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Theorem vieta1lem2 21736
Description: Lemma for vieta1 21737: 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 10335 . . . . . . 7  |-  ( ph  ->  ( D  +  1 )  e.  NN )
52, 4eqeltrrd 2516 . . . . . 6  |-  ( ph  ->  N  e.  NN )
65nnne0d 10362 . . . . 5  |-  ( ph  ->  N  =/=  0 )
71, 6eqnetrd 2624 . . . 4  |-  ( ph  ->  ( # `  R
)  =/=  0 )
8 vieta1.4 . . . . . . . 8  |-  ( ph  ->  F  e.  (Poly `  S ) )
9 vieta1.2 . . . . . . . . . 10  |-  N  =  (deg `  F )
109, 6syl5eqner 2631 . . . . . . . . 9  |-  ( ph  ->  (deg `  F )  =/=  0 )
11 fveq2 5688 . . . . . . . . . . 11  |-  ( F  =  0p  -> 
(deg `  F )  =  (deg `  0p
) )
12 dgr0 21688 . . . . . . . . . . 11  |-  (deg ` 
0p )  =  0
1311, 12syl6eq 2489 . . . . . . . . . 10  |-  ( F  =  0p  -> 
(deg `  F )  =  0 )
1413necon3i 2648 . . . . . . . . 9  |-  ( (deg
`  F )  =/=  0  ->  F  =/=  0p )
1510, 14syl 16 . . . . . . . 8  |-  ( ph  ->  F  =/=  0p )
16 vieta1.3 . . . . . . . . 9  |-  R  =  ( `' F " { 0 } )
1716fta1 21733 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0p )  -> 
( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) )
188, 15, 17syl2anc 656 . . . . . . 7  |-  ( ph  ->  ( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) )
1918simpld 456 . . . . . 6  |-  ( ph  ->  R  e.  Fin )
20 hasheq0 12127 . . . . . 6  |-  ( R  e.  Fin  ->  (
( # `  R )  =  0  <->  R  =  (/) ) )
2119, 20syl 16 . . . . 5  |-  ( ph  ->  ( ( # `  R
)  =  0  <->  R  =  (/) ) )
2221necon3bid 2641 . . . 4  |-  ( ph  ->  ( ( # `  R
)  =/=  0  <->  R  =/=  (/) ) )
237, 22mpbid 210 . . 3  |-  ( ph  ->  R  =/=  (/) )
24 n0 3643 . . 3  |-  ( R  =/=  (/)  <->  E. z  z  e.  R )
2523, 24sylib 196 . 2  |-  ( ph  ->  E. z  z  e.  R )
26 incom 3540 . . . . 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 21735 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  ( Q  e.  (Poly `  CC )  /\  D  =  (deg
`  Q ) ) )
3130simprd 460 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  D  =  (deg `  Q )
)
3230simpld 456 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  Q  e.  (Poly `  CC )
)
33 dgrcl 21660 . . . . . . . . . . 11  |-  ( Q  e.  (Poly `  CC )  ->  (deg `  Q
)  e.  NN0 )
3432, 33syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  Q )  e.  NN0 )
3534nn0red 10633 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  Q )  e.  RR )
3631, 35eqeltrd 2515 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  D  e.  RR )
3736ltp1d 10259 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  D  <  ( D  +  1 ) )
3836, 37gtned 9505 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( D  +  1 )  =/=  D )
39 snssi 4014 . . . . . . . . . . 11  |-  ( z  e.  ( `' Q " { 0 } )  ->  { z } 
C_  ( `' Q " { 0 } ) )
40 ssequn1 3523 . . . . . . . . . . 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 5692 . . . . . . . . 9  |-  ( z  e.  ( `' Q " { 0 } )  ->  ( # `  ( { z }  u.  ( `' Q " { 0 } ) ) )  =  ( # `  ( `' Q " { 0 } ) ) )
438adantr 462 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  F  e.  (Poly `  S )
)
44 cnvimass 5186 . . . . . . . . . . . . . . . . . . . . 21  |-  ( `' F " { 0 } )  C_  dom  F
4516, 44eqsstri 3383 . . . . . . . . . . . . . . . . . . . 20  |-  R  C_  dom  F
46 plyf 21625 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
47 fdm 5560 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F : CC --> CC  ->  dom 
F  =  CC )
488, 46, 473syl 20 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  dom  F  =  CC )
4945, 48syl5sseq 3401 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  R  C_  CC )
5049sselda 3353 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  z  e.  CC )
5116eleq2i 2505 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  e.  R  <->  z  e.  ( `' F " { 0 } ) )
52 ffn 5556 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F : CC --> CC  ->  F  Fn  CC )
53 fniniseg 5821 . . . . . . . . . . . . . . . . . . . . 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 621 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  ( F `  z )  =  0 )
57 eqid 2441 . . . . . . . . . . . . . . . . . . 19  |-  ( Xp  oF  -  ( CC  X.  { z } ) )  =  ( Xp  oF  -  ( CC 
X.  { z } ) )
5857facth 21731 . . . . . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) ) )
6029oveq2i 6101 . . . . . . . . . . . . . . . . 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 2491 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) )
