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Theorem etransc 38187
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 9684 . . . . 5  |-  ( ( k  e.  ZZ  /\  k  =/=  0 )  -> 
1  e.  RR )
2 nn0abscl 13424 . . . . . . 7  |-  ( k  e.  ZZ  ->  ( abs `  k )  e. 
NN0 )
32nn0red 10955 . . . . . 6  |-  ( k  e.  ZZ  ->  ( abs `  k )  e.  RR )
43adantr 471 . . . . 5  |-  ( ( k  e.  ZZ  /\  k  =/=  0 )  -> 
( abs `  k
)  e.  RR )
5 nnabscl 13437 . . . . . 6  |-  ( ( k  e.  ZZ  /\  k  =/=  0 )  -> 
( abs `  k
)  e.  NN )
65nnge1d 10680 . . . . 5  |-  ( ( k  e.  ZZ  /\  k  =/=  0 )  -> 
1  <_  ( abs `  k ) )
71, 4, 6lensymd 9812 . . . 4  |-  ( ( k  e.  ZZ  /\  k  =/=  0 )  ->  -.  ( abs `  k
)  <  1 )
8 nan 588 . . . 4  |-  ( ( k  e.  ZZ  ->  -.  ( k  =/=  0  /\  ( abs `  k
)  <  1 ) )  <->  ( ( k  e.  ZZ  /\  k  =/=  0 )  ->  -.  ( abs `  k )  <  1 ) )
97, 8mpbir 214 . . 3  |-  ( k  e.  ZZ  ->  -.  ( k  =/=  0  /\  ( abs `  k
)  <  1 ) )
109nrex 2854 . 2  |-  -.  E. k  e.  ZZ  (
k  =/=  0  /\  ( abs `  k
)  <  1 )
11 ere 14192 . . . . . . . 8  |-  _e  e.  RR
1211recni 9681 . . . . . . 7  |-  _e  e.  CC
13 neldif 3570 . . . . . . 7  |-  ( ( _e  e.  CC  /\  -.  _e  e.  ( CC 
\  AA ) )  ->  _e  e.  AA )
1412, 13mpan 681 . . . . . 6  |-  ( -.  _e  e.  ( CC 
\  AA )  ->  _e  e.  AA )
15 ene0 14310 . . . . . . . 8  |-  _e  =/=  0
16 elsncg 4003 . . . . . . . . 9  |-  ( _e  e.  CC  ->  (
_e  e.  { 0 }  <->  _e  =  0
) )
1712, 16ax-mp 5 . . . . . . . 8  |-  ( _e  e.  { 0 }  <-> 
_e  =  0 )
1815, 17nemtbir 2731 . . . . . . 7  |-  -.  _e  e.  { 0 }
1918a1i 11 . . . . . 6  |-  ( -.  _e  e.  ( CC 
\  AA )  ->  -.  _e  e.  { 0 } )
2014, 19eldifd 3427 . . . . 5  |-  ( -.  _e  e.  ( CC 
\  AA )  ->  _e  e.  ( AA  \  { 0 } ) )
21 elaa2 38137 . . . . 5  |-  ( _e  e.  ( AA  \  { 0 } )  <-> 
( _e  e.  CC  /\ 
E. q  e.  (Poly `  ZZ ) ( ( (coeff `  q ) `  0 )  =/=  0  /\  ( q `
 _e )  =  0 ) ) )
2220, 21sylib 201 . . . 4  |-  ( -.  _e  e.  ( CC 
\  AA )  -> 
( _e  e.  CC  /\ 
E. q  e.  (Poly `  ZZ ) ( ( (coeff `  q ) `  0 )  =/=  0  /\  ( q `
 _e )  =  0 ) ) )
2322simprd 469 . . 3  |-  ( -.  _e  e.  ( CC 
\  AA )  ->  E. q  e.  (Poly `  ZZ ) ( ( (coeff `  q ) `  0 )  =/=  0  /\  ( q `
 _e )  =  0 ) )
24 simpl 463 . . . . . . 7  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
(coeff `  q ) `  0 )  =/=  0 )  ->  q  e.  (Poly `  ZZ )
)
25 0nn0 10913 . . . . . . . . 9  |-  0  e.  NN0
26 n0p 37414 . . . . . . . . 9  |-  ( ( q  e.  (Poly `  ZZ )  /\  0  e.  NN0  /\  ( (coeff `  q ) `  0
)  =/=  0 )  ->  q  =/=  0p )
2725, 26mp3an2 1361 . . . . . . . 8  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
(coeff `  q ) `  0 )  =/=  0 )  ->  q  =/=  0p )
28 nelsn 4012 . . . . . . . 8  |-  ( q  =/=  0p  ->  -.  q  e.  { 0p } )
2927, 28syl 17 . . . . . . 7  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
(coeff `  q ) `  0 )  =/=  0 )  ->  -.  q  e.  { 0p } )
3024, 29eldifd 3427 . . . . . 6  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
(coeff `  q ) `  0 )  =/=  0 )  ->  q  e.  ( (Poly `  ZZ )  \  { 0p } ) )
3130adantrr 728 . . . . 5  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
( (coeff `  q
) `  0 )  =/=  0  /\  (
q `  _e )  =  0 ) )  ->  q  e.  ( (Poly `  ZZ )  \  { 0p }
