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Theorem plydivlem4 23236
Description: Lemma for plydivex 23237. 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 . . . . . . . 8  |-  ( ph  ->  F  e.  (Poly `  S ) )
2 plybss 23135 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
31, 2syl 17 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
4 plydiv.pl . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
5 plydiv.tm . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
6 plydiv.rc . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  S  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  S )
7 plydiv.m1 . . . . . . . . . . . . 13  |-  ( ph  -> 
-u 1  e.  S
)
84, 5, 6, 7plydivlem1 23233 . . . . . . . . . . . 12  |-  ( ph  ->  0  e.  S )
9 plydiv.a . . . . . . . . . . . . 13  |-  A  =  (coeff `  F )
109coef2 23172 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  0  e.  S )  ->  A : NN0 --> S )
111, 8, 10syl2anc 665 . . . . . . . . . . 11  |-  ( ph  ->  A : NN0 --> S )
12 plydiv.m . . . . . . . . . . . 12  |-  M  =  (deg `  F )
13 dgrcl 23174 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
141, 13syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  F )  e.  NN0 )
1512, 14syl5eqel 2514 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  NN0 )
1611, 15ffvelrnd 6035 . . . . . . . . . 10  |-  ( ph  ->  ( A `  M
)  e.  S )
173, 16sseldd 3465 . . . . . . . . 9  |-  ( ph  ->  ( A `  M
)  e.  CC )
18 plydiv.g . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  (Poly `  S ) )
19 plydiv.b . . . . . . . . . . . . 13  |-  B  =  (coeff `  G )
2019coef2 23172 . . . . . . . . . . . 12  |-  ( ( G  e.  (Poly `  S )  /\  0  e.  S )  ->  B : NN0 --> S )
2118, 8, 20syl2anc 665 . . . . . . . . . . 11  |-  ( ph  ->  B : NN0 --> S )
22 plydiv.n . . . . . . . . . . . 12  |-  N  =  (deg `  G )
23 dgrcl 23174 . . . . . . . . . . . . 13  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
2418, 23syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  G )  e.  NN0 )
2522, 24syl5eqel 2514 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN0 )
2621, 25ffvelrnd 6035 . . . . . . . . . 10  |-  ( ph  ->  ( B `  N
)  e.  S )
273, 26sseldd 3465 . . . . . . . . 9  |-  ( ph  ->  ( B `  N
)  e.  CC )
28 plydiv.z . . . . . . . . . 10  |-  ( ph  ->  G  =/=  0p )
2922, 19dgreq0 23206 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0p  <->  ( B `  N )  =  0 ) )
3018, 29syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( G  =  0p  <->  ( B `  N )  =  0 ) )
3130necon3bid 2682 . . . . . . . . . 10  |-  ( ph  ->  ( G  =/=  0p 
<->  ( B `  N
)  =/=  0 ) )
3228, 31mpbid 213 . . . . . . . . 9  |-  ( ph  ->  ( B `  N
)  =/=  0 )
3317, 27, 32divrecd 10387 . . . . . . . 8  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  =  ( ( A `  M )  x.  ( 1  / 
( B `  N
) ) ) )
34 fvex 5888 . . . . . . . . . . . 12  |-  ( B `
 N )  e. 