6261cnveqd 5011 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  `' F  =  `' (
( Xp  oF  -  ( CC 
X.  { z } ) )  oF  x.  Q ) )
6362imaeq1d 5165 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( `' F " { 0 } )  =  ( `' ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q )
" { 0 } ) )
6416, 63syl5eq 2485 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  R  =  ( `' ( ( Xp  oF  -  ( CC 
X.  { z } ) )  oF  x.  Q ) " { 0 } ) )
65 cnex 9359 . . . . . . . . . . . . . . 15  |-  CC  e.  _V
6665a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  CC  e.  _V )
6757plyremlem 21729 . . . . . . . . . . . . . . . . 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 995 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
Xp  oF  -  ( CC  X.  { z } ) )  e.  (Poly `  CC ) )
70 plyf 21625 . . . . . . . . . . . . . . 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 21625 . . . . . . . . . . . . . . 15  |-  ( Q  e.  (Poly `  CC )  ->  Q : CC --> CC )
7332, 72syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  Q : CC --> CC )
74 ofmulrt 21707 . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  ( `' ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q )
" { 0 } )  =  ( ( `' ( Xp  oF  -  ( CC  X.  { z } ) ) " {
0 } )  u.  ( `' Q " { 0 } ) ) )
7668simp3d 997 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( `' ( Xp  oF  -  ( CC  X.  { z } ) ) " {
0 } )  =  { z } )
7776uneq1d 3506 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( `' ( Xp  oF  -  ( CC  X.  { z } ) ) " { 0 } )  u.  ( `' Q " { 0 } ) )  =  ( { z }  u.  ( `' Q " { 0 } ) ) )
7864, 75, 773eqtrd 2477 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  R  =  ( { z }  u.  ( `' Q " { 0 } ) ) )
7978fveq2d 5692 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 R )  =  ( # `  ( { z }  u.  ( `' Q " { 0 } ) ) ) )
801, 2eqtr4d 2476 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  R
)  =  ( D  +  1 ) )
8180adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 R )  =  ( D  +  1 ) )
8279, 81eqtr3d 2475 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( { z }  u.  ( `' Q " { 0 } ) ) )  =  ( D  + 
1 ) )
8315adantr 462 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  F  =/=  0p )
8461, 83eqnetrrd 2626 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  (
( Xp  oF  -  ( CC 
X.  { z } ) )  oF  x.  Q )  =/=  0p )
85 plymul0or 21706 . . . . . . . . . . . . . . . . . . 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 656 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  (
( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q )  =  0p  <->  ( (
Xp  oF  -  ( CC  X.  { z } ) )  =  0p  \/  Q  =  0p ) ) )
8786necon3abid 2639 . . . . . . . . . . . . . . . . 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 2695 . . . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  Q  =/=  0p )
92 eqid 2441 . . . . . . . . . . . . . . 15  |-  ( `' Q " { 0 } )  =  ( `' Q " { 0 } )
9392fta1 21733 . . . . . . . . . . . . . 14  |-  ( ( Q  e.  (Poly `  CC )  /\  Q  =/=  0p )  -> 
( ( `' Q " { 0 } )  e.  Fin  /\  ( # `
 ( `' Q " { 0 } ) )  <_  (deg `  Q
) ) )
9432, 91, 93syl2anc 656 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( `' Q " { 0 } )  e.  Fin  /\  ( # `
 ( `' Q " { 0 } ) )  <_  (deg `  Q
) ) )
9594simprd 460 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  <_  (deg `  Q
) )
9695, 31breqtrrd 4315 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  <_  D )
97 snfi 7386 . . . . . . . . . . . . . 14  |-  { z }  e.  Fin
9894simpld 456 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( `' Q " { 0 } )  e.  Fin )
99 hashun2 12142 . . . . . . . . . . . . . 14  |-  ( ( { z }  e.  Fin  /\  ( `' Q " { 0 } )  e.  Fin )  -> 
( # `  ( { z }  u.  ( `' Q " { 0 } ) ) )  <_  ( ( # `  { z } )  +  ( # `  ( `' Q " { 0 } ) ) ) )
10097, 98, 99sylancr 658 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( { z }  u.  ( `' Q " { 0 } ) ) )  <_  ( ( # `  { z } )  +  ( # `  ( `' Q " { 0 } ) ) ) )
101 ax-1cn 9336 . . . . . . . . . . . . . . 15  |-  1  e.  CC
1023nncnd 10334 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  D  e.  CC )
103102adantr 462 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  D  e.  CC )
104 addcom 9551 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  CC  /\  D  e.  CC )  ->  ( 1  +  D
