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Theorem vieta1lem2 20181
Description: Lemma for vieta1 20182: inductive step. Let  z be a root of  F. Then  F  =  ( X p  -  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 `  X p  -  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  ( X p  o F  -  ( 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 9973 . . . . . . 7  |-  ( ph  ->  ( D  +  1 )  e.  NN )
52, 4eqeltrrd 2479 . . . . . 6  |-  ( ph  ->  N  e.  NN )
65nnne0d 10000 . . . . 5  |-  ( ph  ->  N  =/=  0 )
71, 6eqnetrd 2585 . . . 4  |-  ( ph  ->  ( # `  R
)  =/=  0 )
8 vieta1.4 . . . . . . . 8  |-  ( ph  ->  F  e.  (Poly `  S ) )
9 vieta1.2 . . . . . . . . . 10  |-  N  =  (deg `  F )
109, 6syl5eqner 2592 . . . . . . . . 9  |-  ( ph  ->  (deg `  F )  =/=  0 )
11 fveq2 5687 . . . . . . . . . . 11  |-  ( F  =  0 p  -> 
(deg `  F )  =  (deg `  0 p
) )
12 dgr0 20133 . . . . . . . . . . 11  |-  (deg ` 
0 p )  =  0
1311, 12syl6eq 2452 . . . . . . . . . 10  |-  ( F  =  0 p  -> 
(deg `  F )  =  0 )
1413necon3i 2606 . . . . . . . . 9  |-  ( (deg
`  F )  =/=  0  ->  F  =/=  0 p )
1510, 14syl 16 . . . . . . . 8  |-  ( ph  ->  F  =/=  0 p )
16 vieta1.3 . . . . . . . . 9  |-  R  =  ( `' F " { 0 } )
1716fta1 20178 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  F  =/=  0 p )  -> 
( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) )
188, 15, 17syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( R  e.  Fin  /\  ( # `  R
)  <_  (deg `  F
) ) )
1918simpld 446 . . . . . 6  |-  ( ph  ->  R  e.  Fin )
20 hasheq0 11599 . . . . . 6  |-  ( R  e.  Fin  ->  (
( # `  R )  =  0  <->  R  =  (/) ) )
2119, 20syl 16 . . . . 5  |-  ( ph  ->  ( ( # `  R
)  =  0  <->  R  =  (/) ) )
2221necon3bid 2602 . . . 4  |-  ( ph  ->  ( ( # `  R
)  =/=  0  <->  R  =/=  (/) ) )
237, 22mpbid 202 . . 3  |-  ( ph  ->  R  =/=  (/) )
24 n0 3597 . . 3  |-  ( R  =/=  (/)  <->  E. z  z  e.  R )
2523, 24sylib 189 . 2  |-  ( ph  ->  E. z  z  e.  R )
26 incom 3493 . . . . 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  ( X p  o F  -  ( CC  X.  { z } ) ) )
3027, 9, 16, 8, 1, 3, 2, 28, 29vieta1lem1 20180 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  ( Q  e.  (Poly `  CC )  /\  D  =  (deg
`  Q ) ) )
3130simprd 450 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  D  =  (deg `  Q )
)
3230simpld 446 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  Q  e.  (Poly `  CC )
)
33 dgrcl 20105 . . . . . . . . . . 11  |-  ( Q  e.  (Poly `  CC )  ->  (deg `  Q
)  e.  NN0 )
3432, 33syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  Q )  e.  NN0 )
3534nn0red 10231 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  Q )  e.  RR )
3631, 35eqeltrd 2478 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  D  e.  RR )
3736ltp1d 9897 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  D  <  ( D  +  1 ) )
3836, 37gtned 9164 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( D  +  1 )  =/=  D )
39 snssi 3902 . . . . . . . . . . 11  |-  ( z  e.  ( `' Q " { 0 } )  ->  { z } 
C_  ( `' Q " { 0 } ) )
40 ssequn1 3477 . . . . . . . . . . 11  |-  ( { z }  C_  ( `' Q " { 0 } )  <->  ( {
z }  u.  ( `' Q " { 0 } ) )  =  ( `' Q " { 0 } ) )
4139, 40sylib 189 . . . . . . . . . 10  |-  ( z  e.  ( `' Q " { 0 } )  ->  ( { z }  u.  ( `' Q " { 0 } ) )  =  ( `' Q " { 0 } ) )
4241fveq2d 5691 . . . . . . . . 9  |-  ( z  e.  ( `' Q " { 0 } )  ->  ( # `  ( { z }  u.  ( `' Q " { 0 } ) ) )  =  ( # `  ( `' Q " { 0 } ) ) )
438adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  F  e.  (Poly `  S )
)
44 cnvimass 5183 . . . . . . . . . . . . . . . . . . . . 21  |-  ( `' F " { 0 } )  C_  dom  F
4516, 44eqsstri 3338 . . . . . . . . . . . . . . . . . . . 20  |-  R  C_  dom  F
46 plyf 20070 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
47 fdm 5554 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F : CC --> CC  ->  dom 
F  =  CC )
488, 46, 473syl 19 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  dom  F  =  CC )
4945, 48syl5sseq 3356 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  R  C_  CC )
5049sselda 3308 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  z  e.  CC )
5116eleq2i 2468 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  e.  R  <->  z  e.  ( `' F " { 0 } ) )
528, 46syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  F : CC --> CC )
53 ffn 5550 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F : CC --> CC  ->  F  Fn  CC )
54 fniniseg 5810 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  Fn  CC  ->  (
z  e.  ( `' F " { 0 } )  <->  ( z  e.  CC  /\  ( F `
 z )  =  0 ) ) )
5552, 53, 543syl 19 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( z  e.  ( `' F " { 0 } )  <->  ( z  e.  CC  /\  ( F `
 z )  =  0 ) ) )
5651, 55syl5bb 249 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( z  e.  R  <->  ( z  e.  CC  /\  ( F `  z )  =  0 ) ) )
5756simplbda 608 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  ( F `  z )  =  0 )
58 eqid 2404 . . . . . . . . . . . . . . . . . . 19  |-  ( X p  o F  -  ( CC  X.  { z } ) )  =  ( X p  o F  -  ( CC  X.  { z } ) )
5958facth 20176 . . . . . . . . . . . . . . . . . 18  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC  /\  ( F `
 z )  =  0 )  ->  F  =  ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { z } ) ) ) ) )
6043, 50, 57, 59syl3anc 1184 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  F  =  ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { z } ) ) ) ) )
6129oveq2i 6051 . . . . . . . . . . . . . . . . 17  |-  ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q )  =  ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  ( F quot  ( X p  o F  -  ( CC  X.  { z } ) ) ) )
6260, 61syl6eqr 2454 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  F  =  ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) )
6362cnveqd 5007 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  `' F  =  `' (
( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) )
6463imaeq1d 5161 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( `' F " { 0 } )  =  ( `' ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q )
" { 0 } ) )
6516, 64syl5eq 2448 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  R  =  ( `' ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) " {
0 } ) )
66 cnex 9027 . . . . . . . . . . . . . . 15  |-  CC  e.  _V
6766a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  CC  e.  _V )
