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Theorem vieta1 20182
Description: The first-order Vieta's formula (see http://en.wikipedia.org/wiki/Vieta%27s_formulas). If a polynomial of degree  N has  N distinct roots, then the sum over these roots can be calculated as  -u A ( N  -  1 )  /  A ( N ). (If the roots are not distinct, then this formula is still true but must double-count some of the roots according to their multiplicities.) (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
vieta1.1  |-  A  =  (coeff `  F )
vieta1.2  |-  N  =  (deg `  F )
vieta1.3  |-  R  =  ( `' F " { 0 } )
vieta1.4  |-  ( ph  ->  F  e.  (Poly `  S ) )
vieta1.5  |-  ( ph  ->  ( # `  R
)  =  N )
vieta1.6  |-  ( ph  ->  N  e.  NN )
Assertion
Ref Expression
vieta1  |-  ( ph  -> 
sum_ x  e.  R  x  =  -u ( ( A `  ( N  -  1 ) )  /  ( A `  N ) ) )
Distinct variable groups:    x, R    ph, x
Allowed substitution hints:    A( x)    S( x)    F( x)    N( x)

Proof of Theorem vieta1
Dummy variables  f 
k  y  z  d  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyssc 20072 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
2 vieta1.4 . . 3  |-  ( ph  ->  F  e.  (Poly `  S ) )
31, 2sseldi 3306 . 2  |-  ( ph  ->  F  e.  (Poly `  CC ) )
4 vieta1.6 . . 3  |-  ( ph  ->  N  e.  NN )
5 eqeq1 2410 . . . . . . 7  |-  ( y  =  1  ->  (
y  =  (deg `  f )  <->  1  =  (deg `  f ) ) )
65anbi1d 686 . . . . . 6  |-  ( y  =  1  ->  (
( y  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( 1  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) ) ) )
76imbi1d 309 . . . . 5  |-  ( y  =  1  ->  (
( ( y  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  <->  ( (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) ) ) )
87ralbidv 2686 . . . 4  |-  ( y  =  1  ->  ( A. f  e.  (Poly `  CC ) ( ( y  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) )  <->  A. f  e.  (Poly `  CC ) ( ( 1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) ) ) )
9 eqeq1 2410 . . . . . . 7  |-  ( y  =  d  ->  (
y  =  (deg `  f )  <->  d  =  (deg `  f ) ) )
109anbi1d 686 . . . . . 6  |-  ( y  =  d  ->  (
( y  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( d  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) ) ) )
1110imbi1d 309 . . . . 5  |-  ( y  =  d  ->  (
( ( y  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  <->  ( (
d  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) ) ) )
1211ralbidv 2686 . . . 4  |-  ( y  =  d  ->  ( A. f  e.  (Poly `  CC ) ( ( y  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) )  <->  A. f  e.  (Poly `  CC ) ( ( d  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) ) ) )
13 eqeq1 2410 . . . . . . 7  |-  ( y  =  ( d  +  1 )  ->  (
y  =  (deg `  f )  <->  ( d  +  1 )  =  (deg `  f )
) )
1413anbi1d 686 . . . . . 6  |-  ( y  =  ( d  +  1 )  ->  (
( y  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( ( d  +  1 )  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) ) ) )
1514imbi1d 309 . . . . 5  |-  ( y  =  ( d  +  1 )  ->  (
( ( y  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  <->  ( (
( d  +  1 )  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) ) ) )
1615ralbidv 2686 . . . 4  |-  ( y  =  ( d  +  1 )  ->  ( A. f  e.  (Poly `  CC ) ( ( y  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) )  <->  A. f  e.  (Poly `  CC ) ( ( ( d  +  1 )  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) ) ) )
17 eqeq1 2410 . . . . . . 7  |-  ( y  =  N  ->  (
y  =  (deg `  f )  <->  N  =  (deg `  f ) ) )
1817anbi1d 686 . . . . . 6  |-  ( y  =  N  ->  (
( y  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( N  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) ) ) )
1918imbi1d 309 . . . . 5  |-  ( y  =  N  ->  (
( ( y  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  <->  ( ( N  =  (deg `  f