)  =  ( D  +  1 ) )
105101, 103, 104sylancr 658 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (
1  +  D )  =  ( D  + 
1 ) )
10682, 105eqtr4d 2476 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( { z }  u.  ( `' Q " { 0 } ) ) )  =  ( 1  +  D ) )
107 hashsng 12132 . . . . . . . . . . . . . . 15  |-  ( z  e.  R  ->  ( # `
 { z } )  =  1 )
108107adantl 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 { z } )  =  1 )
109108oveq1d 6105 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( # `  { z } )  +  (
# `  ( `' Q " { 0 } ) ) )  =  ( 1  +  (
# `  ( `' Q " { 0 } ) ) ) )
110100, 106, 1093brtr3d 4318 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
1  +  D )  <_  ( 1  +  ( # `  ( `' Q " { 0 } ) ) ) )
111 hashcl 12122 . . . . . . . . . . . . . . 15  |-  ( ( `' Q " { 0 } )  e.  Fin  ->  ( # `  ( `' Q " { 0 } ) )  e. 
NN0 )
11298, 111syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  e.  NN0 )
113112nn0red 10633 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  e.  RR )
114 1red 9397 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  1  e.  RR )
11536, 113, 114leadd2d 9930 . . . . . . . . . . . 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 9512 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (
( # `  ( `' Q " { 0 } ) )  =  D  <->  ( ( # `  ( `' Q " { 0 } ) )  <_  D  /\  D  <_  ( # `  ( `' Q " { 0 } ) ) ) ) )
11896, 116, 117mpbir2and 908 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  =  D )
11982, 118eqeq12d 2455 . . . . . . . . 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 2642 . . . . . . 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 3933 . . . . . 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 2485 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  ( { z }  i^i  ( `' Q " { 0 } ) )  =  (/) )
12619adantr 462 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  R  e.  Fin )
12749adantr 462 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  R  C_  CC )
128127sselda 3353 . . . 4  |-  ( ( ( ph  /\  z  e.  R )  /\  x  e.  R )  ->  x  e.  CC )
129125, 78, 126, 128fsumsplit 13212 . . 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 13213 . . . . . 6  |-  ( ( z  e.  CC  /\  z  e.  CC )  -> 
sum_ x  e.  { z } x  =  z )
13250, 50, 131syl2anc 656 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  { z } x  =  z )
13350negnegd 9706 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  -u -u z  =  z )
134132, 133eqtr4d 2476 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  { z } x  =  -u -u z )
135118, 31eqtrd 2473 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  =  (deg `  Q ) )
13628adantr 462 . . . . . . 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 5688 . . . . . . . . . . 11  |-  ( f  =  Q  ->  (deg `  f )  =  (deg
`  Q ) )
138137eqeq2d 2452 . . . . . . . . . 10  |-  ( f  =  Q  ->  ( D  =  (deg `  f
)  <->  D  =  (deg `  Q ) ) )
139 cnveq 5009 . . . . . . . . . . . . 13  |-  ( f  =  Q  ->  `' f  =  `' Q
)
140139imaeq1d 5165 . . . . . . . . . . . 12  |-  ( f  =  Q  ->  ( `' f " {
0 } )  =  ( `' Q " { 0 } ) )
141140fveq2d 5692 . . . . . . . . . . 11  |-  ( f  =  Q  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  ( `' Q " { 0 } ) ) )
142141, 137eqeq12d 2455 . . . . . . . . . 10  |-  ( f  =  Q  ->  (
( # `  ( `' f " { 0 } ) )  =  (deg `  f )  <->  (
# `  ( `' Q " { 0 } ) )  =  (deg
`  Q ) ) )
143138, 142anbi12d 705 . . . . . . . . 9  |-  ( f  =  Q  ->  (
( D  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( D  =  (deg `  Q )  /\  ( # `  ( `' Q " { 0 } ) )  =  (deg `  Q )
) ) )
144140sumeq1d 13174 . . . . . . . . . 10  |-  ( f  =  Q  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  sum_ x  e.  ( `' Q " { 0 } ) x )
145 fveq2 5688 . . . . . . . . . . . . 13  |-  ( f  =  Q  ->  (coeff `  f )  =  (coeff `  Q ) )
146137oveq1d 6105 . . . . . . . . . . . . 13  |-  ( f  =  Q  ->  (
(deg `  f )  -  1 )  =  ( (deg `  Q
)  -  1 ) )
147145, 146fveq12d 5694 . . . . . . . . . . . 12  |-  ( f  =  Q  ->  (
(coeff `  f ) `  ( (deg `  f
)  -  1 ) )  =  ( (coeff `  Q ) `  (
(deg `  Q )  -  1 ) ) )
148145, 137fveq12d 5694 . . . . . . . . . . . 12  |-  ( f  =  Q  ->  (
(coeff `  f ) `  (deg `  f )
)  =  ( (coeff `  Q ) `  (deg `  Q ) ) )
149147, 148oveq12d 6108 . . . . . . . . . . 11  |-  ( f  =  Q  ->  (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) )  =  ( ( (coeff `  Q
