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Theorem plydivlem4 20214
Description: Lemma for plydivex 20215. 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  =/=  0 p )
plydiv.r  |-  R  =  ( F  o F  -  ( G  o F  x.  q )
)
plydiv.d  |-  ( ph  ->  D  e.  NN0 )
plydiv.e  |-  ( ph  ->  ( M  -  N
)  =  D )
plydiv.fz  |-  ( ph  ->  F  =/=  0 p )
plydiv.u  |-  U  =  ( f  o F  -  ( G  o F  x.  p )
)
plydiv.h  |-  H  =  ( z  e.  CC  |->  ( ( ( A `
 M )  / 
( B `  N
) )  x.  (
z ^ D ) ) )
plydiv.al  |-  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  N
)  <  D )  ->  E. p  e.  (Poly `  S ) ( U  =  0 p  \/  (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  =  0 p  \/  (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 20114 . . . . . . 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 20211 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  S )
9 plydiv.a . . . . . . . . . . . 12  |-  A  =  (coeff `  F )
109coef2 20151 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  0  e.  S )  ->  A : NN0 --> S )
111, 8, 10syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  A : NN0 --> S )
12 plydiv.m . . . . . . . . . . 11  |-  M  =  (deg `  F )
13 dgrcl 20153 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
141, 13syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  F )  e.  NN0 )
1512, 14syl5eqel 2521 . . . . . . . . . 10  |-  ( ph  ->  M  e.  NN0 )
1611, 15ffvelrnd 5872 . . . . . . . . 9  |-  ( ph  ->  ( A `  M
)  e.  S )
173, 16sseldd 3350 . . . . . . . 8  |-  ( ph  ->  ( A `  M
)  e.  CC )
18 plydiv.g . . . . . . . . . . 11  |-  ( ph  ->  G  e.  (Poly `  S ) )
19 plydiv.b . . . . . . . . . . . 12  |-  B  =  (coeff `  G )
2019coef2 20151 . . . . . . . . . . 11  |-  ( ( G  e.  (Poly `  S )  /\  0  e.  S )  ->  B : NN0 --> S )
2118, 8, 20syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  B : NN0 --> S )
22 plydiv.n . . . . . . . . . . 11  |-  N  =  (deg `  G )
23 dgrcl 20153 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
2418, 23syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  G )  e.  NN0 )
2522, 24syl5eqel 2521 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
2621, 25ffvelrnd 5872 . . . . . . . . 9  |-  ( ph  ->  ( B `  N
)  e.  S )
273, 26sseldd 3350 . . . . . . . 8  |-  ( ph  ->  ( B `  N
)  e.  CC )
28 plydiv.z . . . . . . . . 9  |-  ( ph  ->  G  =/=  0 p )
2922, 19dgreq0 20184 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0 p  <->  ( B `  N )  =  0 ) )
3018, 29syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( G  =  0 p  <->  ( B `  N )  =  0 ) )
3130necon3bid 2637 . . . . . . . . 9  |-  ( ph  ->  ( G  =/=  0 p 
<->  ( B `  N
)  =/=  0 ) )
3228, 31mpbid 203 . . . . . . . 8  |-  ( ph  ->  ( B `  N
)  =/=  0 )
3317, 27, 32divrecd 9794 . . . . . . 7  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  =  ( ( A `  M )  x.  ( 1  / 
( B `  N
) ) ) )
34 fvex 5743 . . . . . . . . . . 11  |-  ( B `
 N )  e. 