_V
35 eleq1 2494 . . . . . . . . . . . . . . 15  |-  ( x  =  ( B `  N )  ->  (
x  e.  S  <->  ( B `  N )  e.  S
) )
36 neeq1 2705 . . . . . . . . . . . . . . 15  |-  ( x  =  ( B `  N )  ->  (
x  =/=  0  <->  ( B `  N )  =/=  0 ) )
3735, 36anbi12d 715 . . . . . . . . . . . . . 14  |-  ( x  =  ( B `  N )  ->  (
( x  e.  S  /\  x  =/=  0
)  <->  ( ( B `
 N )  e.  S  /\  ( B `
 N )  =/=  0 ) ) )
3837anbi2d 708 . . . . . . . . . . . . 13  |-  ( x  =  ( B `  N )  ->  (
( ph  /\  (
x  e.  S  /\  x  =/=  0 ) )  <-> 
( ph  /\  (
( B `  N
)  e.  S  /\  ( B `  N )  =/=  0 ) ) ) )
39 oveq2 6310 . . . . . . . . . . . . . 14  |-  ( x  =  ( B `  N )  ->  (
1  /  x )  =  ( 1  / 
( B `  N
) ) )
4039eleq1d 2491 . . . . . . . . . . . . 13  |-  ( x  =  ( B `  N )  ->  (
( 1  /  x
)  e.  S  <->  ( 1  /  ( B `  N ) )  e.  S ) )
4138, 40imbi12d 321 . . . . . . . . . . . 12  |-  ( 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 3133 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( B `  N )  e.  S  /\  ( B `  N )  =/=  0 ) )  -> 
( 1  /  ( B `  N )
)  e.  S )
4342ex 435 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( B `
 N )  e.  S  /\  ( B `
 N )  =/=  0 )  ->  (
1  /  ( B `
 N ) )  e.  S ) )
4426, 32, 43mp2and 683 . . . . . . . . 9  |-  ( ph  ->  ( 1  /  ( B `  N )
)  e.  S )
455, 16, 44caovcld 6473 . . . . . . . 8  |-  ( ph  ->  ( ( A `  M )  x.  (
1  /  ( B `
 N ) ) )  e.  S )
4633, 45eqeltrd 2510 . . . . . . 7  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  e.  S )
47 plydiv.d . . . . . . 7  |-  ( ph  ->  D  e.  NN0 )
48 plydiv.h . . . . . . . 8  |-  H  =  ( z  e.  CC  |->  ( ( ( A `
 M )  / 
( B `  N
) )  x.  (
z ^ D ) ) )
4948ply1term 23145 . . . . . . 7  |-  ( ( S  C_  CC  /\  (
( A `  M
)  /  ( B `
 N ) )  e.  S  /\  D  e.  NN0 )  ->  H  e.  (Poly `  S )
)
503, 46, 47, 49syl3anc 1264 . . . . . 6  |-  ( ph  ->  H  e.  (Poly `  S ) )
5150adantr 466 . . . . 5  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  H  e.  (Poly `  S ) )
52 simpr 462 . . . . 5  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  p  e.  (Poly `  S ) )
534adantlr 719 . . . . 5  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
5451, 52, 53plyadd 23158 . . . 4  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( H  oF  +  p )  e.  (Poly `  S )
)
5554adantr 466 . . 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
) )  ->  ( H  oF  +  p
)  e.  (Poly `  S ) )
56 cnex 9621 . . . . . . . . 9  |-  CC  e.  _V
5756a1i 11 . . . . . . . 8  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  CC  e.  _V )
581adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  F  e.  (Poly `  S ) )
59 plyf 23139 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
6058, 59syl 17 . . . . . . . 8  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  F : CC --> CC )
61 mulcl 9624 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
6261adantl 467 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
63 plyf 23139 . . . . . . . . . 10  |-  ( H  e.  (Poly `  S
)  ->  H : CC
--> CC )
6451, 63syl 17 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  H : CC --> CC )
6518adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  G  e.  (Poly `  S ) )
