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Theorem plydivlem4 19692
Description: Lemma for plydivex 19693. 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 19592 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
31, 2syl 15 . . . . . 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 19689 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  S )
9 plydiv.a . . . . . . . . . . . 12  |-  A  =  (coeff `  F )
109coef2 19629 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  0  e.  S )  ->  A : NN0 --> S )
111, 8, 10syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  A : NN0 --> S )
12 plydiv.m . . . . . . . . . . 11  |-  M  =  (deg `  F )
13 dgrcl 19631 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
141, 13syl 15 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  F )  e.  NN0 )
1512, 14syl5eqel 2380 . . . . . . . . . 10  |-  ( ph  ->  M  e.  NN0 )
16 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( A : NN0 --> S  /\  M  e.  NN0 )  -> 
( A `  M
)  e.  S )
1711, 15, 16syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( A `  M
)  e.  S )
183, 17sseldd 3194 . . . . . . . 8  |-  ( ph  ->  ( A `  M
)  e.  CC )
19 plydiv.g . . . . . . . . . . 11  |-  ( ph  ->  G  e.  (Poly `  S ) )
20 plydiv.b . . . . . . . . . . . 12  |-  B  =  (coeff `  G )
2120coef2 19629 . . . . . . . . . . 11  |-  ( ( G  e.  (Poly `  S )  /\  0  e.  S )  ->  B : NN0 --> S )
2219, 8, 21syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  B : NN0 --> S )
23 plydiv.n . . . . . . . . . . 11  |-  N  =  (deg `  G )
24 dgrcl 19631 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
2519, 24syl 15 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  G )  e.  NN0 )
2623, 25syl5eqel 2380 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
27 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( B : NN0 --> S  /\  N  e.  NN0 )  -> 
( B `  N
)  e.  S )
2822, 26, 27syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( B `  N
)  e.  S )
293, 28sseldd 3194 . . . . . . . 8  |-  ( ph  ->  ( B `  N
)  e.  CC )
30 plydiv.z . . . . . . . . 9  |-  ( ph  ->  G  =/=  0 p )
3123, 20dgreq0 19662 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0 p  <->  ( B `  N )  =  0 ) )
3219, 31syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( G  =  0 p  <->  ( B `  N )  =  0 ) )
3332necon3bid 2494 . . . . . . . . 9  |-  ( ph  ->  ( G  =/=  0 p 
<->  ( B `  N
)  =/=  0 ) )
3430, 33mpbid 201 . . . . . . . 8  |-  ( ph  ->  ( B `  N
)  =/=  0 )
3518, 29, 34divrecd 9555 . . . . . . 7  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  =  ( ( A `  M )  x.  ( 1  / 
( B `  N
) ) ) )
36 fvex 5555 . . . . . . . . . . 11  |-  ( B `
 N )  e. 
_V
37 eleq1 2356 . . . . . . . . . . . . . 14  |-  ( x  =  ( B `  N )  ->  (
x  e.  S  <->  ( B `  N )  e.  S
) )
38 neeq1 2467 . . . . . . . . . . . . . 14  |-  ( x  =  ( B `  N )  ->  (
x  =/=  0  <->  ( B `  N )  =/=  0 ) )
3937, 38anbi12d 691 . . . . . . . . . . . . 13  |-  ( x  =  ( B `  N )  ->  (
( x  e.  S  /\  x  =/=  0
)  <->  ( ( B `
 N )  e.  S  /\  ( B `
 N )  =/=  0 ) ) )
4039anbi2d 684 . . . . . . . . . . . 12  |-  ( x  =  ( B `  N )  ->  (
( ph  /\  (
