MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  plydivlem4 Structured version   Unicode version

Theorem plydivlem4 21888
Description: Lemma for plydivex 21889. 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 21788 . . . . . . 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 21885 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  S )
9 plydiv.a . . . . . . . . . . . 12  |-  A  =  (coeff `  F )
109coef2 21825 . . . . . . . . . . 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 21827 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
141, 13syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  F )  e.  NN0 )
1512, 14syl5eqel 2543 . . . . . . . . . 10  |-  ( ph  ->  M  e.  NN0 )
1611, 15ffvelrnd 5946 . . . . . . . . 9  |-  ( ph  ->  ( A `  M
)  e.  S )
173, 16sseldd 3458 . . . . . . . 8  |-  ( ph  ->  ( A `  M
)  e.  CC )
18 plydiv.g . . . . . . . . . . 11  |-  ( ph  ->  G  e.  (Poly `  S ) )
19 plydiv.b . . . . . . . . . . . 12  |-  B  =  (coeff `  G )
2019coef2 21825 . . . . . . . . . . 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 21827 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
2418, 23syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  G )  e.  NN0 )
2522, 24syl5eqel 2543 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
2621, 25ffvelrnd 5946 . . . . . . . . 9  |-  ( ph  ->  ( B `  N
)  e.  S )
273, 26sseldd 3458 . . . . . . . 8  |-  ( ph  ->  ( B `  N
)  e.  CC )
28 plydiv.z . . . . . . . . 9  |-  ( ph  ->  G  =/=  0p )
2922, 19dgreq0 21858 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0p  <->  ( B `  N )  =  0 ) )
3018, 29syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( G  =  0p  <->  ( B `  N )  =  0 ) )
3130necon3bid 2706 . . . . . . . . 9  |-  ( ph  ->  ( G  =/=  0p 
<->  ( B `  N
)  =/=  0 ) )
3228, 31mpbid 210 . . . . . . . 8  |-  ( ph  ->  ( B `  N
)  =/=  0 )
3317, 27, 32divrecd 10214 . . . . . . 7  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  =  ( ( A `  M )  x.  ( 1  / 
( B `  N
) ) ) )
34 fvex 5802 . . . . . . . . . . 11  |-  ( B `
 N )  e. 
_V
35 eleq1 2523 . . . . . . . . . . . . . 14  |-  ( x  =  ( B `  N )  ->  (
x  e.  S  <->  ( B `  N )  e.  S
) )
36 neeq1 2729 . . . . . . . . . . . . . 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 6201 . . . . . . . . . . . . 13  |-  ( x  =  ( B `  N )  ->  (
1  /  x )  =  ( 1  / 
( B `  N
) ) )
4039eleq1d 2520 . . . . . . . . . . . 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 3123 . . . . . . . . . 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 6359 . . . . . . 7  |-  ( ph  ->  ( ( A `  M )  x.  (
1  /  ( B `
 N ) ) )  e.  S )
4633, 45eqeltrd 2539 . . . . . 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 21798 . . . . . 6  |-  ( ( S  C_  CC  /\  (
( A `  M
)  /  ( B `
 N ) )  e.  S  /\  D  e.  NN0 )  ->  H  e.  (Poly `  S )
)
503, 46, 47, 49syl3anc 1219 . . . . 5  |-  ( ph  ->  H  e.  (Poly `  S ) )
5150, 18, 4, 5plymul 21812 . . . 4  |-  ( ph  ->  ( H  oF  x.  G )  e.  (Poly `  S )
)
521, 51, 4, 5, 7plysub 21813 . . 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 2451 . . . . . . 7  |-  (deg `  ( H  oF  x.  G ) )  =  (deg `  ( H  oF  x.  G
) )
5512, 54dgrsub 21865 . . . . . 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 21858 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0p  <->  ( A `  M )  =  0 ) )
591, 58syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  =  0p  <->  ( A `  M )  =  0 ) )
6059necon3bid 2706 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  =/=  0p 
<->  ( A `  M
)  =/=  0 ) )
6157, 60mpbid 210 . . . . . . . . . . . 12  |-  ( ph  ->  ( A `  M
)  =/=  0 )
6217, 27, 61, 32divne0d 10227 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  =/=  0 )
633, 46sseldd 3458 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  e.  CC )