) `  ( (deg `  Q )  -  1 ) )  /  (
(coeff `  Q ) `  (deg `  Q )
) ) )
150149negeqd 9600 . . . . . . . . . 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 2455 . . . . . . . . 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 3066 . . . . . . 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 674 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  ( `' Q " { 0 } ) x  =  -u (
( (coeff `  Q
) `  ( (deg `  Q )  -  1 ) )  /  (
(coeff `  Q ) `  (deg `  Q )
) ) )
15631oveq1d 6105 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  ( D  -  1 )  =  ( (deg `  Q )  -  1 ) )
157156fveq2d 5692 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( D  -  1 ) )  =  ( (coeff `  Q ) `  ( (deg `  Q
)  -  1 ) ) )
15861fveq2d 5692 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  F )  =  (coeff `  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) )
15927, 158syl5eq 2485 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  A  =  (coeff `  ( (
Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) )
16061fveq2d 5692 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  F )  =  (deg
`  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) )
16168simp2d 996 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  1 )
162 ax-1ne0 9347 . . . . . . . . . . . . . . 15  |-  1  =/=  0
163162a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  1  =/=  0 )
164161, 163eqnetrd 2624 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =/=  0 )
165 fveq2 5688 . . . . . . . . . . . . . . 15  |-  ( ( Xp  oF  -  ( CC  X.  { z } ) )  =  0p  ->  (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) )  =  (deg
`  0p ) )
166165, 12syl6eq 2489 . . . . . . . . . . . . . 14  |-  ( ( Xp  oF  -  ( CC  X.  { z } ) )  =  0p  ->  (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) )  =  0 )
167166necon3i 2648 . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . 13  |-  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  (deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) )
170 eqid 2441 . . . . . . . . . . . . 13  |-  (deg `  Q )  =  (deg
`  Q )
171169, 170dgrmul 21696 . . . . . . . . . . . 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 1214 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) )  =  ( (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  +  (deg `  Q )
) )
173160, 172eqtrd 2473 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  F )  =  ( (deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) )  +  (deg `  Q )
) )
1749, 173syl5eq 2485 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  N  =  ( (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  +  (deg `  Q )
) )
175159, 174fveq12d 5694 . . . . . . . 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 2441 . . . . . . . . . 10  |-  (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  (coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) )
177 eqid 2441 . . . . . . . . . 10  |-  (coeff `  Q )  =  (coeff `  Q )
178176, 177, 169, 170coemulhi 21680 . . . . . . . . 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 656 . . . . . . . 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 5692 . . . . . . . . . . 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 3372 . . . . . . . . . . . . . . 15  |-  CC  C_  CC
182 plyid 21636 . . . . . . . . . . . . . . 15  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  Xp  e.  (Poly `  CC ) )
183181, 101, 182mp2an 667 . . . . . . . . . . . . . 14  |-  Xp  e.  (Poly `  CC )
184 plyconst 21633 . . . . . . . . . . . . . . 15  |-  ( ( CC  C_  CC  /\  z  e.  CC )  ->  ( CC  X.  { z } )  e.  (Poly `  CC ) )
185181, 50, 184sylancr 658 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( CC  X.  { z } )  e.  (Poly `  CC ) )
186 eqid 2441 . . . . . . . . . . . . . . 15  |-  (coeff `  Xp )  =  (coeff `  Xp
)
187 eqid 2441 . . . . . . . . . . . . . . 15  |-  (coeff `  ( CC  X.  { z } ) )  =  (coeff `  ( CC  X.  { z } ) )
188186, 187coesub 21683 . . . . . . . . . . . . . 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 658 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  ( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) )
190189fveq1d 5690 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) ` 
1 )  =  ( ( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
1 ) )
191 1nn0 10591 . . . . . . . . . . . . . 14  |-  1  e.  NN0
