {
0 }
55 simpl3r 1086 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  /\  A  =  0 )  ->  ( f `  A )  =  0 )
56 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( A  =  0  ->  (
f `  A )  =  ( f ` 
0 ) )
57 eqid 2471 . . . . . . . . . . . . . . . 16  |-  (coeff `  f )  =  (coeff `  f )
5857coefv0 23281 . . . . . . . . . . . . . . 15  |-  ( f  e.  (Poly `  ZZ )  ->  ( f ` 
0 )  =  ( (coeff `  f ) `  0 ) )
5956, 58sylan9eqr 2527 . . . . . . . . . . . . . 14  |-  ( ( f  e.  (Poly `  ZZ )  /\  A  =  0 )  ->  (
f `  A )  =  ( (coeff `  f ) `  0
) )
6059adantlr 729 . . . . . . . . . . . . 13  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  A  =  0 )  -> 
( f `  A
)  =  ( (coeff `  f ) `  0
) )
61 simplr 770 . . . . . . . . . . . . 13  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  A  =  0 )  -> 
( (coeff `  f
) `  0 )  =/=  0 )
6260, 61eqnetrd 2710 . . . . . . . . . . . 12  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  A  =  0 )  -> 
( f `  A
)  =/=  0 )
6362neneqd 2648 . . . . . . . . . . 11  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
(coeff `  f ) `  0 )  =/=  0 )  /\  A  =  0 )  ->  -.  ( f `  A
)  =  0 )
6463adantlrr 735 . . . . . . . . . 10  |-  ( ( ( f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  /\  A  =  0 )  ->  -.  (
f `  A )  =  0 )
65643adantl1 1186 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  /\  A  =  0 )  ->  -.  (
f `  A )  =  0 )
6655, 65pm2.65da 586 . . . . . . . 8  |-  ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  -.  A  =  0 )
67 elsncg 3983 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  e.  { 0 } 
<->  A  =  0 ) )
6867biimpa 492 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  e.  { 0 } )  ->  A  =  0 )
69683ad2antl1 1192 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  (
( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 ) )  /\  A  e.  {
0 } )  ->  A  =  0 )
7066, 69mtand 671 . . . . . . 7  |-  ( ( A  e.  CC  /\  f  e.  (Poly `  ZZ )  /\  ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  -.  A  e.  { 0 } )
71703exp 1230 . . . . . 6  |-  ( A  e.  CC  ->  (
f  e.  (Poly `  ZZ )  ->  ( ( ( (coeff `  f
) `  0 )  =/=  0  /\  (
f `  A )  =  0 )  ->  -.  A  e.  { 0 } ) ) )
7271adantr 472 . . . . 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 2866 . . . 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 3401 . 2  |-  ( ( A  e.  CC  /\  E. f  e.  (Poly `  ZZ ) ( ( (coeff `  f ) `  0
)  =/=  0  /\  ( f `  A
)  =  0 ) )  ->  A  e.  ( AA  \  { 0 } ) )
7626, 75impbii 192 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   {crab 2760    \ cdif 3387   {csn 3959    |-> cmpt 4454    X. cxp 4837   ` cfv 5589  (class class class)co 6308  infcinf 7973   CCcc 9555   RRcr 9556   0cc0 9557    + caddc 9560    x. cmul 9562    < clt 9693    - cmin 9880   NN0cn0 10893   ZZcz 10961   ...cfz 11810   ^cexp 12310   sum_csu 13829   0pc0p 22706  Polycply 23217  coeffccoe 23219  degcdgr 23220   AAcaa 23346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-0p 22707  df-ply 23221  df-coe 23223  df-dgr 23224  df-aa 23347
This theorem is referenced by:  etransc  38261
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