)  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) ) ) )
2019ralbidv 2686 . . . 4  |-  ( y  =  N  ->  ( A. f  e.  (Poly `  CC ) ( ( y  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) )  <->  A. f  e.  (Poly `  CC ) ( ( N  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) ) ) )
21 eqid 2404 . . . . . . . . . . . . . 14  |-  (coeff `  f )  =  (coeff `  f )
2221coef3 20104 . . . . . . . . . . . . 13  |-  ( f  e.  (Poly `  CC )  ->  (coeff `  f
) : NN0 --> CC )
2322adantr 452 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  (coeff `  f
) : NN0 --> CC )
24 0nn0 10192 . . . . . . . . . . . 12  |-  0  e.  NN0
25 ffvelrn 5827 . . . . . . . . . . . 12  |-  ( ( (coeff `  f ) : NN0 --> CC  /\  0  e.  NN0 )  ->  (
(coeff `  f ) `  0 )  e.  CC )
2623, 24, 25sylancl 644 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  0
)  e.  CC )
27 1nn0 10193 . . . . . . . . . . . 12  |-  1  e.  NN0
28 ffvelrn 5827 . . . . . . . . . . . 12  |-  ( ( (coeff `  f ) : NN0 --> CC  /\  1  e.  NN0 )  ->  (
(coeff `  f ) `  1 )  e.  CC )
2923, 27, 28sylancl 644 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  1
)  e.  CC )
30 simpr 448 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  1  =  (deg `  f ) )
3130fveq2d 5691 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  1
)  =  ( (coeff `  f ) `  (deg `  f ) ) )
32 ax-1ne0 9015 . . . . . . . . . . . . . . . 16  |-  1  =/=  0
3332a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  1  =/=  0 )
3430, 33eqnetrrd 2587 . . . . . . . . . . . . . 14  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  (deg `  f
)  =/=  0 )
35 fveq2 5687 . . . . . . . . . . . . . . . 16  |-  ( f  =  0 p  -> 
(deg `  f )  =  (deg `  0 p
) )
36 dgr0 20133 . . . . . . . . . . . . . . . 16  |-  (deg ` 
0 p )  =  0
3735, 36syl6eq 2452 . . . . . . . . . . . . . . 15  |-  ( f  =  0 p  -> 
(deg `  f )  =  0 )
3837necon3i 2606 . . . . . . . . . . . . . 14  |-  ( (deg
`  f )  =/=  0  ->  f  =/=  0 p )
3934, 38syl 16 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  f  =/=  0 p )
40 eqid 2404 . . . . . . . . . . . . . . . 16  |-  (deg `  f )  =  (deg
`  f )
4140, 21dgreq0 20136 . . . . . . . . . . . . . . 15  |-  ( f  e.  (Poly `  CC )  ->  ( f  =  0 p  <->  ( (coeff `  f ) `  (deg `  f ) )  =  0 ) )
4241necon3bid 2602 . . . . . . . . . . . . . 14  |-  ( f  e.  (Poly `  CC )  ->  ( f  =/=  0 p  <->  ( (coeff `  f ) `  (deg `  f ) )  =/=  0 ) )
4342adantr 452 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( f  =/=  0 p  <->  ( (coeff `  f ) `  (deg `  f ) )  =/=  0 ) )
4439, 43mpbid 202 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  (deg `  f ) )  =/=  0 )
4531, 44eqnetrd 2585 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  1
)  =/=  0 )
4626, 29, 45divcld 9746 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
)  e.  CC )
4746negcld 9354 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
)  e.  CC )
48 id 20 . . . . . . . . . 10  |-  ( x  =  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) )  ->  x  =  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )
4948sumsn 12489 . . . . . . . . 9  |-  ( (
-u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) )  e.  CC  /\  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) )  e.  CC )  ->  sum_ x  e.  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } x  =  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )
5047, 47, 49syl2anc 643 . . . . . . . 8  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  sum_ x  e. 
{ -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } x  =  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )
5150adantrr 698 . . . . . . 7  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  sum_ x  e. 