6858plyremlem 20174 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  CC  ->  (
( X p  o F  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  /\  (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) )  =  1  /\  ( `' ( X p  o F  -  ( CC  X.  { z } ) ) " { 0 } )  =  {
z } ) )
6950, 68syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  (
( X p  o F  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  /\  (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) )  =  1  /\  ( `' ( X p  o F  -  ( CC  X.  { z } ) ) " { 0 } )  =  {
z } ) )
7069simp1d 969 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
X p  o F  -  ( CC  X.  { z } ) )  e.  (Poly `  CC ) )
71 plyf 20070 . . . . . . . . . . . . . . 15  |-  ( ( X p  o F  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  ->  ( X p  o F  -  ( CC  X.  { z } ) ) : CC --> CC )
7270, 71syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (
X p  o F  -  ( CC  X.  { z } ) ) : CC --> CC )
73 plyf 20070 . . . . . . . . . . . . . . 15  |-  ( Q  e.  (Poly `  CC )  ->  Q : CC --> CC )
7432, 73syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  Q : CC --> CC )
75 ofmulrt 20152 . . . . . . . . . . . . . 14  |-  ( ( CC  e.  _V  /\  ( X p  o F  -  ( CC  X.  { z } ) ) : CC --> CC  /\  Q : CC --> CC )  ->  ( `' ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) " {
0 } )  =  ( ( `' ( X p  o F  -  ( CC  X.  { z } ) ) " { 0 } )  u.  ( `' Q " { 0 } ) ) )
7667, 72, 74, 75syl3anc 1184 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  ( `' ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q )
" { 0 } )  =  ( ( `' ( X p  o F  -  ( CC  X.  { z } ) ) " {
0 } )  u.  ( `' Q " { 0 } ) ) )
7769simp3d 971 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( `' ( X p  o F  -  ( CC  X.  { z } ) ) " {
0 } )  =  { z } )
7877uneq1d 3460 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( `' ( X p  o F  -  ( CC  X.  { z } ) ) " { 0 } )  u.  ( `' Q " { 0 } ) )  =  ( { z }  u.  ( `' Q " { 0 } ) ) )
7965, 76, 783eqtrd 2440 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  R  =  ( { z }  u.  ( `' Q " { 0 } ) ) )
8079fveq2d 5691 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 R )  =  ( # `  ( { z }  u.  ( `' Q " { 0 } ) ) ) )
811, 2eqtr4d 2439 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  R
)  =  ( D  +  1 ) )
8281adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 R )  =  ( D  +  1 ) )
8380, 82eqtr3d 2438 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( { z }  u.  ( `' Q " { 0 } ) ) )  =  ( D  + 
1 ) )
8415adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  F  =/=  0 p )
8562, 84eqnetrrd 2587 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  (
( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q )  =/=  0 p )
86 plymul0or 20151 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( X p  o F  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  /\  Q  e.  (Poly `  CC )
)  ->  ( (
( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q )  =  0 p  <->  ( ( X p  o F  -  ( CC  X.  { z } ) )  =  0 p  \/  Q  =  0 p ) ) )
8770, 32, 86syl2anc 643 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  (
( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q )  =  0 p  <->  ( (
X p  o F  -  ( CC  X.  { z } ) )  =  0 p  \/  Q  =  0 p ) ) )
8887necon3abid 2600 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  (
( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q )  =/=  0 p  <->  -.  (
( X p  o F  -  ( CC  X.  { z } ) )  =  0 p  \/  Q  =  0 p ) ) )
8985, 88mpbid 202 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  -.  ( ( X p  o F  -  ( CC  X.  { z } ) )  =  0 p  \/  Q  =  0 p ) )
90 neanior 2652 . . . . . . . . . . . . . . . 16  |-  ( ( ( X p  o F  -  ( CC  X.  { z } ) )  =/=  0 p  /\  Q  =/=  0 p )  <->  -.  (
( X p  o F  -  ( CC  X.  { z } ) )  =  0 p  \/  Q  =  0 p ) )
9189, 90sylibr 204 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
( X p  o F  -  ( CC  X.  { z } ) )  =/=  0 p  /\  Q  =/=  0 p ) )
9291simprd 450 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  Q  =/=  0 p )
93 eqid 2404 . . . . . . . . . . . . . . 15  |-  ( `' Q " { 0 } )  =  ( `' Q " { 0 } )
9493fta1 20178 . . . . . . . . . . . . . 14  |-  ( ( Q  e.  (Poly `  CC )  /\  Q  =/=  0 p )  -> 
( ( `' Q " { 0 } )  e.  Fin  /\  ( # `
 ( `' Q " { 0 } ) )  <_  (deg `  Q
) ) )
9532, 92, 94syl2anc 643 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( `' Q " { 0 } )  e.  Fin  /\  ( # `
 ( `' Q " { 0 } ) )  <_  (deg `  Q
) ) )
9695simprd 450 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  <_  (deg `  Q
) )
9796, 31breqtrrd 4198 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  <_  D )
98 snfi 7146 . . . . . . . . . . . . . 14  |-  { z }  e.  Fin
9995simpld 446 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( `' Q " { 0 } )  e.  Fin )
100 hashun2 11612 . . . . . . . . . . . . . 14  |-  ( ( { z }  e.  Fin  /\  ( `' Q " { 0 } )  e.  Fin )  -> 
( # `  ( { z }  u.  ( `' Q " { 0 } ) ) )  <_  ( ( # `  { z } )  +  ( # `  ( `' Q " { 0 } ) ) ) )
10198, 99, 100sylancr 645 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( { z }  u.  ( `' Q " { 0 } ) ) )  <_  ( ( # `  { z } )  +  ( # `  ( `' Q " { 0 } ) ) ) )
102 ax-1cn 9004 . . . . . . . . . . . . . . 15  |-  1  e.  CC
1033nncnd 9972 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  D  e.  CC )
104103adantr 452 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  D  e.  CC )
105 addcom 9208 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  CC  /\  D  e.  CC )  ->  ( 1  +  D
)  =  ( D  +  1 ) )
106102, 104, 105sylancr 645 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (
1  +  D )  =  ( D  + 
1 ) )
10783, 106eqtr4d 2439 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( { z }  u.  ( `' Q " { 0 } ) ) )  =  ( 1  +  D ) )
108 hashsng 11602 . . . . . . . . . . . . . . 15  |-  ( z  e.  R  ->  ( # `
 { z } )  =  1 )
109108adantl 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 { z } )  =  1 )
110109oveq1d 6055 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( # `  { z } )  +  (
# `  ( `' Q " { 0 } ) ) )  =  ( 1  +  (
# `  ( `' Q " { 0 } ) ) ) )
111101, 107, 1103brtr3d 4201 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
1  +  D )  <_  ( 1  +  ( # `  ( `' Q " { 0 } ) ) ) )
112 hashcl 11594 . . . . . . . . . . . . . . 15  |-  ( ( `' Q " { 0 } )  e.  Fin  ->  ( # `  ( `' Q " { 0 } ) )  e. 
