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Theorem etransc 37968
Description:  _e is transcendental. Section *5 of [Juillerat] p. 11 can be used as a reference for this proof. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Proof shortened by AV, 28-Sep-2020.)
Assertion
Ref Expression
etransc  |-  _e  e.  ( CC  \  AA )

Proof of Theorem etransc
Dummy variables  h  i  l  n  q 
k  j  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1red 9658 . . . . 5  |-  ( ( k  e.  ZZ  /\  k  =/=  0 )  -> 
1  e.  RR )
2 nn0abscl 13363 . . . . . . 7  |-  ( k  e.  ZZ  ->  ( abs `  k )  e. 
NN0 )
32nn0red 10926 . . . . . 6  |-  ( k  e.  ZZ  ->  ( abs `  k )  e.  RR )
43adantr 466 . . . . 5  |-  ( ( k  e.  ZZ  /\  k  =/=  0 )  -> 
( abs `  k
)  e.  RR )
5 nnabscl 13376 . . . . . 6  |-  ( ( k  e.  ZZ  /\  k  =/=  0 )  -> 
( abs `  k
)  e.  NN )
65nnge1d 10652 . . . . 5  |-  ( ( k  e.  ZZ  /\  k  =/=  0 )  -> 
1  <_  ( abs `  k ) )
71, 4, 6lensymd 9786 . . . 4  |-  ( ( k  e.  ZZ  /\  k  =/=  0 )  ->  -.  ( abs `  k
)  <  1 )
8 nan 582 . . . 4  |-  ( ( k  e.  ZZ  ->  -.  ( k  =/=  0  /\  ( abs `  k
)  <  1 ) )  <->  ( ( k  e.  ZZ  /\  k  =/=  0 )  ->  -.  ( abs `  k )  <  1 ) )
97, 8mpbir 212 . . 3  |-  ( k  e.  ZZ  ->  -.  ( k  =/=  0  /\  ( abs `  k
)  <  1 ) )
109nrex 2880 . 2  |-  -.  E. k  e.  ZZ  (
k  =/=  0  /\  ( abs `  k
)  <  1 )
11 ere 14130 . . . . . . . 8  |-  _e  e.  RR
1211recni 9655 . . . . . . 7  |-  _e  e.  CC
13 neldif 3590 . . . . . . 7  |-  ( ( _e  e.  CC  /\  -.  _e  e.  ( CC 
\  AA ) )  ->  _e  e.  AA )
1412, 13mpan 674 . . . . . 6  |-  ( -.  _e  e.  ( CC 
\  AA )  ->  _e  e.  AA )
15 ene0 14248 . . . . . . . 8  |-  _e  =/=  0
16 elsncg 4019 . . . . . . . . 9  |-  ( _e  e.  CC  ->  (
_e  e.  { 0 }  <->  _e  =  0
) )
1712, 16ax-mp 5 . . . . . . . 8  |-  ( _e  e.  { 0 }  <-> 
_e  =  0 )
1815, 17nemtbir 2752 . . . . . . 7  |-  -.  _e  e.  { 0 }
1918a1i 11 . . . . . 6  |-  ( -.  _e  e.  ( CC 
\  AA )  ->  -.  _e  e.  { 0 } )
2014, 19eldifd 3447 . . . . 5  |-  ( -.  _e  e.  ( CC 
\  AA )  ->  _e  e.  ( AA  \  { 0 } ) )
21 elaa2 37918 . . . . 5  |-  ( _e  e.  ( AA  \  { 0 } )  <-> 
( _e  e.  CC  /\ 
E. q  e.  (Poly `  ZZ ) ( ( (coeff `  q ) `  0 )  =/=  0  /\  ( q `
 _e )  =  0 ) ) )
2220, 21sylib 199 . . . 4  |-  ( -.  _e  e.  ( CC 
\  AA )  -> 
( _e  e.  CC  /\ 
E. q  e.  (Poly `  ZZ ) ( ( (coeff `  q ) `  0 )  =/=  0  /\  ( q `
 _e )  =  0 ) ) )
2322simprd 464 . . 3  |-  ( -.  _e  e.  ( CC 
\  AA )  ->  E. q  e.  (Poly `  ZZ ) ( ( (coeff `  q ) `  0 )  =/=  0  /\  ( q `
 _e )  =  0 ) )
24 simpl 458 . . . . . . 7  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
(coeff `  q ) `  0 )  =/=  0 )  ->  q  e.  (Poly `  ZZ )
)
25 0nn0 10884 . . . . . . . . 9  |-  0  e.  NN0
26 n0p 37236 . . . . . . . . 9  |-  ( ( q  e.  (Poly `  ZZ )  /\  0  e.  NN0  /\  ( (coeff `  q ) `  0
)  =/=  0 )  ->  q  =/=  0p )
2725, 26mp3an2 1348 . . . . . . . 8  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
(coeff `  q ) `  0 )  =/=  0 )  ->  q  =/=  0p )