{ -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } x  =  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )
52 eqid 2404 . . . . . . . . . . . . 13  |-  ( `' f " { 0 } )  =  ( `' f " {
0 } )
5352fta1 20178 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  f  =/=  0 p )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )
5439, 53syldan 457 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )
5554simpld 446 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( `' f " { 0 } )  e.  Fin )
5655adantrr 698 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  ( `' f " { 0 } )  e.  Fin )
5721, 40coeid2 20111 . . . . . . . . . . . . . 14  |-  ( ( f  e.  (Poly `  CC )  /\  -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) )  e.  CC )  ->  ( f `  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )  = 
sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  k )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ k ) ) )
5847, 57syldan 457 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( f `  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )  = 
sum_ k  e.  ( 0 ... (deg `  f ) ) ( ( (coeff `  f
) `  k )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ k ) ) )
5930oveq2d 6056 . . . . . . . . . . . . . 14  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( 0 ... 1 )  =  ( 0 ... (deg `  f ) ) )
6059sumeq1d 12450 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  sum_ k  e.  ( 0 ... 1
) ( ( (coeff `  f ) `  k
)  x.  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
) )  =  sum_ k  e.  ( 0 ... (deg `  f
) ) ( ( (coeff `  f ) `  k )  x.  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
) ) )
61 nn0uz 10476 . . . . . . . . . . . . . . 15  |-  NN0  =  ( ZZ>= `  0 )
62 1e0p1 10366 . . . . . . . . . . . . . . 15  |-  1  =  ( 0  +  1 )
63 fveq2 5687 . . . . . . . . . . . . . . . 16  |-  ( k  =  1  ->  (
(coeff `  f ) `  k )  =  ( (coeff `  f ) `  1 ) )
64 oveq2 6048 . . . . . . . . . . . . . . . 16  |-  ( k  =  1  ->  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
)  =  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ 1 ) )
6563, 64oveq12d 6058 . . . . . . . . . . . . . . 15  |-  ( k  =  1  ->  (
( (coeff `  f
) `  k )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ k ) )  =  ( ( (coeff `  f ) `  1 )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ 1 ) ) )
6623ffvelrnda 5829 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  /\  k  e.  NN0 )  ->  ( (coeff `  f ) `  k
)  e.  CC )
67 expcl 11354 . . . . . . . . . . . . . . . . 17  |-  ( (
-u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) )  e.  CC  /\  k  e.  NN0 )  ->  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ k )  e.  CC )
6847, 67sylan 458 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  /\  k  e.  NN0 )  ->  ( -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) ^ k
)  e.  CC )
6966, 68mulcld 9064 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  /\  k  e.  NN0 )  ->  ( (
(coeff `  f ) `  k )  x.  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
) )  e.  CC )
70 0z 10249 . . . . . . . . . . . . . . . . . 18  |-  0  e.  ZZ
7147exp0d 11472 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) ^ 0 )  =  1 )
7271oveq2d 6056 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ 0 ) )  =  ( ( (coeff `  f ) `  0 )  x.  1 ) )
7326mulid1d 9061 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  x.  1 )  =  ( (coeff `  f ) `  0 ) )
7472, 73eqtrd 2436 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ 0 ) )  =  ( (coeff `  f ) `  0
) )
7574, 26eqeltrd 2478 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ 0 ) )  e.  CC )
76 fveq2 5687 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  0  ->  (
(coeff `  f ) `  k )  =  ( (coeff `  f ) `  0 ) )
77 oveq2 6048 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  0  ->  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
)  =  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ 0 ) )
7876, 77oveq12d 6058 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  0  ->  (
( (coeff `  f
) `  k )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ k ) )  =  ( ( (coeff `  f ) `  0 )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ 0 ) ) )
7978fsum1 12490 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  ZZ  /\  ( ( (coeff `  f ) `  0
)  x.  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ 0 ) )  e.  CC )  ->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  f ) `  k
)  x.  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
) )  =  ( ( (coeff `  f
) `  0 )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ 0 ) ) )
8070, 75, 79sylancr 645 . . . . . . . . . . . . . . . . 17  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  f ) `  k
)  x.  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
) )  =  ( ( (coeff `  f
) `  0 )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ 0 ) ) )
8180, 74eqtrd 2436 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  f ) `  k
)  x.  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
) )  =  ( (coeff `  f ) `  0 ) )
8281, 24jctil 524 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( 0  e.  NN0  /\  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  f ) `  k
)  x.  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
) )  =  ( (coeff `  f ) `  0 ) ) )
8347exp1d 11473 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) ^ 1 )  =  -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) )
8483oveq2d 6056 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  1 )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ 1 ) )  =  ( ( (coeff `  f ) `  1 )  x.  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ) )
8529, 46mulneg2d 9443 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  1 )  x.  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )  = 