NN0 )
11399, 112syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  e.  NN0 )
114113nn0red 10231 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  e.  RR )
115 1re 9046 . . . . . . . . . . . . . 14  |-  1  e.  RR
116115a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  1  e.  RR )
11736, 114, 116leadd2d 9577 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  ( D  <_  ( # `  ( `' Q " { 0 } ) )  <->  ( 1  +  D )  <_ 
( 1  +  (
# `  ( `' Q " { 0 } ) ) ) ) )
118111, 117mpbird 224 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  D  <_  ( # `  ( `' Q " { 0 } ) ) )
119114, 36letri3d 9171 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (
( # `  ( `' Q " { 0 } ) )  =  D  <->  ( ( # `  ( `' Q " { 0 } ) )  <_  D  /\  D  <_  ( # `  ( `' Q " { 0 } ) ) ) ) )
12097, 118, 119mpbir2and 889 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  =  D )
12183, 120eqeq12d 2418 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  (
( # `  ( { z }  u.  ( `' Q " { 0 } ) ) )  =  ( # `  ( `' Q " { 0 } ) )  <->  ( D  +  1 )  =  D ) )
12242, 121syl5ib 211 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  (
z  e.  ( `' Q " { 0 } )  ->  ( D  +  1 )  =  D ) )
123122necon3ad 2603 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  (
( D  +  1 )  =/=  D  ->  -.  z  e.  ( `' Q " { 0 } ) ) )
12438, 123mpd 15 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  -.  z  e.  ( `' Q " { 0 } ) )
125 disjsn 3828 . . . . . 6  |-  ( ( ( `' Q " { 0 } )  i^i  { z } )  =  (/)  <->  -.  z  e.  ( `' Q " { 0 } ) )
126124, 125sylibr 204 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  (
( `' Q " { 0 } )  i^i  { z } )  =  (/) )
12726, 126syl5eq 2448 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  ( { z }  i^i  ( `' Q " { 0 } ) )  =  (/) )
12819adantr 452 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  R  e.  Fin )
12949adantr 452 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  R  C_  CC )
130129sselda 3308 . . . 4  |-  ( ( ( ph  /\  z  e.  R )  /\  x  e.  R )  ->  x  e.  CC )
131127, 79, 128, 130fsumsplit 12488 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  R  x  =  ( sum_ x  e.  {
z } x  +  sum_ x  e.  ( `' Q " { 0 } ) x ) )
132 id 20 . . . . . . 7  |-  ( x  =  z  ->  x  =  z )
133132sumsn 12489 . . . . . 6  |-  ( ( z  e.  CC  /\  z  e.  CC )  -> 
sum_ x  e.  { z } x  =  z )
13450, 50, 133syl2anc 643 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  { z } x  =  z )
13550negnegd 9358 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  -u -u z  =  z )
136134, 135eqtr4d 2439 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  { z } x  =  -u -u z )
137120, 31eqtrd 2436 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  ( # `
 ( `' Q " { 0 } ) )  =  (deg `  Q ) )
13828adantr 452 . . . . . . 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 )
) ) ) )
139 fveq2 5687 . . . . . . . . . . 11  |-  ( f  =  Q  ->  (deg `  f )  =  (deg
`  Q ) )
140139eqeq2d 2415 . . . . . . . . . 10  |-  ( f  =  Q  ->  ( D  =  (deg `  f
)  <->  D  =  (deg `  Q ) ) )
141 cnveq 5005 . . . . . . . . . . . . 13  |-  ( f  =  Q  ->  `' f  =  `' Q
)
142141imaeq1d 5161 . . . . . . . . . . . 12  |-  ( f  =  Q  ->  ( `' f " {
0 } )  =  ( `' Q " { 0 } ) )
143142fveq2d 5691 . . . . . . . . . . 11  |-  ( f  =  Q  ->  ( # `
 ( `' f
" { 0 } ) )  =  (
# `  ( `' Q " { 0 } ) ) )
144143, 139eqeq12d 2418 . . . . . . . . . 10  |-  ( f  =  Q  ->  (
( # `  ( `' f " { 0 } ) )  =  (deg `  f )  <->  (
# `  ( `' Q " { 0 } ) )  =  (deg
`  Q ) ) )
145140, 144anbi12d 692 . . . . . . . . 9  |-  ( f  =  Q  ->  (
( D  =  (deg
`  f )  /\  ( # `  ( `' f " { 0 } ) )  =  (deg `  f )
)  <->  ( D  =  (deg `  Q )  /\  ( # `  ( `' Q " { 0 } ) )  =  (deg `  Q )
) ) )
146142sumeq1d 12450 . . . . . . . . . 10  |-  ( f  =  Q  ->  sum_ x  e.  ( `' f " { 0 } ) x  =  sum_ x  e.  ( `' Q " { 0 } ) x )
147 fveq2 5687 . . . . . . . . . . . . 13  |-  ( f  =  Q  ->  (coeff `  f )  =  (coeff `  Q ) )
148139oveq1d 6055 . . . . . . . . . . . . 13  |-  ( f  =  Q  ->  (
(deg `  f )  -  1 )  =  ( (deg `  Q
)  -  1 ) )
149147, 148fveq12d 5693 . . . . . . . . . . . 12  |-  ( f  =  Q  ->  (
(coeff `  f ) `  ( (deg `  f
)  -  1 ) )  =  ( (coeff `  Q ) `  (
(deg `  Q )  -  1 ) ) )
150147, 139fveq12d 5693 . . . . . . . . . . . 12  |-  ( f  =  Q  ->  (
(coeff `  f ) `  (deg `  f )
)  =  ( (coeff `  Q ) `  (deg `  Q ) ) )
151149, 150oveq12d 6058 . . . . . . . . . . 11  |-  ( f  =  Q  ->  (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) )  =  ( ( (coeff `  Q
) `  ( (deg `  Q )  -  1 ) )  /  (
(coeff `  Q ) `  (deg `  Q )
) ) )
152151negeqd 9256 . . . . . . . . . 10  |-  ( f  =  Q  ->  -u (
( (coeff `  f
) `  ( (deg `  f )  -  1 ) )  /  (
(coeff `  f ) `  (deg `  f )
) )  =  -u ( ( (coeff `  Q ) `  (
(deg `  Q )  -  1 ) )  /  ( (coeff `  Q ) `  (deg `  Q ) ) ) )
153146, 152eqeq12d 2418 . . . . . . . . 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 ) ) ) ) )
154145, 153imbi12d 312 . . . . . . . 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 )
) ) ) ) )
155154rspcv 3008 . . . . . . 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 )
) ) ) ) )
15632, 138, 155sylc 58 . . . . . 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 )
) ) ) )
15731, 137, 156mp2and 661 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  ( `' Q " { 0 } ) x  =  -u (
( (coeff `  Q
) `  ( (deg `  Q )  -  1 ) )  /  (
(coeff `  Q ) `  (deg `  Q )
) ) )
15831oveq1d 6055 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  ( D  -  1 )  =  ( (deg `  Q )  -  1 ) )
159158fveq2d 5691 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( D  -  1 ) )  =  ( (coeff `  Q ) `  ( (deg `  Q
)  -  1 ) ) )
16062fveq2d 5691 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  F )  =  (coeff `  ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) ) )
16127, 160syl5eq 2448 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  A  =  (coeff `  ( (
X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) ) )