) )
32 simprr 771 . . . . 5  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
( (coeff `  q
) `  0 )  =/=  0  /\  (
q `  _e )  =  0 ) )  ->  ( q `  _e )  =  0
)
33 eqid 2462 . . . . 5  |-  (coeff `  q )  =  (coeff `  q )
34 simprl 769 . . . . 5  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
( (coeff `  q
) `  0 )  =/=  0  /\  (
q `  _e )  =  0 ) )  ->  ( (coeff `  q ) `  0
)  =/=  0 )
35 eqid 2462 . . . . 5  |-  (deg `  q )  =  (deg
`  q )
36 eqid 2462 . . . . 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 2462 . . . . 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 5888 . . . . . . . . . . . . . . . . . . . 20  |-  ( h  =  l  ->  (
(coeff `  q ) `  h )  =  ( (coeff `  q ) `  l ) )
39 oveq2 6323 . . . . . . . . . . . . . . . . . . . 20  |-  ( h  =  l  ->  (
_e  ^c  h )  =  ( _e  ^c  l ) )
4038, 39oveq12d 6333 . . . . . . . . . . . . . . . . . . 19  |-  ( h  =  l  ->  (
( (coeff `  q
) `  h )  x.  ( _e  ^c 
h ) )  =  ( ( (coeff `  q ) `  l
)  x.  ( _e 
^c  l ) ) )
4140fveq2d 5892 . . . . . . . . . . . . . . . . . 18  |-  ( h  =  l  ->  ( abs `  ( ( (coeff `  q ) `  h
)  x.  ( _e 
^c  h ) ) )  =  ( abs `  ( ( (coeff `  q ) `  l )  x.  (
_e  ^c  l ) ) ) )
4241oveq1d 6330 . . . . . . . . . . . . . . . . 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 13811 . . . . . . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . . 16  |-  ( m  =  n  ->  (
( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
m )  =  ( ( (deg `  q
) ^ ( (deg
`  q )  +  1 ) ) ^
n ) )
46 fveq2 5888 . . . . . . . . . . . . . . . 16  |-  ( m  =  n  ->  ( ! `  m )  =  ( ! `  n ) )
4745, 46oveq12d 6333 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  (
( ( (deg `  q ) ^ (
(deg `  q )  +  1 ) ) ^ m )  / 
( ! `  m
) )  =  ( ( ( (deg `  q ) ^ (
(deg `  q )  +  1 ) ) ^ n )  / 
( ! `  n
) ) )
4844, 47oveq12d 6333 . . . . . . . . . . . . . 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 4509 . . . . . . . . . . . . 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 22 . . . . . . . . . . . 12  |-  ( m  =  n  ->  m  =  n )
5250, 51fveq12d 5894 . . . . . . . . . . 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 5892 . . . . . . . . . 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 4426 . . . . . . . . 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 3031 . . . . . . . 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 5888 . . . . . . . . 9  |-  ( j  =  i  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  i )
)
5756raleqdv 3005 . . . . . . . 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 265 . . . . . . 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 3056 . . . . . 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 8020 . . . . 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 2462 . . . . 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 38186 . . . 4  |-  ( ( q  e.  (Poly `  ZZ )  /\  (
( (coeff `  q
) `  0 )  =/=  0  /\  (
q `  _e )  =  0 ) )  ->  E. k  e.  ZZ  ( k  =/=  0  /\  ( abs `  k
)  <  1 ) )
