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Theorem plydivlem4 21737
Description: Lemma for plydivex 21738. Induction step. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
plydiv.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
plydiv.tm  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
plydiv.rc  |-  ( (
ph  /\  ( x  e.  S  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  S )
plydiv.m1  |-  ( ph  -> 
-u 1  e.  S
)
plydiv.f  |-  ( ph  ->  F  e.  (Poly `  S ) )
plydiv.g  |-  ( ph  ->  G  e.  (Poly `  S ) )
plydiv.z  |-  ( ph  ->  G  =/=  0p )
plydiv.r  |-  R  =  ( F  oF  -  ( G  oF  x.  q )
)
plydiv.d  |-  ( ph  ->  D  e.  NN0 )
plydiv.e  |-  ( ph  ->  ( M  -  N
)  =  D )
plydiv.fz  |-  ( ph  ->  F  =/=  0p )
plydiv.u  |-  U  =  ( f  oF  -  ( G  oF  x.  p )
)
plydiv.h  |-  H  =  ( z  e.  CC  |->  ( ( ( A `
 M )  / 
( B `  N
) )  x.  (
z ^ D ) ) )
plydiv.al  |-  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  N
)  <  D )  ->  E. p  e.  (Poly `  S ) ( U  =  0p  \/  (deg `  U )  <  N ) ) )
plydiv.a  |-  A  =  (coeff `  F )
plydiv.b  |-  B  =  (coeff `  G )
plydiv.m  |-  M  =  (deg `  F )
plydiv.n  |-  N  =  (deg `  G )
Assertion
Ref Expression
plydivlem4  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0p  \/  (deg `  R )  <  N ) )
Distinct variable groups:    x, y,
z, A    f, p, q, x, y, z, F   
f, H, p, q, x, y, z    ph, x, y, z    x, B, y, z    D, f, z    x, M, y, z    f, N, p, q, x, y, z    f, G, p, q, x, y, z    R, f, p, x, y    S, f, p, q, x, y, z    ph, p
Allowed substitution hints:    ph( f, q)    A( f, q, p)    B( f, q, p)    D( x, y, q, p)    R( z,
q)    U( x, y, z, f, q, p)    M( f, q, p)

Proof of Theorem plydivlem4
StepHypRef Expression
1 plydiv.f . . . 4  |-  ( ph  ->  F  e.  (Poly `  S ) )
2 plybss 21637 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  S  C_  CC )
4 plydiv.pl . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
5 plydiv.tm . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
6 plydiv.rc . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  S  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  S )
7 plydiv.m1 . . . . . . . . . . . 12  |-  ( ph  -> 
-u 1  e.  S
)
84, 5, 6, 7plydivlem1 21734 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  S )
9 plydiv.a . . . . . . . . . . . 12  |-  A  =  (coeff `  F )
109coef2 21674 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  0  e.  S )  ->  A : NN0 --> S )
111, 8, 10syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  A : NN0 --> S )
12 plydiv.m . . . . . . . . . . 11  |-  M  =  (deg `  F )
13 dgrcl 21676 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
141, 13syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  F )  e.  NN0 )
1512, 14syl5eqel 2522 . . . . . . . . . 10  |-  ( ph  ->  M  e.  NN0 )
1611, 15ffvelrnd 5839 . . . . . . . . 9  |-  ( ph  ->  ( A `  M
)  e.  S )
173, 16sseldd 3352 . . . . . . . 8  |-  ( ph  ->  ( A `  M
)  e.  CC )
18 plydiv.g . . . . . . . . . . 11  |-  ( ph  ->  G  e.  (Poly `  S ) )
19 plydiv.b . . . . . . . . . . . 12  |-  B  =  (coeff `  G )
2019coef2 21674 . . . . . . . . . . 11  |-  ( ( G  e.  (Poly `  S )  /\  0  e.  S )  ->  B : NN0 --> S )
2118, 8, 20syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  B : NN0 --> S )
22 plydiv.n . . . . . . . . . . 11  |-  N  =  (deg `  G )
23 dgrcl 21676 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
2418, 23syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  G )  e.  NN0 )
2522, 24syl5eqel 2522 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
2621, 25ffvelrnd 5839 . . . . . . . . 9  |-  ( ph  ->  ( B `  N
)  e.  S )
273, 26sseldd 3352 . . . . . . . 8  |-  ( ph  ->  ( B `  N
)  e.  CC )
28 plydiv.z . . . . . . . . 9  |-  ( ph  ->  G  =/=  0p )
2922, 19dgreq0 21707 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0p  <->  ( B `  N )  =  0 ) )
3018, 29syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( G  =  0p  <->  ( B `  N )  =  0 ) )
3130necon3bid 2638 . . . . . . . . 9  |-  ( ph  ->  ( G  =/=  0p 
<->  ( B `  N
)  =/=  0 ) )
3228, 31mpbid 210 . . . . . . . 8  |-  ( ph  ->  ( B `  N
)  =/=  0 )
3317, 27, 32divrecd 10102 . . . . . . 7  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  =  ( ( A `  M )  x.  ( 1  / 
( B `  N
) ) ) )
34 fvex 5696 . . . . . . . . . . 11  |-  ( B `
 N )  e. 