28 nelsn 37262 . . . . . . . 8  |-  ( q  =/=  0p  ->  -.  q  e.  { 0p } )
2927, 28syl 17 . . . . . . 7  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
(coeff `  q ) `  0 )  =/=  0 )  ->  -.  q  e.  { 0p } )
3024, 29eldifd 3447 . . . . . 6  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
(coeff `  q ) `  0 )  =/=  0 )  ->  q  e.  ( (Poly `  ZZ )  \  { 0p } ) )
3130adantrr 721 . . . . 5  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
( (coeff `  q
) `  0 )  =/=  0  /\  (
q `  _e )  =  0 ) )  ->  q  e.  ( (Poly `  ZZ )  \  { 0p }
) )
32 simprr 764 . . . . 5  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
( (coeff `  q
) `  0 )  =/=  0  /\  (
q `  _e )  =  0 ) )  ->  ( q `  _e )  =  0
)
33 eqid 2422 . . . . 5  |-  (coeff `  q )  =  (coeff `  q )
34 simprl 762 . . . . 5  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
( (coeff `  q
) `  0 )  =/=  0  /\  (
q `  _e )  =  0 ) )  ->  ( (coeff `  q ) `  0
)  =/=  0 )
35 eqid 2422 . . . . 5  |-  (deg `  q )  =  (deg
`  q )
36 eqid 2422 . . . . 5  |-  sum_ l  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  l )  x.  ( _e  ^c 
l ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  =  sum_ l  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  l )  x.  ( _e  ^c 
l ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )
37 eqid 2422 . . . . 5  |-  ( n  e.  NN0  |->  ( sum_ l  e.  ( 0 ... (deg `  q
) ) ( ( abs `  ( ( (coeff `  q ) `  l )  x.  (
_e  ^c  l ) ) )  x.  (
(deg `  q )  x.  ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
n )  /  ( ! `  n )
) ) )  =  ( n  e.  NN0  |->  ( sum_ l  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  l )  x.  ( _e  ^c 
l ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q ) ^ ( (deg `  q )  +  1 ) ) ^ n
)  /  ( ! `
 n ) ) ) )
38 fveq2 5877 . . . . . . . . . . . . . . . . . . . 20  |-  ( h  =  l  ->  (
(coeff `  q ) `  h )  =  ( (coeff `  q ) `  l ) )
39 oveq2 6309 . . . . . . . . . . . . . . . . . . . 20  |-  ( h  =  l  ->  (
_e  ^c  h )  =  ( _e  ^c  l ) )
4038, 39oveq12d 6319 . . . . . . . . . . . . . . . . . . 19  |-  ( h  =  l  ->  (
( (coeff `  q
) `  h )  x.  ( _e  ^c 
h ) )  =  ( ( (coeff `  q ) `  l
)  x.  ( _e 
^c  l ) ) )
4140fveq2d 5881 . . . . . . . . . . . . . . . . . 18  |-  ( h  =  l  ->  ( abs `  ( ( (coeff `  q ) `  h
)  x.  ( _e 
^c  h ) ) )  =  ( abs `  ( ( (coeff `  q ) `  l )  x.  (
_e  ^c  l ) ) ) )
4241oveq1d 6316 . . . . . . . . . . . . . . . . 17  |-  ( h  =  l  ->  (
( abs `  (
( (coeff `  q
) `  h )  x.  ( _e  ^c 
h ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  =  ( ( abs `  ( ( (coeff `  q ) `  l
)  x.  ( _e 
^c  l ) ) )  x.  (
(deg `  q )  x.  ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ) ) )
4342cbvsumv 13749 . . . . . . . . . . . . . . . 16  |-  sum_ h  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  h )  x.  ( _e  ^c 
h ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  =  sum_ l  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  l )  x.  ( _e  ^c 
l ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )
4443a1i 11 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  sum_ h  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  h )  x.  ( _e  ^c 
h ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  =  sum_ l  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  l )  x.  ( _e  ^c 
l ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) ) )
45 oveq2 6309 . . . . . . . . . . . . . . . 16  |-  ( m  =  n  ->  (
( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
m )  =  ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
n ) )
46 fveq2 5877 . . . . . . . . . . . . . . . 16  |-  ( m  =  n  ->  ( ! `  m )  =  ( ! `  n ) )
4745, 46oveq12d 6319 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  (
( ( (deg `  q ) ^ (
(deg `  q )  +  1 ) ) ^ m )  / 
( ! `  m
) )  =  ( ( ( (deg `  q ) ^ (
(deg `  q )  +  1 ) ) ^ n )  / 
( ! `  n
) ) )
4844, 47oveq12d 6319 . . . . . . . . . . . . . 14  |-  ( m  =  n  ->  ( sum_ h  e.  ( 0 ... (deg `  q
) ) ( ( abs `  ( ( (coeff `  q ) `  h )  x.  (
_e  ^c  h ) ) )  x.  (
(deg `  q )  x.  ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
m )  /  ( ! `  m )
) )  =  (
sum_ l  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  l )  x.  ( _e  ^c 
l ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q ) ^ ( (deg `  q )  +  1 ) ) ^ n
)  /  ( ! `
 n ) ) ) )
4948cbvmptv 4513 . . . . . . . . . . . . 13  |-  ( m  e.  NN0  |->  ( sum_ h  e.  ( 0 ... (deg `  q )
) ( ( abs `  ( ( (coeff `  q ) `  h
)  x.  ( _e 
^c  h ) ) )  x.  (
(deg `  q )  x.  ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
m )  /  ( ! `  m )
) ) )  =  ( n  e.  NN0  |->  ( sum_ l  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  l )  x.  ( _e  ^c 
l ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q ) ^ ( (deg `  q )  +  1 ) ) ^ n
)  /  ( ! `
 n ) ) ) )
5049a1i 11 . . . . . . . . . . . 12  |-  ( m  =  n  ->  (
m  e.  NN0  |->  ( sum_ h  e.  ( 0 ... (deg `  q )
) ( ( abs `  ( ( (coeff `  q ) `  h
)  x.  ( _e 
^c  h ) ) )  x.  (
(deg `  q )  x.  ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
m )  /  ( ! `  m )
) ) )  =  ( n  e.  NN0  |->  ( sum_ l  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  l )  x.  ( _e  ^c 
l ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q ) ^ ( (deg `  q )  +  1 ) ) ^ n
)  /  ( ! `
 n ) ) ) ) )
51 id 23 . . . . . . . . . . . 12  |-  ( m  =  n  ->  m  =  n )
5250, 51fveq12d 5883 . . . . . . . . . . 11  |-  ( m  =  n  ->  (
( m  e.  NN0  |->  ( sum_ h  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  h )  x.  ( _e  ^c 
h ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q ) ^ ( (deg `  q )  +  1 ) ) ^ m
)  /  ( ! `
 m ) ) ) ) `  m
)  =  ( ( n  e.  NN0  |->  ( sum_ l  e.  ( 0 ... (deg `  q
) ) ( ( abs `  ( ( (coeff `  q ) `  l )  x.  (
_e  ^c  l ) ) )  x.  (
(deg `  q )  x.  ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
n )  /  ( ! `  n )
) ) ) `  n ) )
5352fveq2d 5881 . . . . . . . . . 10  |-  ( m  =  n  ->  ( abs `  ( ( m  e.  NN0  |->  ( sum_ h  e.  ( 0 ... (deg `  q )