_V
35 eleq1 2497 . . . . . . . . . . . . . 14  |-  ( x  =  ( B `  N )  ->  (
x  e.  S  <->  ( B `  N )  e.  S
) )
36 neeq1 2610 . . . . . . . . . . . . . 14  |-  ( x  =  ( B `  N )  ->  (
x  =/=  0  <->  ( B `  N )  =/=  0 ) )
3735, 36anbi12d 693 . . . . . . . . . . . . 13  |-  ( x  =  ( B `  N )  ->  (
( x  e.  S  /\  x  =/=  0
)  <->  ( ( B `
 N )  e.  S  /\  ( B `
 N )  =/=  0 ) ) )
3837anbi2d 686 . . . . . . . . . . . 12  |-  ( x  =  ( B `  N )  ->  (
( ph  /\  (
x  e.  S  /\  x  =/=  0 ) )  <-> 
( ph  /\  (
( B `  N
)  e.  S  /\  ( B `  N )  =/=  0 ) ) ) )
39 oveq2 6090 . . . . . . . . . . . . 13  |-  ( x  =  ( B `  N )  ->  (
1  /  x )  =  ( 1  / 
( B `  N
) ) )
4039eleq1d 2503 . . . . . . . . . . . 12  |-  ( x  =  ( B `  N )  ->  (
( 1  /  x
)  e.  S  <->  ( 1  /  ( B `  N ) )  e.  S ) )
4138, 40imbi12d 313 . . . . . . . . . . 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 3007 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( B `  N )  e.  S  /\  ( B `  N )  =/=  0 ) )  -> 
( 1  /  ( B `  N )
)  e.  S )
4342ex 425 . . . . . . . . 9  |-  ( ph  ->  ( ( ( B `
 N )  e.  S  /\  ( B `
 N )  =/=  0 )  ->  (
1  /  ( B `
 N ) )  e.  S ) )
4426, 32, 43mp2and 662 . . . . . . . 8  |-  ( ph  ->  ( 1  /  ( B `  N )
)  e.  S )
455, 16, 44caovcld 6241 . . . . . . 7  |-  ( ph  ->  ( ( A `  M )  x.  (
1  /  ( B `
 N ) ) )  e.  S )
4633, 45eqeltrd 2511 . . . . . 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 20124 . . . . . 6  |-  ( ( S  C_  CC  /\  (
( A `  M
)  /  ( B `
 N ) )  e.  S  /\  D  e.  NN0 )  ->  H  e.  (Poly `  S )
)
503, 46, 47, 49syl3anc 1185 . . . . 5  |-  ( ph  ->  H  e.  (Poly `  S ) )
5150, 18, 4, 5plymul 20138 . . . 4  |-  ( ph  ->  ( H  o F  x.  G )  e.  (Poly `  S )
)
521, 51, 4, 5, 7plysub 20139 . . 3  |-  ( ph  ->  ( F  o F  -  ( H  o F  x.  G )
)  e.  (Poly `  S ) )
53 plydiv.al . . 3  |-  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  N
)  <  D )  ->  E. p  e.  (Poly `  S ) ( U  =  0 p  \/  (deg `  U )  < 
N ) ) )
54 eqid 2437 . . . . . . 7  |-  (deg `  ( H  o F  x.  G ) )  =  (deg `  ( H  o F  x.  G
) )
5512, 54dgrsub 20191 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  ( H  o F  x.  G
)  e.  (Poly `  S ) )  -> 
(deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  <_  if ( M  <_  (deg `  ( H  o F  x.  G ) ) ,  (deg `  ( H  o F  x.  G
) ) ,  M
) )
561, 51, 55syl2anc 644 . . . . 5  |-  ( ph  ->  (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  <_  if ( M  <_  (deg `  ( H  o F  x.  G ) ) ,  (deg `  ( H  o F  x.  G
) ) ,  M
) )
57 plydiv.fz . . . . . . . . . . . . 13  |-  ( ph  ->  F  =/=  0 p )
5812, 9dgreq0 20184 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0 p  <->  ( A `  M )  =  0 ) )
591, 58syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  =  0 p  <->  ( A `  M )  =  0 ) )
6059necon3bid 2637 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  =/=  0 p 
<->  ( A `  M
)  =/=  0 ) )
6157, 60mpbid 203 . . . . . . . . . . . 12  |-  ( ph  ->  ( A `  M
)  =/=  0 )
6217, 27, 61, 32divne0d 9807 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  =/=  0 )
633, 46sseldd 3350 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  e.  CC )
6448coe1term 20178 . . . . . . . . . . . . 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 1185 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coeff `  H
) `  D )  =  if ( D  =  D ,  ( ( A `  M )  /  ( B `  N ) ) ,  0 ) )
66 eqid 2437 . . . . . . . . . . . . 13  |-  D  =  D
67 iftrue 3746 . . . . . . . . . . . . 13  |-  ( D  =  D  ->  if ( D  =  D ,  ( ( A `
 M )  / 
( B `  N
) ) ,  0 )  =  ( ( A `  M )  /  ( B `  N ) ) )