-u ( ( (coeff `  f ) `  1
)  x.  ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ) )
8626, 29, 45divcan2d 9748 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  1 )  x.  ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )  =  ( (coeff `  f
) `  0 )
)
8786negeqd 9256 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  -u ( ( (coeff `  f ) `  1 )  x.  ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )  = 
-u ( (coeff `  f ) `  0
) )
8884, 85, 873eqtrd 2440 . . . . . . . . . . . . . . . . 17  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  1 )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ 1 ) )  =  -u (
(coeff `  f ) `  0 ) )
8988oveq2d 6056 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  +  ( ( (coeff `  f ) `  1
)  x.  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ 1 ) ) )  =  ( ( (coeff `  f ) `  0
)  +  -u (
(coeff `  f ) `  0 ) ) )
9026negidd 9357 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  + 
-u ( (coeff `  f ) `  0
) )  =  0 )
9189, 90eqtrd 2436 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  +  ( ( (coeff `  f ) `  1
)  x.  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ 1 ) ) )  =  0 )
9261, 62, 65, 69, 82, 91fsump1i 12508 . . . . . . . . . . . . . 14  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( 1  e.  NN0  /\  sum_ k  e.  ( 0 ... 1
) ( ( (coeff `  f ) `  k
)  x.  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
) )  =  0 ) )
9392simprd 450 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  sum_ k  e.  ( 0 ... 1
) ( ( (coeff `  f ) `  k
)  x.  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
) )  =  0 )
9458, 60, 933eqtr2d 2442 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( f `  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )  =  0 )
95 plyf 20070 . . . . . . . . . . . . . . 15  |-  ( f  e.  (Poly `  CC )  ->  f : CC --> CC )
96 ffn 5550 . . . . . . . . . . . . . . 15  |-  ( f : CC --> CC  ->  f  Fn  CC )
9795, 96syl 16 . . . . . . . . . . . . . 14  |-  ( f  e.  (Poly `  CC )  ->  f  Fn  CC )
9897adantr 452 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  f  Fn  CC )
99 fniniseg 5810 . . . . . . . . . . . . 13  |-  ( f  Fn  CC  ->  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) )  e.  ( `' f " {
0 } )  <->  ( -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) )  e.  CC  /\  ( f `  -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) )  =  0 ) ) )
10098, 99syl 16 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) )  e.  ( `' f " {
0 } )  <->  ( -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) )  e.  CC  /\  ( f `  -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) )  =  0 ) ) )
10147, 94, 100mpbir2and 889 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
)  e.  ( `' f " { 0 } ) )
102101snssd 3903 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  { -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) }  C_  ( `' f " {
0 } ) )
103102adantrr 698 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  { -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) }  C_  ( `' f " {
0 } ) )
104 hashsng 11602 . . . . . . . . . . . . . 14  |-  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) )  e.  CC  ->  ( # `  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } )  =  1 )
10547, 104syl 16 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( # `  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } )  =  1 )
106105, 30eqtrd 2436 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( # `  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } )  =  (deg `  f
) )
107106adantrr 698 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  ( # `  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } )  =  (deg `  f
) )
108 simprr 734 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )
109107, 108eqtr4d 2439 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  ( # `  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } )  =  ( # `  ( `' f " {
0 } ) ) )
110 snfi 7146 . . . . . . . . . . . 12  |-  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) }  e.  Fin
111 hashen 11586 . . . . . . . . . . . 12  |-  ( ( { -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) }  e.  Fin  /\  ( `' f " { 0 } )  e.  Fin )  -> 
( ( # `  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } )  =  ( # `  ( `' f " {
0 } ) )  <->  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) }  ~~  ( `' f " {
0 } ) ) )
112110, 55, 111sylancr 645 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( ( # `
 { -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) } )  =  ( # `  ( `' f " {
0 } ) )  <->  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) }  ~~  ( `' f " {
0 } ) ) )
113112adantrr 698 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  ( ( # `
 { -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) } )  =  ( # `  ( `' f " {
0 } ) )  <->  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) }  ~~  ( `' f " {
0 } ) ) )
114109, 113mpbid 202 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  { -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) }  ~~  ( `' f " {
0 } ) )
115 fisseneq 7279 . . . . . . . . 9  |-  ( ( ( `' f " { 0 } )  e.  Fin  /\  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) }  C_  ( `' f " {
0 } )  /\  {
-u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) }  ~~  ( `' f " {
0 } ) )  ->  { -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) }  =  ( `' f " {
0 } ) )
11656, 103, 114, 115syl3anc 1184 . . . . . . . 8  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  { -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) }  =  ( `' f " {
0 } ) )
117116sumeq1d 12450 . . . . . . 7  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  sum_ x  e. 