) ( ( abs `  ( ( (coeff `  q ) `  h
)  x.  ( _e 
^c  h ) ) )  x.  (
(deg `  q )  x.  ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
m )  /  ( ! `  m )
) ) ) `  m ) )  =  ( abs `  (
( n  e.  NN0  |->  ( sum_ l  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  l )  x.  ( _e  ^c 
l ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q ) ^ ( (deg `  q )  +  1 ) ) ^ n
)  /  ( ! `
 n ) ) ) ) `  n
) ) )
5453breq1d 4430 . . . . . . . . 9  |-  ( m  =  n  ->  (
( abs `  (
( m  e.  NN0  |->  ( sum_ h  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  h )  x.  ( _e  ^c 
h ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q ) ^ ( (deg `  q )  +  1 ) ) ^ m
)  /  ( ! `
 m ) ) ) ) `  m
) )  <  1  <->  ( abs `  ( ( n  e.  NN0  |->  ( sum_ l  e.  ( 0 ... (deg `  q
) ) ( ( abs `  ( ( (coeff `  q ) `  l )  x.  (
_e  ^c  l ) ) )  x.  (
(deg `  q )  x.  ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
n )  /  ( ! `  n )
) ) ) `  n ) )  <  1 ) )
5554cbvralv 3055 . . . . . . . 8  |-  ( A. m  e.  ( ZZ>= `  j ) ( abs `  ( ( m  e. 
NN0  |->  ( sum_ h  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  h )  x.  ( _e  ^c 
h ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q ) ^ ( (deg `  q )  +  1 ) ) ^ m
)  /  ( ! `
 m ) ) ) ) `  m
) )  <  1  <->  A. n  e.  ( ZZ>= `  j ) ( abs `  ( ( n  e. 
NN0  |->  ( sum_ l  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  l )  x.  ( _e  ^c 
l ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q ) ^ ( (deg `  q )  +  1 ) ) ^ n
)  /  ( ! `
 n ) ) ) ) `  n
) )  <  1
)
56 fveq2 5877 . . . . . . . . 9  |-  ( j  =  i  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  i )
)
5756raleqdv 3031 . . . . . . . 8  |-  ( j  =  i  ->  ( A. n  e.  ( ZZ>=
`  j ) ( abs `  ( ( n  e.  NN0  |->  ( sum_ l  e.  ( 0 ... (deg `  q
) ) ( ( abs `  ( ( (coeff `  q ) `  l )  x.  (
_e  ^c  l ) ) )  x.  (
(deg `  q )  x.  ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
n )  /  ( ! `  n )
) ) ) `  n ) )  <  1  <->  A. n  e.  (
ZZ>= `  i ) ( abs `  ( ( n  e.  NN0  |->  ( sum_ l  e.  ( 0 ... (deg `  q
) ) ( ( abs `  ( ( (coeff `  q ) `  l )  x.  (
_e  ^c  l ) ) )  x.  (
(deg `  q )  x.  ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
n )  /  ( ! `  n )
) ) ) `  n ) )  <  1 ) )
5855, 57syl5bb 260 . . . . . . 7  |-  ( j  =  i  ->  ( A. m  e.  ( ZZ>=
`  j ) ( abs `  ( ( m  e.  NN0  |->  ( sum_ h  e.  ( 0 ... (deg `  q )
) ( ( abs `  ( ( (coeff `  q ) `  h
)  x.  ( _e 
^c  h ) ) )  x.  (
(deg `  q )  x.  ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
m )  /  ( ! `  m )
) ) ) `  m ) )  <  1  <->  A. n  e.  (
ZZ>= `  i ) ( abs `  ( ( n  e.  NN0  |->  ( sum_ l  e.  ( 0 ... (deg `  q
) ) ( ( abs `  ( ( (coeff `  q ) `  l )  x.  (
_e  ^c  l ) ) )  x.  (
(deg `  q )  x.  ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
n )  /  ( ! `  n )
) ) ) `  n ) )  <  1 ) )
5958cbvrabv 3080 . . . . . 6  |-  { j  e.  NN0  |  A. m  e.  ( ZZ>= `  j ) ( abs `  ( ( m  e. 