66 plyf 23139 . . . . . . . . . 10  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
6765, 66syl 17 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  G : CC --> CC )
68 inidm 3671 . . . . . . . . 9  |-  ( CC 
i^i  CC )  =  CC
6962, 64, 67, 57, 57, 68off 6557 . . . . . . . 8  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( H  oF  x.  G ) : CC --> CC )
70 plyf 23139 . . . . . . . . . 10  |-  ( p  e.  (Poly `  S
)  ->  p : CC
--> CC )
7170adantl 467 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  p : CC --> CC )
7262, 67, 71, 57, 57, 68off 6557 . . . . . . . 8  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( G  oF  x.  p ) : CC --> CC )
73 subsub4 9908 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  -  y
)  -  z )  =  ( x  -  ( y  +  z ) ) )
7473adantl 467 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( ( x  -  y )  -  z
)  =  ( x  -  ( y  +  z ) ) )
7557, 60, 69, 72, 74caofass 6576 . . . . . . 7  |-  ( (
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
) ) ) )
76 mulcom 9626 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
7776adantl 467 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  =  ( y  x.  x ) )
7857, 64, 67, 77caofcom 6574 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( H  oF  x.  G )  =  ( G  oF  x.  H )
)
7978oveq1d 6317 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( H  oF  x.  G
)  oF  +  ( G  oF  x.  p ) )  =  ( ( G  oF  x.  H )  oF  +  ( G  oF  x.  p
) ) )
80 adddi 9629 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z
) ) )
8180adantl 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e.  (Poly `  S )
)  /\  ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( x  x.  (
y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z ) ) )
8257, 67, 64, 71, 81caofdi 6578 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( G  oF  x.  ( H  oF  +  p
) )  =  ( ( G  oF  x.  H )  oF  +  ( G  oF  x.  p
) ) )
8379, 82eqtr4d 2466 . . . . . . . 8  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( H  oF  x.  G
)  oF  +  ( G  oF  x.  p ) )  =  ( G  oF  x.  ( H  oF  +  p )
) )
8483oveq2d 6318 . . . . . . 7  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( F  oF  -  ( ( H  oF  x.  G
)  oF  +  ( G  oF  x.  p ) ) )  =  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )
8575, 84eqtrd 2463 . . . . . 6  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
)  =  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )
8685eqeq1d 2424 . . . . 5  |-  ( (
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 ) )
8785fveq2d 5882 . . . . . 6  |-  ( (
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
) ) ) ) )
8887breq1d 4430 . . . . 5  |-  ( (
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 ) )
8986, 88orbi12d 714 . . . 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
)  <->  ( ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )  <  N ) ) )
9089biimpa 486 . . 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
) )  ->  (
( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )  <  N ) )
91 plydiv.r . . . . . . 7  |-  R  =  ( F  oF  -  ( G  oF  x.  q )
)