{ -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } x  =  sum_ x  e.  ( `' f " {
0 } ) x )
118 1m1e0 10024 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
11930oveq1d 6055 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( 1  -  1 )  =  ( (deg `  f
)  -  1 ) )
120118, 119syl5eqr 2450 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  0  =  ( (deg `  f )  -  1 ) )
121120fveq2d 5691 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  0
)  =  ( (coeff `  f ) `  (
(deg `  f )  -  1 ) ) )
122121, 31oveq12d 6058 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
)  =  ( ( (coeff `  f ) `  ( (deg `  f
)  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) )
123122negeqd 9256 . . . . . . . 8  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
)  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )
124123adantrr 698 . . . . . . 7  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
)  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )
12551, 117, 1243eqtr3d 2444 . . . . . 6  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )
126125ex 424 . . . . 5  |-  ( f  e.  (Poly `  CC )  ->  ( ( 1  =  (deg `  f
)  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) ) )
127126rgen 2731 . . . 4  |-  A. f  e.  (Poly `  CC )
( ( 1  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )
128 id 20 . . . . . . . . . . 11  |-  ( y  =  x  ->  y  =  x )
129128cbvsumv 12445 . . . . . . . . . 10  |-  sum_ y  e.  ( `' f " { 0 } ) y  =  sum_ x  e.  ( `' f " { 0 } ) x
130129eqeq1i 2411 . . . . . . . . 9  |-  ( sum_ y  e.  ( `' f " { 0 } ) y  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) )  <->  sum_ x  e.  ( `' f " { 0 } ) x  = 
-u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) )
131130imbi2i 304 . . . . . . . 8  |-  ( ( ( d  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  ->  sum_ y  e.  ( `' f " { 0 } ) y  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  <->  ( (
d  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) ) )
132131ralbii 2690 . . . . . . 7  |-  ( A. f  e.  (Poly `  CC ) ( ( d  =  (deg `  f
)  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ y  e.  ( `' f " { 0 } ) y  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  <->  A. f  e.  (Poly `  CC )
( ( d  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) ) )
133 eqid 2404 . . . . . . . . 9  |-  (coeff `  g )  =  (coeff `  g )
134 eqid 2404 . . . . . . . . 9  |-  (deg `  g )  =  (deg
`  g )
135 eqid 2404 . . . . . . . . 9  |-  ( `' g " { 0 } )  =  ( `' g " {
0 } )
136 simp1r 982 . . . . . . . . 9  |-  ( ( ( d  e.  NN  /\  g  e.  (Poly `  CC ) )  /\  A. f  e.  (Poly `  CC ) ( ( d  =  (deg `  f
)  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ y  e.  ( `' f " { 0 } ) y  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  /\  ( ( d  +  1 )  =  (deg
`  g )  /\  ( # `  ( `' g " { 0 } ) )  =  (deg `  g )
) )  ->  g  e.  (Poly `  CC )
)
137 simp3r 986 . . . . . . . . 9  |-  ( ( ( d  e.  NN  /\  g  e.  (Poly `  CC ) )  /\  A. f  e.  (Poly `  CC ) ( ( d  =  (deg `  f
)  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ y  e.  ( `' f " { 0 } ) y  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  /\  ( ( d  +  1 )  =  (deg
`  g )  /\  ( # `  ( `' g " { 0 } ) )  =  (deg `  g )
) )  ->  ( # `
 ( `' g
" { 0 } ) )  =  (deg
`  g ) )
138 simp1l 981 . . . . . . . . 9  |-  ( ( ( d  e.  NN  /\  g  e.  (Poly `  CC ) )  /\  A. f  e.  (Poly `  CC ) ( ( d  =  (deg `  f
)  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ y  e.  ( `' f " { 0 } ) y  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  /\  ( ( d  +  1 )  =  (deg