16262fveq2d 5691 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  F )  =  (deg
`  ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) ) )
16369simp2d 970 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) )  =  1 )
164 ax-1ne0 9015 . . . . . . . . . . . . . . 15  |-  1  =/=  0
165164a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  1  =/=  0 )
166163, 165eqnetrd 2585 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) )  =/=  0
)
167 fveq2 5687 . . . . . . . . . . . . . . 15  |-  ( ( X p  o F  -  ( CC  X.  { z } ) )  =  0 p  ->  (deg `  (
X p  o F  -  ( CC  X.  { z } ) ) )  =  (deg
`  0 p ) )
168167, 12syl6eq 2452 . . . . . . . . . . . . . 14  |-  ( ( X p  o F  -  ( CC  X.  { z } ) )  =  0 p  ->  (deg `  (
X p  o F  -  ( CC  X.  { z } ) ) )  =  0 )
169168necon3i 2606 . . . . . . . . . . . . 13  |-  ( (deg
`  ( X p  o F  -  ( CC  X.  { z } ) ) )  =/=  0  ->  ( X p  o F  -  ( CC  X.  { z } ) )  =/=  0 p )
170166, 169syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
X p  o F  -  ( CC  X.  { z } ) )  =/=  0 p )
171 eqid 2404 . . . . . . . . . . . . 13  |-  (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) )  =  (deg
`  ( X p  o F  -  ( CC  X.  { z } ) ) )
172 eqid 2404 . . . . . . . . . . . . 13  |-  (deg `  Q )  =  (deg
`  Q )
173171, 172dgrmul 20141 . . . . . . . . . . . 12  |-  ( ( ( ( X p  o F  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  /\  (
X p  o F  -  ( CC  X.  { z } ) )  =/=  0 p )  /\  ( Q  e.  (Poly `  CC )  /\  Q  =/=  0 p ) )  -> 
(deg `  ( (
X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) )  =  ( (deg `  (
X p  o F  -  ( CC  X.  { z } ) ) )  +  (deg
`  Q ) ) )
17470, 170, 32, 92, 173syl22anc 1185 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) )  =  ( (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) )  +  (deg
`  Q ) ) )
175162, 174eqtrd 2436 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  F )  =  ( (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) )  +  (deg `  Q )
) )
1769, 175syl5eq 2448 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  N  =  ( (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) )  +  (deg
`  Q ) ) )
177161, 176fveq12d 5693 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  ( A `  N )  =  ( (coeff `  ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) ) `
 ( (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) )  +  (deg
`  Q ) ) ) )
178 eqid 2404 . . . . . . . . . 10  |-  (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) )  =  (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) )
179 eqid 2404 . . . . . . . . . 10  |-  (coeff `  Q )  =  (coeff `  Q )
180178, 179, 171, 172coemulhi 20125 . . . . . . . . 9  |-  ( ( ( X p  o F  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  /\  Q  e.  (Poly `  CC )
)  ->  ( (coeff `  ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) ) `
 ( (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) )  +  (deg
`  Q ) ) )  =  ( ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) ) )  x.  ( (coeff `  Q ) `  (deg `  Q ) ) ) )
18170, 32, 180syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( (
X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) ) `  ( (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) )  +  (deg `  Q )
) )  =  ( ( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) ) )  x.  ( (coeff `  Q
) `  (deg `  Q
) ) ) )
182163fveq2d 5691 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) ) )  =  ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  1
) )
183 ssid 3327 . . . . . . . . . . . . . . 15  |-  CC  C_  CC
184 plyid 20081 . . . . . . . . . . . . . . 15  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  X p  e.  (Poly `  CC ) )
185183, 102, 184mp2an 654 . . . . . . . . . . . . . 14  |-  X p  e.  (Poly `  CC )
186 plyconst 20078 . . . . . . . . . . . . . . 15  |-  ( ( CC  C_  CC  /\  z  e.  CC )  ->  ( CC  X.  { z } )  e.  (Poly `  CC ) )
187183, 50, 186sylancr 645 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  ( CC  X.  { z } )  e.  (Poly `  CC ) )
188 eqid 2404 . . . . . . . . . . . . . . 15  |-  (coeff `  X p )  =  (coeff `  X p )
189 eqid 2404 . . . . . . . . . . . . . . 15  |-  (coeff `  ( CC  X.  { z } ) )  =  (coeff `  ( CC  X.  { z } ) )
190188, 189coesub 20128 . . . . . . . . . . . . . 14  |-  ( ( X p  e.  (Poly `  CC )  /\  ( CC  X.  { z } )  e.  (Poly `  CC ) )  ->  (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) )  =  ( (coeff `  X p
)  o F  -  (coeff `  ( CC  X.  { z } ) ) ) )
191185, 187, 190sylancr 645 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) )  =  ( (coeff `  X p
)  o F  -  (coeff `  ( CC  X.  { z } ) ) ) )
192191fveq1d 5689 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) ` 
1 )  =  ( ( (coeff `  X p )  o F  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
1 ) )
193 1nn0 10193 . . . . . . . . . . . . . 14  |-  1  e.  NN0
194188coef3 20104 . . . . . . . . . . . . . . . . 17  |-  ( X p  e.  (Poly `  CC )  ->  (coeff `  X p ) : NN0 --> CC )
195 ffn 5550 . . . . . . . . . . . . . . . . 17  |-  ( (coeff `  X p ) : NN0 --> CC  ->  (coeff `  X p )  Fn 
NN0 )
196185, 194, 195mp2b 10 . . . . . . . . . . . . . . . 16  |-  (coeff `  X p )  Fn  NN0
197196a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  X p )  Fn 
NN0 )
198189coef3 20104 . . . . . . . . . . . . . . . 16  |-  ( ( CC  X.  { z } )  e.  (Poly `  CC )  ->  (coeff `  ( CC  X.  {
z } ) ) : NN0 --> CC )
199 ffn 5550 . . . . . . . . . . . . . . . 16  |-  ( (coeff `  ( CC  X.  {
z } ) ) : NN0 --> CC  ->  (coeff `  ( CC  X.  {
z } ) )  Fn  NN0 )
200187, 198, 1993syl 19 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  ( CC  X.  {
z } ) )  Fn  NN0 )
201 nn0ex 10183 . . . . . . . . . . . . . . . 16  |-  NN0  e.  _V
202201a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  NN0  e.  _V )
203 inidm 3510 . . . . . . . . . . . . . . 15  |-  ( NN0 
i^i  NN0 )  =  NN0