192186coef3 21659 . . . . . . . . . . . . . . . . 17  |-  ( Xp  e.  (Poly `  CC )  ->  (coeff `  Xp ) : NN0 --> CC )
193 ffn 5556 . . . . . . . . . . . . . . . . 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 21659 . . . . . . . . . . . . . . . 16  |-  ( ( CC  X.  { z } )  e.  (Poly `  CC )  ->  (coeff `  ( CC  X.  {
z } ) ) : NN0 --> CC )
197 ffn 5556 . . . . . . . . . . . . . . . 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 10581 . . . . . . . . . . . . . . . 16  |-  NN0  e.  _V
200199a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  NN0  e.  _V )
201 inidm 3556 . . . . . . . . . . . . . . 15  |-  ( NN0 
i^i  NN0 )  =  NN0
202 coeidp 21689 . . . . . . . . . . . . . . . . 17  |-  ( 1  e.  NN0  ->  ( (coeff `  Xp ) ` 
1 )  =  if ( 1  =  1 ,  1 ,  0 ) )
203202adantl 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
(coeff `  Xp
) `  1 )  =  if ( 1  =  1 ,  1 ,  0 ) )
204 eqid 2441 . . . . . . . . . . . . . . . . 17  |-  1  =  1
205204iftruei 3795 . . . . . . . . . . . . . . . 16  |-  if ( 1  =  1 ,  1 ,  0 )  =  1
206203, 205syl6eq 2489 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
(coeff `  Xp
) `  1 )  =  1 )
207 0lt1 9858 . . . . . . . . . . . . . . . . . 18  |-  0  <  1
208 0re 9382 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR
209 1re 9381 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  RR
210208, 209ltnlei 9491 . . . . . . . . . . . . . . . . . 18  |-  ( 0  <  1  <->  -.  1  <_  0 )
211207, 210mpbi 208 . . . . . . . . . . . . . . . . 17  |-  -.  1  <_  0
21250adantr 462 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  z  e.  CC )
213 0dgr 21672 . . . . . . . . . . . . . . . . . . 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 4301 . . . . . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . . . . . . . 20  |-  (deg `  ( CC  X.  { z } ) )  =  (deg `  ( CC  X.  { z } ) )
218187, 217dgrub 21661 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( CC  X.  {
z } )  e.  (Poly `  CC )  /\  1  e.  NN0  /\  ( (coeff `  ( CC  X.  { z } ) ) `  1
)  =/=  0 )  ->  1  <_  (deg `  ( CC  X.  {
z } ) ) )
2192183expia 1184 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( CC  X.  {
z } )  e.  (Poly `  CC )  /\  1  e.  NN0 )  ->  ( ( (coeff `  ( CC  X.  {
z } ) ) `
 1 )  =/=  0  ->  1  <_  (deg
`  ( CC  X.  { z } ) ) ) )
220185, 219sylan 468 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
( (coeff `  ( CC  X.  { z } ) ) `  1
)  =/=  0  -> 
1  <_  (deg `  ( CC  X.  { z } ) ) ) )
221220necon1bd 2677 . . . . . . . . . . . . . . . 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 6328 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
1 )  =  ( 1  -  0 ) )
224191, 223mpan2 666 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
1 )  =  ( 1  -  0 ) )
225 1m0e1 10428 . . . . . . . . . . . . 13  |-  ( 1  -  0 )  =  1
226224, 225syl6eq 2489 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
1 )  =  1 )
227190, 226eqtrd 2473 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) ` 
1 )  =  1 )
228180, 227eqtrd 2473 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  (deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) )  =  1 )
229228oveq1d 6105 . . . . . . . . 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 21659 . . . . . . . . . . . 12  |-  ( Q  e.  (Poly `  CC )  ->  (coeff `  Q
) : NN0 --> CC )
23132, 230syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  Q ) : NN0 --> CC )
232231, 34ffvelrnd 5841 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  (deg `  Q )
)  e.  CC )
233232mulid2d 9400 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  (
1  x.  ( (coeff `  Q ) `  (deg `  Q ) ) )  =  ( (coeff `  Q ) `  (deg `  Q ) ) )
234229, 233eqtrd 2473 . . . . . . . 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 2477 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( A `  N )  =  ( (coeff `  Q ) `  (deg `  Q ) ) )
236157, 235oveq12d 6108 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) )  =  ( ( (coeff `  Q ) `  (
(deg `  Q )  -  1 ) )  /  ( (coeff `  Q ) `  (deg `  Q ) ) ) )
237236negeqd 9600 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  -u (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) )  = 
-u ( ( (coeff `  Q ) `  (
(deg `  Q )  -  1 ) )  /  ( (coeff `  Q ) `  (deg `  Q ) ) ) )
