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Theorem vieta1lem2 21789
Description: Lemma for vieta1 21790: inductive step. Let  z be a root of  F. Then  F  =  ( Xp  -  z
)  x.  Q for some  Q by the factor theorem, and  Q is a degree-  D polynomial, so by the induction hypothesis  sum_ x  e.  ( `' Q "
0 ) x  = 
-u (coeff `  Q
) `  ( D  -  1 )  /  (coeff `  Q
) `  D, so  sum_ x  e.  R x  =  z  -  (coeff `  Q
) `  ( D  - 
1 )  /  (coeff `  Q ) `  D. Now the coefficients of  F are  A `  ( D  +  1 )  =  (coeff `  Q
) `  D and  A `  D  =  sum_ k  e.  ( 0 ... D
) (coeff `  Xp  -  z ) `  k  x.  (coeff `  Q )  `  ( D  -  k ), which works out to  -u z  x.  (coeff `  Q ) `  D  +  (coeff `  Q ) `  ( D  -  1 ), so putting it all together we have  sum_ x  e.  R x  =  -u A `  D  /  A `  ( D  +  1 ) as we wanted to show. (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 )
vieta1lem.6  |-  ( ph  ->  D  e.  NN )
vieta1lem.7  |-  ( ph  ->  ( D  +  1 )  =  N )
vieta1lem.8  |-  ( ph  ->  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 ) ) ) ) )
vieta1lem.9  |-  Q  =  ( F quot  ( Xp  oF  -  ( CC  X.  { z } ) ) )
Assertion
Ref Expression
vieta1lem2  |-  ( ph  -> 
sum_ x  e.  R  x  =  -u ( ( A `  ( N  -  1 ) )  /  ( A `  N ) ) )
Distinct variable groups:    D, f    f, F    z, f, N   
x, f, Q    R, f    x, z, R    A, f, z    ph, x, z
Allowed substitution hints:    ph( f)    A( x)    D( x, z)    Q( z)    S( x, z, f)    F( x, z)    N( x)

Proof of Theorem vieta1lem2
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 vieta1.5 . . . . 5  |-  ( ph  ->  ( # `  R
)  =  N )
2 vieta1lem.7 . . . . . . 7  |-  ( ph  ->  ( D  +  1 )  =  N )
3 vieta1lem.6 . . . . . . . 8  |-  ( ph  ->  D  e.  NN )
43peano2nnd 10351 . . . . . . 7  |-  ( ph  ->  ( D  +  1 )  e.  NN )
52, 4eqeltrrd 2518 . . . . . 6  |-  ( ph  ->  N  e.  NN )
65nnne0d 10378 . . . . 5  |-  ( ph  ->  N  =/=  0 )
71, 6eqnetrd 2638 . . . 4  |-  ( ph  ->  ( # `  R
)  =/=  0 )
8 vieta1.4 . . . . . . . 8  |-  ( ph  ->  F  e.  (Poly `  S ) )
9 vieta1.2 . . . . . . . . . 10  |-  N  =  (deg `  F )
109, 6syl5eqner 2645 . . . . . . . . 9  |-  ( ph  ->  (deg `  F )  =/=  0 )
11 fveq2 5703 . . . . . . . . . . 11  |-  ( F  =  0p  -> 
(deg `  F )  =  (deg `  0p
) )
12 dgr0 21741 . . . . . . . . . . 11  |-  (deg ` 
0p )  =  0
1311, 12syl6eq 2491 . . . . . . . . . 10  |-  ( F  =  0p  -> 
(deg `  F )  =  0 )
1413necon3i 2662 . . . . . . . . 9  |-  ( (deg
`  F )  =/=  0  ->  F  =/=  0p )
1510, 14syl 16 . . . . . . . 8  |-  ( ph  ->  F  =/=  0p )
16 vieta1.3 . . . . . . . . 9  |-  R  =  ( `' F " { 0 } )
1716fta1 21786 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  -> 
( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) )
188, 15, 17syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) )
1918simpld 459 . . . . . 6  |-  ( ph  ->  R  e.  Fin )
20 hasheq0 12143 . . . . . 6  |-  ( R  e.  Fin  ->  (
( # `  R )  =  0  <->  R  =  (/) ) )
2119, 20syl 16 . . . . 5  |-  ( ph  ->  ( ( # `  R
)  =  0  <->  R  =  (/) ) )
2221necon3bid 2655 . . . 4  |-  ( ph  ->  ( ( # `  R
)  =/=  0  <->  R  =/=  (/) ) )
237, 22mpbid 210 . . 3  |-  ( ph  ->  R  =/=  (/) )
24 n0 3658 . . 3  |-  ( R  =/=  (/)  <->  E. z  z  e.  R )
2523, 24sylib 196 . 2  |-  ( ph  ->  E. z  z  e.  R )
26 incom 3555 . . . . 5  |-  ( { z }  i^i  ( `' Q " { 0 } ) )  =  ( ( `' Q " { 0 } )  i^i  { z } )
27 vieta1.1 . . . . . . . . . . 11  |-  A  =  (coeff `  F )
28 vieta1lem.8 . . . . . . . . . . 11  |-  ( ph  ->  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 ) ) ) ) )
29 vieta1lem.9 . . . . . . . . . . 11  |-  Q  =  ( F quot  ( Xp  oF  -  ( CC  X.  { z } ) ) )
3027, 9, 16, 8, 1, 3, 2, 28, 29vieta1lem1 21788 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  ( Q  e.  (Poly `  CC )  /\  D  =  (deg
`  Q ) ) )
3130simprd 463 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  D  =  (deg `  Q )
)
3230simpld 459 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  Q  e.  (Poly `  CC )
)
33 dgrcl 21713 . . . . . . . . . . 11  |-  ( Q  e.  (Poly `  CC )  ->  (deg `  Q
)  e.  NN0 )
3432, 33syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  Q )  e.  NN0 )
3534nn0red 10649 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  Q )  e.  RR )
3631, 35eqeltrd 2517 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  D  e.  RR )
3736ltp1d 10275 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  D  <  ( D  +  1 ) )
3836, 37gtned 9521 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( D  +  1 )  =/=  D )
39 snssi 4029 . . . . . . . . . . 11  |-  ( z  e.  ( `' Q " { 0 } )  ->  { z } 
C_  ( `' Q " { 0 } ) )
40 ssequn1 3538 . . . . . . . . . . 11  |-  ( { z }  C_  ( `' Q " { 0 } )  <->  ( {
z }  u.  ( `' Q " { 0 } ) )  =  ( `' Q " { 0 } ) )
4139, 40sylib 196 . . . . . . . . . 10  |-  ( z  e.  ( `' Q " { 0 } )  ->  ( { z }  u.  ( `' Q " { 0 } ) )  =  ( `' Q " { 0 } ) )
4241fveq2d 5707 . . . . . . . . 9  |-  ( z  e.  ( `' Q " { 0 } )  ->  ( # `  ( { z }  u.  ( `' Q " { 0 } ) ) )  =  ( # `  ( `' Q " { 0 } ) ) )
438adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  F  e.  (Poly `  S )
)
44 cnvimass 5201 . . . . . . . . . . . . . . . . . . . . 21  |-  ( `' F " { 0 } )  C_  dom  F
4516, 44eqsstri 3398 . . . . . . . . . . . . . . . . . . . 20  |-  R  C_  dom  F
46 plyf 21678 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
47 fdm 5575 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F : CC --> CC  ->  dom 
F  =  CC )
488, 46, 473syl 20 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  dom  F  =  CC )
4945, 48syl5sseq 3416 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  R  C_  CC )
5049sselda 3368 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  z  e.  CC )
5116eleq2i 2507 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  e.  R  <->  z  e.  ( `' F " { 0 } ) )
52 ffn 5571 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F : CC --> CC  ->  F  Fn  CC )
53 fniniseg 5836 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  Fn  CC  ->  (
z  e.  ( `' F " { 0 } )  <->  ( z  e.  CC  /\  ( F `
 z )  =  0 ) ) )
548, 46, 52, 534syl 21 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( z  e.  ( `' F " { 0 } )  <->  ( z  e.  CC  /\  ( F `
 z )  =  0 ) ) )