204 coeidp 20134 . . . . . . . . . . . . . . . . 17  |-  ( 1  e.  NN0  ->  ( (coeff `  X p ) ` 
1 )  =  if ( 1  =  1 ,  1 ,  0 ) )
205204adantl 453 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
(coeff `  X p
) `  1 )  =  if ( 1  =  1 ,  1 ,  0 ) )
206 eqid 2404 . . . . . . . . . . . . . . . . 17  |-  1  =  1
207 iftrue 3705 . . . . . . . . . . . . . . . . 17  |-  ( 1  =  1  ->  if ( 1  =  1 ,  1 ,  0 )  =  1 )
208206, 207ax-mp 8 . . . . . . . . . . . . . . . 16  |-  if ( 1  =  1 ,  1 ,  0 )  =  1
209205, 208syl6eq 2452 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
(coeff `  X p
) `  1 )  =  1 )
210 0lt1 9506 . . . . . . . . . . . . . . . . . 18  |-  0  <  1
211 0re 9047 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR
212211, 115ltnlei 9150 . . . . . . . . . . . . . . . . . 18  |-  ( 0  <  1  <->  -.  1  <_  0 )
213210, 212mpbi 200 . . . . . . . . . . . . . . . . 17  |-  -.  1  <_  0
21450adantr 452 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  z  e.  CC )
215 0dgr 20117 . . . . . . . . . . . . . . . . . . 19  |-  ( z  e.  CC  ->  (deg `  ( CC  X.  {
z } ) )  =  0 )
216214, 215syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (deg `  ( CC  X.  {
z } ) )  =  0 )
217216breq2d 4184 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
1  <_  (deg `  ( CC  X.  { z } ) )  <->  1  <_  0 ) )
218213, 217mtbiri 295 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  -.  1  <_  (deg `  ( CC  X.  { z } ) ) )
219 eqid 2404 . . . . . . . . . . . . . . . . . . . 20  |-  (deg `  ( CC  X.  { z } ) )  =  (deg `  ( CC  X.  { z } ) )
220189, 219dgrub 20106 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( CC  X.  {
z } )  e.  (Poly `  CC )  /\  1  e.  NN0  /\  ( (coeff `  ( CC  X.  { z } ) ) `  1
)  =/=  0 )  ->  1  <_  (deg `  ( CC  X.  {
z } ) ) )
2212203expia 1155 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( CC  X.  {
z } )  e.  (Poly `  CC )  /\  1  e.  NN0 )  ->  ( ( (coeff `  ( CC  X.  {
z } ) ) `
 1 )  =/=  0  ->  1  <_  (deg
`  ( CC  X.  { z } ) ) ) )
222187, 221sylan 458 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
( (coeff `  ( CC  X.  { z } ) ) `  1
)  =/=  0  -> 
1  <_  (deg `  ( CC  X.  { z } ) ) ) )
223222necon1bd 2635 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  ( -.  1  <_  (deg `  ( CC  X.  { z } ) )  -> 
( (coeff `  ( CC  X.  { z } ) ) `  1
)  =  0 ) )
224218, 223mpd 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
(coeff `  ( CC  X.  { z } ) ) `  1 )  =  0 )
225197, 200, 202, 202, 203, 209, 224ofval 6273 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  R )  /\  1  e.  NN0 )  ->  (
( (coeff `  X p )  o F  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
1 )  =  ( 1  -  0 ) )
226193, 225mpan2 653 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  X p )  o F  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
1 )  =  ( 1  -  0 ) )
227102subid1i 9328 . . . . . . . . . . . . 13  |-  ( 1  -  0 )  =  1
228226, 227syl6eq 2452 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  X p )  o F  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
1 )  =  1 )
229192, 228eqtrd 2436 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) ` 
1 )  =  1 )
230182, 229eqtrd 2436 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) ) )  =  1 )
231230oveq1d 6055 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) ) )  x.  ( (coeff `  Q
) `  (deg `  Q
) ) )  =  ( 1  x.  (
(coeff `  Q ) `  (deg `  Q )
) ) )
232179coef3 20104 . . . . . . . . . . . 12  |-  ( Q  e.  (Poly `  CC )  ->  (coeff `  Q
) : NN0 --> CC )
23332, 232syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  Q ) : NN0 --> CC )
234233, 34ffvelrnd 5830 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  (deg `  Q )
)  e.  CC )
235234mulid2d 9062 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  (
1  x.  ( (coeff `  Q ) `  (deg `  Q ) ) )  =  ( (coeff `  Q ) `  (deg `  Q ) ) )
236231, 235eqtrd 2436 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) ) )  x.  ( (coeff `  Q
) `  (deg `  Q
) ) )  =  ( (coeff `  Q
) `  (deg `  Q
) ) )
237177, 181, 2363eqtrd 2440 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( A `  N )  =  ( (coeff `  Q ) `  (deg `  Q ) ) )
238159, 237oveq12d 6058 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) )  =  ( ( (coeff `  Q ) `  (
(deg `  Q )  -  1 ) )  /  ( (coeff `  Q ) `  (deg `  Q ) ) ) )
239238negeqd 9256 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  -u (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) )  = 
-u ( ( (coeff `  Q ) `  (
(deg `  Q )  -  1 ) )  /  ( (coeff `  Q ) `  (deg `  Q ) ) ) )
240157, 239eqtr4d 2439 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  ( `' Q " { 0 } ) x  =  -u (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) ) )
241136, 240oveq12d 6058 . . 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 ) ) ) )
24250negcld 9354 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  -u z  e.  CC )
243 nnm1nn0 10217 . . . . . . . . 9  |-  ( D  e.  NN  ->  ( D  -  1 )  e.  NN0 )
2443, 243syl 16 . . . . . . . 8  |-  ( ph  ->  ( D  -  1 )  e.  NN0 )
245244adantr 452 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( D  -  1 )  e.  NN0 )
246233, 245ffvelrnd 5830 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( D  -  1 ) )  e.  CC )
247237, 234eqeltrd 2478 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  ( A `  N )  e.  CC )
2489, 27dgreq0 20136 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0 p  <->  ( A `  N )  =  0 ) )
24943, 248syl 16 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  ( F  =  0 p  <->  ( A `  N )  =  0 ) )
250249necon3bid 2602 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( F  =/=  0 p  <->  ( A `  N )  =/=  0
) )
25184, 250mpbid 202 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  ( A `  N )  =/=  0 )
252246, 247, 251divcld 9746 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) )  e.  CC )