92 oveq2 6310 . . . . . . . 8  |-  ( q  =  ( H  oF  +  p )  ->  ( G  oF  x.  q )  =  ( G  oF  x.  ( H  oF  +  p )
) )
9392oveq2d 6318 . . . . . . 7  |-  ( q  =  ( H  oF  +  p )  ->  ( F  oF  -  ( G  oF  x.  q )
)  =  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )
9491, 93syl5eq 2475 . . . . . 6  |-  ( q  =  ( H  oF  +  p )  ->  R  =  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )
9594eqeq1d 2424 . . . . 5  |-  ( q  =  ( H  oF  +  p )  ->  ( R  =  0p  <->  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) )  =  0p ) )
9694fveq2d 5882 . . . . . 6  |-  ( q  =  ( H  oF  +  p )  ->  (deg `  R )  =  (deg `  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) ) )
9796breq1d 4430 . . . . 5  |-  ( q  =  ( H  oF  +  p )  ->  ( (deg `  R
)  <  N  <->  (deg `  ( F  oF  -  ( G  oF  x.  ( H  oF  +  p
) ) ) )  <  N ) )
9895, 97orbi12d 714 . . . 4  |-  ( 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 ) ) )
9998rspcev 3182 . . 3  |-  ( ( ( 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 ) )
10055, 90, 99syl2anc 665 . 2  |-  ( ( ( 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
) )
10150, 18, 4, 5plymul 23159 . . . 4  |-  ( ph  ->  ( H  oF  x.  G )  e.  (Poly `  S )
)
1021, 101, 4, 5, 7plysub 23160 . . 3  |-  ( ph  ->  ( F  oF  -  ( H  oF  x.  G )
)  e.  (Poly `  S ) )
103 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 ) ) )
104 eqid 2422 . . . . . . 7  |-  (deg `  ( H  oF  x.  G ) )  =  (deg `  ( H  oF  x.  G
) )
10512, 104dgrsub 23213 . . . . . 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
) )
1061, 101, 105syl2anc 665 . . . . 5  |-  ( ph  ->  (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  <_  if ( M  <_  (deg `  ( H  oF  x.  G ) ) ,  (deg `  ( H  oF  x.  G
) ) ,  M
) )
107 plydiv.fz . . . . . . . . . . . . 13  |-  ( ph  ->  F  =/=  0p )
10812, 9dgreq0 23206 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0p  <->  ( A `  M )  =  0 ) )
1091, 108syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  =  0p  <->  ( A `  M )  =  0 ) )
110109necon3bid 2682 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  =/=  0p 
<->  ( A `  M
)  =/=  0 ) )
111107, 110mpbid 213 . . . . . . . . . . . 12  |-  ( ph  ->  ( A `  M
)  =/=  0 )
11217, 27, 111, 32divne0d 10400 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  =/=  0 )
1133, 46sseldd 3465 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A `  M )  /  ( B `  N )
)  e.  CC )
11448coe1term 23200 . . . . . . . . . . . . 13  |-  ( ( ( ( A `  M )  /  ( B `  N )
)  e.  CC  /\  D  e.  NN0  /\  D  e.  NN0 )  ->  (
(coeff `  H ) `  D )  =  if ( D  =  D ,  ( ( A `
 M )  / 
( B `  N
) ) ,  0 ) )
115113, 47, 47, 114syl3anc 1264 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coeff `  H
) `  D )  =  if ( D  =  D ,  ( ( A `  M )  /  ( B `  N ) ) ,  0 ) )
116 eqid 2422 . . . . . . . . . . . . 13  |-  D  =  D
117116iftruei 3916 . . . . . . . . . . . 12  |-  if ( D  =  D , 
( ( A `  M )  /  ( B `  N )
) ,  0 )  =  ( ( A `
 M )  / 
( B `  N
) )
118115, 117syl6eq 2479 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  H
) `  D )  =  ( ( A `
 M )  / 
( B `  N
) ) )
119 c0ex 9638 . . . . . . . . . . . . 13  |-  0  e.  _V
120119fvconst2 6132 . . . . . . . . . . . 12  |-  ( D  e.  NN0  ->  ( ( NN0  X.  { 0 } ) `  D
