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Theorem vieta1 21737
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 21627 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
2 vieta1.4 . . 3  |-  ( ph  ->  F  e.  (Poly `  S ) )
31, 2sseldi 3351 . 2  |-  ( ph  ->  F  e.  (Poly `  CC ) )
4 vieta1.6 . . 3  |-  ( ph  ->  N  e.  NN )
5 eqeq1 2447 . . . . . . 7  |-  ( y  =  1  ->  (
y  =  (deg `  f )  <->  1  =  (deg `  f ) ) )
65anbi1d 699 . . . . . 6  |-  ( y  =  1  ->  (
( y  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( 1  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) ) ) )
76imbi1d 317 . . . . 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 2733 . . . 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 2447 . . . . . . 7  |-  ( y  =  d  ->  (
y  =  (deg `  f )  <->  d  =  (deg `  f ) ) )
109anbi1d 699 . . . . . 6  |-  ( y  =  d  ->  (
( y  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( d  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) ) ) )
1110imbi1d 317 . . . . 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 2733 . . . 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 2447 . . . . . . 7  |-  ( y  =  ( d  +  1 )  ->  (
y  =  (deg `  f )  <->  ( d  +  1 )  =  (deg `  f )
) )
1413anbi1d 699 . . . . . 6  |-  ( y  =  ( d  +  1 )  ->  (
( y  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( ( d  +  1 )  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) ) ) )
1514imbi1d 317 . . . . 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 2733 . . . 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 2447 . . . . . . 7  |-  ( y  =  N  ->  (
y  =  (deg `  f )  <->  N  =  (deg `  f ) ) )
1817anbi1d 699 . . . . . 6  |-  ( y  =  N  ->  (
( y  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( N  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) ) ) )
1918imbi1d 317 . . . . 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 2733 . . . 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 2441 . . . . . . . . . . . . . 14  |-  (coeff `  f )  =  (coeff `  f )
2221coef3 21659 . . . . . . . . . . . . 13  |-  ( f  e.  (Poly `  CC )  ->  (coeff `  f
) : NN0 --> CC )
2322adantr 462 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  (coeff `  f
) : NN0 --> CC )
24 0nn0 10590 . . . . . . . . . . . 12  |-  0  e.  NN0
25 ffvelrn 5838 . . . . . . . . . . . 12  |-  ( ( (coeff `  f ) : NN0 --> CC  /\  0  e.  NN0 )  ->  (
(coeff `  f ) `  0 )  e.  CC )
2623, 24, 25sylancl 657 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  0
)  e.  CC )
27 1nn0 10591 . . . . . . . . . . . 12  |-  1  e.  NN0
28 ffvelrn 5838 . . . . . . . . . . . 12  |-  ( ( (coeff `  f ) : NN0 --> CC  /\  1  e.  NN0 )  ->  (
(coeff `  f ) `  1 )  e.  CC )
2923, 27, 28sylancl 657 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  1
)  e.  CC )
30 simpr 458 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  1  =  (deg `  f ) )
3130fveq2d 5692 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  1
)  =  ( (coeff `  f ) `  (deg `  f ) ) )
32 ax-1ne0 9347 . . . . . . . . . . . . . . . 16  |-  1  =/=  0
3332a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  1  =/=  0 )
3430, 33eqnetrrd 2626 . . . . . . . . . . . . . 14  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  (deg `  f
)  =/=  0 )
35 fveq2 5688 . . . . . . . . . . . . . . . 16  |-  ( f  =  0p  -> 
(deg `  f )  =  (deg `  0p
) )
36 dgr0 21688 . . . . . . . . . . . . . . . 16  |-  (deg ` 
0p )  =  0
3735, 36syl6eq 2489 . . . . . . . . . . . . . . 15  |-  ( f  =  0p  -> 
(deg `  f )  =  0 )
3837necon3i 2648 . . . . . . . . . . . . . 14  |-  ( (deg
`  f )  =/=  0  ->  f  =/=  0p )
3934, 38syl 16 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  f  =/=  0p )
40 eqid 2441 . . . . . . . . . . . . . . . 16  |-  (deg `  f )  =  (deg
`  f )
4140, 21dgreq0 21691 . . . . . . . . . . . . . . 15  |-  ( f  e.  (Poly `  CC )  ->  ( f  =  0p  <->  ( (coeff `  f ) `  (deg `  f ) )  =  0 ) )
4241necon3bid 2641 . . . . . . . . . . . . . 14  |-  ( f  e.  (Poly `  CC )  ->  ( f  =/=  0p  <->  ( (coeff `  f ) `  (deg `  f ) )  =/=  0 ) )
4342adantr 462 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( f  =/=  0p  <->  ( (coeff `  f ) `  (deg `  f ) )  =/=  0 ) )
4439, 43mpbid 210 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  (deg `  f ) )  =/=  0 )
4531, 44eqnetrd 2624 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  1
)  =/=  0 )
4626, 29, 45divcld 10103 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
)  e.  CC )
4746negcld 9702 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
)  e.  CC )
48 id 22 . . . . . . . . . 10  |-  ( x  =  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) )  ->  x  =  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )
4948sumsn 13213 . . . . . . . . 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 656 . . . . . . . 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 711 . . . . . . 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 2441 . . . . . . . . . . . . 13  |-  ( `' f " { 0 } )  =  ( `' f " {
0 } )
5352fta1 21733 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  f  =/=  0p )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )
5439, 53syldan 467 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )
5554simpld 456 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( `' f " { 0 } )  e.  Fin )
5655adantrr 711 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  ( `' f " { 0 } )  e.  Fin )
5721, 40coeid2 21666 . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . 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 6106 . . . . . . . . . . . . . 14  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( 0 ... 1 )  =  ( 0 ... (deg `  f ) ) )
6059sumeq1d 13174 . . . . . . . . . . . . 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 10891 . . . . . . . . . . . . . . 15  |-  NN0  =  ( ZZ>= `  0 )
62 1e0p1 10779 . . . . . . . . . . . . . . 15  |-  1  =  ( 0  +  1 )
63 fveq2 5688 . . . . . . . . . . . . . . . 16  |-  ( k  =  1  ->  (
(coeff `  f ) `  k )  =  ( (coeff `  f ) `  1 ) )
64 oveq2 6098 . . . . . . . . . . . . . . . 16  |-  ( k  =  1  ->  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
)  =  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ 1 ) )
6563, 64oveq12d 6108 . . . . . . . . . . . . . . 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 5840 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  /\  k  e.  NN0 )  ->  ( (coeff `  f ) `  k
)  e.  CC )
67 expcl 11879 . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  /\  k  e.  NN0 )  ->  ( -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) ^ k
)  e.  CC )
6966, 68mulcld 9402 . . . . . . . . . . . . . . 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 10653 . . . . . . . . . . . . . . . . . 18  |-  0  e.  ZZ
7147exp0d 11998 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) ^ 0 )  =  1 )
7271oveq2d 6106 . . . . . . . . . . . . . . . . . . . 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 9399 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  x.  1 )  =  ( (coeff `  f ) `  0 ) )
7472, 73eqtrd 2473 . . . . . . . . . . . . . . . . . . 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 2515 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ 0 ) )  e.  CC )
76 fveq2 5688 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  0  ->  (
(coeff `  f ) `  k )  =  ( (coeff `  f ) `  0 ) )
77 oveq2 6098 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  0  ->  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
)  =  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ 0 ) )
7876, 77oveq12d 6108 . . . . . . . . . . . . . . . . . . 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 13214 . . . . . . . . . . . . . . . . . 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 658 . . . . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . . . . . 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 534 . . . . . . . . . . . . . . 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 11999 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) ^ 1 )  =  -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) )
8483oveq2d 6106 . . . . . . . . . . . . . . . . . 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 9794 . . . . . . . . . . . . . . . . . 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 10105 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  1 )  x.  ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )  =  ( (coeff `  f
) `  0 )
)
8786negeqd 9600 . . . . . . . . . . . . . . . . . 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 2477 . . . . . . . . . . . . . . . . 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 6106 . . . . . . . . . . . . . . . 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 9705 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  + 
-u ( (coeff `  f ) `  0
) )  =  0 )
9189, 90eqtrd 2473 . . . . . . . . . . . . . . 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 13232 . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . 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 2479 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( f `  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )  =  0 )
95 plyf 21625 . . . . . . . . . . . . . . 15  |-  ( f  e.  (Poly `  CC )  ->  f : CC --> CC )
96 ffn 5556 . . . . . . . . . . . . . . 15  |-  ( f : CC --> CC  ->  f  Fn  CC )
9795, 96syl 16 . . . . . . . . . . . . . 14  |-  ( f  e.  (Poly `  CC )  ->  f  Fn  CC )
9897adantr 462 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  f  Fn  CC )
99 fniniseg 5821 . . . . . . . . . . . . 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 908 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