238155, 237eqtr4d 2476 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  ( `' Q " { 0 } ) x  =  -u (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) ) )
239134, 238oveq12d 6108 . . 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 9702 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  -u z  e.  CC )
241 nnm1nn0 10617 . . . . . . . . 9  |-  ( D  e.  NN  ->  ( D  -  1 )  e.  NN0 )
2423, 241syl 16 . . . . . . . 8  |-  ( ph  ->  ( D  -  1 )  e.  NN0 )
243242adantr 462 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( D  -  1 )  e.  NN0 )
244231, 243ffvelrnd 5841 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( D  -  1 ) )  e.  CC )
245235, 232eqeltrd 2515 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  ( A `  N )  e.  CC )
2469, 27dgreq0 21691 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0p  <->  ( A `  N )  =  0 ) )
24743, 246syl 16 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  ( F  =  0p  <->  ( A `  N )  =  0 ) )
248247necon3bid 2641 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( F  =/=  0p  <->  ( A `  N )  =/=  0
) )
24983, 248mpbid 210 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  ( A `  N )  =/=  0 )
250244, 245, 249divcld 10103 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) )  e.  CC )
251240, 250negdid 9728 . . . 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 9402 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( -u z  x.  ( A `
 N ) )  e.  CC )
253252, 244, 245, 249divdird 10141 . . . . . 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 10617 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
2555, 254syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( N  -  1 )  e.  NN0 )
256255adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  ( N  -  1 )  e.  NN0 )
257176, 177coemul 21678 . . . . . . . . 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 1213 . . . . . . . 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 5690 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  ( A `  ( N  -  1 ) )  =  ( (coeff `  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) `
 ( N  - 
1 ) ) )
260 1e0p1 10779 . . . . . . . . . . . 12  |-  1  =  ( 0  +  1 )
261260oveq2i 6101 . . . . . . . . . . 11  |-  ( 0 ... 1 )  =  ( 0 ... (
0  +  1 ) )
262261sumeq1i 13171 . . . . . . . . . 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 10590 . . . . . . . . . . . . 13  |-  0  e.  NN0
264 nn0uz 10891 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
265263, 264eleqtri 2513 . . . . . . . . . . . 12  |-  0  e.  ( ZZ>= `  0 )
266265a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  0  e.  ( ZZ>= `  0 )
)
267261eleq2i 2505 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... 1 )  <->  k  e.  ( 0 ... (
0  +  1 ) ) )
268176coef3 21659 . . . . . . . . . . . . . . 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 11477 . . . . . . . . . . . . . 14  |-  ( k  e.  ( 0 ... 1 )  ->  k  e.  NN0 )
271 ffvelrn 5838 . . . . . . . . . . . . . 14  |-  ( ( (coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) : NN0 --> CC  /\  k  e.  NN0 )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  k )  e.  CC )
272269, 270, 271syl2an 474 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... 1
) )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  k )  e.  CC )
2732oveq1d 6105 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( D  + 
1 )  -  1 )  =  ( N  -  1 ) )
274 pncan 9612 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( D  e.  CC  /\  1  e.  CC )  ->  ( ( D  + 
1 )  -  1 )  =  D )
275102, 101, 274sylancl 657 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( D  + 
1 )  -  1 )  =  D )
276273, 275eqtr3d 2475 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( N  -  1 )  =  D )
277276adantr 462 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  ( N  -  1 )  =  D )
2783adantr 462 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  D  e.  NN )
279277, 278eqeltrd 2515 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  ( N  -  1 )  e.  NN )
280 nnuz 10892 . . . . . . . . . . . . . . . . 17  |-  NN  =  ( ZZ>= `  1 )
281279, 280syl6eleq 2531 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  ( N  -  1 )  e.  ( ZZ>= `  1
) )
282 fzss2 11494 . . . . . . . . . . . . . . . 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 3353 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... 1
) )  ->  k  e.  ( 0 ... ( N  -  1 ) ) )