x  e.  S  /\  x  =/=  0 ) )  <-> 
( ph  /\  (
( B `  N
)  e.  S  /\  ( B `  N )  =/=  0 ) ) ) )
41 oveq2 5882 . . . . . . . . . . . . 13  |-  ( x  =  ( B `  N )  ->  (
1  /  x )  =  ( 1  / 
( B `  N
) ) )
4241eleq1d 2362 . . . . . . . . . . . 12  |-  ( x  =  ( B `  N )  ->  (
( 1  /  x
)  e.  S  <->  ( 1  /  ( B `  N ) )  e.  S ) )
4340, 42imbi12d 311 . . . . . . . . . . 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 ) ) )
4436, 43, 6vtocl 2851 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( B `  N )  e.  S  /\  ( B `  N )  =/=  0 ) )  -> 
( 1  /  ( B `  N )
)  e.  S )
4544ex 423 . . . . . . . . 9  |-  ( ph  ->  ( ( ( B `
 N )  e.  S  /\  ( B `
 N )  =/=  0 )  ->  (
1  /  ( B `
 N ) )  e.  S ) )
4628, 34, 45mp2and 660 . . . . . . . 8  |-  ( ph  ->  ( 1  /  ( B `  N )
)  e.  S )
475, 17, 46caovcld 6029 . . . . . . 7  |-  ( ph  ->  ( ( A `  M )  x.  (
1  /  ( B `
 N ) ) )  e.  S )
4835, 47eqeltrd 2370 . . . . . 6  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  e.  S )
49 plydiv.d . . . . . 6  |-  ( ph  ->  D  e.  NN0 )
50 plydiv.h . . . . . . 7  |-  H  =  ( z  e.  CC  |->  ( ( ( A `
 M )  / 
( B `  N
) )  x.  (
z ^ D ) ) )
5150ply1term 19602 . . . . . 6  |-  ( ( S  C_  CC  /\  (
( A `  M
)  /  ( B `
 N ) )  e.  S  /\  D  e.  NN0 )  ->  H  e.  (Poly `  S )
)
523, 48, 49, 51syl3anc 1182 . . . . 5  |-  ( ph  ->  H  e.  (Poly `  S ) )
5352, 19, 4, 5plymul 19616 . . . 4  |-  ( ph  ->  ( H  o F  x.  G )  e.  (Poly `  S )
)
541, 53, 4, 5, 7plysub 19617 . . 3  |-  ( ph  ->  ( F  o F  -  ( H  o F  x.  G )
)  e.  (Poly `  S ) )
55 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 ) ) )
56 eqid 2296 . . . . . . 7  |-  (deg `  ( H  o F  x.  G ) )  =  (deg `  ( H  o F  x.  G
) )
5712, 56dgrsub 19669 . . . . . 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
) )
581, 53, 57syl2anc 642 . . . . 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
) )
59 plydiv.fz . . . . . . . . . . . . 13  |-  ( ph  ->  F  =/=  0 p )
6012, 9dgreq0 19662 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0 p  <->  ( A `  M )  =  0 ) )
611, 60syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  =  0 p  <->  ( A `  M )  =  0 ) )
6261necon3bid 2494 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  =/=  0 p 
<->  ( A `  M
)  =/=  0 ) )
6359, 62mpbid 201 . . . . . . . . . . . 12  |-  ( ph  ->  ( A `  M
)  =/=  0 )
6418, 29, 63, 34divne0d 9568 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  =/=  0 )
653, 48sseldd 3194 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  e.  CC )
6650coe1term 19656 . . . . . . . . . . . . 13  |-  ( ( ( ( A `  M )  /  ( B `  N )
)  e.  CC  /\  D  e.  NN0  /\  D  e.  NN0 )  ->  (
(coeff `  H ) `  D )  =  if ( D  =  D ,  ( ( A `
 M )  / 
( B `  N
) ) ,  0 ) )
6765, 49, 49, 66syl3anc 1182 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coeff `  H
) `  D )  =  if ( D  =  D ,  ( ( A `  M )  /  ( B `  N ) ) ,  0 ) )
68 eqid 2296 . . . . . . . . . . . . 13  |-  D  =  D
69 iftrue 3584 . . . . . . . . . . . . 13  |-  ( D  =  D  ->  if ( D  =  D ,  ( ( A `
 M )  / 
( B `  N
) ) ,  0 )  =  ( ( A `  M )  /  ( B `  N ) ) )
7068, 69ax-mp 8 . . . . . . . . . . . 12  |-  if ( D  =  D , 