_V
35 eleq1 2498 . . . . . . . . . . . . . 14  |-  ( x  =  ( B `  N )  ->  (
x  e.  S  <->  ( B `  N )  e.  S
) )
36 neeq1 2611 . . . . . . . . . . . . . 14  |-  ( x  =  ( B `  N )  ->  (
x  =/=  0  <->  ( B `  N )  =/=  0 ) )
3735, 36anbi12d 710 . . . . . . . . . . . . 13  |-  ( x  =  ( B `  N )  ->  (
( x  e.  S  /\  x  =/=  0
)  <->  ( ( B `
 N )  e.  S  /\  ( B `
 N )  =/=  0 ) ) )
3837anbi2d 703 . . . . . . . . . . . 12  |-  ( x  =  ( B `  N )  ->  (
( ph  /\  (
x  e.  S  /\  x  =/=  0 ) )  <-> 
( ph  /\  (
( B `  N
)  e.  S  /\  ( B `  N )  =/=  0 ) ) ) )
39 oveq2 6094 . . . . . . . . . . . . 13  |-  ( x  =  ( B `  N )  ->  (
1  /  x )  =  ( 1  / 
( B `  N
) ) )
4039eleq1d 2504 . . . . . . . . . . . 12  |-  ( x  =  ( B `  N )  ->  (
( 1  /  x
)  e.  S  <->  ( 1  /  ( B `  N ) )  e.  S ) )
4138, 40imbi12d 320 . . . . . . . . . . 11  |-  ( x  =  ( B `  N )  ->  (
( ( ph  /\  ( x  e.  S  /\  x  =/=  0
) )  ->  (
1  /  x )  e.  S )  <->  ( ( ph  /\  ( ( B `
 N )  e.  S  /\  ( B `
 N )  =/=  0 ) )  -> 
( 1  /  ( B `  N )
)  e.  S ) ) )
4234, 41, 6vtocl 3019 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( B `  N )  e.  S  /\  ( B `  N )  =/=  0 ) )  -> 
( 1  /  ( B `  N )
)  e.  S )
4342ex 434 . . . . . . . . 9  |-  ( ph  ->  ( ( ( B `
 N )  e.  S  /\  ( B `
 N )  =/=  0 )  ->  (
1  /  ( B `
 N ) )  e.  S ) )
4426, 32, 43mp2and 679 . . . . . . . 8  |-  ( ph  ->  ( 1  /  ( B `  N )
)  e.  S )
455, 16, 44caovcld 6251 . . . . . . 7  |-  ( ph  ->  ( ( A `  M )  x.  (
1  /  ( B `
 N ) ) )  e.  S )
4633, 45eqeltrd 2512 . . . . . 6  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  e.  S )
47 plydiv.d . . . . . 6  |-  ( ph  ->  D  e.  NN0 )
48 plydiv.h . . . . . . 7  |-  H  =  ( z  e.  CC  |->  ( ( ( A `
 M )  / 
( B `  N
) )  x.  (
z ^ D ) ) )
4948ply1term 21647 . . . . . 6  |-  ( ( S  C_  CC  /\  (
( A `  M
)  /  ( B `
 N ) )  e.  S  /\  D  e.  NN0 )  ->  H  e.  (Poly `  S )
)
503, 46, 47, 49syl3anc 1218 . . . . 5  |-  ( ph  ->  H  e.  (Poly `  S ) )
5150, 18, 4, 5plymul 21661 . . . 4  |-  ( ph  ->  ( H  oF  x.  G )  e.  (Poly `  S )
)
521, 51, 4, 5, 7plysub 21662 . . 3  |-  ( ph  ->  ( F  oF  -  ( H  oF  x.  G )
)  e.  (Poly `  S ) )
53 plydiv.al . . 3  |-  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  N
)  <  D )  ->  E. p  e.  (Poly `  S ) ( U  =  0p  \/  (deg `  U )  <  N ) ) )
54 eqid 2438 . . . . . . 7  |-  (deg `  ( H  oF  x.  G ) )  =  (deg `  ( H  oF  x.  G
) )
5512, 54dgrsub 21714 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  ( H  oF  x.  G
)  e.  (Poly `  S ) )  -> 
(deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  <_  if ( M  <_  (deg `  ( H  oF  x.  G ) ) ,  (deg `  ( H  oF  x.  G