285 fznn0sub 11483 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  (
( N  -  1 )  -  k )  e.  NN0 )
286 ffvelrn 5838 . . . . . . . . . . . . . . 15  |-  ( ( (coeff `  Q ) : NN0 --> CC  /\  (
( N  -  1 )  -  k )  e.  NN0 )  -> 
( (coeff `  Q
) `  ( ( N  -  1 )  -  k ) )  e.  CC )
287231, 285, 286syl2an 474 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) )  e.  CC )
288284, 287syldan 467 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... 1
) )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) )  e.  CC )
289272, 288mulcld 9402 . . . . . . . . . . . 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 473 . . . . . . . . . . 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 2491 . . . . . . . . . . . . 13  |-  ( k  =  ( 0  +  1 )  ->  k  =  1 )
293292fveq2d 5692 . . . . . . . . . . . 12  |-  ( k  =  ( 0  +  1 )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  k )  =  ( (coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) ` 
1 ) )
294292oveq2d 6106 . . . . . . . . . . . . 13  |-  ( k  =  ( 0  +  1 )  ->  (
( N  -  1 )  -  k )  =  ( ( N  -  1 )  - 
1 ) )
295294fveq2d 5692 . . . . . . . . . . . 12  |-  ( k  =  ( 0  +  1 )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) )  =  ( (coeff `  Q ) `  ( ( N  - 
1 )  -  1 ) ) )
296293, 295oveq12d 6108 . . . . . . . . . . 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 13219 . . . . . . . . . 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 2485 . . . . . . . . 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 3476 . . . . . . . . . . . . . 14  |-  ( k  e.  ( ( 0 ... ( N  - 
1 ) )  \ 
( 0 ... 1
) )  ->  -.  k  e.  ( 0 ... 1 ) )
300299adantl 463 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  -.  k  e.  ( 0 ... 1
) )
301 eldifi 3475 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( ( 0 ... ( N  - 
1 ) )  \ 
( 0 ... 1
) )  ->  k  e.  ( 0 ... ( N  -  1 ) ) )
302 elfznn0 11477 . . . . . . . . . . . . . . . . 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 21661 . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . 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 11445 . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  k  e.  ( ZZ>= `  0 )
)
310 1z 10672 . . . . . . . . . . . . . . . . 17  |-  1  e.  ZZ
311 elfz5 11441 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  ( ZZ>= ` 
0 )  /\  1  e.  ZZ )  ->  (
k  e.  ( 0 ... 1 )  <->  k  <_  1 ) )
312309, 310, 311sylancl 657 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( k  e.  ( 0 ... 1
)  <->  k  <_  1
) )
313161breq2d 4301 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  (
k  <_  (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) )  <->  k  <_  1 ) )
314313adantr 462 . . . . . . . . . . . . . . . 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 2677 . . . . . . . . . . . . 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 6105 . . . . . . . . . . 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 471 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) )  e.  CC )
321320mul02d 9563 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( 0  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  =  0 )
322319, 321eqtrd 2473 . . . . . . . . . 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 11791 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (
0 ... ( N  - 
1 ) )  e. 
Fin )
324283, 289, 322, 323fsumss 13198 . . . . . . . . 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 10653 . . . . . . . . . . . 12  |-  0  e.  ZZ
326189fveq1d 5690 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) ` 
0 )  =  ( ( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
0 ) )
327 coeidp 21689 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0  e.  NN0  ->  ( (coeff `  Xp ) ` 
0 )  =  if ( 0  =  1 ,  1 ,  0 ) )
328162nesymi 2646 . . . . . . . . . . . . . . . . . . . . 21  |-  -.  0  =  1
329328iffalsei 3797 . . . . . . . . . . . . . . . . . . . 20  |-  if ( 0  =  1 ,  1 ,  0 )  =  0
330327, 329syl6eq 2489 . . . . . . . . . . . . . . . . . . 19  |-  ( 0  e.  NN0  ->  ( (coeff `  Xp ) ` 
0 )  =  0 )
331330adantl 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  R )  /\  0  e.  NN0 )  ->  (
(coeff `  Xp
) `  0 )  =  0 )
332 0cn 9374 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  CC
333 vex 2973 . . . . . . . . . . . . . . . . . . . . . 22  |-  z  e. 