( ( A `  M )  /  ( B `  N )
) ,  0 )  =  ( ( A `
 M )  / 
( B `  N
) )
7167, 70syl6eq 2344 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  H
) `  D )  =  ( ( A `
 M )  / 
( B `  N
) ) )
72 c0ex 8848 . . . . . . . . . . . . 13  |-  0  e.  _V
7372fvconst2 5745 . . . . . . . . . . . 12  |-  ( D  e.  NN0  ->  ( ( NN0  X.  { 0 } ) `  D
)  =  0 )
7449, 73syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( ( NN0  X.  { 0 } ) `
 D )  =  0 )
7564, 71, 743netr4d 2486 . . . . . . . . . 10  |-  ( ph  ->  ( (coeff `  H
) `  D )  =/=  ( ( NN0  X.  { 0 } ) `
 D ) )
76 fveq2 5541 . . . . . . . . . . . . 13  |-  ( H  =  0 p  -> 
(coeff `  H )  =  (coeff `  0 p
) )
77 coe0 19653 . . . . . . . . . . . . 13  |-  (coeff ` 
0 p )  =  ( NN0  X.  {
0 } )
7876, 77syl6eq 2344 . . . . . . . . . . . 12  |-  ( H  =  0 p  -> 
(coeff `  H )  =  ( NN0  X.  { 0 } ) )
7978fveq1d 5543 . . . . . . . . . . 11  |-  ( H  =  0 p  -> 
( (coeff `  H
) `  D )  =  ( ( NN0 
X.  { 0 } ) `  D ) )
8079necon3i 2498 . . . . . . . . . 10  |-  ( ( (coeff `  H ) `  D )  =/=  (
( NN0  X.  { 0 } ) `  D
)  ->  H  =/=  0 p )
8175, 80syl 15 . . . . . . . . 9  |-  ( ph  ->  H  =/=  0 p )
82 eqid 2296 . . . . . . . . . 10  |-  (deg `  H )  =  (deg
`  H )
8382, 23dgrmul 19667 . . . . . . . . 9  |-  ( ( ( H  e.  (Poly `  S )  /\  H  =/=  0 p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0 p ) )  ->  (deg `  ( H  o F  x.  G
) )  =  ( (deg `  H )  +  N ) )
8452, 81, 19, 30, 83syl22anc 1183 . . . . . . . 8  |-  ( ph  ->  (deg `  ( H  o F  x.  G
) )  =  ( (deg `  H )  +  N ) )
8550dgr1term 19657 . . . . . . . . . . . 12  |-  ( ( ( ( A `  M )  /  ( B `  N )
)  e.  CC  /\  ( ( A `  M )  /  ( B `  N )
)  =/=  0  /\  D  e.  NN0 )  ->  (deg `  H )  =  D )
8665, 64, 49, 85syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  H )  =  D )
87 plydiv.e . . . . . . . . . . 11  |-  ( ph  ->  ( M  -  N
)  =  D )
8886, 87eqtr4d 2331 . . . . . . . . . 10  |-  ( ph  ->  (deg `  H )  =  ( M  -  N ) )
8988oveq1d 5889 . . . . . . . . 9  |-  ( ph  ->  ( (deg `  H
)  +  N )  =  ( ( M  -  N )  +  N ) )
9015nn0cnd 10036 . . . . . . . . . 10  |-  ( ph  ->  M  e.  CC )
9126nn0cnd 10036 . . . . . . . . . 10  |-  ( ph  ->  N  e.  CC )
9290, 91npcand 9177 . . . . . . . . 9  |-  ( ph  ->  ( ( M  -  N )  +  N
)  =  M )
9389, 92eqtrd 2328 . . . . . . . 8  |-  ( ph  ->  ( (deg `  H
)  +  N )  =  M )
9484, 93eqtrd 2328 . . . . . . 7  |-  ( ph  ->  (deg `  ( H  o F  x.  G
) )  =  M )
9594ifeq1d 3592 . . . . . 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 ) )
96 ifid 3610 . . . . . 6  |-  if ( M  <_  (deg `  ( H  o F  x.  G
) ) ,  M ,  M )  =  M
9795, 96syl6eq 2344 . . . . 5  |-  ( ph  ->  if ( M  <_ 
(deg `  ( H  o F  x.  G
) ) ,  (deg
`  ( H  o F  x.  G )
) ,  M )  =  M )
9858, 97breqtrd 4063 . . . 4  |-  ( ph  ->  (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  <_  M )
99 eqid 2296 . . . . . . . 8  |-  (coeff `  ( H  o F  x.  G ) )  =  (coeff `  ( H  o F  x.  G
) )
1009, 99coesub 19654 . . . . . . 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
) ) ) )
1011, 53, 100syl2anc 642 . . . . . 6  |-  ( ph  ->  (coeff `  ( F  o F  -  ( H  o F  x.  G