) ) ,  M
) )
561, 51, 55syl2anc 661 . . . . 5  |-  ( ph  ->  (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  <_  if ( M  <_  (deg `  ( H  oF  x.  G ) ) ,  (deg `  ( H  oF  x.  G
) ) ,  M
) )
57 plydiv.fz . . . . . . . . . . . . 13  |-  ( ph  ->  F  =/=  0p )
5812, 9dgreq0 21707 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0p  <->  ( A `  M )  =  0 ) )
591, 58syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  =  0p  <->  ( A `  M )  =  0 ) )
6059necon3bid 2638 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  =/=  0p 
<->  ( A `  M
)  =/=  0 ) )
6157, 60mpbid 210 . . . . . . . . . . . 12  |-  ( ph  ->  ( A `  M
)  =/=  0 )
6217, 27, 61, 32divne0d 10115 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  =/=  0 )
633, 46sseldd 3352 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  e.  CC )
6448coe1term 21701 . . . . . . . . . . . . 13  |-  ( ( ( ( A `  M )  /  ( B `  N )
)  e.  CC  /\  D  e.  NN0  /\  D  e.  NN0 )  ->  (
(coeff `  H ) `  D )  =  if ( D  =  D ,  ( ( A `
 M )  / 
( B `  N
) ) ,  0 ) )
6563, 47, 47, 64syl3anc 1218 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coeff `  H
) `  D )  =  if ( D  =  D ,  ( ( A `  M )  /  ( B `  N ) ) ,  0 ) )
66 eqid 2438 . . . . . . . . . . . . 13  |-  D  =  D
6766iftruei 3793 . . . . . . . . . . . 12  |-  if ( D  =  D , 
( ( A `  M )  /  ( B `  N )
) ,  0 )  =  ( ( A `
 M )  / 
( B `  N
) )
6865, 67syl6eq 2486 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  H
) `  D )  =  ( ( A `
 M )  / 
( B `  N
) ) )
69 c0ex 9372 . . . . . . . . . . . . 13  |-  0  e.  _V
7069fvconst2 5928 . . . . . . . . . . . 12  |-  ( D  e.  NN0  ->  ( ( NN0  X.  { 0 } ) `  D
)  =  0 )
7147, 70syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ( NN0  X.  { 0 } ) `
 D )  =  0 )
7262, 68, 713netr4d 2630 . . . . . . . . . 10  |-  ( ph  ->  ( (coeff `  H
) `  D )  =/=  ( ( NN0  X.  { 0 } ) `
 D ) )
73 fveq2 5686 . . . . . . . . . . . . 13  |-  ( H  =  0p  -> 
(coeff `  H )  =  (coeff `  0p
) )
74 coe0 21698 . . . . . . . . . . . . 13  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
7573, 74syl6eq 2486 . . . . . . . . . . . 12  |-  ( H  =  0p  -> 
(coeff `  H )  =  ( NN0  X.  { 0 } ) )
7675fveq1d 5688 . . . . . . . . . . 11  |-  ( H  =  0p  -> 
( (coeff `  H
) `  D )  =  ( ( NN0 
X.  { 0 } ) `  D ) )
7776necon3i 2645 . . . . . . . . . 10  |-  ( ( (coeff `  H ) `  D )  =/=  (
( NN0  X.  { 0 } ) `  D
)  ->  H  =/=  0p )
7872, 77syl 16 . . . . . . . . 9  |-  ( ph  ->  H  =/=  0p )
79 eqid 2438 . . . . . . . . . 10  |-  (deg `  H )  =  (deg
`  H )
8079, 22dgrmul 21712 . . . . . . . . 9  |-  ( ( ( H  e.  (Poly `  S )  /\  H  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  (deg `  ( H  oF  x.  G