253242, 252negdid 9380 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  -u ( -u z  +  ( ( (coeff `  Q ) `  ( D  -  1 ) )  /  ( A `  N )
) )  =  (
-u -u z  +  -u ( ( (coeff `  Q ) `  ( D  -  1 ) )  /  ( A `
 N ) ) ) )
254242, 247mulcld 9064 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  ( -u z  x.  ( A `
 N ) )  e.  CC )
255254, 246, 247, 251divdird 9784 . . . . . 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 )
) ) )
256 nnm1nn0 10217 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
2575, 256syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( N  -  1 )  e.  NN0 )
258257adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  ( N  -  1 )  e.  NN0 )
259178, 179coemul 20123 . . . . . . . . 9  |-  ( ( ( X p  o F  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  /\  Q  e.  (Poly `  CC )  /\  ( N  -  1 )  e.  NN0 )  ->  ( (coeff `  (
( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) ) `  ( N  -  1
) )  =  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) ) )
26070, 32, 258, 259syl3anc 1184 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( (
X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) ) `  ( N  -  1
) )  =  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) ) )
261161fveq1d 5689 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  ( A `  ( N  -  1 ) )  =  ( (coeff `  ( ( X p  o F  -  ( CC  X.  { z } ) )  o F  x.  Q ) ) `
 ( N  - 
1 ) ) )
262 1e0p1 10366 . . . . . . . . . . . 12  |-  1  =  ( 0  +  1 )
263262oveq2i 6051 . . . . . . . . . . 11  |-  ( 0 ... 1 )  =  ( 0 ... (
0  +  1 ) )
264263sumeq1i 12447 . . . . . . . . . 10  |-  sum_ k  e.  ( 0 ... 1
) ( ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  =  sum_ k  e.  ( 0 ... ( 0  +  1 ) ) ( ( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )
265 0nn0 10192 . . . . . . . . . . . . 13  |-  0  e.  NN0
266 nn0uz 10476 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
267265, 266eleqtri 2476 . . . . . . . . . . . 12  |-  0  e.  ( ZZ>= `  0 )
268267a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  0  e.  ( ZZ>= `  0 )
)
269263eleq2i 2468 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... 1 )  <->  k  e.  ( 0 ... (
0  +  1 ) ) )
270178coef3 20104 . . . . . . . . . . . . . . 15  |-  ( ( X p  o F  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  ->  (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) : NN0 --> CC )
27170, 270syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) : NN0 --> CC )
272 elfznn0 11039 . . . . . . . . . . . . . 14  |-  ( k  e.  ( 0 ... 1 )  ->  k  e.  NN0 )
273 ffvelrn 5827 . . . . . . . . . . . . . 14  |-  ( ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) : NN0 --> CC  /\  k  e.  NN0 )  ->  (
(coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k )  e.  CC )
274271, 272, 273syl2an 464 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... 1
) )  ->  (
(coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k )  e.  CC )
2752oveq1d 6055 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( D  + 
1 )  -  1 )  =  ( N  -  1 ) )
276 pncan 9267 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( D  e.  CC  /\  1  e.  CC )  ->  ( ( D  + 
1 )  -  1 )  =  D )
277103, 102, 276sylancl 644 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( D  + 
1 )  -  1 )  =  D )
278275, 277eqtr3d 2438 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( N  -  1 )  =  D )
279278adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  ( N  -  1 )  =  D )
2803adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  R )  ->  D  e.  NN )
281279, 280eqeltrd 2478 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  ( N  -  1 )  e.  NN )
282 nnuz 10477 . . . . . . . . . . . . . . . . 17  |-  NN  =  ( ZZ>= `  1 )
283281, 282syl6eleq 2494 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  ( N  -  1 )  e.  ( ZZ>= `  1
) )
284 fzss2 11048 . . . . . . . . . . . . . . . 16  |-  ( ( N  -  1 )  e.  ( ZZ>= `  1
)  ->  ( 0 ... 1 )  C_  ( 0 ... ( N  -  1 ) ) )
285283, 284syl 16 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
0 ... 1 )  C_  ( 0 ... ( N  -  1 ) ) )
286285sselda 3308 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... 1
) )  ->  k  e.  ( 0 ... ( N  -  1 ) ) )
287 fznn0sub 11041 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  (
( N  -  1 )  -  k )  e.  NN0 )
288 ffvelrn 5827 . . . . . . . . . . . . . . 15  |-  ( ( (coeff `  Q ) : NN0 --> CC  /\  (
( N  -  1 )  -  k )  e.  NN0 )  -> 
( (coeff `  Q
) `  ( ( N  -  1 )  -  k ) )  e.  CC )
289233, 287, 288syl2an 464 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) )  e.  CC )
290286, 289syldan 457 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... 1
) )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) )  e.  CC )
291274, 290mulcld 9064 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... 1
) )  ->  (
( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  e.  CC )
292269, 291sylan2br 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( 0 ... (
0  +  1 ) ) )  ->  (
( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  e.  CC )
293 id 20 . . . . . . . . . . . . . 14  |-  ( k  =  ( 0  +  1 )  ->  k  =  ( 0  +  1 ) )
294293, 262syl6eqr 2454 . . . . . . . . . . . . 13  |-  ( k  =  ( 0  +  1 )  ->  k  =  1 )
295294fveq2d 5691 . . . . . . . . . . . 12  |-  ( k  =  ( 0  +  1 )  ->  (
(coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k )  =  ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) ` 
1 ) )
296294oveq2d 6056 . . . . . . . . . . . . 13  |-  ( k  =  ( 0  +  1 )  ->  (
( N  -  1 )  -  k )  =  ( ( N  -  1 )  - 
1 ) )
297296fveq2d 5691 . . . . . . . . . . . 12  |-  ( k  =  ( 0  +  1 )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) )  =  ( (coeff `  Q ) `  ( ( N  - 
1 )  -  1 ) ) )
298295, 297oveq12d 6058 . . . . . . . . . . 11  |-  ( k  =  ( 0  +  1 )  ->  (
( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  =  ( ( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  1