NN0  |->  ( sum_ h  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  h )  x.  ( _e  ^c 
h ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q ) ^ ( (deg `  q )  +  1 ) ) ^ m
)  /  ( ! `
 m ) ) ) ) `  m
) )  <  1 }  =  { i  e.  NN0  |  A. n  e.  ( ZZ>= `  i )
( abs `  (
( n  e.  NN0  |->  ( sum_ l  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  l )  x.  ( _e  ^c 
l ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q ) ^ ( (deg `  q )  +  1 ) ) ^ n
)  /  ( ! `
 n ) ) ) ) `  n
) )  <  1 }
6059infeq1i 7996 . . . . 5  |- inf ( { j  e.  NN0  |  A. m  e.  ( ZZ>=
`  j ) ( abs `  ( ( m  e.  NN0  |->  ( sum_ h  e.  ( 0 ... (deg `  q )
) ( ( abs `  ( ( (coeff `  q ) `  h
)  x.  ( _e 
^c  h ) ) )  x.  (
(deg `  q )  x.  ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
m )  /  ( ! `  m )
) ) ) `  m ) )  <  1 } ,  RR ,  <  )  = inf ( { i  e.  NN0  | 
A. n  e.  (
ZZ>= `  i ) ( abs `  ( ( n  e.  NN0  |->  ( sum_ l  e.  ( 0 ... (deg `  q
) ) ( ( abs `  ( ( (coeff `  q ) `  l )  x.  (
_e  ^c  l ) ) )  x.  (
(deg `  q )  x.  ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
n )  /  ( ! `  n )
) ) ) `  n ) )  <  1 } ,  RR ,  <  )
61 eqid 2422 . . . . 5  |-  sup ( { ( abs `  (
(coeff `  q ) `  0 ) ) ,  ( ! `  (deg `  q ) ) , inf ( { j  e.  NN0  |  A. m  e.  ( ZZ>= `  j ) ( abs `  ( ( m  e. 
NN0  |->  ( sum_ h  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  h )  x.  ( _e  ^c 
h ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q ) ^ ( (deg `  q )  +  1 ) ) ^ m
)  /  ( ! `
 m ) ) ) ) `  m
) )  <  1 } ,  RR ,  <  ) } ,  RR* ,  <  )  =  sup ( { ( abs `  (
(coeff `  q ) `  0 ) ) ,  ( ! `  (deg `  q ) ) , inf ( { j  e.  NN0  |  A. m  e.  ( ZZ>= `  j ) ( abs `  ( ( m  e. 