)  e.  ( `' f " { 0 } ) )
102101snssd 4015 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  { -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) }  C_  ( `' f " {
0 } ) )
103102adantrr 711 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  { -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) }  C_  ( `' f " {
0 } ) )
104 hashsng 12132 . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( # `  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } )  =  (deg `  f
) )
107106adantrr 711 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  ( # `  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } )  =  (deg `  f
) )
108 simprr 751 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )
109107, 108eqtr4d 2476 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  ( # `  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } )  =  ( # `  ( `' f " {
0 } ) ) )
110 snfi 7386 . . . . . . . . . . . 12  |-  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) }  e.  Fin
111 hashen 12114 . . . . . . . . . . . 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 658 . . . . . . . . . . 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 711 . . . . . . . . . 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 210 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  { -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) }  ~~  ( `' f " {
0 } ) )
115 fisseneq 7520 . . . . . . . . 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 1213 . . . . . . . 8  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  { -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) }  =  ( `' f " {
0 } ) )
117116sumeq1d 13174 . . . . . . 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 10386 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
11930oveq1d 6105 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( 1  -  1 )  =  ( (deg `  f
)  -  1 ) )
120118, 119syl5eqr 2487 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  0  =  ( (deg `  f )  -  1 ) )
121120fveq2d 5692 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  0
)  =  ( (coeff `  f ) `  (
(deg `  f )  -  1 ) ) )
122121, 31oveq12d 6108 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
)  =  ( ( (coeff `  f ) `  ( (deg `  f
)  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) )
123122negeqd 9600 . . . . . . . 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 711 . . . . . . 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 2481 . . . . . 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 434 . . . . 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 2779 . . . 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 22 . . . . . . . . . . 11  |-  ( y  =  x  ->  y  =  x )
129128cbvsumv 13169 . . . . . . . . . 10  |-  sum_ y  e.  ( `' f " { 0 } ) y  =  sum_ x  e.  ( `' f " { 0 } ) x
130129eqeq1i 2448 . . . . . . . . 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 312 . . . . . . . 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 2737 . . . . . . 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 2441 . . . . . . . . 9  |-  (coeff `  g )  =  (coeff `  g )
134 eqid 2441 . . . . . . . . 9  |-  (deg `  g )  =  (deg
`  g )
135 eqid 2441 . . . . . . . . 9  |-  ( `' g " { 0 } )  =  ( `' g " {
0 } )
136 simp1r 1008 . . . . . . . . 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 1012 . . . . . . . . 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 1007 . . . . . . . . 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 1011 . . . . . . . . 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 984 . . . . . . . . . 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 196 . . . . . . . . 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 2441 . . . . . . . . 9  |-  ( g quot  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  ( g quot  ( Xp  oF  -  ( CC  X.  { z } ) ) )
143133, 134, 135, 136, 137, 138, 139, 141, 142vieta1lem2 21736 . . . . . . . 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 1181 . . . . . . 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 218 . . . . . 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 2804 . . . . 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 5688 . . . . . . . . 9  |-  ( g  =  f  ->  (deg `  g )  =  (deg
`  f ) )
148147eqeq2d 2452 . . . . . . . 8  |-  ( g  =  f  ->  (
( d  +  1 )  =  (deg `  g )  <->  ( d  +  1 )  =  (deg `  f )
) )
149 cnveq 5009 . . . . . . . . . . 11  |-  ( g  =  f  ->  `' g  =  `' f
)
150149imaeq1d 5165 . . . . . . . . . 10  |-  ( g  =  f  ->  ( `' g " {
0 } )  =  ( `' f " { 0 } ) )
151150fveq2d 5692 . . . . . . . . 9  |-  ( g  =  f  ->  ( # `
 ( `' g
" { 0 } ) )  =  (
# `  ( `' f " { 0 } ) ) )
152151, 147eqeq12d 2455 . . . . . . . 8  |-  ( g  =  f  ->  (
( # `  ( `' g " { 0 } ) )  =  (deg `  g )  <->  (