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  1 ) ) ) )
299268, 292, 298fsump1 12495 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  sum_ k  e.  ( 0 ... (
0  +  1 ) ) ( ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  =  (
sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  +  ( ( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  1
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  1 ) ) ) ) )
300264, 299syl5eq 2448 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  sum_ k  e.  ( 0 ... 1
) ( ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  =  (
sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  +  ( ( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  1
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  1 ) ) ) ) )
301 eldifn 3430 . . . . . . . . . . . . . 14  |-  ( k  e.  ( ( 0 ... ( N  - 
1 ) )  \ 
( 0 ... 1
) )  ->  -.  k  e.  ( 0 ... 1 ) )
302301adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  -.  k  e.  ( 0 ... 1
) )
303 eldifi 3429 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( ( 0 ... ( N  - 
1 ) )  \ 
( 0 ... 1
) )  ->  k  e.  ( 0 ... ( N  -  1 ) ) )
304 elfznn0 11039 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
305303, 304syl 16 . . . . . . . . . . . . . . . 16  |-  ( k  e.  ( ( 0 ... ( N  - 
1 ) )  \ 
( 0 ... 1
) )  ->  k  e.  NN0 )
306178, 171dgrub 20106 . . . . . . . . . . . . . . . . 17  |-  ( ( ( X p  o F  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  /\  k  e.  NN0  /\  ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  =/=  0 )  ->  k  <_  (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) ) )
3073063expia 1155 . . . . . . . . . . . . . . . 16  |-  ( ( ( X p  o F  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  /\  k  e.  NN0 )  ->  (
( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  =/=  0  -> 
k  <_  (deg `  (
X p  o F  -  ( CC  X.  { z } ) ) ) ) )
30870, 305, 307syl2an 464 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( (
(coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k )  =/=  0  ->  k  <_  (deg `  (
X p  o F  -  ( CC  X.  { z } ) ) ) ) )
309 elfzuz 11011 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  ( ZZ>= `  0 )
)
310303, 309syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  ( ( 0 ... ( N  - 
1 ) )  \ 
( 0 ... 1
) )  ->  k  e.  ( ZZ>= `  0 )
)
311310adantl 453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  k  e.  ( ZZ>= `  0 )
)
312 1z 10267 . . . . . . . . . . . . . . . . 17  |-  1  e.  ZZ
313 elfz5 11007 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  ( ZZ>= ` 
0 )  /\  1  e.  ZZ )  ->  (
k  e.  ( 0 ... 1 )  <->  k  <_  1 ) )
314311, 312, 313sylancl 644 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( k  e.  ( 0 ... 1
)  <->  k  <_  1
) )
315163breq2d 4184 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  (
k  <_  (deg `  (
X p  o F  -  ( CC  X.  { z } ) ) )  <->  k  <_  1 ) )
316315adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( k  <_  (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) )  <->  k  <_  1 ) )
317314, 316bitr4d 248 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( k  e.  ( 0 ... 1
)  <->  k  <_  (deg `  ( X p  o F  -  ( CC  X.  { z } ) ) ) ) )
318308, 317sylibrd 226 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( (
(coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k )  =/=  0  ->  k  e.  ( 0 ... 1 ) ) )
319318necon1bd 2635 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( -.  k  e.  ( 0 ... 1 )  -> 
( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  =  0 ) )
320302, 319mpd 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  =  0 )
321320oveq1d 6055 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( (
(coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) )  =  ( 0  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) ) )
322303, 289sylan2 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) )  e.  CC )
323322mul02d 9220 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( 0  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  =  0 )
324321, 323eqtrd 2436 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  R )  /\  k  e.  ( ( 0 ... ( N  -  1 ) )  \  (
0 ... 1 ) ) )  ->  ( (
(coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k )  x.  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) ) )  =  0 )
325 fzfid 11267 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (
0 ... ( N  - 
1 ) )  e. 
Fin )
326285, 291, 324, 325fsumss 12474 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  sum_ k  e.  ( 0 ... 1
) ( ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  =  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) ) )
327 0z 10249 . . . . . . . . . . . 12  |-  0  e.  ZZ
328191fveq1d 5689 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) ` 
0 )  =  ( ( (coeff `  X p )  o F  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
0 ) )
329 coeidp 20134 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0  e.  NN0  ->  ( (coeff `  X p ) ` 
0 )  =  if ( 0  =  1 ,  1 ,  0 ) )
330164necomi 2649 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  =/=  1
331 df-ne 2569 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  =/=  1  <->  -.  0  =  1 )
332330, 331mpbi 200 . . . . . . . . . . . . . . . . . . . . 21  |-  -.  0  =  1
333 iffalse 3706 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  0  =  1  ->  if ( 0  =  1 ,  1 ,  0 )  =  0 )
334332, 333ax-mp 8 . . . . . . . . . . . . . . . . . . . 20  |-  if ( 0  =  1 ,  1 ,  0 )  =  0
335329, 334syl6eq 2452 . . . . . . . . . . . . . . . . . . 19  |-  ( 0  e.  NN0  ->  ( (coeff `  X p ) ` 
0 )  =  0 )
336335adantl 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  R )  /\  0  e.  NN0 )  ->  (
(coeff `  X p
) `  0 )  =  0 )
337 0cn 9040 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  CC
338 vex 2919 . . . . . . . . . . . . . . . . . . . . . 22  |-  z  e. 