6866, 67ax-mp 8 . . . . . . . . . . . 12  |-  if ( D  =  D , 
( ( A `  M )  /  ( B `  N )
) ,  0 )  =  ( ( A `
 M )  / 
( B `  N
) )
6965, 68syl6eq 2485 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  H
) `  D )  =  ( ( A `
 M )  / 
( B `  N
) ) )
70 c0ex 9086 . . . . . . . . . . . . 13  |-  0  e.  _V
7170fvconst2 5948 . . . . . . . . . . . 12  |-  ( D  e.  NN0  ->  ( ( NN0  X.  { 0 } ) `  D
)  =  0 )
7247, 71syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ( NN0  X.  { 0 } ) `
 D )  =  0 )
7362, 69, 723netr4d 2629 . . . . . . . . . 10  |-  ( ph  ->  ( (coeff `  H
) `  D )  =/=  ( ( NN0  X.  { 0 } ) `
 D ) )
74 fveq2 5729 . . . . . . . . . . . . 13  |-  ( H  =  0 p  -> 
(coeff `  H )  =  (coeff `  0 p
) )
75 coe0 20175 . . . . . . . . . . . . 13  |-  (coeff ` 
0 p )  =  ( NN0  X.  {
0 } )
7674, 75syl6eq 2485 . . . . . . . . . . . 12  |-  ( H  =  0 p  -> 
(coeff `  H )  =  ( NN0  X.  { 0 } ) )
7776fveq1d 5731 . . . . . . . . . . 11  |-  ( H  =  0 p  -> 
( (coeff `  H
) `  D )  =  ( ( NN0 
X.  { 0 } ) `  D ) )
7877necon3i 2644 . . . . . . . . . 10  |-  ( ( (coeff `  H ) `  D )  =/=  (
( NN0  X.  { 0 } ) `  D
)  ->  H  =/=  0 p )
7973, 78syl 16 . . . . . . . . 9  |-  ( ph  ->  H  =/=  0 p )
80 eqid 2437 . . . . . . . . . 10  |-  (deg `  H )  =  (deg
`  H )
8180, 22dgrmul 20189 . . . . . . . . 9  |-  ( ( ( H  e.  (Poly `  S )  /\  H  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  (deg `  ( H  o F  x.  G
) )  =  ( (deg `  H )  +  N ) )
8250, 79, 18, 28, 81syl22anc 1186 . . . . . . . 8  |-  ( ph  ->  (deg `  ( H  o F  x.  G
) )  =  ( (deg `  H )  +  N ) )
8348dgr1term 20179 . . . . . . . . . . . 12  |-  ( ( ( ( A `  M )  /  ( B `  N )
)  e.  CC  /\  ( ( A `  M )  /  ( B `  N )
)  =/=  0  /\  D  e.  NN0 )  ->  (deg `  H )  =  D )
8463, 62, 47, 83syl3anc 1185 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  H )  =  D )
85 plydiv.e . . . . . . . . . . 11  |-  ( ph  ->  ( M  -  N
)  =  D )
8684, 85eqtr4d 2472 . . . . . . . . . 10  |-  ( ph  ->  (deg `  H )  =  ( M  -  N ) )
8786oveq1d 6097 . . . . . . . . 9  |-  ( ph  ->  ( (deg `  H
)  +  N )  =  ( ( M  -  N )  +  N ) )
8815nn0cnd 10277 . . . . . . . . . 10  |-  ( ph  ->  M  e.  CC )
8925nn0cnd 10277 . . . . . . . . . 10  |-  ( ph  ->  N  e.  CC )
9088, 89npcand 9416 . . . . . . . . 9  |-  ( ph  ->  ( ( M  -  N )  +  N
)  =  M )
9187, 90eqtrd 2469 . . . . . . . 8  |-  ( ph  ->  ( (deg `  H
)  +  N )  =  M )
9282, 91eqtrd 2469 . . . . . . 7  |-  ( ph  ->  (deg `  ( H  o F  x.  G
) )  =  M )
9392ifeq1d 3754 . . . . . 6  |-  ( ph  ->  if ( M  <_ 
(deg `  ( H  o F  x.  G
) ) ,  (deg
`  ( H  o F  x.  G )
) ,  M )  =  if ( M  <_  (deg `  ( H  o F  x.  G
) ) ,  M ,  M ) )
94 ifid 3772 . . . . . 6  |-  if ( M  <_  (deg `  ( H  o F  x.  G
) ) ,  M ,  M )  =  M
9593, 94syl6eq 2485 . . . . 5  |-  ( ph  ->  if ( M  <_ 
(deg `  ( H  o F  x.  G
) ) ,  (deg
`  ( H  o F  x.  G )
) ,  M )  =  M )
9656, 95breqtrd 4237 . . . 4  |-  ( ph  ->  (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  <_  M )
97 eqid 2437 . . . . . . . 8  |-  (coeff `  ( H  o F  x.  G ) )  =  (coeff `  ( H  o F  x.  G
) )
989, 97coesub 20176 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  ( H  o F  x.  G
)  e.  (Poly `  S ) )  -> 
(coeff `  ( F  o F  -  ( H  o F  x.  G
) ) )  =  ( A  o F  -  (coeff `  ( H  o F  x.  G
) ) ) )
991, 51, 98syl2anc 644 . . . . . 6  |-  ( ph  ->  (coeff `  ( F  o F  -  ( H  o F  x.  G
) ) )  =  ( A  o F  -  (coeff `  ( H  o F  x.  G
) ) ) )
10099fveq1d 5731 . . . . 5  |-  ( ph  ->  ( (coeff `  ( F  o F  -  ( H  o F  x.  G
) ) ) `  M )  =  ( ( A  o F  -  (coeff `  ( H  o F  x.  G