`  g )  /\  ( # `  ( `' g " { 0 } ) )  =  (deg `  g )
) )  ->  d  e.  NN )
139 simp3l 985 . . . . . . . . 9  |-  ( ( ( d  e.  NN  /\  g  e.  (Poly `  CC ) )  /\  A. f  e.  (Poly `  CC ) ( ( d  =  (deg `  f
)  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ y  e.  ( `' f " { 0 } ) y  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  /\  ( ( d  +  1 )  =  (deg
`  g )  /\  ( # `  ( `' g " { 0 } ) )  =  (deg `  g )
) )  ->  (
d  +  1 )  =  (deg `  g
) )
140 simp2 958 . . . . . . . . . 10  |-  ( ( ( d  e.  NN  /\  g  e.  (Poly `  CC ) )  /\  A. f  e.  (Poly `  CC ) ( ( d  =  (deg `  f
)  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ y  e.  ( `' f " { 0 } ) y  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  /\  ( ( d  +  1 )  =  (deg
`  g )  /\  ( # `  ( `' g " { 0 } ) )  =  (deg `  g )
) )  ->  A. f  e.  (Poly `  CC )
( ( d  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ y  e.  ( `' f " { 0 } ) y  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) ) )
141140, 132sylib 189 . . . . . . . . 9  |-  ( ( ( d  e.  NN  /\  g  e.  (Poly `  CC ) )  /\  A. f  e.  (Poly `  CC ) ( ( d  =  (deg `  f
)  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ y  e.  ( `' f " { 0 } ) y  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  /\  ( ( d  +  1 )  =  (deg
`  g )  /\  ( # `  ( `' g " { 0 } ) )  =  (deg `  g )
) )  ->  A. f  e.  (Poly `  CC )
( ( d  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) ) )
142 eqid 2404 . . . . . . . . 9  |-  ( g quot  ( X p  o F  -  ( CC  X.  { z } ) ) )  =  ( g quot  ( X p  o F  -  ( CC  X.  { z } ) ) )
143133, 134, 135, 136, 137, 138, 139, 141, 142vieta1lem2 20181 . . . . . . . 8  |-  ( ( ( d  e.  NN  /\  g  e.  (Poly `  CC ) )  /\  A. f  e.  (Poly `  CC ) ( ( d  =  (deg `  f
)  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ y  e.  ( `' f " { 0 } ) y  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  /\  ( ( d  +  1 )  =  (deg
`  g )  /\  ( # `  ( `' g " { 0 } ) )  =  (deg `  g )
) )  ->  sum_ x  e.  ( `' g " { 0 } ) x  =  -u (
( (coeff `  g
) `  ( (deg `  g )  -  1 ) )  /  (
(coeff `  g ) `  (deg `  g )
) ) )
1441433exp 1152 . . . . . . 7  |-  ( ( d  e.  NN  /\  g  e.  (Poly `  CC ) )  ->  ( A. f  e.  (Poly `  CC ) ( ( d  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ y  e.  ( `' f " {
0 } ) y  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) )  ->  ( (
( d  +  1 )  =  (deg `  g )  /\  ( # `
 ( `' g
" { 0 } ) )  =  (deg
`  g ) )  ->  sum_ x  e.  ( `' g " {
0 } ) x  =  -u ( ( (coeff `  g ) `  (
(deg `  g )  -  1 ) )  /  ( (coeff `  g ) `  (deg `  g ) ) ) ) ) )
145132, 144syl5bir 210 . . . . . 6  |-  ( ( d  e.  NN  /\  g  e.  (Poly `  CC ) )  ->  ( A. f  e.  (Poly `  CC ) ( ( d  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) )  ->  ( (
( d  +  1 )  =  (deg `  g )  /\  ( # `
 ( `' g
" { 0 } ) )  =  (deg
`  g ) )  ->  sum_ x  e.  ( `' g " {
0 } ) x  =  -u ( ( (coeff `  g ) `  (
(deg `  g )  -  1 ) )  /  ( (coeff `  g ) `  (deg `  g ) ) ) ) ) )
146145ralrimdva 2756 . . . . 5  |-  ( d  e.  NN  ->  ( A. f  e.  (Poly `  CC ) ( ( d  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) )  ->  A. g  e.  (Poly `  CC )
( ( ( d  +  1 )  =  (deg `  g )  /\  ( # `  ( `' g " {
0 } ) )  =  (deg `  g
) )  ->  sum_ x  e.  ( `' g " { 0 } ) x  =  -u (
( (coeff `  g
) `  ( (deg `  g )  -  1 ) )  /  (
(coeff `  g ) `  (deg `  g )
) ) ) ) )
147 fveq2 5687 . . . . . . . . 9  |-  ( g  =  f  ->  (deg `  g )  =  (deg
`  f ) )
148147eqeq2d 2415 . . . . . . . 8  |-  ( g  =  f  ->  (