5551, 54syl5bb 257 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( z  e.  R  <->  ( z  e.  CC  /\  ( F `  z )  =  0 ) ) )
5655simplbda 624 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  ( F `  z )  =  0 )
57 eqid 2443 . . . . . . . . . . . . . . . . . . 19  |-  ( Xp  oF  -  ( CC  X.  { z } ) )  =  ( Xp  oF  -  ( CC 
X.  { z } ) )
5857facth 21784 . . . . . . . . . . . . . . . . . 18  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC  /\  ( F `
 z )  =  0 )  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) ) )
5943, 50, 56, 58syl3anc 1218 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) ) )
6029oveq2i 6114 . . . . . . . . . . . . . . . . 17  |-  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q )  =  ( ( Xp  oF  -  ( CC 
X.  { z } ) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { z } ) ) ) )
6159, 60syl6eqr 2493 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) )
6261cnveqd 5027 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  `' F  =  `' (
( Xp  oF  -  ( CC 
X.  { z } ) )  oF  x.  Q ) )
6362imaeq1d 5180 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( `' F " { 0 } )  =  ( `' ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q )
" { 0 } ) )
6416, 63syl5eq 2487 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  R  =  ( `' ( ( Xp  oF  -  ( CC 
X.  { z } ) )  oF  x.  Q ) " { 0 } ) )
65 cnex 9375 . . . . . . . . . . . . . . 15  |-  CC  e.  _V
6665a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  CC  e.  _V )
6757plyremlem 21782 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  CC  ->  (
( Xp  oF  -  ( CC 
X.  { z } ) )  e.  (Poly `  CC )  /\  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  1  /\  ( `' ( Xp  oF  -  ( CC 
X.  { z } ) ) " {
0 } )  =  { z } ) )
6850, 67syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  (
( Xp  oF  -  ( CC 
X.  { z } ) )  e.  (Poly `  CC )  /\  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  1  /\  ( `' ( Xp  oF  -  ( CC 
X.  { z } ) ) " {
0 } )  =  { z } ) )
6968simp1d 1000 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
Xp  oF  -  ( CC  X.  { z } ) )  e.  (Poly `  CC ) )
70 plyf 21678 . . . . . . . . . . . . . . 15  |-  ( ( Xp  oF  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  ->  ( Xp  oF  -  ( CC  X.  { z } ) ) : CC --> CC )
7169, 70syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (
Xp  oF  -  ( CC  X.  { z } ) ) : CC --> CC )
72 plyf 21678 . . . . . . . . . . . . . . 15  |-  ( Q  e.  (Poly `  CC )  ->  Q : CC --> CC )
7332, 72syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  Q : CC --> CC )
74 ofmulrt 21760 . . . . . . . . . . . . . 14  |-  ( ( CC  e.  _V  /\  ( Xp  oF  -  ( CC 
X.  { z } ) ) : CC --> CC  /\  Q : CC --> CC )  ->  ( `' ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) " { 0 } )  =  ( ( `' ( Xp  oF  -  ( CC 
X.  { z } ) ) " {
0 } )  u.  ( `' Q " { 0 } ) ) )
7566, 71, 73, 74syl3anc 1218 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  ( `' ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q )
" { 0 } )  =  ( ( `' ( Xp  oF  -  ( CC  X.  { z } ) ) " {
0 } )  u.  ( `' Q " { 0 } ) ) )
7668simp3d 1002 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( `' ( Xp  oF  -  ( CC  X.  { z } ) ) " {
0 } )  =  { z } )
7776uneq1d 3521 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( `' ( Xp  oF  -  ( CC  X.  { z } ) ) " { 0 } )  u.  ( `' Q " { 0 } ) )  =  ( { z }  u.  ( `' Q " { 0 } ) ) )
7864, 75, 773eqtrd 2479 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  R  =  ( { z }  u.  ( `' Q " { 0 } ) ) )
7978fveq2d 5707 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 R )  =  ( # `  ( { z }  u.  ( `' Q " { 0 } ) ) ) )
801, 2eqtr4d 2478 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  R
)  =  ( D  +  1 ) )
8180adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 R )  =  ( D  +  1 ) )
8279, 81eqtr3d 2477 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( { z }  u.  ( `' Q " { 0 } ) ) )  =  ( D  + 
1 ) )
8315adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  F  =/=  0p )
8461, 83eqnetrrd 2640 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  (
( Xp  oF  -  ( CC 
X.  { z } ) )  oF  x.  Q )  =/=  0p )
85 plymul0or 21759 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( Xp  oF  -  ( CC 
X.  { z } ) )  e.  (Poly `  CC )  /\  Q  e.  (Poly `  CC )
)  ->  ( (
( Xp  oF  -  ( CC 
X.  { z } ) )  oF  x.  Q )  =  0p  <->  ( (
Xp  oF  -  ( CC  X.  { z } ) )  =  0p  \/  Q  =  0p ) ) )
8669, 32, 85syl2anc 661 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  (
( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q )  =  0p  <->  ( (
Xp  oF  -  ( CC  X.  { z } ) )  =  0p  \/  Q  =  0p ) ) )
8786necon3abid 2653 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  (
( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q )  =/=  0p  <->  -.  (
( Xp  oF  -  ( CC 
X.  { z } ) )  =  0p  \/  Q  =  0p ) ) )
8884, 87mpbid 210 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  -.  ( ( Xp  oF  -  ( CC  X.  { z } ) )  =  0p  \/  Q  =  0p ) )
89 neanior 2709 . . . . . . . . . . . . . . . 16  |-  ( ( ( Xp  oF  -  ( CC 
X.  { z } ) )  =/=  0p  /\  Q  =/=  0p )  <->  -.  (
( Xp  oF  -  ( CC 
X.  { z } ) )  =  0p  \/  Q  =  0p ) )
9088, 89sylibr 212 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
( Xp  oF  -  ( CC 
X.  { z } ) )  =/=  0p  /\  Q  =/=  0p ) )
9190simprd 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  Q  =/=  0p )
92 eqid 2443 . . . . . . . . . . . . . . 15  |-  ( `' Q " { 0 } )  =  ( `' Q " { 0 } )
9392fta1 21786 . . . . . . . . . . . . . 14  |-  ( ( Q  e.  (Poly `  CC )  /\  Q  =/=  0p )  -> 
( ( `' Q " { 0 } )  e.  Fin  /\  ( # `
 ( `' Q " { 0 } ) )  <_  (deg `  Q
) ) )
9432, 91, 93syl2anc 661 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( `' Q " { 0 } )  e.  Fin  /\  ( # `
 ( `' Q " { 0 } ) )  <_  (deg `  Q
) ) )
9594simprd 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  <_  (deg `  Q
) )
9695, 31breqtrrd 4330 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  <_  D )
97 snfi 7402 . . . . . . . . . . . . . 14  |-  { z }  e.  Fin
9894simpld 459 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( `' Q " { 0 } )  e.  Fin )
99 hashun2 12158 . . . . . . . . . . . . . 14  |-  ( ( { z }  e.  Fin  /\  ( `' Q " { 0 } )  e.  Fin )  -> 
