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Theorem vieta1 21790
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 21680 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
2 vieta1.4 . . 3  |-  ( ph  ->  F  e.  (Poly `  S ) )
31, 2sseldi 3366 . 2  |-  ( ph  ->  F  e.  (Poly `  CC ) )
4 vieta1.6 . . 3  |-  ( ph  ->  N  e.  NN )
5 eqeq1 2449 . . . . . . 7  |-  ( y  =  1  ->  (
y  =  (deg `  f )  <->  1  =  (deg `  f ) ) )
65anbi1d 704 . . . . . 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 2747 . . . 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 2449 . . . . . . 7  |-  ( y  =  d  ->  (
y  =  (deg `  f )  <->  d  =  (deg `  f ) ) )
109anbi1d 704 . . . . . 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 2747 . . . 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 2449 . . . . . . 7  |-  ( y  =  ( d  +  1 )  ->  (
y  =  (deg `  f )  <->  ( d  +  1 )  =  (deg `  f )
) )
1413anbi1d 704 . . . . . 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 2747 . . . 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 2449 . . . . . . 7  |-  ( y  =  N  ->  (
y  =  (deg `  f )  <->  N  =  (deg `  f ) ) )
1817anbi1d 704 . . . . . 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 2747 . . . 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 2443 . . . . . . . . . . . . . 14  |-  (coeff `  f )  =  (coeff `  f )
2221coef3 21712 . . . . . . . . . . . . 13  |-  ( f  e.  (Poly `  CC )  ->  (coeff `  f
) : NN0 --> CC )
2322adantr 465 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  (coeff `  f
) : NN0 --> CC )
24 0nn0 10606 . . . . . . . . . . . 12  |-  0  e.  NN0
25 ffvelrn 5853 . . . . . . . . . . . 12  |-  ( ( (coeff `  f ) : NN0 --> CC  /\  0  e.  NN0 )  ->  (
(coeff `  f ) `  0 )  e.  CC )
2623, 24, 25sylancl 662 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  0
)  e.  CC )
27 1nn0 10607 . . . . . . . . . . . 12  |-  1  e.  NN0
28 ffvelrn 5853 . . . . . . . . . . . 12  |-  ( ( (coeff `  f ) : NN0 --> CC  /\  1  e.  NN0 )  ->  (
(coeff `  f ) `  1 )  e.  CC )
2923, 27, 28sylancl 662 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  1
)  e.  CC )
30 simpr 461 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  1  =  (deg `  f ) )
3130fveq2d 5707 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  1
)  =  ( (coeff `  f ) `  (deg `  f ) ) )
32 ax-1ne0 9363 . . . . . . . . . . . . . . . 16  |-  1  =/=  0
3332a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  1  =/=  0 )
3430, 33eqnetrrd 2640 . . . . . . . . . . . . . 14  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  (deg `  f
)  =/=  0 )
35 fveq2 5703 . . . . . . . . . . . . . . . 16  |-  ( f  =  0p  -> 
(deg `  f )  =  (deg `  0p
) )
36 dgr0 21741 . . . . . . . . . . . . . . . 16  |-  (deg ` 
0p )  =  0
3735, 36syl6eq 2491 . . . . . . . . . . . . . . 15  |-  ( f  =  0p  -> 
(deg `  f )  =  0 )
3837necon3i 2662 . . . . . . . . . . . . . 14  |-  ( (deg
`  f )  =/=  0  ->  f  =/=  0p )
3934, 38syl 16 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  f  =/=  0p )
40 eqid 2443 . . . . . . . . . . . . . . . 16  |-  (deg `  f )  =  (deg
`  f )
4140, 21dgreq0 21744 . . . . . . . . . . . . . . 15  |-  ( f  e.  (Poly `  CC )  ->  ( f  =  0p  <->  ( (coeff `  f ) `  (deg `  f ) )  =  0 ) )
4241necon3bid 2655 . . . . . . . . . . . . . 14  |-  ( f  e.  (Poly `  CC )  ->  ( f  =/=  0p  <->  ( (coeff `  f ) `  (deg `  f ) )  =/=  0 ) )
4342adantr 465 . . . . . . . . . . . . 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 2638 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  1
)  =/=  0 )
4626, 29, 45divcld 10119 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
)  e.  CC )
4746negcld 9718 . . . . . . . . 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 13229 . . . . . . . . 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 661 . . . . . . . 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 716 . . . . . . 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 2443 . . . . . . . . . . . . 13  |-  ( `' f " { 0 } )  =  ( `' f " {
0 } )
5352fta1 21786 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  f  =/=  0p )  -> 
( ( `' f
" { 0 } )  e.  Fin  /\  ( # `  ( `' f " { 0 } ) )  <_ 
(deg `  f )
) )
5439, 53syldan 470 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( ( `' f " {
0 } )  e. 