NN0  |->  ( sum_ h  e.  ( 0 ... (deg `  q ) ) ( ( abs `  (
( (coeff `  q
) `  h )  x.  ( _e  ^c 
h ) ) )  x.  ( (deg `  q )  x.  (
(deg `  q ) ^ ( (deg `  q )  +  1 ) ) ) )  x.  ( ( ( (deg `  q ) ^ ( (deg `  q )  +  1 ) ) ^ m
)  /  ( ! `
 m ) ) ) ) `  m
) )  <  1 } ,  RR ,  <  ) } ,  RR* ,  <  )
6231, 32, 33, 34, 35, 36, 37, 60, 61etransclem48 37967 . . . 4  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
( (coeff `  q
) `  0 )  =/=  0  /\  (
q `  _e )  =  0 ) )  ->  E. k  e.  ZZ  ( k  =/=  0  /\  ( abs `  k
)  <  1 ) )
6362rexlimiva 2913 . . 3  |-  ( E. q  e.  (Poly `  ZZ ) ( ( (coeff `  q ) `  0
)  =/=  0  /\  ( q `  _e )  =  0 )  ->  E. k  e.  ZZ  ( k  =/=  0  /\  ( abs `  k
)  <  1 ) )
6423, 63syl 17 . 2  |-  ( -.  _e  e.  ( CC 
\  AA )  ->  E. k  e.  ZZ  ( k  =/=  0  /\  ( abs `  k
)  <  1 ) )
6510, 64mt3 183 1  |-  _e  e.  ( CC  \  AA )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   E.wrex 2776   {crab 2779    \ cdif 3433   {csn 3996   {ctp 4000   class class class wbr 4420    |-> cmpt 4479   ` cfv 5597  (class class class)co 6301   supcsup 7956  infcinf 7957   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   RR*cxr 9674    < clt 9675    / cdiv 10269   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784   ^cexp 12271   !cfa 12458   abscabs 13285   sum_csu 13739   _eceu 14102   0pc0p 22613  Polycply 23124  coeffccoe 23126  degcdgr 23127   AAcaa 23253    ^c ccxp 23491
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 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cc 8865  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  ax-mulf 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-iin 4299  df-disj 4392  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-ofr 6542  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-omul 7191  df-er 7367  df-map 7478  df-pm 7479  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-acn 8377  df-cda 8598  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-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-fac 12459  df-bc 12487  df-hash 12515  df-shft 13118  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-limsup 13513  df-clim 13539  df-rlim 13540  df-sum 13740  df-prod 13947  df-ef 14108  df-e 14109  df-sin 14110  df-cos 14111  df-tan 14112  df-pi 14113  df-dvds 14293  df-gcd 14456  df-prm 14610  df-struct 15110  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-mulr 15191  df-starv 15192  df-sca 15193  df-vsca 15194  df-ip 15195  df-tset 15196  df-ple 15197  df-ds 15199  df-unif 15200  df-hom 15201  df-cco 15202  df-rest 15308  df-topn 15309  df-0g 15327  df-gsum 15328  df-topgen 15329  df-pt 15330  df-prds 15333  df-xrs 15387  df-qtop 15393  df-imas 15394  df-xps 15397  df-mre 15479  df-mrc 15480  df-acs 15482  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-submnd 16570  df-mulg 16663  df-cntz 16958  df-cmn 17419  df-psmet 18949  df-xmet 18950  df-met 18951  df-bl 18952  df-mopn 18953  df-fbas 18954  df-fg 18955  df-cnfld 18958  df-top 19907  df-bases 19908  df-topon 19909  df-topsp 19910  df-cld 20020  df-ntr 20021  df-cls 20022  df-nei 20100  df-lp 20138  df-perf 20139  df-cn 20229  df-cnp 20230  df-haus 20317  df-cmp 20388  df-tx 20563  df-hmeo 20756  df-fil 20847  df-fm 20939  df-flim 20940  df-flf 20941  df-xms 21321  df-ms 21322  df-tms 21323  df-cncf 21896  df-ovol 22402  df-vol 22404  df-mbf 22563  df-itg1 22564  df-itg2 22565  df-ibl 22566  df-itg 22567  df-0p 22614  df-limc 22807  df-dv 22808  df-dvn 22809  df-ply 23128  df-coe 23130  df-dgr 23131  df-aa 23254  df-log 23492  df-cxp 23493
This theorem is referenced by: (None)
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