) ) ) `  M ) )
1019coef3 20152 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
102 ffn 5592 . . . . . . . 8  |-  ( A : NN0 --> CC  ->  A  Fn  NN0 )
1031, 101, 1023syl 19 . . . . . . 7  |-  ( ph  ->  A  Fn  NN0 )
10497coef3 20152 . . . . . . . 8  |-  ( ( H  o F  x.  G )  e.  (Poly `  S )  ->  (coeff `  ( H  o F  x.  G ) ) : NN0 --> CC )
105 ffn 5592 . . . . . . . 8  |-  ( (coeff `  ( H  o F  x.  G ) ) : NN0 --> CC  ->  (coeff `  ( H  o F  x.  G ) )  Fn  NN0 )
10651, 104, 1053syl 19 . . . . . . 7  |-  ( ph  ->  (coeff `  ( H  o F  x.  G
) )  Fn  NN0 )
107 nn0ex 10228 . . . . . . . 8  |-  NN0  e.  _V
108107a1i 11 . . . . . . 7  |-  ( ph  ->  NN0  e.  _V )
109 inidm 3551 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
110 eqidd 2438 . . . . . . 7  |-  ( (
ph  /\  M  e.  NN0 )  ->  ( A `  M )  =  ( A `  M ) )
111 eqid 2437 . . . . . . . . . . 11  |-  (coeff `  H )  =  (coeff `  H )
112111, 19, 80, 22coemulhi 20173 . . . . . . . . . 10  |-  ( ( H  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( H  o F  x.  G ) ) `
 ( (deg `  H )  +  N
) )  =  ( ( (coeff `  H
) `  (deg `  H
) )  x.  ( B `  N )
) )
11350, 18, 112syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( (coeff `  ( H  o F  x.  G
) ) `  (
(deg `  H )  +  N ) )  =  ( ( (coeff `  H ) `  (deg `  H ) )  x.  ( B `  N
) ) )
11491fveq2d 5733 . . . . . . . . 9  |-  ( ph  ->  ( (coeff `  ( H  o F  x.  G
) ) `  (
(deg `  H )  +  N ) )  =  ( (coeff `  ( H  o F  x.  G
) ) `  M
) )
11584fveq2d 5733 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coeff `  H
) `  (deg `  H
) )  =  ( (coeff `  H ) `  D ) )
116115, 69eqtrd 2469 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  H
) `  (deg `  H
) )  =  ( ( A `  M
)  /  ( B `
 N ) ) )
117116oveq1d 6097 . . . . . . . . . 10  |-  ( ph  ->  ( ( (coeff `  H ) `  (deg `  H ) )  x.  ( B `  N
) )  =  ( ( ( A `  M )  /  ( B `  N )
)  x.  ( B `
 N ) ) )
11817, 27, 32divcan1d 9792 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A `
 M )  / 
( B `  N
) )  x.  ( B `  N )
)  =  ( A `
 M ) )
119117, 118eqtrd 2469 . . . . . . . . 9  |-  ( ph  ->  ( ( (coeff `  H ) `  (deg `  H ) )  x.  ( B `  N
) )  =  ( A `  M ) )
120113, 114, 1193eqtr3d 2477 . . . . . . . 8  |-  ( ph  ->  ( (coeff `  ( H  o F  x.  G
) ) `  M
)  =  ( A `
 M ) )
121120adantr 453 . . . . . . 7  |-  ( (
ph  /\  M  e.  NN0 )  ->  ( (coeff `  ( H  o F  x.  G ) ) `
 M )  =  ( A `  M
) )
122103, 106, 108, 108, 109, 110, 121ofval 6315 . . . . . 6  |-  ( (
ph  /\  M  e.  NN0 )  ->  ( ( A  o F  -  (coeff `  ( H  o F  x.  G ) ) ) `  M )  =  ( ( A `
 M )  -  ( A `  M ) ) )
12315, 122mpdan 651 . . . . 5  |-  ( ph  ->  ( ( A  o F  -  (coeff `  ( H  o F  x.  G
) ) ) `  M )  =  ( ( A `  M
)  -  ( A `
 M ) ) )
12417subidd 9400 . . . . 5  |-  ( ph  ->  ( ( A `  M )  -  ( A `  M )
)  =  0 )
125100, 123, 1243eqtrd 2473 . . . 4  |-  ( ph  ->  ( (coeff `  ( F  o F  -  ( H  o F  x.  G
) ) ) `  M )  =  0 )
126 dgrcl 20153 . . . . . . . . . 10  |-  ( ( F  o F  -  ( H  o F  x.  G ) )  e.  (Poly `  S )  ->  (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  e. 
NN0 )
12752, 126syl 16 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  e. 
NN0 )
128127nn0red 10276 . . . . . . . 8  |-  ( ph  ->  (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  e.  RR )
12915nn0red 10276 . . . . . . . 8  |-  ( ph  ->  M  e.  RR )
13025nn0red 10276 . . . . . . . 8  |-  ( ph  ->  N  e.  RR )
131128, 129, 130ltsub1d 9636 . . . . . . 7  |-  ( ph  ->  ( (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  < 
M  <->  ( (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  -  N