) )  =  ( (deg `  H )  +  N ) )
8150, 78, 18, 28, 80syl22anc 1219 . . . . . . . 8  |-  ( ph  ->  (deg `  ( H  oF  x.  G
) )  =  ( (deg `  H )  +  N ) )
8248dgr1term 21702 . . . . . . . . . . . 12  |-  ( ( ( ( A `  M )  /  ( B `  N )
)  e.  CC  /\  ( ( A `  M )  /  ( B `  N )
)  =/=  0  /\  D  e.  NN0 )  ->  (deg `  H )  =  D )
8363, 62, 47, 82syl3anc 1218 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  H )  =  D )
84 plydiv.e . . . . . . . . . . 11  |-  ( ph  ->  ( M  -  N
)  =  D )
8583, 84eqtr4d 2473 . . . . . . . . . 10  |-  ( ph  ->  (deg `  H )  =  ( M  -  N ) )
8685oveq1d 6101 . . . . . . . . 9  |-  ( ph  ->  ( (deg `  H
)  +  N )  =  ( ( M  -  N )  +  N ) )
8715nn0cnd 10630 . . . . . . . . . 10  |-  ( ph  ->  M  e.  CC )
8825nn0cnd 10630 . . . . . . . . . 10  |-  ( ph  ->  N  e.  CC )
8987, 88npcand 9715 . . . . . . . . 9  |-  ( ph  ->  ( ( M  -  N )  +  N
)  =  M )
9086, 89eqtrd 2470 . . . . . . . 8  |-  ( ph  ->  ( (deg `  H
)  +  N )  =  M )
9181, 90eqtrd 2470 . . . . . . 7  |-  ( ph  ->  (deg `  ( H  oF  x.  G
) )  =  M )
9291ifeq1d 3802 . . . . . 6  |-  ( ph  ->  if ( M  <_ 
(deg `  ( H  oF  x.  G
) ) ,  (deg
`  ( H  oF  x.  G )
) ,  M )  =  if ( M  <_  (deg `  ( H  oF  x.  G
) ) ,  M ,  M ) )
93 ifid 3821 . . . . . 6  |-  if ( M  <_  (deg `  ( H  oF  x.  G
) ) ,  M ,  M )  =  M
9492, 93syl6eq 2486 . . . . 5  |-  ( ph  ->  if ( M  <_ 
(deg `  ( H  oF  x.  G
) ) ,  (deg
`  ( H  oF  x.  G )
) ,  M )  =  M )
9556, 94breqtrd 4311 . . . 4  |-  ( ph  ->  (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  <_  M )
96 eqid 2438 . . . . . . . 8  |-  (coeff `  ( H  oF  x.  G ) )  =  (coeff `  ( H  oF  x.  G
) )
979, 96coesub 21699 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  ( H  oF  x.  G
)  e.  (Poly `  S ) )  -> 
(coeff `  ( F  oF  -  ( H  oF  x.  G
) ) )  =  ( A  oF  -  (coeff `  ( H  oF  x.  G
) ) ) )
981, 51, 97syl2anc 661 . . . . . 6  |-  ( ph  ->  (coeff `  ( F  oF  -  ( H  oF  x.  G
) ) )  =  ( A  oF  -  (coeff `  ( H  oF  x.  G
) ) ) )
9998fveq1d 5688 . . . . 5  |-  ( ph  ->  ( (coeff `  ( F  oF  -  ( H  oF  x.  G
) ) ) `  M )  =  ( ( A  oF  -  (coeff `  ( H  oF  x.  G
) ) ) `  M ) )
1009coef3 21675 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
101 ffn 5554 . . . . . . . 8  |-  ( A : NN0 --> CC  ->  A  Fn  NN0 )
1021, 100, 1013syl 20 . . . . . . 7  |-  ( ph  ->  A  Fn  NN0 )
10396coef3 21675 . . . . . . . 8  |-  ( ( H  oF  x.  G )  e.  (Poly `  S )  ->  (coeff `  ( H  oF  x.  G ) ) : NN0 --> CC )
104 ffn 5554 . . . . . . . 8  |-  ( (coeff `  ( H  oF  x.  G ) ) : NN0 --> CC  ->  (coeff `  ( H  oF  x.  G ) )  Fn  NN0 )