)  =  0 )
12147, 120syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( ( NN0  X.  { 0 } ) `
 D )  =  0 )
122112, 118, 1213netr4d 2729 . . . . . . . . . 10  |-  ( ph  ->  ( (coeff `  H
) `  D )  =/=  ( ( NN0  X.  { 0 } ) `
 D ) )
123 fveq2 5878 . . . . . . . . . . . . 13  |-  ( H  =  0p  -> 
(coeff `  H )  =  (coeff `  0p
) )
124 coe0 23197 . . . . . . . . . . . . 13  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
125123, 124syl6eq 2479 . . . . . . . . . . . 12  |-  ( H  =  0p  -> 
(coeff `  H )  =  ( NN0  X.  { 0 } ) )
126125fveq1d 5880 . . . . . . . . . . 11  |-  ( H  =  0p  -> 
( (coeff `  H
) `  D )  =  ( ( NN0 
X.  { 0 } ) `  D ) )
127126necon3i 2664 . . . . . . . . . 10  |-  ( ( (coeff `  H ) `  D )  =/=  (
( NN0  X.  { 0 } ) `  D
)  ->  H  =/=  0p )
128122, 127syl 17 . . . . . . . . 9  |-  ( ph  ->  H  =/=  0p )
129 eqid 2422 . . . . . . . . . 10  |-  (deg `  H )  =  (deg
`  H )
130129, 22dgrmul 23211 . . . . . . . . 9  |-  ( ( ( H  e.  (Poly `  S )  /\  H  =/=  0p )  /\  ( G  e.  (Poly `  S )  /\  G  =/=  0p ) )  ->  (deg `  ( H  oF  x.  G
) )  =  ( (deg `  H )  +  N ) )
13150, 128, 18, 28, 130syl22anc 1265 . . . . . . . 8  |-  ( ph  ->  (deg `  ( H  oF  x.  G
) )  =  ( (deg `  H )  +  N ) )
13248dgr1term 23201 . . . . . . . . . . . 12  |-  ( ( ( ( A `  M )  /  ( B `  N )
)  e.  CC  /\  ( ( A `  M )  /  ( B `  N )
)  =/=  0  /\  D  e.  NN0 )  ->  (deg `  H )  =  D )
133113, 112, 47, 132syl3anc 1264 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  H )  =  D )
134 plydiv.e . . . . . . . . . . 11  |-  ( ph  ->  ( M  -  N
)  =  D )
135133, 134eqtr4d 2466 . . . . . . . . . 10  |-  ( ph  ->  (deg `  H )  =  ( M  -  N ) )
136135oveq1d 6317 . . . . . . . . 9  |-  ( ph  ->  ( (deg `  H
)  +  N )  =  ( ( M  -  N )  +  N ) )
13715nn0cnd 10928 . . . . . . . . . 10  |-  ( ph  ->  M  e.  CC )
13825nn0cnd 10928 . . . . . . . . . 10  |-  ( ph  ->  N  e.  CC )
139137, 138npcand 9991 . . . . . . . . 9  |-  ( ph  ->  ( ( M  -  N )  +  N
)  =  M )
140136, 139eqtrd 2463 . . . . . . . 8  |-  ( ph  ->  ( (deg `  H
)  +  N )  =  M )
141131, 140eqtrd 2463 . . . . . . 7  |-  ( ph  ->  (deg `  ( H  oF  x.  G
) )  =  M )
142141ifeq1d 3927 . . . . . 6  |-  ( ph  ->  if ( M  <_ 
(deg `  ( H  oF  x.  G
) ) ,  (deg
`  ( H  oF  x.  G )
) ,  M )  =  if ( M  <_  (deg `  ( H  oF  x.  G
) ) ,  M ,  M ) )
143 ifid 3946 . . . . . 6  |-  if ( M  <_  (deg `  ( H  oF  x.  G
) ) ,  M ,  M )  =  M
144142, 143syl6eq 2479 . . . . 5  |-  ( ph  ->  if ( M  <_ 
(deg `  ( H  oF  x.  G
) ) ,  (deg
`  ( H  oF  x.  G )
) ,  M )  =  M )
145106, 144breqtrd 4445 . . . 4  |-  ( ph  ->  (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  <_  M )
146 eqid 2422 . . . . . . . 8  |-  (coeff `  ( H  oF  x.  G ) )  =  (coeff `  ( H  oF  x.  G
) )
1479, 146coesub 23198 . . . . . . 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
) ) ) )
1481, 101, 147syl2anc 665 . . . . . 6  |-  ( ph  ->  (coeff `  ( F  oF  -  ( H  oF  x.  G
) ) )  =  ( A  oF  -  (coeff `  ( H  oF  x.  G
) ) ) )
149148fveq1d 5880 . . . . 5  |-  ( ph  ->  ( (coeff `  ( F  oF  -  ( H  oF  x.  G
) ) ) `  M )  =  ( ( A  oF  -  (coeff `  ( H  oF  x.  G