Fin  /\  ( # `  ( `' f " {
0 } ) )  <_  (deg `  f
) ) )
5554simpld 459 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( `' f " { 0 } )  e.  Fin )
5655adantrr 716 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  ( `' f " { 0 } )  e.  Fin )
5721, 40coeid2 21719 . . . . . . . . . . . . . 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 470 . . . . . . . . . . . . 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 6119 . . . . . . . . . . . . . 14  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( 0 ... 1 )  =  ( 0 ... (deg `  f ) ) )
6059sumeq1d 13190 . . . . . . . . . . . . 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 10907 . . . . . . . . . . . . . . 15  |-  NN0  =  ( ZZ>= `  0 )
62 1e0p1 10795 . . . . . . . . . . . . . . 15  |-  1  =  ( 0  +  1 )
63 fveq2 5703 . . . . . . . . . . . . . . . 16  |-  ( k  =  1  ->  (
(coeff `  f ) `  k )  =  ( (coeff `  f ) `  1 ) )
64 oveq2 6111 . . . . . . . . . . . . . . . 16  |-  ( k  =  1  ->  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
)  =  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ 1 ) )
6563, 64oveq12d 6121 . . . . . . . . . . . . . . 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 5855 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  /\  k  e.  NN0 )  ->  ( (coeff `  f ) `  k
)  e.  CC )
67 expcl 11895 . . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  /\  k  e.  NN0 )  ->  ( -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) ^ k
)  e.  CC )
6966, 68mulcld 9418 . . . . . . . . . . . . . . 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 10669 . . . . . . . . . . . . . . . . . 18  |-  0  e.  ZZ
7147exp0d 12014 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) ^ 0 )  =  1 )
7271oveq2d 6119 . . . . . . . . . . . . . . . . . . . 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 9415 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  x.  1 )  =  ( (coeff `  f ) `  0 ) )
7472, 73eqtrd 2475 . . . . . . . . . . . . . . . . . . 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 2517 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  x.  ( -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
) ^ 0 ) )  e.  CC )
76 fveq2 5703 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  0  ->  (
(coeff `  f ) `  k )  =  ( (coeff `  f ) `  0 ) )
77 oveq2 6111 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  0  ->  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ k
)  =  ( -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) ^ 0 ) )
7876, 77oveq12d 6121 . . . . . . . . . . . . . . . . . . 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 13230 . . . . . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . . . . 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 537 . . . . . . . . . . . . . . 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 12015 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) ^ 1 )  =  -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) )
8483oveq2d 6119 . . . . . . . . . . . . . . . . . 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 9810 . . . . . . . . . . . . . . . . . 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 10121 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  1 )  x.  ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )  =  ( (coeff `  f
) `  0 )
)
8786negeqd 9616 . . . . . . . . . . . . . . . . . 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 2479 . . . . . . . . . . . . . . . . 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 6119 . . . . . . . . . . . . . . . 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 9721 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  + 
-u ( (coeff `  f ) `  0
) )  =  0 )
9189, 90eqtrd 2475 . . . . . . . . . . . . . . 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 13248 . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . 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 2481 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( f `  -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) )  =  0 )
95 plyf 21678 . . . . . . . . . . . . . . 15  |-  ( f  e.  (Poly `  CC )  ->  f : CC --> CC )
96 ffn 5571 . . . . . . . . . . . . . . 15  |-  ( f : CC --> CC  ->  f  Fn  CC )
9795, 96syl 16 . . . . . . . . . . . . . 14  |-  ( f  e.  (Poly `  CC )  ->  f  Fn  CC )
9897adantr 465 . . . . . . . . . . . . 13  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  f  Fn  CC )
99 fniniseg 5836 . . . . . . . . . . . . 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 913 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  -u ( ( (coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