6448coe1term 21852 . . . . . . . . . . . . 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 1219 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coeff `  H
) `  D )  =  if ( D  =  D ,  ( ( A `  M )  /  ( B `  N ) ) ,  0 ) )
66 eqid 2451 . . . . . . . . . . . . 13  |-  D  =  D
6766iftruei 3899 . . . . . . . . . . . 12  |-  if ( D  =  D , 
( ( A `  M )  /  ( B `  N )
) ,  0 )  =  ( ( A `
 M )  / 
( B `  N
) )
6865, 67syl6eq 2508 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  H
) `  D )  =  ( ( A `
 M )  / 
( B `  N
) ) )
69 c0ex 9484 . . . . . . . . . . . . 13  |-  0  e.  _V
7069fvconst2 6035 . . . . . . . . . . . 12  |-  ( D  e.  NN0  ->  ( ( NN0  X.  { 0 } ) `  D
)  =  0 )
7147, 70syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ( NN0  X.  { 0 } ) `
 D )  =  0 )
7262, 68, 713netr4d 2753 . . . . . . . . . 10  |-  ( ph  ->  ( (coeff `  H
) `  D )  =/=  ( ( NN0  X.  { 0 } ) `
 D ) )
73 fveq2 5792 . . . . . . . . . . . . 13  |-  ( H  =  0p  -> 
(coeff `  H )  =  (coeff `  0p
) )
74 coe0 21849 . . . . . . . . . . . . 13  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
7573, 74syl6eq 2508 . . . . . . . . . . . 12  |-  ( H  =  0p  -> 
(coeff `  H )  =  ( NN0  X.  { 0 } ) )
7675fveq1d 5794 . . . . . . . . . . 11  |-  ( H  =  0p  -> 
( (coeff `  H
) `  D )  =  ( ( NN0 
X.  { 0 } ) `  D ) )
7776necon3i 2688 . . . . . . . . . 10  |-  ( ( (coeff `  H ) `  D )  =/=  (
( NN0  X.  { 0 } ) `  D
)  ->  H  =/=  0p )
7872, 77syl 16 . . . . . . . . 9  |-  ( ph  ->  H  =/=  0p )
79 eqid 2451 . . . . . . . . . 10  |-  (deg `  H )  =  (deg
`  H )
8079, 22dgrmul 21863 . . . . . . . . 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 1220 . . . . . . . 8  |-  ( ph  ->  (deg `  ( H  oF  x.  G
) )  =  ( (deg `  H )  +  N ) )
8248dgr1term 21853 . . . . . . . . . . . 12  |-  ( ( ( ( A `  M )  /  ( B `  N )
)  e.  CC  /\  ( ( A `  M )  /  ( B `  N )
)  =/=  0  /\  D  e.  NN0 )  ->  (deg `  H )  =  D )
8363, 62, 47, 82syl3anc 1219 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  H )  =  D )
84 plydiv.e . . . . . . . . . . 11  |-  ( ph  ->  ( M  -  N
)  =  D )
8583, 84eqtr4d 2495 . . . . . . . . . 10  |-  ( ph  ->  (deg `  H )  =  ( M  -  N ) )
8685oveq1d 6208 . . . . . . . . 9  |-  ( ph  ->  ( (deg `  H
)  +  N )  =  ( ( M  -  N )  +  N ) )
8715nn0cnd 10742 . . . . . . . . . 10  |-  ( ph  ->  M  e.  CC )
8825nn0cnd 10742 . . . . . . . . . 10  |-  ( ph  ->  N  e.  CC )
8987, 88npcand 9827 . . . . . . . . 9  |-  ( ph  ->  ( ( M  -  N )  +  N
)  =  M )
9086, 89eqtrd 2492 . . . . . . . 8  |-  ( ph  ->  ( (deg `  H
)  +  N )  =  M )
9181, 90eqtrd 2492 . . . . . . 7  |-  ( ph  ->  (deg `  ( H  oF  x.  G
) )  =  M )
9291ifeq1d 3908 . . . . . 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 3927 . . . . . 6  |-  if ( M  <_  (deg `  ( H  oF  x.  G
) ) ,  M ,  M )  =  M
9492, 93syl6eq 2508 . . . . 5  |-  ( ph  ->  if ( M  <_ 
(deg `  ( H  oF  x.  G
) ) ,  (deg
`  ( H  oF  x.  G )
) ,  M )  =  M )
9556, 94breqtrd 4417 . . . 4  |-  ( ph  ->  (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  <_  M )
96 eqid 2451 . . . . . . . 8  |-  (coeff `  ( H  oF  x.  G ) )  =  (coeff `  ( H  oF  x.  G
) )
979, 96coesub 21850 . . . . . . 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 5794 . . . . 5  |-  ( ph  ->  ( (coeff `  ( F  oF  -  ( H  oF  x.  G
) ) ) `  M )  =  ( ( A  oF  -  (coeff `  ( H  oF  x.  G
) ) ) `  M ) )
1009coef3 21826 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
101 ffn 5660 . . . . . . . 8  |-  ( A : NN0 --> CC  ->  A  Fn  NN0 )
1021, 100, 1013syl 20 . . . . . . 7  |-  ( ph  ->  A  Fn  NN0 )
10396coef3 21826 . . . . . . . 8  |-  ( ( H  oF  x.  G )  e.  (Poly `  S )  ->  (coeff `  ( H  oF  x.  G ) ) : NN0 --> CC )
104 ffn 5660 . . . . . . . 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 10689 . . . . . . . 8  |-  NN0  e.  _V
107106a1i 11 . . . . . . 7  |-  ( ph  ->  NN0  e.  _V )