) ) ) `  M ) )
1509coef3 23173 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
151 ffn 5743 . . . . . . . 8  |-  ( A : NN0 --> CC  ->  A  Fn  NN0 )
1521, 150, 1513syl 18 . . . . . . 7  |-  ( ph  ->  A  Fn  NN0 )
153146coef3 23173 . . . . . . . 8  |-  ( ( H  oF  x.  G )  e.  (Poly `  S )  ->  (coeff `  ( H  oF  x.  G ) ) : NN0 --> CC )
154 ffn 5743 . . . . . . . 8  |-  ( (coeff `  ( H  oF  x.  G ) ) : NN0 --> CC  ->  (coeff `  ( H  oF  x.  G ) )  Fn  NN0 )
155101, 153, 1543syl 18 . . . . . . 7  |-  ( ph  ->  (coeff `  ( H  oF  x.  G
) )  Fn  NN0 )
156 nn0ex 10876 . . . . . . . 8  |-  NN0  e.  _V
157156a1i 11 . . . . . . 7  |-  ( ph  ->  NN0  e.  _V )
158 inidm 3671 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
159 eqidd 2423 . . . . . . 7  |-  ( (
ph  /\  M  e.  NN0 )  ->  ( A `  M )  =  ( A `  M ) )
160 eqid 2422 . . . . . . . . . . 11  |-  (coeff `  H )  =  (coeff `  H )
161160, 19, 129, 22coemulhi 23195 . . . . . . . . . 10  |-  ( ( H  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( H  oF  x.  G ) ) `
 ( (deg `  H )  +  N
) )  =  ( ( (coeff `  H
) `  (deg `  H
) )  x.  ( B `  N )
) )
16250, 18, 161syl2anc 665 . . . . . . . . 9  |-  ( ph  ->  ( (coeff `  ( H  oF  x.  G
) ) `  (
(deg `  H )  +  N ) )  =  ( ( (coeff `  H ) `  (deg `  H ) )  x.  ( B `  N
) ) )
163140fveq2d 5882 . . . . . . . . 9  |-  ( ph  ->  ( (coeff `  ( H  oF  x.  G
) ) `  (
(deg `  H )  +  N ) )  =  ( (coeff `  ( H  oF  x.  G
) ) `  M
) )
164133fveq2d 5882 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coeff `  H
) `  (deg `  H
) )  =  ( (coeff `  H ) `  D ) )
165164, 118eqtrd 2463 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  H
) `  (deg `  H
) )  =  ( ( A `  M
)  /  ( B `
 N ) ) )
166165oveq1d 6317 . . . . . . . . . 10  |-  ( ph  ->  ( ( (coeff `  H ) `  (deg `  H ) )  x.  ( B `  N
) )  =  ( ( ( A `  M )  /  ( B `  N )
)  x.  ( B `
 N ) ) )
16717, 27, 32divcan1d 10385 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A `
 M )  / 
( B `  N
) )  x.  ( B `  N )
)  =  ( A `
 M ) )
168166, 167eqtrd 2463 . . . . . . . . 9  |-  ( ph  ->  ( ( (coeff `  H ) `  (deg `  H ) )  x.  ( B `  N
) )  =  ( A `  M ) )
169162, 163, 1683eqtr3d 2471 . . . . . . . 8  |-  ( ph  ->  ( (coeff `  ( H  oF  x.  G
) ) `  M
)  =  ( A `
 M ) )
170169adantr 466 . . . . . . 7  |-  ( (
ph  /\  M  e.  NN0 )  ->  ( (coeff `  ( H  oF  x.  G ) ) `
 M )  =  ( A `  M
) )
171152, 155, 157, 157, 158, 159, 170ofval 6551 . . . . . 6  |-  ( (
ph  /\  M  e.  NN0 )  ->  ( ( A  oF  -  (coeff `  ( H  oF  x.  G ) ) ) `  M )  =  ( ( A `
 M )  -  ( A `  M ) ) )
17215, 171mpdan 672 . . . . 5  |-  ( ph  ->  ( ( A  oF  -  (coeff `  ( H  oF  x.  G
) ) ) `  M )  =  ( ( A `  M
)  -  ( A `
 M ) ) )
17317subidd 9975 . . . . 5  |-  ( ph  ->  ( ( A `  M )  -  ( A `  M )
)  =  0 )
174149, 172, 1733eqtrd 2467 . . . 4  |-  ( ph  ->  ( (coeff `  ( F  oF  -  ( H  oF  x.  G
) ) ) `  M )  =  0 )
175 dgrcl 23174 . . . . . . . . . 10  |-  ( ( F  oF  -  ( H  oF  x.  G ) )  e.  (Poly `  S )  ->  (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  e. 
NN0 )
176102, 175syl 17 . . . . . . . . 9  |-  ( ph  ->  (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  e. 