( # `  ( { z }  u.  ( `' Q " { 0 } ) ) )  <_  ( ( # `  { z } )  +  ( # `  ( `' Q " { 0 } ) ) ) )
10097, 98, 99sylancr 663 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( { z }  u.  ( `' Q " { 0 } ) ) )  <_  ( ( # `  { z } )  +  ( # `  ( `' Q " { 0 } ) ) ) )
101 ax-1cn 9352 . . . . . . . . . . . . . . 15  |-  1  e.  CC
1023nncnd 10350 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  D  e.  CC )
103102adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  D  e.  CC )
104 addcom 9567 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  CC  /\  D  e.  CC )  ->  ( 1  +  D
)  =  ( D  +  1 ) )
105101, 103, 104sylancr 663 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (
1  +  D )  =  ( D  + 
1 ) )
10682, 105eqtr4d 2478 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( { z }  u.  ( `' Q " { 0 } ) ) )  =  ( 1  +  D ) )
107 hashsng 12148 . . . . . . . . . . . . . . 15  |-  ( z  e.  R  ->  ( # `
 { z } )  =  1 )
108107adantl 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 { z } )  =  1 )
109108oveq1d 6118 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( # `  { z } )  +  (
# `  ( `' Q " { 0 } ) ) )  =  ( 1  +  (
# `  ( `' Q " { 0 } ) ) ) )
110100, 106, 1093brtr3d 4333 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
1  +  D )  <_  ( 1  +  ( # `  ( `' Q " { 0 } ) ) ) )
111 hashcl 12138 . . . . . . . . . . . . . . 15  |-  ( ( `' Q " { 0 } )  e.  Fin  ->  ( # `  ( `' Q " { 0 } ) )  e. 
NN0 )
11298, 111syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  e.  NN0 )
113112nn0red 10649 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  e.  RR )
114 1red 9413 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  1  e.  RR )
11536, 113, 114leadd2d 9946 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  ( D  <_  ( # `  ( `' Q " { 0 } ) )  <->  ( 1  +  D )  <_ 
( 1  +  (
# `  ( `' Q " { 0 } ) ) ) ) )
116110, 115mpbird 232 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  D  <_  ( # `  ( `' Q " { 0 } ) ) )
117113, 36letri3d 9528 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (
( # `  ( `' Q " { 0 } ) )  =  D  <->  ( ( # `  ( `' Q " { 0 } ) )  <_  D  /\  D  <_  ( # `  ( `' Q " { 0 } ) ) ) ) )
11896, 116, 117mpbir2and 913 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  =  D )
11982, 118eqeq12d 2457 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  (
( # `  ( { z }  u.  ( `' Q " { 0 } ) ) )  =  ( # `  ( `' Q " { 0 } ) )  <->  ( D  +  1 )  =  D ) )
12042, 119syl5ib 219 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  (
z  e.  ( `' Q " { 0 } )  ->  ( D  +  1 )  =  D ) )
121120necon3ad 2656 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  (
( D  +  1 )  =/=  D  ->  -.  z  e.  ( `' Q " { 0 } ) ) )
12238, 121mpd 15 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  -.  z  e.  ( `' Q " { 0 } ) )
123 disjsn 3948 . . . . . 6  |-  ( ( ( `' Q " { 0 } )  i^i  { z } )  =  (/)  <->  -.  z  e.  ( `' Q " { 0 } ) )
124122, 123sylibr 212 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  (
( `' Q " { 0 } )  i^i  { z } )  =  (/) )
12526, 124syl5eq 2487 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  ( { z }  i^i  ( `' Q " { 0 } ) )  =  (/) )
12619adantr 465 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  R  e.  Fin )
12749adantr 465 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  R  C_  CC )
128127sselda 3368 . . . 4  |-  ( ( ( ph  /\  z  e.  R )  /\  x  e.  R )  ->  x  e.  CC )
129125, 78, 126, 128fsumsplit 13228 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  R  x  =  ( sum_ x  e.  {
z } x  +  sum_ x  e.  ( `' Q " { 0 } ) x ) )
130 id 22 . . . . . . 7  |-  ( x  =  z  ->  x  =  z )
131130sumsn 13229 . . . . . 6  |-  ( ( z  e.  CC  /\  z  e.  CC )  -> 
sum_ x  e.  { z } x  =  z )
13250, 50, 131syl2anc 661 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  { z } x  =  z )
13350negnegd 9722 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  -u -u z  =  z )
134132, 133eqtr4d 2478 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  { z } x  =  -u -u z )
135118, 31eqtrd 2475 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  =  (deg `  Q ) )
13628adantr 465 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  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 )
) ) ) )
137 fveq2 5703 . . . . . . . . . . 11  |-  ( f  =  Q  ->  (deg `  f )  =  (deg
`  Q ) )
138137eqeq2d 2454 . . . . . . . . . 10  |-  ( f  =  Q  ->  ( D  =  (deg `  f
)  <->  D  =  (deg `  Q ) ) )
139 cnveq 5025 . . . . . . . . . . . . 13  |-  ( f  =  Q  ->  `' f  =  `' Q
)
140139imaeq1d 5180 . . . . . . . . . . . 12  |-  ( f  =  Q  ->  ( `' f " {
0 } )  =  ( `' Q " { 0 } ) )
141140fveq2d 5707 . . . . . . . . . . 11  |-  ( f  =  Q  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  ( `' Q " { 0 } ) ) )
142141, 137eqeq12d 2457 . . . . . . . . . 10  |-  ( f  =  Q  ->  (
( # `  ( `' f " { 0 } ) )  =  (deg `  f )  <->  (
# `  ( `' Q " { 0 } ) )  =  (deg
`  Q ) ) )
143138, 142anbi12d 710 . . . . . . . . 9  |-  ( f  =  Q  ->  (
( D  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( D  =  (deg `  Q )  /\  ( # `  ( `' Q " { 0 } ) )  =  (deg `  Q )
) ) )
144140sumeq1d 13190 . . . . . . . . . 10  |-  ( f  =  Q  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  sum_ x  e.  ( `' Q " { 0 } ) x )
145 fveq2 5703 . . . . . . . . . . . . 13  |-  ( f  =  Q  ->  (coeff `  f )  =  (coeff `  Q ) )
146137oveq1d 6118 . . . . . . . . . . . . 13  |-  ( f  =  Q  ->  (
(deg `  f )  -  1 )  =  ( (deg `  Q
)  -  1 ) )
147145, 146fveq12d 5709 . . . . . . . . . . . 12  |-  ( f  =  Q  ->  (
(coeff `  f ) `  ( (deg `  f
)  -  1 ) )  =  ( (coeff `  Q ) `  (
(deg `  Q )  -  1 ) ) )
148145, 137fveq12d 5709 . . . . . . . . . . . 12  |-  ( f  =  Q  ->  (
(coeff `  f ) `  (deg `  f )
)  =  ( (coeff `  Q ) `  (deg `  Q ) ) )
149147, 148oveq12d 6121 . . . . . . . . . . 11  |-  ( f  =  Q  ->  (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) )  =  ( ( (coeff `  Q
) `  ( (deg `  Q )  -  1 ) )  /  (
(coeff `  Q ) `  (deg `  Q )
) ) )
150149negeqd 9616 . . . . . . . . . 10  |-  ( f  =  Q  ->  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) )  =  -u ( ( (coeff `  Q ) `  (
(deg `  Q )  -  1 ) )  /  ( (coeff `  Q ) `  (deg `  Q ) ) ) )