10551, 103, 1043syl 20 . . . . . . 7  |-  ( ph  ->  (coeff `  ( H  oF  x.  G
) )  Fn  NN0 )
106 nn0ex 10577 . . . . . . . 8  |-  NN0  e.  _V
107106a1i 11 . . . . . . 7  |-  ( ph  ->  NN0  e.  _V )
108 inidm 3554 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
109 eqidd 2439 . . . . . . 7  |-  ( (
ph  /\  M  e.  NN0 )  ->  ( A `  M )  =  ( A `  M ) )
110 eqid 2438 . . . . . . . . . . 11  |-  (coeff `  H )  =  (coeff `  H )
111110, 19, 79, 22coemulhi 21696 . . . . . . . . . 10  |-  ( ( H  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( H  oF  x.  G ) ) `
 ( (deg `  H )  +  N
) )  =  ( ( (coeff `  H
) `  (deg `  H
) )  x.  ( B `  N )
) )
11250, 18, 111syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( (coeff `  ( H  oF  x.  G
) ) `  (
(deg `  H )  +  N ) )  =  ( ( (coeff `  H ) `  (deg `  H ) )  x.  ( B `  N
) ) )
11390fveq2d 5690 . . . . . . . . 9  |-  ( ph  ->  ( (coeff `  ( H  oF  x.  G
) ) `  (
(deg `  H )  +  N ) )  =  ( (coeff `  ( H  oF  x.  G
) ) `  M
) )
11483fveq2d 5690 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coeff `  H
) `  (deg `  H
) )  =  ( (coeff `  H ) `  D ) )
115114, 68eqtrd 2470 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  H
) `  (deg `  H
) )  =  ( ( A `  M
)  /  ( B `
 N ) ) )
116115oveq1d 6101 . . . . . . . . . 10  |-  ( ph  ->  ( ( (coeff `  H ) `  (deg `  H ) )  x.  ( B `  N
) )  =  ( ( ( A `  M )  /  ( B `  N )
)  x.  ( B `
 N ) ) )
11717, 27, 32divcan1d 10100 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A `
 M )  / 
( B `  N
) )  x.  ( B `  N )
)  =  ( A `
 M ) )
118116, 117eqtrd 2470 . . . . . . . . 9  |-  ( ph  ->  ( ( (coeff `  H ) `  (deg `  H ) )  x.  ( B `  N
) )  =  ( A `  M ) )
119112, 113, 1183eqtr3d 2478 . . . . . . . 8  |-  ( ph  ->  ( (coeff `  ( H  oF  x.  G
) ) `  M
)  =  ( A `
 M ) )
120119adantr 465 . . . . . . 7  |-  ( (
ph  /\  M  e.  NN0 )  ->  ( (coeff `  ( H  oF  x.  G ) ) `
 M )  =  ( A `  M
) )
121102, 105, 107, 107, 108, 109, 120ofval 6324 . . . . . 6  |-  ( (
ph  /\  M  e.  NN0 )  ->  ( ( A  oF  -  (coeff `  ( H  oF  x.  G ) ) ) `  M )  =  ( ( A `
 M )  -  ( A `  M ) ) )
12215, 121mpdan 668 . . . . 5  |-  ( ph  ->  ( ( A  oF  -  (coeff `  ( H  oF  x.  G
) ) ) `  M )  =  ( ( A `  M
)  -  ( A `
 M ) ) )
12317subidd 9699 . . . . 5  |-  ( ph  ->  ( ( A `  M )  -  ( A `  M )
)  =  0 )
12499, 122, 1233eqtrd 2474 . . . 4  |-  ( ph  ->  ( (coeff `  ( F  oF  -  ( H  oF  x.  G
) ) ) `  M )  =  0 )
125 dgrcl 21676 . . . . . . . . . 10  |-  ( ( F  oF  -  ( H  oF  x.  G ) )  e.  (Poly `  S )  ->  (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  e. 
NN0 )
12652, 125syl 16 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  e. 