) ) )  =  ( A  o F  -  (coeff `  ( H  o F  x.  G
) ) ) )
102101fveq1d 5543 . . . . 5  |-  ( ph  ->  ( (coeff `  ( F  o F  -  ( H  o F  x.  G
) ) ) `  M )  =  ( ( A  o F  -  (coeff `  ( H  o F  x.  G
) ) ) `  M ) )
1039coef3 19630 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
104 ffn 5405 . . . . . . . 8  |-  ( A : NN0 --> CC  ->  A  Fn  NN0 )
1051, 103, 1043syl 18 . . . . . . 7  |-  ( ph  ->  A  Fn  NN0 )
10699coef3 19630 . . . . . . . 8  |-  ( ( H  o F  x.  G )  e.  (Poly `  S )  ->  (coeff `  ( H  o F  x.  G ) ) : NN0 --> CC )
107 ffn 5405 . . . . . . . 8  |-  ( (coeff `  ( H  o F  x.  G ) ) : NN0 --> CC  ->  (coeff `  ( H  o F  x.  G ) )  Fn  NN0 )
10853, 106, 1073syl 18 . . . . . . 7  |-  ( ph  ->  (coeff `  ( H  o F  x.  G
) )  Fn  NN0 )
109 nn0ex 9987 . . . . . . . 8  |-  NN0  e.  _V
110109a1i 10 . . . . . . 7  |-  ( ph  ->  NN0  e.  _V )
111 inidm 3391 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
112 eqidd 2297 . . . . . . 7  |-  ( (
ph  /\  M  e.  NN0 )  ->  ( A `  M )  =  ( A `  M ) )
113 eqid 2296 . . . . . . . . . . 11  |-  (coeff `  H )  =  (coeff `  H )
114113, 20, 82, 23coemulhi 19651 . . . . . . . . . 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 )
) )
11552, 19, 114syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( (coeff `  ( H  o F  x.  G
) ) `  (
(deg `  H )  +  N ) )  =  ( ( (coeff `  H ) `  (deg `  H ) )  x.  ( B `  N
) ) )
11693fveq2d 5545 . . . . . . . . 9  |-  ( ph  ->  ( (coeff `  ( H  o F  x.  G
) ) `  (
(deg `  H )  +  N ) )  =  ( (coeff `  ( H  o F  x.  G
) ) `  M
) )
11786fveq2d 5545 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coeff `  H
) `  (deg `  H
) )  =  ( (coeff `  H ) `  D ) )
118117, 71eqtrd 2328 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  H
) `  (deg `  H
) )  =  ( ( A `  M
)  /  ( B `
 N ) ) )
119118oveq1d 5889 . . . . . . . . . 10  |-  ( ph  ->  ( ( (coeff `  H ) `  (deg `  H ) )  x.  ( B `  N
) )  =  ( ( ( A `  M )  /  ( B `  N )
)  x.  ( B `
 N ) ) )
12018, 29, 34divcan1d 9553 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A `
 M )  / 
( B `  N
) )  x.  ( B `  N )
)  =  ( A `
 M ) )
121119, 120eqtrd 2328 . . . . . . . . 9  |-  ( ph  ->  ( ( (coeff `  H ) `  (deg `  H ) )  x.  ( B `  N
) )  =  ( A `  M ) )
122115, 116, 1213eqtr3d 2336 . . . . . . . 8  |-  ( ph  ->  ( (coeff `  ( H  o F  x.  G
) ) `  M
)  =  ( A `
 M ) )
123122adantr 451 . . . . . . 7  |-  ( (
ph  /\  M  e.  NN0 )  ->  ( (coeff `  ( H  o F  x.  G ) ) `
 M )  =  ( A `  M
) )
124105, 108, 110, 110, 111, 112, 123ofval 6103 . . . . . 6  |-  ( (
ph  /\  M  e.  NN0 )  ->  ( ( A  o F  -  (coeff `  ( H  o F  x.  G ) ) ) `  M )  =  ( ( A `
 M )  -  ( A `  M ) ) )
12515, 124mpdan 649 . . . . 5  |-  ( ph  ->  ( ( A  o F  -  (coeff `  ( H  o F  x.  G
) ) ) `  M )  =  ( ( A `  M
)  -  ( A `
 M ) ) )
12618subidd 9161 . . . . 5  |-  ( ph  ->  ( ( A `  M )  -  ( A `  M )
)  =  0 )
127102, 125, 1263eqtrd 2332 . . . 4  |-  ( ph  ->  ( (coeff `  ( F  o F  -  ( H  o F  x.  G
) ) ) `  M )  =  0 )
128 dgrcl 19631 . . . . . . . . . 10  |-  ( ( F  o F  -  ( H  o F  x.  G ) )  e.  (Poly `  S )  ->  (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  e. 