)  e.  ( `' f " { 0 } ) )
102101snssd 4030 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  { -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) }  C_  ( `' f " {
0 } ) )
103102adantrr 716 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  { -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) }  C_  ( `' f " {
0 } ) )
104 hashsng 12148 . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( # `  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } )  =  (deg `  f
) )
107106adantrr 716 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  ( # `  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } )  =  (deg `  f
) )
108 simprr 756 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )
109107, 108eqtr4d 2478 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  ( # `  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) } )  =  ( # `  ( `' f " {
0 } ) ) )
110 snfi 7402 . . . . . . . . . . . 12  |-  { -u ( ( (coeff `  f ) `  0
)  /  ( (coeff `  f ) `  1
) ) }  e.  Fin
111 hashen 12130 . . . . . . . . . . . 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 663 . . . . . . . . . . 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 716 . . . . . . . . . 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 7536 . . . . . . . . 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 1218 . . . . . . . 8  |-  ( ( f  e.  (Poly `  CC )  /\  (
1  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) ) )  ->  { -u (
( (coeff `  f
) `  0 )  /  ( (coeff `  f ) `  1
) ) }  =  ( `' f " {
0 } ) )
117116sumeq1d 13190 . . . . . . 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 10402 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
11930oveq1d 6118 . . . . . . . . . . . 12  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( 1  -  1 )  =  ( (deg `  f
)  -  1 ) )
120118, 119syl5eqr 2489 . . . . . . . . . . 11  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  0  =  ( (deg `  f )  -  1 ) )
121120fveq2d 5707 . . . . . . . . . 10  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (coeff `  f ) `  0
)  =  ( (coeff `  f ) `  (
(deg `  f )  -  1 ) ) )
122121, 31oveq12d 6121 . . . . . . . . 9  |-  ( ( f  e.  (Poly `  CC )  /\  1  =  (deg `  f )
)  ->  ( (
(coeff `  f ) `  0 )  / 
( (coeff `  f
) `  1 )
)  =  ( ( (coeff `  f ) `  ( (deg `  f
)  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) )
123122negeqd 9616 . . . . . . . 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 716 . . . . . . 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 2483 . . . . . 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 2793 . . . 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 13185 . . . . . . . . . 10  |-  sum_ y  e.  ( `' f " { 0 } ) y  =  sum_ x  e.  ( `' f " { 0 } ) x
130129eqeq1i 2450 . . . . . . . . 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 2751 . . . . . . 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 2443 . . . . . . . . 9  |-  (coeff `  g )  =  (coeff `  g )
134 eqid 2443 . . . . . . . . 9  |-  (deg `  g )  =  (deg
`  g )
135 eqid 2443 . . . . . . . . 9  |-  ( `' g " { 0 } )  =  ( `' g " {
0 } )
136 simp1r 1013 . . . . . . . . 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 1017 . . . . . . . . 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 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 )
) )  ->  d  e.  NN )
139 simp3l 1016 . . . . . . . . 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 989 . . . . . . . . . 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 2443 . . . . . . . . 9  |-  ( g quot  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  ( g quot  ( Xp  oF  -  ( CC  X.  { z } ) ) )
143133, 134, 135, 136, 137, 138, 139, 141, 142vieta1lem2 21789 . . . . . . . 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 1186 . . . . . . 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 2818 . . . . 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 5703 . . . . . . . . 9  |-  ( g  =  f  ->  (deg `  g )  =  (deg
`  f ) )
148147eqeq2d 2454 . . . . . . . 8  |-  ( g  =  f  ->  (
( d  +  1 )  =  (deg `  g )  <->  ( d  +  1 )  =  (deg `  f )
) )
149 cnveq 5025 . . . . . . . . . . 11  |-  ( g  =  f  ->  `' g  =  `' f
)
150149imaeq1d 5180 . . . . . . . . . 10  |-  ( g  =  f  ->  ( `' g " {
0 } )  =  ( `' f " { 0 } ) )
151150fveq2d 5707 . . . . . . . . 9  |-  ( g  =  f  ->  ( # `
 ( `' g
" { 0 } ) )  =  (
# `  ( `' f " { 0 } ) ) )
152151, 147eqeq12d 2457 . . . . . . . 8  |-  ( g  =  f  ->  (
( # `  ( `' g " { 0 } ) )  =  (deg `  g )  <->  (
# `  ( `' f " { 0 } ) )  =  (deg
`  f ) ) )
153148, 152anbi12d 710 . . . . . . 7  |-  ( g  =  f  ->  (
( ( d  +  1 )  =  (deg