6362rexlimiva 2887 . . 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 185 1  |-  _e  e.  ( CC  \  AA )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750   {crab 2753    \ cdif 3413   {csn 3980   {ctp 3984   class class class wbr 4416    |-> cmpt 4475   ` cfv 5601  (class class class)co 6315   supcsup 7980  infcinf 7981   CCcc 9563   RRcr 9564   0cc0 9565   1c1 9566    + caddc 9568    x. cmul 9570   RR*cxr 9700    < clt 9701    / cdiv 10297   NN0cn0 10898   ZZcz 10966   ZZ>=cuz 11188   ...cfz 11813   ^cexp 12304   !cfa 12491   abscabs 13346   sum_csu 13801   _eceu 14164   0pc0p 22676  Polycply 23187  coeffccoe 23189  degcdgr 23190   AAcaa 23316    ^c ccxp 23554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cc 8891  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643  ax-addf 9644  ax-mulf 9645
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-disj 4388  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-of 6558  df-ofr 6559  df-om 6720  df-1st 6820  df-2nd 6821  df-supp 6942  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-2o 7209  df-oadd 7212  df-omul 7213  df-er 7389  df-map 7500  df-pm 7501  df-ixp 7549  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-fsupp 7910  df-fi 7951  df-sup 7982  df-inf 7983  df-oi 8051  df-card 8399  df-acn 8402  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-5 10699  df-6 10700  df-7 10701  df-8 10702  df-9 10703  df-10 10704  df-n0 10899  df-z 10967  df-dec 11081  df-uz 11189  df-q 11294  df-rp 11332  df-xneg 11438  df-xadd 11439  df-xmul 11440  df-ioo 11668  df-ioc 11669  df-ico 11670  df-icc 11671  df-fz 11814  df-fzo 11947  df-fl 12060  df-mod 12129  df-seq 12246  df-exp 12305  df-fac 12492  df-bc 12520  df-hash 12548  df-shft 13179  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-limsup 13575  df-clim 13601  df-rlim 13602  df-sum 13802  df-prod 14009  df-ef 14170  df-e 14171  df-sin 14172  df-cos 14173  df-tan 14174  df-pi 14175  df-dvds 14355  df-gcd 14518  df-prm 14672  df-struct 15172  df-ndx 15173  df-slot 15174  df-base 15175  df-sets 15176  df-ress 15177  df-plusg 15252  df-mulr 15253  df-starv 15254  df-sca 15255  df-vsca 15256  df-ip 15257  df-tset 15258  df-ple 15259  df-ds 15261  df-unif 15262  df-hom 15263  df-cco 15264  df-rest 15370  df-topn 15371  df-0g 15389  df-gsum 15390  df-topgen 15391  df-pt 15392  df-prds 15395  df-xrs 15449  df-qtop 15455  df-imas 15456  df-xps 15459  df-mre 15541  df-mrc 15542  df-acs 15544  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-submnd 16632  df-mulg 16725  df-cntz 17020  df-cmn 17481  df-psmet 19011  df-xmet 19012  df-met 19013  df-bl 19014  df-mopn 19015  df-fbas 19016  df-fg 19017  df-cnfld 19020  df-top 19970  df-bases 19971  df-topon 19972  df-topsp 19973  df-cld 20083  df-ntr 20084  df-cls 20085  df-nei 20163  df-lp 20201  df-perf 20202  df-cn 20292  df-cnp 20293  df-haus 20380  df-cmp 20451  df-tx 20626  df-hmeo 20819  df-fil 20910  df-fm 21002  df-flim 21003  df-flf 21004  df-xms 21384  df-ms 21385  df-tms 21386  df-cncf 21959  df-ovol 22465  df-vol 22467  df-mbf 22626  df-itg1 22627  df-itg2 22628  df-ibl 22629  df-itg 22630  df-0p 22677  df-limc 22870  df-dv 22871  df-dvn 22872  df-ply 23191  df-coe 23193  df-dgr 23194  df-aa 23317  df-log 23555  df-cxp 23556
This theorem is referenced by: (None)
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