151144, 150eqeq12d 2457 . . . . . . . . 9  |-  ( f  =  Q  ->  ( sum_ x  e.  ( `' f " { 0 } ) x  = 
-u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) )  <->  sum_ x  e.  ( `' Q " { 0 } ) x  = 
-u ( ( (coeff `  Q ) `  (
(deg `  Q )  -  1 ) )  /  ( (coeff `  Q ) `  (deg `  Q ) ) ) ) )
152143, 151imbi12d 320 . . . . . . . 8  |-  ( f  =  Q  ->  (
( ( D  =  (deg `  f )  /\  ( # `  ( `' f " {
0 } ) )  =  (deg `  f
) )  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) ) )  <->  ( ( D  =  (deg `  Q
)  /\  ( # `  ( `' Q " { 0 } ) )  =  (deg `  Q )
)  ->  sum_ x  e.  ( `' Q " { 0 } ) x  =  -u (
( (coeff `  Q
) `  ( (deg `  Q )  -  1 ) )  /  (
(coeff `  Q ) `  (deg `  Q )
) ) ) ) )
153152rspcv 3081 . . . . . . 7  |-  ( Q  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  =  (deg `  Q )  /\  ( # `  ( `' Q " { 0 } ) )  =  (deg `  Q )
)  ->  sum_ x  e.  ( `' Q " { 0 } ) x  =  -u (
( (coeff `  Q
) `  ( (deg `  Q )  -  1 ) )  /  (
(coeff `  Q ) `  (deg `  Q )
) ) ) ) )
15432, 136, 153sylc 60 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (
( D  =  (deg
`  Q )  /\  ( # `  ( `' Q " { 0 } ) )  =  (deg `  Q )
)  ->  sum_ x  e.  ( `' Q " { 0 } ) x  =  -u (
( (coeff `  Q
) `  ( (deg `  Q )  -  1 ) )  /  (
(coeff `  Q ) `  (deg `  Q )
) ) ) )
15531, 135, 154mp2and 679 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  ( `' Q " { 0 } ) x  =  -u (
( (coeff `  Q
) `  ( (deg `  Q )  -  1 ) )  /  (
(coeff `  Q ) `  (deg `  Q )
) ) )
15631oveq1d 6118 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  ( D  -  1 )  =  ( (deg `  Q )  -  1 ) )
157156fveq2d 5707 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( D  -  1 ) )  =  ( (coeff `  Q ) `  ( (deg `  Q
)  -  1 ) ) )
15861fveq2d 5707 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  F )  =  (coeff `  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) )
15927, 158syl5eq 2487 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  A  =  (coeff `  ( (
Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) )
16061fveq2d 5707 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  F )  =  (deg
`  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) )
16168simp2d 1001 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  1 )
162 ax-1ne0 9363 . . . . . . . . . . . . . . 15  |-  1  =/=  0
163162a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  1  =/=  0 )
164161, 163eqnetrd 2638 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =/=  0 )
165 fveq2 5703 . . . . . . . . . . . . . . 15  |-  ( ( Xp  oF  -  ( CC  X.  { z } ) )  =  0p  ->  (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) )  =  (deg
`  0p ) )
166165, 12syl6eq 2491 . . . . . . . . . . . . . 14  |-  ( ( Xp  oF  -  ( CC  X.  { z } ) )  =  0p  ->  (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) )  =  0 )
167166necon3i 2662 . . . . . . . . . . . . 13  |-  ( (deg
`  ( Xp  oF  -  ( CC  X.  { z } ) ) )  =/=  0  ->  ( Xp  oF  -  ( CC  X.  { z } ) )  =/=  0p )
168164, 167syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
Xp  oF  -  ( CC  X.  { z } ) )  =/=  0p )
169 eqid 2443 . . . . . . . . . . . . 13  |-  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  (deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) )
170 eqid 2443 . . . . . . . . . . . . 13  |-  (deg `  Q )  =  (deg
`  Q )
171169, 170dgrmul 21749 . . . . . . . . . . . 12  |-  ( ( ( ( Xp  oF  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  /\  (
Xp  oF  -  ( CC  X.  { z } ) )  =/=  0p )  /\  ( Q  e.  (Poly `  CC )  /\  Q  =/=  0p ) )  -> 
(deg `  ( (
Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) )  =  ( (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) )  +  (deg
`  Q ) ) )
17269, 168, 32, 91, 171syl22anc 1219 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) )  =  ( (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  +  (deg `  Q )
) )
173160, 172eqtrd 2475 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  F )  =  ( (deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) )  +  (deg `  Q )
) )
1749, 173syl5eq 2487 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  N  =  ( (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  +  (deg `  Q )
) )
175159, 174fveq12d 5709 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  ( A `  N )  =  ( (coeff `  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) `
 ( (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  +  (deg `  Q )
) ) )
176 eqid 2443 . . . . . . . . . 10  |-  (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  (coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) )
177 eqid 2443 . . . . . . . . . 10  |-  (coeff `  Q )  =  (coeff `  Q )
178176, 177, 169, 170coemulhi 21733 . . . . . . . . 9  |-  ( ( ( Xp  oF  -  ( CC 
X.  { z } ) )  e.  (Poly `  CC )  /\  Q  e.  (Poly `  CC )
)  ->  ( (coeff `  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) `
 ( (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  +  (deg `  Q )
) )  =  ( ( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) )  x.  ( (coeff `  Q ) `  (deg `  Q ) ) ) )
17969, 32, 178syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( (
Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) `  ( (deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) )  +  (deg `  Q )
) )  =  ( ( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) )  x.  ( (coeff `  Q ) `  (deg `  Q ) ) ) )
180161fveq2d 5707 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  (deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) )  =  ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) ` 
1 ) )
181 ssid 3387 . . . . . . . . . . . . . . 15  |-  CC  C_  CC
182 plyid 21689 . . . . . . . . . . . . . . 15  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  Xp  e.  (Poly `  CC ) )
183181, 101, 182mp2an 672 . . . . . . . . . . . . . 14  |-  Xp  e.  (Poly `  CC )
184 plyconst 21686 . . . . . . . . . . . . . . 15  |-  ( ( CC  C_  CC  /\  z  e.  CC )  ->  ( CC  X.  { z } )  e.  (Poly `  CC ) )
185181, 50, 184sylancr 663 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( CC  X.  { z } )  e.  (Poly `  CC ) )
186 eqid 2443 . . . . . . . . . . . . . . 15  |-  (coeff `  Xp )  =  (coeff `  Xp
)
187 eqid 2443 . . . . . . . . . . . . . . 15  |-  (coeff `  ( CC  X.  { z } ) )  =  (coeff `  ( CC  X.  { z } ) )