NN0 )
12954, 128syl 15 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  e. 
NN0 )
130129nn0red 10035 . . . . . . . 8  |-  ( ph  ->  (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  e.  RR )
13115nn0red 10035 . . . . . . . 8  |-  ( ph  ->  M  e.  RR )
13226nn0red 10035 . . . . . . . 8  |-  ( ph  ->  N  e.  RR )
133130, 131, 132ltsub1d 9397 . . . . . . 7  |-  ( ph  ->  ( (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  < 
M  <->  ( (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  -  N
)  <  ( M  -  N ) ) )
13487breq2d 4051 . . . . . . 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 )
)
135133, 134bitrd 244 . . . . . 6  |-  ( ph  ->  ( (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  < 
M  <->  ( (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  -  N
)  <  D )
)
136135orbi2d 682 . . . . 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
) ) )
137 eqid 2296 . . . . . . 7  |-  (deg `  ( F  o F  -  ( H  o F  x.  G )
) )  =  (deg
`  ( F  o F  -  ( H  o F  x.  G
) ) )
138 eqid 2296 . . . . . . 7  |-  (coeff `  ( F  o F  -  ( H  o F  x.  G )
) )  =  (coeff `  ( F  o F  -  ( H  o F  x.  G )
) )
139137, 138dgrlt 19663 . . . . . 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 ) ) )
14054, 15, 139syl2anc 642 . . . . 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 ) ) )
141136, 140bitr3d 246 . . . 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 ) ) )
14298, 127, 141mpbir2and 888 . . 3  |-  ( ph  ->  ( ( F  o F  -  ( H  o F  x.  G
) )  =  0 p  \/  ( (deg
`  ( F  o F  -  ( H  o F  x.  G
) ) )  -  N )  <  D
) )
143 eqeq1 2302 . . . . . 6  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  (
f  =  0 p  <-> 
( F  o F  -  ( H  o F  x.  G )
)  =  0 p ) )
144 fveq2 5541 . . . . . . . 8  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  (deg `  f )  =  (deg
`  ( F  o F  -  ( H  o F  x.  G
) ) ) )
145144oveq1d 5889 . . . . . . 7  |-  ( f  =  ( F  o F  -  ( H  o F  x.  G
) )  ->  (
(deg `  f )  -  N )  =  ( (deg `  ( F  o F  -  ( H  o F  x.  G
) ) )  -  N ) )
146145breq1d 4049 . . . . . 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 )
)
147143, 146orbi12d 690 . . . . 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
) ) )
148 plydiv.u . . . . . . . . 9  |-  U  =  ( f  o F  -  ( G  o F  x.  p )
)
149 oveq1 5881 . . . . . . . . 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 )
) )
150148, 149syl5eq 2340 . . . . . . . 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 )
) )
151150eqeq1d 2304 . . . . . . 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 ) )
152150fveq2d 5545 . . . . . . . 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 )
) ) )
153152breq1d 4049 . . . . . . 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
) )
154151, 153orbi12d 690 . . . . . 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
) ) )
155154rexbidv 2577 . . . . 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 ) ) )
156147, 155imbi12d 311 . . . 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
) ) ) )
157156rspcv 2893 . . 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 ) ) ) )
15854, 55, 142, 157syl3c 57 . 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
) )
15952adantr 451 . . . . . . 7  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  H  e.  (Poly `  S ) )
160 simpr 447 . . . . . . 7  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  p  e.  (Poly `  S ) )
1614adantlr 695 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
162159, 160, 161plyadd 19615 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( H  o F  +  p )  e.  (Poly `  S )
)
163162adantr 451 . . . . 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 ) )
164 cnex 8834 . . . . . . . . . . 11  |-  CC  e.  _V
165164a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  CC  e.  _V )
1661adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  F  e.  (Poly `  S ) )
167 plyf 19596 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
168166, 167syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  F : CC --> CC )
169 mulcl 8837 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
170169adantl 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
171 plyf 19596 . . . . . . . . . . . 12  |-  ( H  e.  (Poly `  S
)  ->  H : CC
--> CC )
172159, 171syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  H : CC --> CC )