`  g )  /\  ( # `  ( `' g " { 0 } ) )  =  (deg `  g )
)  <->  ( ( d  +  1 )  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) ) ) )
154150sumeq1d 13190 . . . . . . . 8  |-  ( g  =  f  ->  sum_ x  e.  ( `' g " { 0 } ) x  =  sum_ x  e.  ( `' f " { 0 } ) x )
155 fveq2 5703 . . . . . . . . . . 11  |-  ( g  =  f  ->  (coeff `  g )  =  (coeff `  f ) )
156147oveq1d 6118 . . . . . . . . . . 11  |-  ( g  =  f  ->  (
(deg `  g )  -  1 )  =  ( (deg `  f
)  -  1 ) )
157155, 156fveq12d 5709 . . . . . . . . . 10  |-  ( g  =  f  ->  (
(coeff `  g ) `  ( (deg `  g
)  -  1 ) )  =  ( (coeff `  f ) `  (
(deg `  f )  -  1 ) ) )
158155, 147fveq12d 5709 . . . . . . . . . 10  |-  ( g  =  f  ->  (
(coeff `  g ) `  (deg `  g )
)  =  ( (coeff `  f ) `  (deg `  f ) ) )
159157, 158oveq12d 6121 . . . . . . . . 9  |-  ( g  =  f  ->  (
( (coeff `  g
) `  ( (deg `  g )  -  1 ) )  /  (
(coeff `  g ) `  (deg `  g )
) )  =  ( ( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )
160159negeqd 9616 . . . . . . . 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 2457 . . . . . . 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 2959 . . . . 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 10352 . . 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 5703 . . . . . . 7  |-  ( f  =  F  ->  (deg `  f )  =  (deg
`  F ) )
169168eqeq2d 2454 . . . . . 6  |-  ( f  =  F  ->  ( N  =  (deg `  f
)  <->  N  =  (deg `  F ) ) )
170 cnveq 5025 . . . . . . . . . 10  |-  ( f  =  F  ->  `' f  =  `' F
)
171170imaeq1d 5180 . . . . . . . . 9  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  ( `' F " { 0 } ) )
172 vieta1.3 . . . . . . . . 9  |-  R  =  ( `' F " { 0 } )
173171, 172syl6eqr 2493 . . . . . . . 8  |-  ( f  =  F  ->  ( `' f " {
0 } )  =  R )
174173fveq2d 5707 . . . . . . 7  |-  ( f  =  F  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  R )
)
175 vieta1.2 . . . . . . . 8  |-  N  =  (deg `  F )
176168, 175syl6eqr 2493 . . . . . . 7  |-  ( f  =  F  ->  (deg `  f )  =  N )
177174, 176eqeq12d 2457 . . . . . 6  |-  ( f  =  F  ->  (
( # `  ( `' f " { 0 } ) )  =  (deg `  f )  <->  (
# `  R )  =  N ) )
178169, 177anbi12d 710 . . . . 5  |-  ( f  =  F  ->  (
( N  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( N  =  (deg `  F )  /\  ( # `  R
)  =  N ) ) )
179175biantrur 506 . . . . 5  |-  ( (
# `  R )  =  N  <->  ( N  =  (deg `  F )  /\  ( # `  R
)  =  N ) )
180178, 179syl6bbr 263 . . . 4  |-  ( f  =  F  ->  (
( N  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( # `  R
)  =  N ) )
181173sumeq1d 13190 . . . . 5  |-  ( f  =  F  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  sum_ x  e.  R  x )
182 fveq2 5703 . . . . . . . . 9  |-  ( f  =  F  ->  (coeff `  f )  =  (coeff `  F ) )
183 vieta1.1 . . . . . . . . 9  |-  A  =  (coeff `  F )
184182, 183syl6eqr 2493 . . . . . . . 8  |-  ( f  =  F  ->  (coeff `  f )  =  A )
185176oveq1d 6118 . . . . . . . 8  |-  ( f  =  F  ->  (
(deg `  f )  -  1 )  =  ( N  -  1 ) )
186184, 185fveq12d 5709 . . . . . . 7  |-  ( f  =  F  ->  (
(coeff `  f ) `  ( (deg `  f
)  -  1 ) )  =  ( A `
 ( N  - 
1 ) ) )
187184, 176fveq12d 5709 . . . . . . 7  |-  ( f  =  F  ->  (
(coeff `  f ) `  (deg `  f )
)  =  ( A `
 N ) )
188186, 187oveq12d 6121 . . . . . 6  |-  ( f  =  F  ->  (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) )  =  ( ( A `  ( N  -  1 ) )  /  ( A `
 N ) ) )
189188negeqd 9616 . . . . 5  |-  ( f  =  F  ->  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) )  =  -u ( ( A `  ( N  -  1
) )  /  ( A `  N )
) )
190181, 189eqeq12d 2457 . . . 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 3081 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727    C_ wss 3340   {csn 3889   class class class wbr 4304    X. cxp 4850   `'ccnv 4851   "cima 4855    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103    oFcof 6330    ~~ cen 7319   Fincfn 7322   CCcc 9292   0cc0 9294   1c1 9295    + caddc 9297    x. cmul 9299    <_ cle 9431    - cmin 9607   -ucneg 9608    / cdiv 10005   NNcn 10334   NN0cn0 10591   ZZcz 10658   ...cfz 11449   ^cexp 11877   #chash 12115   sum_csu 13175   0pc0p 21159  Polycply 21664   Xpcidp 21665  coeffccoe 21666  degcdgr 21667   quot cquot 21768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-oi 7736  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-fz 11450  df-fzo 11561  df-fl 11654  df-seq 11819  df-exp 11878  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-clim 12978  df-rlim 12979  df-sum 13176  df-0p 21160  df-ply 21668  df-idp 21669  df-coe 21670  df-dgr 21671  df-quot 21769
This theorem is referenced by:  basellem5  22434
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