_V
334333fvconst2 5930 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0  e.  CC  ->  (
( CC  X.  {
z } ) ` 
0 )  =  z )
335332, 334ax-mp 5 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( CC  X.  { z } ) `  0
)  =  z
336187coefv0 21674 . . . . . . . . . . . . . . . . . . . . 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 2488 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( CC  X.  { z } ) ) `  0 )  =  z )
339338adantr 462 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  R )  /\  0  e.  NN0 )  ->  (
(coeff `  ( CC  X.  { z } ) ) `  0 )  =  z )
340195, 198, 200, 200, 201, 331, 339ofval 6328 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  R )  /\  0  e.  NN0 )  ->  (
( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
0 )  =  ( 0  -  z ) )
341263, 340mpan2 666 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
0 )  =  ( 0  -  z ) )
342 df-neg 9594 . . . . . . . . . . . . . . . 16  |-  -u z  =  ( 0  -  z )
343341, 342syl6eqr 2491 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  Xp )  oF  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
0 )  =  -u z )
344326, 343eqtrd 2473 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) ` 
0 )  =  -u z )
345277oveq1d 6105 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  (
( N  -  1 )  -  0 )  =  ( D  - 
0 ) )
346103subid1d 9704 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  ( D  -  0 )  =  D )
347345, 346, 313eqtrd 2477 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  (
( N  -  1 )  -  0 )  =  (deg `  Q
) )
348347fveq2d 5692 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  0 ) )  =  ( (coeff `  Q ) `  (deg `  Q )
) )
349348, 235eqtr4d 2476 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  0 ) )  =  ( A `  N ) )
350344, 349oveq12d 6108 . . . . . . . . . . . . 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 2515 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  0
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  0 ) ) )  e.  CC )
352 fveq2 5688 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
(coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) `  k )  =  ( (coeff `  ( Xp  oF  -  ( CC  X.  { z } ) ) ) ` 
0 ) )
353 oveq2 6098 . . . . . . . . . . . . . . 15  |-  ( k  =  0  ->  (
( N  -  1 )  -  k )  =  ( ( N  -  1 )  - 
0 ) )
354353fveq2d 5692 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) )  =  ( (coeff `  Q ) `  ( ( N  - 
1 )  -  0 ) ) )
355352, 354oveq12d 6108 . . . . . . . . . . . . 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 13214 . . . . . . . . . . . 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 658 . . . . . . . . . . 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 2473 . . . . . . . . . 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 6105 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( N  -  1 )  -  1 )  =  ( D  - 
1 ) )
360359fveq2d 5692 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  1 ) )  =  ( (coeff `  Q ) `  ( D  -  1 ) ) )
361227, 360oveq12d 6108 . . . . . . . . . . 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 9400 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (
1  x.  ( (coeff `  Q ) `  ( D  -  1 ) ) )  =  ( (coeff `  Q ) `  ( D  -  1 ) ) )
363361, 362eqtrd 2473 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  (
Xp  oF  -  ( CC  X.  { z } ) ) ) `  1
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  1 ) ) )  =  ( (coeff `  Q ) `  ( D  -  1 ) ) )
364358, 363oveq12d 6108 . . . . . . . . 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 2482 . . . . . . . 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 2484 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  (
( -u z  x.  ( A `  N )
)  +  ( (coeff `  Q ) `  ( D  -  1 ) ) )  =  ( A `  ( N  -  1 ) ) )
367366oveq1d 6105 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (
( ( -u z  x.  ( A `  N
) )  +  ( (coeff `  Q ) `  ( D  -  1 ) ) )  / 
( A `  N
) )  =  ( ( A `  ( N  -  1 ) )  /  ( A `
 N ) ) )
368240, 245, 249divcan4d 10109 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  (
( -u z  x.  ( A `  N )
)  /  ( A `
 N ) )  =  -u z )
369368oveq1d 6105 . . . . . 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 2482 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  ( -u z  +  ( ( (coeff `  Q ) `  ( D  -  1 ) )  /  ( A `  N )
) )  =  ( ( A `  ( N  -  1 ) )  /  ( A `
 N ) ) )
371370negeqd 9600 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  -u ( -u z  +  ( ( (coeff `  Q ) `  ( D  -  1 ) )  /  ( A `  N )
) )  =  -u ( ( A `  ( N  -  1
) )  /  ( A `  N )
) )
372251, 371eqtr3d 2475 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  ( -u -u z  +  -u (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) ) )  =  -u ( ( A `
 ( N  - 
1 ) )  / 
( A `  N
) ) )
373129, 239, 3723eqtrd 2477 . 2  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  R  x  =  -u ( ( A `  ( N  -  1
) )  /  ( A `  N )
) )
37425, 373exlimddv 1697 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 960    = wceq 1364   E.wex 1591    e. wcel 1761    =/= wne 2604   A.wral 2713   _Vcvv 2970    \ cdif 3322    u. cun 3323    i^i cin 3324    C_ wss 3325   (/)c0 3634   ifcif 3788   {csn 3874   class class class wbr 4289    X. cxp 4834   `'ccnv 4835   dom cdm 4836   "cima 4839    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   Fincfn 7306   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    < clt 9414    <_ cle 9415    - cmin 9591   -ucneg 9592    / cdiv 9989   NNcn 10318   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433   #chash 12099   sum_csu 13159   0pc0p 21106  Polycply 21611   Xpcidp 21612  coeffccoe 21613  degcdgr 21614   quot cquot 21715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-0p 21107  df-ply 21615  df-idp 21616  df-coe 21617  df-dgr 21618  df-quot 21716
This theorem is referenced by:  vieta1  21737
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