( d  +  1 )  =  (deg `  g )  <->  ( d  +  1 )  =  (deg `  f )
) )
149 cnveq 5005 . . . . . . . . . . 11  |-  ( g  =  f  ->  `' g  =  `' f
)
150149imaeq1d 5161 . . . . . . . . . 10  |-  ( g  =  f  ->  ( `' g " {
0 } )  =  ( `' f " { 0 } ) )
151150fveq2d 5691 . . . . . . . . 9  |-  ( g  =  f  ->  ( # `
 ( `' g
" { 0 } ) )  =  (
# `  ( `' f " { 0 } ) ) )
152151, 147eqeq12d 2418 . . . . . . . 8  |-  ( g  =  f  ->  (
( # `  ( `' g " { 0 } ) )  =  (deg `  g )  <->  (
# `  ( `' f " { 0 } ) )  =  (deg
`  f ) ) )
153148, 152anbi12d 692 . . . . . . 7  |-  ( g  =  f  ->  (
( ( d  +  1 )  =  (deg
`  g )  /\  ( # `  ( `' g " { 0 } ) )  =  (deg `  g )
)  <->  ( ( d  +  1 )  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) ) ) )
154150sumeq1d 12450 . . . . . . . 8  |-  ( g  =  f  ->  sum_ x  e.  ( `' g " { 0 } ) x  =  sum_ x  e.  ( `' f " { 0 } ) x )
155 fveq2 5687 . . . . . . . . . . 11  |-  ( g  =  f  ->  (coeff `  g )  =  (coeff `  f ) )
156147oveq1d 6055 . . . . . . . . . . 11  |-  ( g  =  f  ->  (
(deg `  g )  -  1 )  =  ( (deg `  f
)  -  1 ) )
157155, 156fveq12d 5693 . . . . . . . . . 10  |-  ( g  =  f  ->  (
(coeff `  g ) `  ( (deg `  g
)  -  1 ) )  =  ( (coeff `  f ) `  (
(deg `  f )  -  1 ) ) )
158155, 147fveq12d 5693 . . . . . . . . . 10  |-  ( g  =  f  ->  (
(coeff `  g ) `  (deg `  g )
)  =  ( (coeff `  f ) `  (deg `  f ) ) )
159157, 158oveq12d 6058 . . . . . . . . 9  |-  ( g  =  f  ->  (
( (coeff `  g
) `  ( (deg `  g )  -  1 ) )  /  (
(coeff `  g ) `  (deg `  g )
) )  =  ( ( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )
160159negeqd 9256 . . . . . . . 8  |-  ( g  =  f  ->  -u (
( (coeff `  g
) `  ( (deg `  g )  -  1 ) )  /  (
(coeff `  g ) `  (deg `  g )
) )  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) )
161154, 160eqeq12d 2418 . . . . . . 7  |-  ( g  =  f  ->  ( sum_ x  e.  ( `' g " { 0 } ) x  = 
-u ( ( (coeff `  g ) `  (
(deg `  g )  -  1 ) )  /  ( (coeff `  g ) `  (deg `  g ) ) )  <->  sum_ x  e.  ( `' f " { 0 } ) x  = 
-u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) ) )
162153, 161imbi12d 312 . . . . . 6  |-  ( g  =  f  ->  (
( ( ( d  +  1 )  =  (deg `  g )  /\  ( # `  ( `' g " {
0 } ) )  =  (deg `  g
) )  ->  sum_ x  e.  ( `' g " { 0 } ) x  =  -u (
( (coeff `  g
) `  ( (deg `  g )  -  1 ) )  /  (
(coeff `  g ) `  (deg `  g )
) ) )  <->  ( (
( d  +  1 )  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) ) ) )
163162cbvralv 2892 . . . . 5  |-  ( A. g  e.  (Poly `  CC ) ( ( ( d  +  1 )  =  (deg `  g
)  /\  ( # `  ( `' g " {
0 } ) )  =  (deg `  g
) )  ->  sum_ x  e.  ( `' g " { 0 } ) x  =  -u (
( (coeff `  g
) `  ( (deg `  g )  -  1 ) )  /  (
(coeff `  g ) `  (deg `  g )
) ) )  <->  A. f  e.  (Poly `  CC )
( ( ( d  +  1 )  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) ) )
164146, 163syl6ib 218 . . . 4  |-  ( d  e.  NN  ->  ( A. f  e.  (Poly `  CC ) ( ( d  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) )  ->  A. f  e.  (Poly `  CC )
( ( ( d  +  1 )  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) ) ) )
1658, 12, 16, 20, 127, 164nnind 9974 . . 3  |-  ( N  e.  NN  ->  A. f  e.  (Poly `  CC )
( ( N  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) ) )
1664, 165syl 16 . 2  |-  ( ph  ->  A. f  e.  (Poly `  CC ) ( ( N  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) ) )
167 vieta1.5 . 2  |-  ( ph  ->  ( # `  R
)  =  N )
168 fveq2 5687 . . . . . . 7  |-  ( f  =  F  ->  (deg `  f )  =  (deg
`  F ) )
169168eqeq2d 2415 . . . . . 6  |-  ( f  =  F  ->  ( N  =  (deg `  f
)  <->  N  =  (deg `  F ) ) )
170 cnveq 5005 . . . . . . . . . 10  |-  ( f  =  F  ->  `' f  =  `' F