188186, 187coesub 21736 . . . . . . . . . . . . . 14  |-  ( ( Xp  e.  (Poly `  CC )  /\  ( CC  X.  { z } )  e.  (Poly `  CC ) )  ->  (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  ( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) )
189183, 185, 188sylancr 663 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  ( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) )
190189fveq1d 5705 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) ` 
1 )  =  ( ( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
1 ) )
191 1nn0 10607 . . . . . . . . . . . . . 14  |-  1  e.  NN0
192186coef3 21712 . . . . . . . . . . . . . . . . 17  |-  ( Xp  e.  (Poly `  CC )  ->  (coeff `  Xp ) : NN0 --> CC )
193 ffn 5571 . . . . . . . . . . . . . . . . 17  |-  ( (coeff `  Xp ) : NN0 --> CC  ->  (coeff `  Xp )  Fn 
NN0 )
194183, 192, 193mp2b 10 . . . . . . . . . . . . . . . 16  |-  (coeff `  Xp )  Fn 
NN0
195194a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  Xp )  Fn 
NN0 )
196187coef3 21712 . . . . . . . . . . . . . . . 16  |-  ( ( CC  X.  { z } )  e.  (Poly `  CC )  ->  (coeff `  ( CC  X.  {
z } ) ) : NN0 --> CC )
197 ffn 5571 . . . . . . . . . . . . . . . 16  |-  ( (coeff `  ( CC  X.  {
z } ) ) : NN0 --> CC  ->  (coeff `  ( CC  X.  {
z } ) )  Fn  NN0 )
198185, 196, 1973syl 20 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  ( CC  X.  {
z } ) )  Fn  NN0 )
199 nn0ex 10597 . . . . . . . . . . . . . . . 16  |-  NN0  e.  _V
200199a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  NN0  e.  _V )
201 inidm 3571 . . . . . . . . . . . . . . 15  |-  ( NN0 
i^i  NN0 )  =  NN0
202 coeidp 21742 . . . . . . . . . . . . . . . . 17  |-  ( 1  e.  NN0  ->  ( (coeff `  Xp ) ` 
1 )  =  if ( 1  =  1 ,  1 ,  0 ) )
203202adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
(coeff `  Xp
) `  1 )  =  if ( 1  =  1 ,  1 ,  0 ) )
204 eqid 2443 . . . . . . . . . . . . . . . . 17  |-  1  =  1
205204iftruei 3810 . . . . . . . . . . . . . . . 16  |-  if ( 1  =  1 ,  1 ,  0 )  =  1
206203, 205syl6eq 2491 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
(coeff `  Xp
) `  1 )  =  1 )
207 0lt1 9874 . . . . . . . . . . . . . . . . . 18  |-  0  <  1
208 0re 9398 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR
209 1re 9397 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  RR
210208, 209ltnlei 9507 . . . . . . . . . . . . . . . . . 18  |-  ( 0  <  1  <->  -.  1  <_  0 )
211207, 210mpbi 208 . . . . . . . . . . . . . . . . 17  |-  -.  1  <_  0
21250adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  z  e.  CC )
213 0dgr 21725 . . . . . . . . . . . . . . . . . . 19  |-  ( z  e.  CC  ->  (deg `  ( CC  X.  {
z } ) )  =  0 )
214212, 213syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (deg `  ( CC  X.  {
z } ) )  =  0 )
215214breq2d 4316 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
1  <_  (deg `  ( CC  X.  { z } ) )  <->  1  <_  0 ) )
216211, 215mtbiri 303 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  -.  1  <_  (deg `  ( CC  X.  { z } ) ) )
217 eqid 2443 . . . . . . . . . . . . . . . . . . . 20  |-  (deg `  ( CC  X.  { z } ) )  =  (deg `  ( CC  X.  { z } ) )
218187, 217dgrub 21714 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( CC  X.  {
z } )  e.  (Poly `  CC )  /\  1  e.  NN0  /\  ( (coeff `  ( CC  X.  { z } ) ) `  1
)  =/=  0 )  ->  1  <_  (deg `  ( CC  X.  {
z } ) ) )
2192183expia 1189 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( CC  X.  {
z } )  e.  (Poly `  CC )  /\  1  e.  NN0 )  ->  ( ( (coeff `  ( CC  X.  {
z } ) ) `
 1 )  =/=  0  ->  1  <_  (deg
`  ( CC  X.  { z } ) ) ) )
220185, 219sylan 471 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
( (coeff `  ( CC  X.  { z } ) ) `  1
)  =/=  0  -> 
1  <_  (deg `  ( CC  X.  { z } ) ) ) )
221220necon1bd 2691 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  ( -.  1  <_  (deg `  ( CC  X.  { z } ) )  -> 
( (coeff `  ( CC  X.  { z } ) ) `  1
)  =  0 ) )
222216, 221mpd 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
(coeff `  ( CC  X.  { z } ) ) `  1 )  =  0 )
223195, 198, 200, 200, 201, 206, 222ofval 6341 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
1 )  =  ( 1  -  0 ) )
224191, 223mpan2 671 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
1 )  =  ( 1  -  0 ) )
225 1m0e1 10444 . . . . . . . . . . . . 13  |-  ( 1  -  0 )  =  1
226224, 225syl6eq 2491 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
1 )  =  1 )
227190, 226eqtrd 2475 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) ` 
1 )  =  1 )
228180, 227eqtrd 2475 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  (deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) )  =  1 )
229228oveq1d 6118 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) )  x.  ( (coeff `  Q ) `  (deg `  Q ) ) )  =  ( 1  x.  ( (coeff `  Q
) `  (deg `  Q
) ) ) )
230177coef3 21712 . . . . . . . . . . . 12  |-  ( Q  e.  (Poly `  CC )  ->  (coeff `  Q
) : NN0 --> CC )
23132, 230syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  Q ) : NN0 --> CC )
232231, 34ffvelrnd 5856 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  (deg `  Q )
)  e.  CC )
233232mulid2d 9416 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  (
1  x.  ( (coeff `  Q ) `  (deg `  Q ) ) )  =  ( (coeff `  Q ) `  (deg `  Q ) ) )
234229, 233eqtrd 2475 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) )  x.  ( (coeff `  Q ) `  (deg `  Q ) ) )  =  ( (coeff `  Q ) `  (deg `  Q ) ) )
235175, 179, 2343eqtrd 2479 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( A `  N )  =  ( (coeff `  Q ) `  (deg `  Q ) ) )
236157, 235oveq12d 6121 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) )  =  ( ( (coeff `  Q ) `  (
(deg `  Q )  -  1 ) )  /  ( (coeff `  Q ) `  (deg `  Q ) ) ) )
237236negeqd 9616 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  -u (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) )  = 
-u ( ( (coeff `  Q ) `  (
(deg `  Q )  -  1 ) )  /  ( (coeff `  Q ) `  (deg `  Q ) ) ) )
238155, 237eqtr4d 2478 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  ( `' Q " { 0 } ) x  =  -u (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) ) )
239134, 238oveq12d 6121 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  ( sum_ x  e.  { z } x  +  sum_ x  e.  ( `' Q " { 0 } ) x )  =  (