108 inidm 3660 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
109 eqidd 2452 . . . . . . 7  |-  ( (
ph  /\  M  e.  NN0 )  ->  ( A `  M )  =  ( A `  M ) )
110 eqid 2451 . . . . . . . . . . 11  |-  (coeff `  H )  =  (coeff `  H )
111110, 19, 79, 22coemulhi 21847 . . . . . . . . . 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 5796 . . . . . . . . 9  |-  ( ph  ->  ( (coeff `  ( H  oF  x.  G
) ) `  (
(deg `  H )  +  N ) )  =  ( (coeff `  ( H  oF  x.  G
) ) `  M
) )
11483fveq2d 5796 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coeff `  H
) `  (deg `  H
) )  =  ( (coeff `  H ) `  D ) )
115114, 68eqtrd 2492 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  H
) `  (deg `  H
) )  =  ( ( A `  M
)  /  ( B `
 N ) ) )
116115oveq1d 6208 . . . . . . . . . 10  |-  ( ph  ->  ( ( (coeff `  H ) `  (deg `  H ) )  x.  ( B `  N
) )  =  ( ( ( A `  M )  /  ( B `  N )
)  x.  ( B `
 N ) ) )
11717, 27, 32divcan1d 10212 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A `
 M )  / 
( B `  N
) )  x.  ( B `  N )
)  =  ( A `
 M ) )
118116, 117eqtrd 2492 . . . . . . . . 9  |-  ( ph  ->  ( ( (coeff `  H ) `  (deg `  H ) )  x.  ( B `  N
) )  =  ( A `  M ) )
119112, 113, 1183eqtr3d 2500 . . . . . . . 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 6432 . . . . . 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 9811 . . . . 5  |-  ( ph  ->  ( ( A `  M )  -  ( A `  M )
)  =  0 )
12499, 122, 1233eqtrd 2496 . . . 4  |-  ( ph  ->  ( (coeff `  ( F  oF  -  ( H  oF  x.  G
) ) ) `  M )  =  0 )
125 dgrcl 21827 . . . . . . . . . 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 10741 . . . . . . . 8  |-  ( ph  ->  (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  e.  RR )
12815nn0red 10741 . . . . . . . 8  |-  ( ph  ->  M  e.  RR )
12925nn0red 10741 . . . . . . . 8  |-  ( ph  ->  N  e.  RR )
130127, 128, 129ltsub1d 10052 . . . . . . 7  |-  ( ph  ->  ( (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  < 
M  <->  ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  -  N
)  <  ( M  -  N ) ) )
13184breq2d 4405 . . . . . . 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 2451 . . . . . . 7  |-  (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  =  (deg
`  ( F  oF  -  ( H  oF  x.  G
) ) )
135 eqid 2451 . . . . . . 7  |-  (coeff `  ( F  oF  -  ( H  oF  x.  G )
) )  =  (coeff `  ( F  oF  -  ( H  oF  x.  G )
) )
136134, 135dgrlt 21859 . . . . . 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 2455 . . . . . 6  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (
f  =  0p  <-> 
( F  oF  -  ( H  oF  x.  G )
)  =  0p ) )
141 fveq2 5792 . . . . . . . 8  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (deg `  f )  =  (deg
`  ( F  oF  -  ( H  oF  x.  G
) ) ) )
142141oveq1d 6208 . . . . . . 7  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (
(deg `  f )  -  N )  =  ( (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  -  N ) )
143142breq1d 4403 . . . . . 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 6200 . . . . . . . . 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 2504 . . . . . . . 8  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  U  =  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )
148147eqeq1d 2453 . . . . . . 7  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  ( U  =  0p  <->  ( ( F  oF  -  ( H  oF  x.  G )
)  oF  -  ( G  oF  x.  p ) )  =  0p ) )
149147fveq2d 5796 . . . . . . . 8  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (deg `  U )  =  (deg
`  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) ) )
150149breq1d 4403 . . . . . . 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 2855 . . . . 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 3168 . . 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 21811 . . . . . 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 9467 . . . . . . . . . . 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 21792 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