)  <  ( M  -  N ) ) )
13285breq2d 4225 . . . . . . 7  |-  ( ph  ->  ( ( (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  -  N
)  <  ( M  -  N )  <->  ( (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  -  N
)  <  D )
)
133131, 132bitrd 246 . . . . . 6  |-  ( ph  ->  ( (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  < 
M  <->  ( (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  -  N
)  <  D )
)
134133orbi2d 684 . . . . 5  |-  ( ph  ->  ( ( ( F  o F  -  ( H  o F  x.  G
) )  =  0 p  \/  (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  <  M
)  <->  ( ( F  o F  -  ( H  o F  x.  G
) )  =  0 p  \/  ( (deg
`  ( F  o F  -  ( H  o F  x.  G
) ) )  -  N )  <  D
) ) )
135 eqid 2437 . . . . . . 7  |-  (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  =  (deg
`  ( F  o F  -  ( H  o F  x.  G
) ) )
136 eqid 2437 . . . . . . 7  |-  (coeff `  ( F  o F  -  ( H  o F  x.  G )
) )  =  (coeff `  ( F  o F  -  ( H  o F  x.  G )
) )
137135, 136dgrlt 20185 . . . . . 6  |-  ( ( ( F  o F  -  ( H  o F  x.  G )
)  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  (
( ( F  o F  -  ( H  o F  x.  G
) )  =  0 p  \/  (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  <  M
)  <->  ( (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  <_  M  /\  ( (coeff `  ( F  o F  -  ( H  o F  x.  G
) ) ) `  M )  =  0 ) ) )
13852, 15, 137syl2anc 644 . . . . 5  |-  ( ph  ->  ( ( ( F  o F  -  ( H  o F  x.  G
) )  =  0 p  \/  (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  <  M
)  <->  ( (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  <_  M  /\  ( (coeff `  ( F  o F  -  ( H  o F  x.  G
) ) ) `  M )  =  0 ) ) )
139134, 138bitr3d 248 . . . 4  |-  ( ph  ->  ( ( ( F  o F  -  ( H  o F  x.  G
) )  =  0 p  \/  ( (deg
`  ( F  o F  -  ( H  o F  x.  G
) ) )  -  N )  <  D
)  <->  ( (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  <_  M  /\  ( (coeff `  ( F  o F  -  ( H  o F  x.  G
) ) ) `  M )  =  0 ) ) )
14096, 125, 139mpbir2and 890 . . 3  |-  ( ph  ->  ( ( F  o F  -  ( H  o F  x.  G
) )  =  0 p  \/  ( (deg
`  ( F  o F  -  ( H  o F  x.  G
) ) )  -  N )  <  D
) )
141 eqeq1 2443 . . . . . 6  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  (
f  =  0 p  <-> 
( F  o F  -  ( H  o F  x.  G )
)  =  0 p ) )
142 fveq2 5729 . . . . . . . 8  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  (deg `  f )  =  (deg
`  ( F  o F  -  ( H  o F  x.  G
) ) ) )
143142oveq1d 6097 . . . . . . 7  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  (
(deg `  f )  -  N )  =  ( (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  -  N ) )
144143breq1d 4223 . . . . . 6  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  (
( (deg `  f
)  -  N )  <  D  <->  ( (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  -  N
)  <  D )
)
145141, 144orbi12d 692 . . . . 5  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  (
( f  =  0 p  \/  ( (deg
`  f )  -  N )  <  D
)  <->  ( ( F  o F  -  ( H  o F  x.  G
) )  =  0 p  \/  ( (deg
`  ( F  o F  -  ( H  o F  x.  G
) ) )  -  N )  <  D
) ) )
146 plydiv.u . . . . . . . . 9  |-  U  =  ( f  o F  -  ( G  o F  x.  p )
)
147 oveq1 6089 . . . . . . . . 9  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  (
f  o F  -  ( G  o F  x.  p ) )  =  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
) )
148146, 147syl5eq 2481 . . . . . . . 8  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  U  =  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
) )
149148eqeq1d 2445 . . . . . . 7  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  ( U  =  0 p  <->  ( ( F  o F  -  ( H  o F  x.  G )