-u -u z  +  -u ( ( (coeff `  Q ) `  ( D  -  1 ) )  /  ( A `
 N ) ) ) )
24050negcld 9718 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  -u z  e.  CC )
241 nnm1nn0 10633 . . . . . . . . 9  |-  ( D  e.  NN  ->  ( D  -  1 )  e.  NN0 )
2423, 241syl 16 . . . . . . . 8  |-  ( ph  ->  ( D  -  1 )  e.  NN0 )
243242adantr 465 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( D  -  1 )  e.  NN0 )
244231, 243ffvelrnd 5856 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( D  -  1 ) )  e.  CC )
245235, 232eqeltrd 2517 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  ( A `  N )  e.  CC )
2469, 27dgreq0 21744 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0p  <->  ( A `  N )  =  0 ) )
24743, 246syl 16 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  ( F  =  0p  <->  ( A `  N )  =  0 ) )
248247necon3bid 2655 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( F  =/=  0p  <->  ( A `  N )  =/=  0
) )
24983, 248mpbid 210 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  ( A `  N )  =/=  0 )
250244, 245, 249divcld 10119 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) )  e.  CC )
251240, 250negdid 9744 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  -u ( -u z  +  ( ( (coeff `  Q ) `  ( D  -  1 ) )  /  ( A `  N )
) )  =  (
-u -u z  +  -u ( ( (coeff `  Q ) `  ( D  -  1 ) )  /  ( A `
 N ) ) ) )
252240, 245mulcld 9418 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( -u z  x.  ( A `
 N ) )  e.  CC )
253252, 244, 245, 249divdird 10157 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (
( ( -u z  x.  ( A `  N
) )  +  ( (coeff `  Q ) `  ( D  -  1 ) ) )  / 
( A `  N
) )  =  ( ( ( -u z  x.  ( A `  N
) )  /  ( A `  N )
)  +  ( ( (coeff `  Q ) `  ( D  -  1 ) )  /  ( A `  N )
) ) )
254 nnm1nn0 10633 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
2555, 254syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( N  -  1 )  e.  NN0 )
256255adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  ( N  -  1 )  e.  NN0 )
257176, 177coemul 21731 . . . . . . . . 9  |-  ( ( ( Xp  oF  -  ( CC 
X.  { z } ) )  e.  (Poly `  CC )  /\  Q  e.  (Poly `  CC )  /\  ( N  -  1 )  e.  NN0 )  ->  ( (coeff `  (
( Xp  oF  -  ( CC 
X.  { z } ) )  oF  x.  Q ) ) `
 ( N  - 
1 ) )  = 
sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) ) )
25869, 32, 256, 257syl3anc 1218 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( (
Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) `  ( N  -  1
) )  =  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) ) )
259159fveq1d 5705 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  ( A `  ( N  -  1 ) )  =  ( (coeff `  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) `
 ( N  - 
1 ) ) )
260 1e0p1 10795 . . . . . . . . . . . 12  |-  1  =  ( 0  +  1 )
261260oveq2i 6114 . . . . . . . . . . 11  |-  ( 0 ... 1 )  =  ( 0 ... (
0  +  1 ) )
262261sumeq1i 13187 . . . . . . . . . 10  |-  sum_ k  e.  ( 0 ... 1
) ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) )  = 
sum_ k  e.  ( 0 ... ( 0  +  1 ) ) ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) )
263 0nn0 10606 . . . . . . . . . . . . 13  |-  0  e.  NN0
264 nn0uz 10907 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
265263, 264eleqtri 2515 . . . . . . . . . . . 12  |-  0  e.  ( ZZ>= `  0 )
266265a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  0  e.  ( ZZ>= `  0 )
)
267261eleq2i 2507 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... 1 )  <->  k  e.  ( 0 ... (
0  +  1 ) ) )
268176coef3 21712 . . . . . . . . . . . . . . 15  |-  ( ( Xp  oF  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  ->  (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) : NN0 --> CC )
26969, 268syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) : NN0 --> CC )
270 elfznn0 11493 . . . . . . . . . . . . . 14  |-  ( k  e.  ( 0 ... 1 )  ->  k  e.  NN0 )
271 ffvelrn 5853 . . . . . . . . . . . . . 14  |-  ( ( (coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) : NN0 --> CC  /\  k  e.  NN0 )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  k )  e.  CC )
272269, 270, 271syl2an 477 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... 1
) )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  k )  e.  CC )
2732oveq1d 6118 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( D  + 
1 )  -  1 )  =  ( N  -  1 ) )
274 pncan 9628 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( D  e.  CC  /\  1  e.  CC )  ->  ( ( D  + 
1 )  -  1 )  =  D )
275102, 101, 274sylancl 662 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( D  + 
1 )  -  1 )  =  D )
276273, 275eqtr3d 2477 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( N  -  1 )  =  D )
277276adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  ( N  -  1 )  =  D )
2783adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  D  e.  NN )
279277, 278eqeltrd 2517 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  ( N  -  1 )  e.  NN )
280 nnuz 10908 . . . . . . . . . . . . . . . . 17  |-  NN  =  ( ZZ>= `  1 )
281279, 280syl6eleq 2533 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  ( N  -  1 )  e.  ( ZZ>= `  1
) )
282 fzss2 11510 . . . . . . . . . . . . . . . 16  |-  ( ( N  -  1 )  e.  ( ZZ>= `  1
)  ->  ( 0 ... 1 )  C_  ( 0 ... ( N  -  1 ) ) )
283281, 282syl 16 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
0 ... 1 )  C_  ( 0 ... ( N  -  1 ) ) )
284283sselda 3368 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... 1
) )  ->  k  e.  ( 0 ... ( N  -  1 ) ) )
285 fznn0sub 11499 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  (
( N  -  1 )  -  k )  e.  NN0 )
286 ffvelrn 5853 . . . . . . . . . . . . . . 15  |-  ( ( (coeff `  Q ) : NN0 --> CC  /\  (
( N  -  1 )  -  k )  e.  NN0 )  -> 
( (coeff `  Q
) `  ( ( N  -  1 )  -  k ) )  e.  CC )
287231, 285, 286syl2an 477 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) )  e.  CC )
288284, 287syldan 470 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... 1
) )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) )  e.  CC )
289272, 288mulcld 9418 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... 1
) )  ->  (
( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  e.  CC )
290267, 289sylan2br 476 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... (