# `  ( `' f " { 0 } ) )  =  (deg
`  f ) ) )
153148, 152anbi12d 705 . . . . . . 7  |-  ( g  =  f  ->  (
( ( d  +  1 )  =  (deg
`  g )  /\  ( # `  ( `' g " { 0 } ) )  =  (deg `  g )
)  <->  ( ( d  +  1 )  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) ) ) )
154150sumeq1d 13174 . . . . . . . 8  |-  ( g  =  f  ->  sum_ x  e.  ( `' g " { 0 } ) x  =  sum_ x  e.  ( `' f " { 0 } ) x )
155 fveq2 5688 . . . . . . . . . . 11  |-  ( g  =  f  ->  (coeff `  g )  =  (coeff `  f ) )
156147oveq1d 6105 . . . . . . . . . . 11  |-  ( g  =  f  ->  (
(deg `  g )  -  1 )  =  ( (deg `  f
)  -  1 ) )
157155, 156fveq12d 5694 . . . . . . . . . 10  |-  ( g  =  f  ->  (
(coeff `  g ) `  ( (deg `  g
)  -  1 ) )  =  ( (coeff `  f ) `  (
(deg `  f )  -  1 ) ) )
158155, 147fveq12d 5694 . . . . . . . . . 10  |-  ( g  =  f  ->  (
(coeff `  g ) `  (deg `  g )
)  =  ( (coeff `  f ) `  (deg `  f ) ) )
159157, 158oveq12d 6108 . . . . . . . . 9  |-  ( g  =  f  ->  (
( (coeff `  g
) `  ( (deg `  g )  -  1 ) )  /  (
(coeff `  g ) `  (deg `  g )
) )  =  ( ( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )
160159negeqd 9600 . . . . . . . 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 2455 . . . . . . 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 320 . . . . . 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 2945 . . . . 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 226 . . . 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 10336 . . 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 5688 . . . . . . 7  |-  ( f  =  F  ->  (deg `  f )  =  (deg
`  F ) )
169168eqeq2d 2452 . . . . . 6  |-  ( f  =  F  ->  ( N  =  (deg `  f
)  <->  N  =  (deg `  F ) ) )
170 cnveq 5009 . . . . . . . . . 10  |-  ( f  =  F  ->  `' f  =  `' F
)
171170imaeq1d 5165 . . . . . . . . 9  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  ( `' F " { 0 } ) )
172 vieta1.3 . . . . . . . . 9  |-  R  =  ( `' F " { 0 } )
173171, 172syl6eqr 2491 . . . . . . . 8  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  R )
174173fveq2d 5692 . . . . . . 7  |-  ( f  =  F  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  R )
)
175 vieta1.2 . . . . . . . 8  |-  N  =  (deg `  F )
176168, 175syl6eqr 2491 . . . . . . 7  |-  ( f  =  F  ->  (deg `  f )  =  N )
177174, 176eqeq12d 2455 . . . . . 6  |-  ( f  =  F  ->  (
( # `  ( `' f " { 0 } ) )  =  (deg `  f )  <->  (
# `  R )  =  N ) )
178169, 177anbi12d 705 . . . . 5  |-  ( f  =  F  ->  (
( N  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( N  =  (deg `  F )  /\  ( # `  R
)  =  N ) ) )
179175biantrur 503 . . . . 5  |-  ( (
# `  R )  =  N  <->  ( N  =  (deg `  F )  /\  ( # `  R
)  =  N ) )
180178, 179syl6bbr 263 . . . 4  |-  ( f  =  F  ->  (
( N  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( # `  R
)  =  N ) )
181173sumeq1d 13174 . . . . 5  |-  ( f  =  F  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  sum_ x  e.  R  x )
182 fveq2 5688 . . . . . . . . 9  |-  ( f  =  F  ->  (coeff `  f )  =  (coeff `  F ) )
183 vieta1.1 . . . . . . . . 9  |-  A  =  (coeff `  F )
184182, 183syl6eqr 2491 . . . . . . . 8  |-  ( f  =  F  ->  (coeff `  f )  =  A )
185176oveq1d 6105 . . . . . . . 8  |-  ( f  =  F  ->  (
(deg `  f )  -  1 )  =  ( N  -  1 ) )
186184, 185fveq12d 5694 . . . . . . 7  |-  ( f  =  F  ->  (
(coeff `  f ) `  ( (deg `  f
)  -  1 ) )  =  ( A `
 ( N  - 
1 ) ) )
187184, 176fveq12d 5694 . . . . . . 7  |-  ( f  =  F  ->  (
(coeff `  f ) `  (deg `  f )
)  =  ( A `
 N ) )
188186, 187oveq12d 6108 . . . . . 6  |-  ( f  =  F  ->  (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) )  =  ( ( A `  ( N  -  1 ) )  /  ( A `
 N ) ) )
189188negeqd 9600 . . . . 5  |-  ( f  =  F  ->  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) )  =  -u ( ( A `  ( N  -  1
) )  /  ( A `  N )
) )
190181, 189eqeq12d 2455 . . . 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 320 . . 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 3066 . 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 61 1  |-  ( ph  -> 
sum_ x  e.  R  x  =  -u ( ( A `  ( N  -  1 ) )  /  ( A `  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713    C_ wss 3325   {csn 3874   class class class wbr 4289    X. cxp 4834   `'ccnv 4835   "cima 4839    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317    ~~ cen 7303   Fincfn 7306   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    <_ cle 9415    - cmin 9591   -ucneg 9592    / cdiv 9989   NNcn 10318   NN0cn0 10575   ZZcz 10642   ...cfz 11433   ^cexp 11861   #chash 12099   sum_csu 13159   0pc0p 21106  Polycply 21611   Xpcidp 21612  coeffccoe 21613  degcdgr 21614   quot cquot 21715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-0p 21107  df-ply 21615  df-idp 21616  df-coe 21617  df-dgr 21618  df-quot 21716
This theorem is referenced by:  basellem5  22381
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