)  o F  -  ( G  o F  x.  p ) )  =  0 p ) )
150148fveq2d 5733 . . . . . . . 8  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  (deg `  U )  =  (deg
`  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
) ) )
151150breq1d 4223 . . . . . . 7  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  (
(deg `  U )  <  N  <->  (deg `  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
) )  <  N
) )
152149, 151orbi12d 692 . . . . . 6  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  (
( U  =  0 p  \/  (deg `  U )  <  N
)  <->  ( ( ( F  o F  -  ( H  o F  x.  G ) )  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
) )  <  N
) ) )
153152rexbidv 2727 . . . . 5  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  ( E. p  e.  (Poly `  S ) ( U  =  0 p  \/  (deg `  U )  < 
N )  <->  E. p  e.  (Poly `  S )
( ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
)  =  0 p  \/  (deg `  (
( F  o F  -  ( H  o F  x.  G )
)  o F  -  ( G  o F  x.  p ) ) )  <  N ) ) )
154145, 153imbi12d 313 . . . 4  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  (
( ( f  =  0 p  \/  (
(deg `  f )  -  N )  <  D
)  ->  E. p  e.  (Poly `  S )
( U  =  0 p  \/  (deg `  U )  <  N
) )  <->  ( (
( F  o F  -  ( H  o F  x.  G )
)  =  0 p  \/  ( (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  -  N
)  <  D )  ->  E. p  e.  (Poly `  S ) ( ( ( F  o F  -  ( H  o F  x.  G )
)  o F  -  ( G  o F  x.  p ) )  =  0 p  \/  (deg `  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
) )  <  N
) ) ) )
155154rspcv 3049 . . 3  |-  ( ( F  o F  -  ( H  o F  x.  G ) )  e.  (Poly `  S )  ->  ( A. f  e.  (Poly `  S )
( ( f  =  0 p  \/  (
(deg `  f )  -  N )  <  D
)  ->  E. p  e.  (Poly `  S )
( U  =  0 p  \/  (deg `  U )  <  N
) )  ->  (
( ( F  o F  -  ( H  o F  x.  G
) )  =  0 p  \/  ( (deg
`  ( F  o F  -  ( H  o F  x.  G
) ) )  -  N )  <  D
)  ->  E. p  e.  (Poly `  S )
( ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
)  =  0 p  \/  (deg `  (
( F  o F  -  ( H  o F  x.  G )
)  o F  -  ( G  o F  x.  p ) ) )  <  N ) ) ) )
15652, 53, 140, 155syl3c 60 . 2  |-  ( ph  ->  E. p  e.  (Poly `  S ) ( ( ( F  o F  -  ( H  o F  x.  G )
)  o F  -  ( G  o F  x.  p ) )  =  0 p  \/  (deg `  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
) )  <  N
) )
15750adantr 453 . . . . . . 7  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  H  e.  (Poly `  S ) )
158 simpr 449 . . . . . . 7  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  p  e.  (Poly `  S ) )
1594adantlr 697 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
160157, 158, 159plyadd 20137 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( H  o F  +  p )  e.  (Poly `  S )
)
161160adantr 453 . . . . 5  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( (
( F  o F  -  ( H  o F  x.  G )
)  o F  -  ( G  o F  x.  p ) )  =  0 p  \/  (deg `  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
) )  <  N
) )  ->  ( H  o F  +  p
)  e.  (Poly `  S ) )
162 cnex 9072 . . . . . . . . . . 11  |-  CC  e.  _V
163162a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  CC  e.  _V )
1641adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  F  e.  (Poly `  S ) )
165 plyf 20118 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
166164, 165syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  F : CC --> CC )
167 mulcl 9075 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
168167adantl 454 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
169 plyf 20118 . . . . . . . . . . . 12  |-  ( H  e.  (Poly `  S
)  ->  H : CC
--> CC )
170157, 169syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  H : CC --> CC )