0  +  1 ) ) )  ->  (
( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  e.  CC )
291 id 22 . . . . . . . . . . . . . 14  |-  ( k  =  ( 0  +  1 )  ->  k  =  ( 0  +  1 ) )
292291, 260syl6eqr 2493 . . . . . . . . . . . . 13  |-  ( k  =  ( 0  +  1 )  ->  k  =  1 )
293292fveq2d 5707 . . . . . . . . . . . 12  |-  ( k  =  ( 0  +  1 )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  k )  =  ( (coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) ` 
1 ) )
294292oveq2d 6119 . . . . . . . . . . . . 13  |-  ( k  =  ( 0  +  1 )  ->  (
( N  -  1 )  -  k )  =  ( ( N  -  1 )  - 
1 ) )
295294fveq2d 5707 . . . . . . . . . . . 12  |-  ( k  =  ( 0  +  1 )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) )  =  ( (coeff `  Q ) `  ( ( N  - 
1 )  -  1 ) ) )
296293, 295oveq12d 6121 . . . . . . . . . . 11  |-  ( k  =  ( 0  +  1 )  ->  (
( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  =  ( ( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  1
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  1 ) ) ) )
297266, 290, 296fsump1 13235 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  sum_ k  e.  ( 0 ... (
0  +  1 ) ) ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) )  =  ( sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) )  +  ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) ` 
1 )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  1 ) ) ) ) )
298262, 297syl5eq 2487 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  sum_ k  e.  ( 0 ... 1
) ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) )  =  ( sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) )  +  ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) ` 
1 )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  1 ) ) ) ) )
299 eldifn 3491 . . . . . . . . . . . . . 14  |-  ( k  e.  ( ( 0 ... ( N  - 
1 ) )  \ 
( 0 ... 1
) )  ->  -.  k  e.  ( 0 ... 1 ) )
300299adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  -.  k  e.  ( 0 ... 1
) )
301 eldifi 3490 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( ( 0 ... ( N  - 
1 ) )  \ 
( 0 ... 1
) )  ->  k  e.  ( 0 ... ( N  -  1 ) ) )
302 elfznn0 11493 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
303301, 302syl 16 . . . . . . . . . . . . . . . 16  |-  ( k  e.  ( ( 0 ... ( N  - 
1 ) )  \ 
( 0 ... 1
) )  ->  k  e.  NN0 )
304176, 169dgrub 21714 . . . . . . . . . . . . . . . . 17  |-  ( ( ( Xp  oF  -  ( CC 
X.  { z } ) )  e.  (Poly `  CC )  /\  k  e.  NN0  /\  ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) `  k )  =/=  0
)  ->  k  <_  (deg
`  ( Xp  oF  -  ( CC  X.  { z } ) ) ) )
3053043expia 1189 . . . . . . . . . . . . . . . 16  |-  ( ( ( Xp  oF  -  ( CC 
X.  { z } ) )  e.  (Poly `  CC )  /\  k  e.  NN0 )  ->  (
( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  k
)  =/=  0  -> 
k  <_  (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) ) )
30669, 303, 305syl2an 477 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  k )  =/=  0  ->  k  <_  (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) ) )
307 elfzuz 11461 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  ( ZZ>= `  0 )
)
308301, 307syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  ( ( 0 ... ( N  - 
1 ) )  \ 
( 0 ... 1
) )  ->  k  e.  ( ZZ>= `  0 )
)
309308adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  k  e.  ( ZZ>= `  0 )
)
310 1z 10688 . . . . . . . . . . . . . . . . 17  |-  1  e.  ZZ
311 elfz5 11457 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  ( ZZ>= ` 
0 )  /\  1  e.  ZZ )  ->  (
k  e.  ( 0 ... 1 )  <->  k  <_  1 ) )
312309, 310, 311sylancl 662 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( k  e.  ( 0 ... 1
)  <->  k  <_  1
) )
313161breq2d 4316 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  (
k  <_  (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) )  <->  k  <_  1 ) )
314313adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( k  <_  (deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) )  <->  k  <_  1 ) )
315312, 314bitr4d 256 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( k  e.  ( 0 ... 1
)  <->  k  <_  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) ) )
316306, 315sylibrd 234 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  k )  =/=  0  ->  k  e.  ( 0 ... 1 ) ) )
317316necon1bd 2691 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( -.  k  e.  ( 0 ... 1 )  -> 
( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  k
)  =  0 ) )
318300, 317mpd 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) `  k )  =  0 )
319318oveq1d 6118 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) )  =  ( 0  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) ) )
320301, 287sylan2 474 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) )  e.  CC )
321320mul02d 9579 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( 0  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  =  0 )
322319, 321eqtrd 2475 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) )  =  0 )
323 fzfid 11807 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (
0 ... ( N  - 
1 ) )  e. 
Fin )
324283, 289, 322, 323fsumss 13214 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  sum_ k  e.  ( 0 ... 1
) ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) )  = 
sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) ) )
325 0z 10669 . . . . . . . . . . . 12  |-  0  e.  ZZ
326189fveq1d 5705 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) ` 
0 )  =  ( ( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
0 ) )
327 coeidp 21742 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0  e.  NN0  ->  ( (coeff `  Xp ) ` 
0 )  =  if ( 0  =  1 ,  1 ,  0 ) )
328162nesymi 2660 . . . . . . . . . . . . . . . . . . . . 21  |-  -.  0  =  1
329328iffalsei 3812 . . . . . . . . . . . . . . . . . . . 20  |-  if ( 0  =  1 ,  1 ,  0 )  =  0
330327, 329syl6eq 2491 . . . . . . . . . . . . . . . . . . 19  |-  ( 0  e.  NN0  ->  ( (coeff `  Xp ) ` 
0 )  =  0 )
331330adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  R )  /\  0  e.  NN0 )  ->  (
(coeff `  Xp
) `  0 )  =  0 )
332 0cn 9390 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  CC
333 vex 2987 . . . . . . . . . . . . . . . . . . . . . 22  |-  z  e. 