NN0 )
127126nn0red 10629 . . . . . . . 8  |-  ( ph  ->  (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  e.  RR )
12815nn0red 10629 . . . . . . . 8  |-  ( ph  ->  M  e.  RR )
12925nn0red 10629 . . . . . . . 8  |-  ( ph  ->  N  e.  RR )
130127, 128, 129ltsub1d 9940 . . . . . . 7  |-  ( ph  ->  ( (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  < 
M  <->  ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  -  N
)  <  ( M  -  N ) ) )
13184breq2d 4299 . . . . . . 7  |-  ( ph  ->  ( ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  -  N
)  <  ( M  -  N )  <->  ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  -  N
)  <  D )
)
132130, 131bitrd 253 . . . . . 6  |-  ( ph  ->  ( (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  < 
M  <->  ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  -  N
)  <  D )
)
133132orbi2d 701 . . . . 5  |-  ( ph  ->  ( ( ( F  oF  -  ( H  oF  x.  G
) )  =  0p  \/  (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  <  M
)  <->  ( ( F  oF  -  ( H  oF  x.  G
) )  =  0p  \/  ( (deg
`  ( F  oF  -  ( H  oF  x.  G
) ) )  -  N )  <  D
) ) )
134 eqid 2438 . . . . . . 7  |-  (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  =  (deg
`  ( F  oF  -  ( H  oF  x.  G
) ) )
135 eqid 2438 . . . . . . 7  |-  (coeff `  ( F  oF  -  ( H  oF  x.  G )
) )  =  (coeff `  ( F  oF  -  ( H  oF  x.  G )
) )
136134, 135dgrlt 21708 . . . . . 6  |-  ( ( ( F  oF  -  ( H  oF  x.  G )
)  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  (
( ( F  oF  -  ( H  oF  x.  G
) )  =  0p  \/  (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  <  M
)  <->  ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  <_  M  /\  ( (coeff `  ( F  oF  -  ( H  oF  x.  G
) ) ) `  M )  =  0 ) ) )
13752, 15, 136syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( ( F  oF  -  ( H  oF  x.  G
) )  =  0p  \/  (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  <  M
)  <->  ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  <_  M  /\  ( (coeff `  ( F  oF  -  ( H  oF  x.  G
) ) ) `  M )  =  0 ) ) )
138133, 137bitr3d 255 . . . 4  |-  ( ph  ->  ( ( ( F  oF  -  ( H  oF  x.  G
) )  =  0p  \/  ( (deg
`  ( F  oF  -  ( H  oF  x.  G
) ) )  -  N )  <  D
)  <->  ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  <_  M  /\  ( (coeff `  ( F  oF  -  ( H  oF  x.  G
) ) ) `  M )  =  0 ) ) )
13995, 124, 138mpbir2and 913 . . 3  |-  ( ph  ->  ( ( F  oF  -  ( H  oF  x.  G
) )  =  0p  \/  ( (deg
`  ( F  oF  -  ( H  oF  x.  G
) ) )  -  N )  <  D
) )
140 eqeq1 2444 . . . . . 6  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (
f  =  0p  <-> 
( F  oF  -  ( H  oF  x.  G )
)  =  0p ) )
141 fveq2 5686 . . . . . . . 8  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (deg `  f )  =  (deg
`  ( F  oF  -  ( H  oF  x.  G
) ) ) )
142141oveq1d 6101 . . . . . . 7  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (
(deg `  f )  -  N )  =  ( (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  -  N ) )
143142breq1d 4297 . . . . . 6  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (
( (deg `  f
)  -  N )  <  D  <->  ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  -  N
)  <  D )
)
144140, 143orbi12d 709 . . . . 5  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  N )  <  D
)  <->  ( ( F  oF  -  ( H  oF  x.  G
) )  =  0p  \/  ( (deg
`  ( F  oF  -  ( H  oF  x.  G
) ) )  -  N )  <  D
) ) )
145 plydiv.u . . . . . . . . 9  |-  U  =  ( f  oF  -  ( G  oF  x.  p )
)
146 oveq1 6093 . . . . . . . . 9  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (
f  oF  -  ( G  oF  x.  p ) )  =  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )
147145, 146syl5eq 2482 . . . . . . . 8  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  U  =  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )
148147eqeq1d 2446 . . . . . . 7  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  ( U  =  0p  <->  ( ( F  oF  -  ( H  oF  x.  G )
)  oF  -  ( G  oF  x.  p ) )  =  0p ) )
149147fveq2d 5690 . . . . . . . 8  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (deg `  U )  =  (deg
`  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) ) )
150149breq1d 4297 . . . . . . 7  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (
(deg `  U )  <  N  <->  (deg `  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )  <  N
) )
151148, 150orbi12d 709 . . . . . 6  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (
( U  =  0p  \/  (deg `  U )  <  N
)  <->  ( ( ( F  oF  -  ( H  oF  x.  G ) )  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )  <  N
) ) )
152151rexbidv 2731 . . . . 5  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  ( E. p  e.  (Poly `  S ) ( U  =  0p  \/  (deg `  U )  <  N )  <->  E. p  e.  (Poly `  S )
( ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
)  =  0p  \/  (deg `  (
( F  oF  -  ( H  oF  x.  G )
)  oF  -  ( G  oF  x.  p ) ) )  <  N ) ) )
153144, 152imbi12d 320 . . . 4  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (
( ( f  =  0p  \/  (
(deg `  f )  -  N )  <  D
)  ->  E. p  e.  (Poly `  S )
( U  =  0p  \/  (deg `  U )  <  N
) )  <->  ( (
( F  oF  -  ( H  oF  x.  G )
)  =  0p  \/  ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  -  N
)  <  D )  ->  E. p  e.  (Poly `  S ) ( ( ( F  oF  -  ( H  oF  x.  G )
)  oF  -  ( G  oF  x.  p ) )  =  0p  \/  (deg `  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )  <  N
) ) ) )
154153rspcv 3064 . . 3  |-  ( ( F  oF  -  ( H  oF  x.  G ) )  e.  (Poly `  S )  ->  ( A. f  e.  (Poly `  S )
( ( f  =  0p  \/  (
(deg `  f )  -  N )  <  D
)  ->  E. p  e.  (Poly `  S )
( U  =  0p  \/  (deg `  U )  <  N
) )  ->  (
( ( F  oF  -  ( H  oF  x.  G
) )  =  0p  \/  ( (deg
`  ( F  oF  -  ( H  oF  x.  G
) ) )  -  N )  <  D
)  ->  E. p  e.  (Poly `  S )
( ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
)  =  0p  \/  (deg `  (
( F  oF  -  ( H  oF  x.  G )
)  oF  -  ( G  oF  x.  p ) ) )  <  N ) ) ) )
15552, 53, 139, 154syl3c 61 . 2  |-  ( ph  ->  E. p  e.  (Poly `  S ) ( ( ( F  oF  -  ( H  oF  x.  G )
)  oF  -  ( G  oF  x.  p ) )  =  0p  \/  (deg `  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )  <  N
) )
15650adantr 465 . . . . . . 7  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  H  e.  (Poly `  S ) )
157 simpr 461 . . . . . . 7  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  p  e.  (Poly `  S ) )
1584adantlr 714 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
159156, 157, 158plyadd 21660 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( H  oF  +  p )  e.  (Poly `  S )
)
160159adantr 465 . . . . 5  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( (
( F  oF  -  ( H  oF  x.  G )
)  oF  -  ( G  oF  x.  p ) )  =  0p  \/  (deg `  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )  <  N
) )  ->  ( H  oF  +  p
)  e.  (Poly `  S ) )
161 cnex 9355 . . . . . . . . . . 11  |-  CC  e.  _V
162161a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  CC  e.  _V )
1631adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  F  e.  (Poly `  S ) )
164 plyf 21641 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
165163, 164syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  F : CC --> CC )
166 mulcl 9358 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
167166adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
168 plyf 21641 . . . . . . . . . . . 12  |-  ( H  e.  (Poly `  S
)  ->  H : CC
--> CC )
169156, 168syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  H : CC --> CC )
17018adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  G  e.  (Poly `  S ) )
171 plyf 21641 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
172170, 171syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  G : CC --> CC )
173 inidm 3554 . . . . . . . . . . 11  |-  ( CC 
i^i  CC )  =  CC
174167, 169, 172, 162, 162, 173off 6329 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( H  oF  x.  G ) : CC --> CC )
175 plyf 21641 . . . . . . . . . . . 12  |-  ( p  e.  (Poly `  S
)  ->  p : CC
--> CC )
176175adantl 466 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  p : CC --> CC )
177167, 172, 176, 162, 162, 173off 6329 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( G  oF  x.  p ) : CC --> CC )
178 subsub4 9634 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  -  y
)  -  z )  =  ( x  -  ( y  +  z ) ) )
179178adantl 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( ( x  -  y )  -  z
)  =  ( x  -  ( y  +  z ) ) )
180162, 165, 174, 177, 179caofass 6349 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
)  =  ( F  oF  -  (
( H  oF  x.  G )  oF  +  ( G  oF  x.  p
) ) ) )
181 mulcom 9360 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
182181adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  =  ( y  x.  x ) )
183162, 169, 172, 182caofcom 6347 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( H  oF  x.  G )  =  ( G  oF  x.  H )
)
184183oveq1d 6101 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( H  oF  x.  G
)  oF  +  ( G  oF  x.  p ) )  =  ( ( G  oF  x.  H )  oF  +  ( G  oF  x.  p
) ) )
185 adddi 9363 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z
) ) )
186185adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( x  x.  (
y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z ) ) )
187162, 172, 169, 176, 186caofdi 6351 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( G  oF  x.  ( H  oF  +  p
) )  =  ( ( G  oF  x.  H )  oF  +  ( G  oF  x.  p
) ) )
188184, 187eqtr4d 2473 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( H  oF  x.  G