17319adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  G  e.  (Poly `  S ) )
174 plyf 19596 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
175173, 174syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  G : CC --> CC )
176 inidm 3391 . . . . . . . . . . 11  |-  ( CC 
i^i  CC )  =  CC
177170, 172, 175, 165, 165, 176off 6109 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( H  o F  x.  G ) : CC --> CC )
178 plyf 19596 . . . . . . . . . . . 12  |-  ( p  e.  (Poly `  S
)  ->  p : CC
--> CC )
179178adantl 452 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  p : CC --> CC )
180170, 175, 179, 165, 165, 176off 6109 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( G  o F  x.  p ) : CC --> CC )
181 subsub4 9096 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  -  y
)  -  z )  =  ( x  -  ( y  +  z ) ) )
182181adantl 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( ( x  -  y )  -  z
)  =  ( x  -  ( y  +  z ) ) )
183165, 168, 177, 180, 182caofass 6127 . . . . . . . . 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
) ) ) )
184 mulcom 8839 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
185184adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  =  ( y  x.  x ) )
186165, 172, 175, 185caofcom 6125 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( H  o F  x.  G )  =  ( G  o F  x.  H )
)
187186oveq1d 5889 . . . . . . . . . . 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
) ) )
188 adddi 8842 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z
) ) )
189188adantl 452 . . . . . . . . . . . 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 ) ) )
190165, 175, 172, 179, 189caofdi 6129 . . . . . . . . . . 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
) ) )
191187, 190eqtr4d 2331 . . . . . . . . . 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 )
) )
192191oveq2d 5890 . . . . . . . . 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
) ) ) )
193183, 192eqtrd 2328 . . . . . . . 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
) ) ) )
194193eqeq1d 2304 . . . . . . 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 ) )
195193fveq2d 5545 . . . . . . . 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
) ) ) ) )
196195breq1d 4049 . . . . . . 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 ) )
197194, 196orbi12d 690 . . . . . 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 ) ) )
198197biimpa 470 . . . . 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 ) )
199 plydiv.r . . . . . . . . 9  |-  R  =  ( F  o F  -  ( G  o F  x.  q )
)
200 oveq2 5882 . . . . . . . . . 10  |-  ( q  =  ( H  o F  +  p )  ->  ( G  o F  x.  q )  =  ( G  o F  x.  ( H  o F  +  p )
) )
201200oveq2d 5890 . . . . . . . . 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
) ) ) )
202199, 201syl5eq 2340 . . . . . . . 8  |-  ( q  =  ( H  o F  +  p )  ->  R  =  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) )
203202eqeq1d 2304 . . . . . . 7  |-  ( q  =  ( H  o F  +  p )  ->  ( R  =  0 p  <->  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) )  =  0 p ) )
204202fveq2d 5545 . . . . . . . 8  |-  ( q  =  ( H  o F  +  p )  ->  (deg `  R )  =  (deg `  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) ) )
205204breq1d 4049 . . . . . . 7  |-  ( q  =  ( H  o F  +  p )  ->  ( (deg `  R
)  <  N  <->  (deg `  ( F  o F  -  ( G  o F  x.  ( H  o F  +  p
) ) ) )  <  N ) )
206203, 205orbi12d 690 . . . . . 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 ) ) )
207206rspcev 2897 . . . . 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 ) )
208163, 198, 207syl2anc 642 . . . 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
) )
209208ex 423 . . 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
) ) )
210209rexlimdva 2680 . 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 ) ) )
211158, 210mpd 14 1  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   ifcif 3578   {csn 3653   class class class wbr 4039    e. cmpt 4093    X. cxp 4703    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   NN0cn0 9981   ^cexp 11120   0 pc0p 19040  Polycply 19582  coeffccoe 19584  degcdgr 19585
This theorem is referenced by:  plydivex  19693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588  df-dgr 19589
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