_V
334333fvconst2 5945 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0  e.  CC  ->  (
( CC  X.  {
z } ) ` 
0 )  =  z )
335332, 334ax-mp 5 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( CC  X.  { z } ) `  0
)  =  z
336187coefv0 21727 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( CC  X.  { z } )  e.  (Poly `  CC )  ->  (
( CC  X.  {
z } ) ` 
0 )  =  ( (coeff `  ( CC  X.  { z } ) ) `  0 ) )
337185, 336syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  z  e.  R )  ->  (
( CC  X.  {
z } ) ` 
0 )  =  ( (coeff `  ( CC  X.  { z } ) ) `  0 ) )
338335, 337syl5reqr 2490 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( CC  X.  { z } ) ) `  0 )  =  z )
339338adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  R )  /\  0  e.  NN0 )  ->  (
(coeff `  ( CC  X.  { z } ) ) `  0 )  =  z )
340195, 198, 200, 200, 201, 331, 339ofval 6341 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  R )  /\  0  e.  NN0 )  ->  (
( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
0 )  =  ( 0  -  z ) )
341263, 340mpan2 671 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
0 )  =  ( 0  -  z ) )
342 df-neg 9610 . . . . . . . . . . . . . . . 16  |-  -u z  =  ( 0  -  z )
343341, 342syl6eqr 2493 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
0 )  =  -u z )
344326, 343eqtrd 2475 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) ` 
0 )  =  -u z )
345277oveq1d 6118 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  (
( N  -  1 )  -  0 )  =  ( D  - 
0 ) )
346103subid1d 9720 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  ( D  -  0 )  =  D )
347345, 346, 313eqtrd 2479 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  (
( N  -  1 )  -  0 )  =  (deg `  Q
) )
348347fveq2d 5707 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  0 ) )  =  ( (coeff `  Q ) `  (deg `  Q )
) )
349348, 235eqtr4d 2478 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  0 ) )  =  ( A `  N ) )
350344, 349oveq12d 6121 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  0
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  0 ) ) )  =  (
-u z  x.  ( A `  N )
) )
351350, 252eqeltrd 2517 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  0
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  0 ) ) )  e.  CC )
352 fveq2 5703 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  k )  =  ( (coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) ` 
0 ) )
353 oveq2 6111 . . . . . . . . . . . . . . 15  |-  ( k  =  0  ->  (
( N  -  1 )  -  k )  =  ( ( N  -  1 )  - 
0 ) )
354353fveq2d 5707 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) )  =  ( (coeff `  Q ) `  ( ( N  - 
1 )  -  0 ) ) )
355352, 354oveq12d 6121 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  =  ( ( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  0
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  0 ) ) ) )
356355fsum1 13230 . . . . . . . . . . . 12  |-  ( ( 0  e.  ZZ  /\  ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) ` 
0 )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  0 ) ) )  e.  CC )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) )  =  ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) ` 
0 )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  0 ) ) ) )
357325, 351, 356sylancr 663 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) )  =  ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) ` 
0 )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  0 ) ) ) )
358357, 350eqtrd 2475 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) )  =  ( -u z  x.  ( A `  N
) ) )
359277oveq1d 6118 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( N  -  1 )  -  1 )  =  ( D  - 
1 ) )
360359fveq2d 5707 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  1 ) )  =  ( (coeff `  Q ) `  ( D  -  1 ) ) )
361227, 360oveq12d 6121 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  1
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  1 ) ) )  =  ( 1  x.  ( (coeff `  Q ) `  ( D  -  1 ) ) ) )
362244mulid2d 9416 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (
1  x.  ( (coeff `  Q ) `  ( D  -  1 ) ) )  =  ( (coeff `  Q ) `  ( D  -  1 ) ) )
363361, 362eqtrd 2475 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  1
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  1 ) ) )  =  ( (coeff `  Q ) `  ( D  -  1 ) ) )
364358, 363oveq12d 6121 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  ( sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  +  ( ( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  1
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  1 ) ) ) )  =  ( ( -u z  x.  ( A `  N
) )  +  ( (coeff `  Q ) `  ( D  -  1 ) ) ) )
365298, 324, 3643eqtr3rd 2484 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  (
( -u z  x.  ( A `  N )
)  +  ( (coeff `  Q ) `  ( D  -  1 ) ) )  =  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) ) )
366258, 259, 3653eqtr4rd 2486 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  (
( -u z  x.  ( A `  N )
)  +  ( (coeff `  Q ) `  ( D  -  1 ) ) )  =  ( A `  ( N  -  1 ) ) )
367366oveq1d 6118 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (
( ( -u z  x.  ( A `  N
) )  +  ( (coeff `  Q ) `  ( D  -  1 ) ) )  / 
( A `  N
) )  =  ( ( A `  ( N  -  1 ) )  /  ( A `
 N ) ) )
368240, 245, 249divcan4d 10125 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  (
( -u z  x.  ( A `  N )
)  /  ( A `
 N ) )  =  -u z )
369368oveq1d 6118 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (
( ( -u z  x.  ( A `  N
) )  /  ( A `  N )
)  +  ( ( (coeff `  Q ) `  ( D  -  1 ) )  /  ( A `  N )
) )  =  (
-u z  +  ( ( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) ) ) )
370253, 367, 3693eqtr3rd 2484 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  ( -u z  +  ( ( (coeff `  Q ) `  ( D  -  1 ) )  /  ( A `  N )
) )  =  ( ( A `  ( N  -  1 ) )  /  ( A `
 N ) ) )
371370negeqd 9616 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  -u ( -u z  +  ( ( (coeff `  Q ) `  ( D  -  1 ) )  /  ( A `  N )
) )  =  -u ( ( A `  ( N  -  1
) )  /  ( A `  N )
) )
372251, 371eqtr3d 2477 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  ( -u -u z  +  -u (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) ) )  =  -u ( ( A `
 ( N  - 
1 ) )  / 
( A `  N
) ) )
373129, 239, 3723eqtrd 2479 . 2  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  R  x  =  -u ( ( A `  ( N  -  1
) )  /  ( A `  N )
) )
37425, 373exlimddv 1692 1  |-  ( ph  -> 
sum_ x  e.  R  x  =  -u ( ( A `  ( N  -  1 ) )  /  ( A `  N ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2618   A.wral 2727   _Vcvv 2984    \ cdif 3337    u. cun 3338    i^i cin 3339    C_ wss 3340   (/)c0 3649   ifcif 3803   {csn 3889   class class class wbr 4304    X. cxp 4850   `'ccnv 4851   dom cdm 4852   "cima 4855    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103    oFcof 6330   Fincfn 7322   CCcc 9292   RRcr 9293   0cc0 9294   1c1 9295    + caddc 9297    x. cmul 9299    < clt 9430    <_ cle 9431    - cmin 9607   -ucneg 9608    / cdiv 10005   NNcn 10334   NN0cn0 10591   ZZcz 10658   ZZ>=cuz 10873   ...cfz 11449   #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:  vieta1  21790
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