)  oF  +  ( G  oF  x.  p ) )  =  ( G  oF  x.  ( H  oF  +  p )
) )
189188oveq2d 6102 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( F  oF  -  ( ( H  oF  x.  G
)  oF  +  ( G  oF  x.  p ) ) )  =  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )
190180, 189eqtrd 2470 . . . . . . . 8  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
)  =  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )
191190eqeq1d 2446 . . . . . . 7  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( ( F  oF  -  ( H  oF  x.  G ) )  oF  -  ( G  oF  x.  p
) )  =  0p  <->  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) )  =  0p ) )
192190fveq2d 5690 . . . . . . . 8  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  (deg `  (
( F  oF  -  ( H  oF  x.  G )
)  oF  -  ( G  oF  x.  p ) ) )  =  (deg `  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) ) )
193192breq1d 4297 . . . . . . 7  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( (deg `  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )  <  N  <->  (deg
`  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )  <  N ) )
194191, 193orbi12d 709 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( ( ( F  oF  -  ( H  oF  x.  G )
)  oF  -  ( G  oF  x.  p ) )  =  0p  \/  (deg `  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )  <  N
)  <->  ( ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )  <  N ) ) )
195194biimpa 484 . . . . 5  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( (
( F  oF  -  ( H  oF  x.  G )
)  oF  -  ( G  oF  x.  p ) )  =  0p  \/  (deg `  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )  <  N
) )  ->  (
( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )  <  N ) )
196 plydiv.r . . . . . . . . 9  |-  R  =  ( F  oF  -  ( G  oF  x.  q )
)
197 oveq2 6094 . . . . . . . . . 10  |-  ( q  =  ( H  oF  +  p )  ->  ( G  oF  x.  q )  =  ( G  oF  x.  ( H  oF  +  p )
) )
198197oveq2d 6102 . . . . . . . . 9  |-  ( q  =  ( H  oF  +  p )  ->  ( F  oF  -  ( G  oF  x.  q )
)  =  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )
199196, 198syl5eq 2482 . . . . . . . 8  |-  ( q  =  ( H  oF  +  p )  ->  R  =  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )
200199eqeq1d 2446 . . . . . . 7  |-  ( q  =  ( H  oF  +  p )  ->  ( R  =  0p  <->  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) )  =  0p ) )
201199fveq2d 5690 . . . . . . . 8  |-  ( q  =  ( H  oF  +  p )  ->  (deg `  R )  =  (deg `  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) ) )
202201breq1d 4297 . . . . . . 7  |-  ( q  =  ( H  oF  +  p )  ->  ( (deg `  R
)  <  N  <->  (deg `  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )  <  N ) )
203200, 202orbi12d 709 . . . . . 6  |-  ( q  =  ( H  oF  +  p )  ->  ( ( R  =  0p  \/  (deg `  R )  <  N
)  <->  ( ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )  <  N ) ) )
204203rspcev 3068 . . . . 5  |-  ( ( ( H  oF  +  p )  e.  (Poly `  S )  /\  ( ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )  <  N ) )  ->  E. q  e.  (Poly `  S ) ( R  =  0p  \/  (deg `  R )  <  N ) )
205160, 195, 204syl2anc 661 . . . 4  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( (
( F  oF  -  ( H  oF  x.  G )
)  oF  -  ( G  oF  x.  p ) )  =  0p  \/  (deg `  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )  <  N
) )  ->  E. q  e.  (Poly `  S )
( R  =  0p  \/  (deg `  R )  <  N
) )
206205ex 434 . . 3  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( ( ( F  oF  -  ( H  oF  x.  G )
)  oF  -  ( G  oF  x.  p ) )  =  0p  \/  (deg `  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )  <  N
)  ->  E. q  e.  (Poly `  S )
( R  =  0p  \/  (deg `  R )  <  N
) ) )
207206rexlimdva 2836 . 2  |-  ( ph  ->  ( E. p  e.  (Poly `  S )
( ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
)  =  0p  \/  (deg `  (
( F  oF  -  ( H  oF  x.  G )
)  oF  -  ( G  oF  x.  p ) ) )  <  N )  ->  E. q  e.  (Poly `  S ) ( R  =  0p  \/  (deg `  R )  <  N ) ) )
208155, 207mpd 15 1  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0p  \/  (deg `  R )  <  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   E.wrex 2711   _Vcvv 2967    C_ wss 3323   ifcif 3786   {csn 3872   class class class wbr 4287    e. cmpt 4345    X. cxp 4833    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086    oFcof 6313   CCcc 9272   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    < clt 9410    <_ cle 9411    - cmin 9587   -ucneg 9588    / cdiv 9985   NN0cn0 10571   ^cexp 11857   0pc0p 21122  Polycply 21627  coeffccoe 21629  degcdgr 21630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-0p 21123  df-ply 21631  df-coe 21633  df-dgr 21634
This theorem is referenced by:  plydivex  21738
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