17118adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  G  e.  (Poly `  S ) )
172 plyf 20118 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
173171, 172syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  G : CC --> CC )
174 inidm 3551 . . . . . . . . . . 11  |-  ( CC 
i^i  CC )  =  CC
175168, 170, 173, 163, 163, 174off 6321 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( H  o F  x.  G ) : CC --> CC )
176 plyf 20118 . . . . . . . . . . . 12  |-  ( p  e.  (Poly `  S
)  ->  p : CC
--> CC )
177176adantl 454 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  p : CC --> CC )
178168, 173, 177, 163, 163, 174off 6321 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( G  o F  x.  p ) : CC --> CC )
179 subsub4 9335 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  -  y
)  -  z )  =  ( x  -  ( y  +  z ) ) )
180179adantl 454 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( ( x  -  y )  -  z
)  =  ( x  -  ( y  +  z ) ) )
181163, 166, 175, 178, 180caofass 6339 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
)  =  ( F  o F  -  (
( H  o F  x.  G )  o F  +  ( G  o F  x.  p
) ) ) )
182 mulcom 9077 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
183182adantl 454 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  =  ( y  x.  x ) )
184163, 170, 173, 183caofcom 6337 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( H  o F  x.  G )  =  ( G  o F  x.  H )
)
185184oveq1d 6097 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( H  o F  x.  G
)  o F  +  ( G  o F  x.  p ) )  =  ( ( G  o F  x.  H )  o F  +  ( G  o F  x.  p
) ) )
186 adddi 9080 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z
) ) )
187186adantl 454 . . . . . . . . . . . 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 ) ) )
188163, 173, 170, 177, 187caofdi 6341 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( G  o F  x.  ( H  o F  +  p
) )  =  ( ( G  o F  x.  H )  o F  +  ( G  o F  x.  p
) ) )
189185, 188eqtr4d 2472 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( H  o F  x.  G
)  o F  +  ( G  o F  x.  p ) )  =  ( G  o F  x.  ( H  o F  +  p )
) )
190189oveq2d 6098 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( F  o F  -  ( ( H  o F  x.  G
)  o F  +  ( G  o F  x.  p ) ) )  =  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) )
191181, 190eqtrd 2469 . . . . . . . 8  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
)  =  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) )
192191eqeq1d 2445 . . . . . . 7  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( ( F  o F  -  ( H  o F  x.  G ) )  o F  -  ( G  o F  x.  p
) )  =  0 p  <->  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) )  =  0 p ) )
193191fveq2d 5733 . . . . . . . 8  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  (deg `  (
( F  o F  -  ( H  o F  x.  G )
)  o F  -  ( G  o F  x.  p ) ) )  =  (deg `  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) ) )
194193breq1d 4223 . . . . . . 7  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( (deg `  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
) )  <  N  <->  (deg
`  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) )  <  N ) )
195192, 194orbi12d 692 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( ( ( F  o F  -  ( H  o F  x.  G )
)  o F  -  ( G  o F  x.  p ) )  =  0 p  \/  (deg `  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
) )  <  N
)  <->  ( ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) )  <  N ) ) )
196195biimpa 472 . . . . 5  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( (