_V
339338fvconst2 5906 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0  e.  CC  ->  (
( CC  X.  {
z } ) ` 
0 )  =  z )
340337, 339ax-mp 8 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( CC  X.  { z } ) `  0
)  =  z
341189coefv0 20119 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( CC  X.  { z } )  e.  (Poly `  CC )  ->  (
( CC  X.  {
z } ) ` 
0 )  =  ( (coeff `  ( CC  X.  { z } ) ) `  0 ) )
342187, 341syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  z  e.  R )  ->  (
( CC  X.  {
z } ) ` 
0 )  =  ( (coeff `  ( CC  X.  { z } ) ) `  0 ) )
343340, 342syl5reqr 2451 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( CC  X.  { z } ) ) `  0 )  =  z )
344343adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  z  e.  R )  /\  0  e.  NN0 )  ->  (
(coeff `  ( CC  X.  { z } ) ) `  0 )  =  z )
345197, 200, 202, 202, 203, 336, 344ofval 6273 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  z  e.  R )  /\  0  e.  NN0 )  ->  (
( (coeff `  X p )  o F  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
0 )  =  ( 0  -  z ) )
346265, 345mpan2 653 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  X p )  o F  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
0 )  =  ( 0  -  z ) )
347 df-neg 9250 . . . . . . . . . . . . . . . 16  |-  -u z  =  ( 0  -  z )
348346, 347syl6eqr 2454 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  X p )  o F  -  (coeff `  ( CC  X.  { z } ) ) ) ` 
0 )  =  -u z )
349328, 348eqtrd 2436 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) ` 
0 )  =  -u z )
350279oveq1d 6055 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  (
( N  -  1 )  -  0 )  =  ( D  - 
0 ) )
351104subid1d 9356 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  R )  ->  ( D  -  0 )  =  D )
352350, 351, 313eqtrd 2440 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  R )  ->  (
( N  -  1 )  -  0 )  =  (deg `  Q
) )
353352fveq2d 5691 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  0 ) )  =  ( (coeff `  Q ) `  (deg `  Q )
) )
354353, 237eqtr4d 2439 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  0 ) )  =  ( A `  N ) )
355349, 354oveq12d 6058 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  0
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  0 ) ) )  =  (
-u z  x.  ( A `  N )
) )
356355, 254eqeltrd 2478 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  0
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  0 ) ) )  e.  CC )
357 fveq2 5687 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
(coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k )  =  ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) ` 
0 ) )
358 oveq2 6048 . . . . . . . . . . . . . . 15  |-  ( k  =  0  ->  (
( N  -  1 )  -  k )  =  ( ( N  -  1 )  - 
0 ) )
359358fveq2d 5691 . . . . . . . . . . . . . 14  |-  ( k  =  0  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  k
) )  =  ( (coeff `  Q ) `  ( ( N  - 
1 )  -  0 ) ) )
360357, 359oveq12d 6058 . . . . . . . . . . . . 13  |-  ( k  =  0  ->  (
( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  =  ( ( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  0
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  0 ) ) ) )
361360fsum1 12490 . . . . . . . . . . . 12  |-  ( ( 0  e.  ZZ  /\  ( ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  0
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  0 ) ) )  e.  CC )  ->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  =  ( ( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  0
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  0 ) ) ) )
362327, 356, 361sylancr 645 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  =  ( ( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  0
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  0 ) ) ) )
363362, 355eqtrd 2436 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  ( X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  =  (
-u z  x.  ( A `  N )
) )
364279oveq1d 6055 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  R )  ->  (
( N  -  1 )  -  1 )  =  ( D  - 
1 ) )
365364fveq2d 5691 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  R )  ->  (
(coeff `  Q ) `  ( ( N  - 
1 )  -  1 ) )  =  ( (coeff `  Q ) `  ( D  -  1 ) ) )
366229, 365oveq12d 6058 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  1
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  1 ) ) )  =  ( 1  x.  ( (coeff `  Q ) `  ( D  -  1 ) ) ) )
367246mulid2d 9062 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  R )  ->  (
1  x.  ( (coeff `  Q ) `  ( D  -  1 ) ) )  =  ( (coeff `  Q ) `  ( D  -  1 ) ) )
368366, 367eqtrd 2436 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (
( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  1
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  1 ) ) )  =  ( (coeff `  Q ) `  ( D  -  1 ) ) )
369363, 368oveq12d 6058 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  ( sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) )  +  ( ( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  1
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  1 ) ) ) )  =  ( ( -u z  x.  ( A `  N
) )  +  ( (coeff `  Q ) `  ( D  -  1 ) ) ) )
370300, 326, 3693eqtr3rd 2445 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  (
( -u z  x.  ( A `  N )
)  +  ( (coeff `  Q ) `  ( D  -  1 ) ) )  =  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( (coeff `  (
X p  o F  -  ( CC  X.  { z } ) ) ) `  k
)  x.  ( (coeff `  Q ) `  (
( N  -  1 )  -  k ) ) ) )
371260, 261, 3703eqtr4rd 2447 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  (
( -u z  x.  ( A `  N )
)  +  ( (coeff `  Q ) `  ( D  -  1 ) ) )  =  ( A `  ( N  -  1 ) ) )
372371oveq1d 6055 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (
( ( -u z  x.  ( A `  N
) )  +  ( (coeff `  Q ) `  ( D  -  1 ) ) )  / 
( A `  N
) )  =  ( ( A `  ( N  -  1 ) )  /  ( A `
 N ) ) )
373242, 247, 251divcan4d 9752 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  (
( -u z  x.  ( A `  N )
)  /  ( A `
 N ) )  =  -u z )
374373oveq1d 6055 . . . . . 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 ) ) ) )
375255, 372, 3743eqtr3rd 2445 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  ( -u z  +  ( ( (coeff `  Q ) `  ( D  -  1 ) )  /  ( A `  N )
) )  =  ( ( A `  ( N  -  1 ) )  /  ( A `
 N ) ) )
376375negeqd 9256 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  -u ( -u z  +  ( ( (coeff `  Q ) `  ( D  -  1 ) )  /  ( A `  N )
) )  =  -u ( ( A `  ( N  -  1
) )  /  ( A `  N )
) )
377253, 376eqtr3d 2438 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  ( -u -u z  +  -u (
( (coeff `  Q
) `  ( D  -  1 ) )  /  ( A `  N ) ) )  =  -u ( ( A `
 ( N  - 
1 ) )  / 
( A `  N
) ) )
378131, 241, 3773eqtrd 2440 . 2  |-  ( (
ph  /\  z  e.  R )  ->  sum_ x  e.  R  x  =  -u ( ( A `  ( N  -  1
) )  /  ( A `  N )
) )
37925, 378exlimddv 1645 1  |-  ( ph  -> 
sum_ x  e.  R  x  =  -u ( ( A `  ( N  -  1 ) )  /  ( A `  N ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   _Vcvv 2916    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   ifcif 3699   {csn 3774   class class class wbr 4172    X. cxp 4835   `'ccnv 4836   dom cdm 4837   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   #chash 11573   sum_csu 12434   0 pc0p 19514  Polycply 20056   X pcidp 20057  coeffccoe 20058  degcdgr 20059   quot cquot 20160
This theorem is referenced by:  vieta1  20182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-idp 20061  df-coe 20062  df-dgr 20063  df-quot 20161
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