165163, 164syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  F : CC --> CC )
166 mulcl 9470 . . . . . . . . . . . 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 21792 . . . . . . . . . . . 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 21792 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
172170, 171syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  G : CC --> CC )
173 inidm 3660 . . . . . . . . . . 11  |-  ( CC 
i^i  CC )  =  CC
174167, 169, 172, 162, 162, 173off 6437 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( H  oF  x.  G ) : CC --> CC )
175 plyf 21792 . . . . . . . . . . . 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 6437 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( G  oF  x.  p ) : CC --> CC )
178 subsub4 9746 . . . . . . . . . . 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 6457 . . . . . . . . 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 9472 . . . . . . . . . . . . . 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 6455 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( H  oF  x.  G )  =  ( G  oF  x.  H )
)
184183oveq1d 6208 . . . . . . . . . . 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 9475 . . . . . . . . . . . . 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 6459 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( G  oF  x.  ( H  oF  +  p
) )  =  ( ( G  oF  x.  H )  oF  +  ( G  oF  x.  p
) ) )
188184, 187eqtr4d 2495 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( H  oF  x.  G
)  oF  +  ( G  oF  x.  p ) )  =  ( G  oF  x.  ( H  oF  +  p )
) )
189188oveq2d 6209 . . . . . . . . 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 2492 . . . . . . . 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 2453 . . . . . . 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 5796 . . . . . . . 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 4403 . . . . . . 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 6201 . . . . . . . . . 10  |-  ( q  =  ( H  oF  +  p )  ->  ( G  oF  x.  q )  =  ( G  oF  x.  ( H  oF  +  p )
) )
198197oveq2d 6209 . . . . . . . . 9  |-  ( q  =  ( H  oF  +  p )  ->  ( F  oF  -  ( G  oF  x.  q )
)  =  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )
199196, 198syl5eq 2504 . . . . . . . 8  |-  ( q  =  ( H  oF  +  p )  ->  R  =  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )
200199eqeq1d 2453 . . . . . . 7  |-  ( q  =  ( H  oF  +  p )  ->  ( R  =  0p  <->  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) )  =  0p ) )
201199fveq2d 5796 . . . . . . . 8  |-  ( q  =  ( H  oF  +  p )  ->  (deg `  R )  =  (deg `  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) ) )
202201breq1d 4403 . . . . . . 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 3172 . . . . 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 2940 . 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 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796   _Vcvv 3071    C_ wss 3429   ifcif 3892   {csn 3978   class class class wbr 4393    |-> cmpt 4451    X. cxp 4939    Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193    oFcof 6421   CCcc 9384   0cc0 9386   1c1 9387    + caddc 9389    x. cmul 9391    < clt 9522    <_ cle 9523    - cmin 9699   -ucneg 9700    / cdiv 10097   NN0cn0 10683   ^cexp 11975   0pc0p 21273  Polycply 21778  coeffccoe 21780  degcdgr 21781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-fz 11548  df-fzo 11659  df-fl 11752  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-rlim 13078  df-sum 13275  df-0p 21274  df-ply 21782  df-coe 21784  df-dgr 21785
This theorem is referenced by:  plydivex  21889
  Copyright terms: Public domain W3C validator