( F  o F  -  ( H  o F  x.  G )
)  o F  -  ( G  o F  x.  p ) )  =  0 p  \/  (deg `  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
) )  <  N
) )  ->  (
( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) )  <  N ) )
197 plydiv.r . . . . . . . . 9  |-  R  =  ( F  o F  -  ( G  o F  x.  q )
)
198 oveq2 6090 . . . . . . . . . 10  |-  ( q  =  ( H  o F  +  p )  ->  ( G  o F  x.  q )  =  ( G  o F  x.  ( H  o F  +  p )
) )
199198oveq2d 6098 . . . . . . . . 9  |-  ( q  =  ( H  o F  +  p )  ->  ( F  o F  -  ( G  o F  x.  q )
)  =  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) )
200197, 199syl5eq 2481 . . . . . . . 8  |-  ( q  =  ( H  o F  +  p )  ->  R  =  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) )
201200eqeq1d 2445 . . . . . . 7  |-  ( q  =  ( H  o F  +  p )  ->  ( R  =  0 p  <->  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) )  =  0 p ) )
202200fveq2d 5733 . . . . . . . 8  |-  ( q  =  ( H  o F  +  p )  ->  (deg `  R )  =  (deg `  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) ) )
203202breq1d 4223 . . . . . . 7  |-  ( q  =  ( H  o F  +  p )  ->  ( (deg `  R
)  <  N  <->  (deg `  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) )  <  N ) )
204201, 203orbi12d 692 . . . . . 6  |-  ( q  =  ( H  o F  +  p )  ->  ( ( R  =  0 p  \/  (deg `  R )  <  N
)  <->  ( ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) )  <  N ) ) )
205204rspcev 3053 . . . . 5  |-  ( ( ( H  o F  +  p )  e.  (Poly `  S )  /\  ( ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) )  <  N ) )  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
N ) )
206161, 196, 205syl2anc 644 . . . 4  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( (
( F  o F  -  ( H  o F  x.  G )
)  o F  -  ( G  o F  x.  p ) )  =  0 p  \/  (deg `  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
) )  <  N
) )  ->  E. q  e.  (Poly `  S )
( R  =  0 p  \/  (deg `  R )  <  N
) )
207206ex 425 . . 3  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( ( ( F  o F  -  ( H  o F  x.  G )
)  o F  -  ( G  o F  x.  p ) )  =  0 p  \/  (deg `  ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
) )  <  N
)  ->  E. q  e.  (Poly `  S )
( R  =  0 p  \/  (deg `  R )  <  N
) ) )
208207rexlimdva 2831 . 2  |-  ( ph  ->  ( E. p  e.  (Poly `  S )
( ( ( F  o F  -  ( H  o F  x.  G
) )  o F  -  ( G  o F  x.  p )
)  =  0 p  \/  (deg `  (
( F  o F  -  ( H  o F  x.  G )
)  o F  -  ( G  o F  x.  p ) ) )  <  N )  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
N ) ) )
209156, 208mpd 15 1  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706   E.wrex 2707   _Vcvv 2957    C_ wss 3321   ifcif 3740   {csn 3815   class class class wbr 4213    e. cmpt 4267    X. cxp 4877    Fn wfn 5450   -->wf 5451   ` cfv 5455  (class class class)co 6082    o Fcof 6304   CCcc 8989   0cc0 8991   1c1 8992    + caddc 8994    x. cmul 8996    < clt 9121    <_ cle 9122    - cmin 9292   -ucneg 9293    / cdiv 9678   NN0cn0 10222   ^cexp 11383   0 pc0p 19562  Polycply 20104  coeffccoe 20106  degcdgr 20107
This theorem is referenced by:  plydivex  20215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069  ax-addf 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-pm 7022  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-oi 7480  df-card 7827  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-fz 11045  df-fzo 11137  df-fl 11203  df-seq 11325  df-exp 11384  df-hash 11620  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-clim 12283  df-rlim 12284  df-sum 12481  df-0p 19563  df-ply 20108  df-coe 20110  df-dgr 20111
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