)
171170imaeq1d 5161 . . . . . . . . 9  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  ( `' F " { 0 } ) )
172 vieta1.3 . . . . . . . . 9  |-  R  =  ( `' F " { 0 } )
173171, 172syl6eqr 2454 . . . . . . . 8  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  R )
174173fveq2d 5691 . . . . . . 7  |-  ( f  =  F  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  R )
)
175 vieta1.2 . . . . . . . 8  |-  N  =  (deg `  F )
176168, 175syl6eqr 2454 . . . . . . 7  |-  ( f  =  F  ->  (deg `  f )  =  N )
177174, 176eqeq12d 2418 . . . . . 6  |-  ( f  =  F  ->  (
( # `  ( `' f " { 0 } ) )  =  (deg `  f )  <->  (
# `  R )  =  N ) )
178169, 177anbi12d 692 . . . . 5  |-  ( f  =  F  ->  (
( N  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( N  =  (deg `  F )  /\  ( # `  R
)  =  N ) ) )
179175biantrur 493 . . . . 5  |-  ( (
# `  R )  =  N  <->  ( N  =  (deg `  F )  /\  ( # `  R
)  =  N ) )
180178, 179syl6bbr 255 . . . 4  |-  ( f  =  F  ->  (
( N  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( # `  R
)  =  N ) )
181173sumeq1d 12450 . . . . 5  |-  ( f  =  F  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  sum_ x  e.  R  x )
182 fveq2 5687 . . . . . . . . 9  |-  ( f  =  F  ->  (coeff `  f )  =  (coeff `  F ) )
183 vieta1.1 . . . . . . . . 9  |-  A  =  (coeff `  F )
184182, 183syl6eqr 2454 . . . . . . . 8  |-  ( f  =  F  ->  (coeff `  f )  =  A )
185176oveq1d 6055 . . . . . . . 8  |-  ( f  =  F  ->  (
(deg `  f )  -  1 )  =  ( N  -  1 ) )
186184, 185fveq12d 5693 . . . . . . 7  |-  ( f  =  F  ->  (
(coeff `  f ) `  ( (deg `  f
)  -  1 ) )  =  ( A `
 ( N  - 
1 ) ) )
187184, 176fveq12d 5693 . . . . . . 7  |-  ( f  =  F  ->  (
(coeff `  f ) `  (deg `  f )
)  =  ( A `
 N ) )
188186, 187oveq12d 6058 . . . . . 6  |-  ( f  =  F  ->  (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) )  =  ( ( A `  ( N  -  1 ) )  /  ( A `
 N ) ) )
189188negeqd 9256 . . . . 5  |-  ( f  =  F  ->  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) )  =  -u ( ( A `  ( N  -  1
) )  /  ( A `  N )
) )
190181, 189eqeq12d 2418 . . . 4  |-  ( f  =  F  ->  ( sum_ x  e.  ( `' f " { 0 } ) x  = 
-u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) )  <->  sum_ x  e.  R  x  =  -u ( ( A `
 ( N  - 
1 ) )  / 
( A `  N
) ) ) )
191180, 190imbi12d 312 . . 3  |-  ( f  =  F  ->  (
( ( N  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  <->  ( ( # `
 R )  =  N  ->  sum_ x  e.  R  x  =  -u ( ( A `  ( N  -  1
) )  /  ( A `  N )
) ) ) )
192191rspcv 3008 . 2  |-  ( F  e.  (Poly `  CC )  ->  ( A. f  e.  (Poly `  CC )
( ( N  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  -> 
( ( # `  R
)  =  N  ->  sum_ x  e.  R  x  =  -u ( ( A `
 ( N  - 
1 ) )  / 
( A `  N
) ) ) ) )
1933, 166, 167, 192syl3c 59 1  |-  ( ph  -> 
sum_ x  e.  R  x  =  -u ( ( A `  ( N  -  1 ) )  /  ( A `  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666    C_ wss 3280   {csn 3774   class class class wbr 4172    X. cxp 4835   `'ccnv 4836   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262    ~~ cen 7065   Fincfn 7068   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    <_ cle 9077    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZcz 10238   ...cfz 10999   ^cexp 11337   #chash 11573   sum_csu 12434   0 pc0p 19514  Polycply 20056   X pcidp 20057  coeffccoe 20058  degcdgr 20059   quot cquot 20160
This theorem is referenced by:  basellem5  20820
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-idp 20061  df-coe 20062  df-dgr 20063  df-quot 20161
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