NN0 )
177176nn0red 10927 . . . . . . . 8  |-  ( ph  ->  (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  e.  RR )
17815nn0red 10927 . . . . . . . 8  |-  ( ph  ->  M  e.  RR )
17925nn0red 10927 . . . . . . . 8  |-  ( ph  ->  N  e.  RR )
180177, 178, 179ltsub1d 10223 . . . . . . 7  |-  ( ph  ->  ( (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  < 
M  <->  ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  -  N
)  <  ( M  -  N ) ) )
181134breq2d 4432 . . . . . . 7  |-  ( ph  ->  ( ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  -  N
)  <  ( M  -  N )  <->  ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  -  N
)  <  D )
)
182180, 181bitrd 256 . . . . . 6  |-  ( ph  ->  ( (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  < 
M  <->  ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  -  N
)  <  D )
)
183182orbi2d 706 . . . . 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
) ) )
184 eqid 2422 . . . . . . 7  |-  (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  =  (deg
`  ( F  oF  -  ( H  oF  x.  G
) ) )
185 eqid 2422 . . . . . . 7  |-  (coeff `  ( F  oF  -  ( H  oF  x.  G )
) )  =  (coeff `  ( F  oF  -  ( H  oF  x.  G )
) )
186184, 185dgrlt 23207 . . . . . 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 ) ) )
187102, 15, 186syl2anc 665 . . . . 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 ) ) )
188183, 187bitr3d 258 . . . 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 ) ) )
189145, 174, 188mpbir2and 930 . . 3  |-  ( ph  ->  ( ( F  oF  -  ( H  oF  x.  G
) )  =  0p  \/  ( (deg
`  ( F  oF  -  ( H  oF  x.  G
) ) )  -  N )  <  D
) )
190 eqeq1 2426 . . . . . 6  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (
f  =  0p  <-> 
( F  oF  -  ( H  oF  x.  G )
)  =  0p ) )
191 fveq2 5878 . . . . . . . 8  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (deg `  f )  =  (deg
`  ( F  oF  -  ( H  oF  x.  G
) ) ) )
192191oveq1d 6317 . . . . . . 7  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (
(deg `  f )  -  N )  =  ( (deg `  ( F  oF  -  ( H  oF  x.  G
) ) )  -  N ) )
193192breq1d 4430 . . . . . 6  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (
( (deg `  f
)  -  N )  <  D  <->  ( (deg `  ( F  oF  -  ( H  oF  x.  G )
) )  -  N
)  <  D )
)
194190, 193orbi12d 714 . . . . 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
) ) )
195 plydiv.u . . . . . . . . 9  |-  U  =  ( f  oF  -  ( G  oF  x.  p )
)
196 oveq1 6309 . . . . . . . . 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 )
) )
197195, 196syl5eq 2475 . . . . . . . 8  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  U  =  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )
198197eqeq1d 2424 . . . . . . 7  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  ( U  =  0p  <->  ( ( F  oF  -  ( H  oF  x.  G )
)  oF  -  ( G  oF  x.  p ) )  =  0p ) )
199197fveq2d 5882 . . . . . . . 8  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (deg `  U )  =  (deg
`  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) ) )
200199breq1d 4430 . . . . . . 7  |-  ( f  =  ( F  oF  -  ( H  oF  x.  G
) )  ->  (
(deg `  U )  <  N  <->  (deg `  ( ( F  oF  -  ( H  oF  x.  G
) )  oF  -  ( G  oF  x.  p )
) )  <  N
) )
201198, 200orbi12d 714 . . . . . 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
) ) )
202201rexbidv 2939 . . . . 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 ) ) )
203194, 202imbi12d 321 . . . 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
) ) ) )
204203rspcv 3178 . . 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 ) ) ) )
205102, 103, 189, 204syl3c 63 . 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
) )
206100, 205r19.29a 2970 1  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0p  \/  (deg `  R )  <  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   E.wrex 2776   _Vcvv 3081    C_ wss 3436   ifcif 3909   {csn 3996   class class class wbr 4420    |-> cmpt 4479    X. cxp 4848    Fn wfn 5593   -->wf 5594   ` cfv 5598  (class class class)co 6302    oFcof 6540   CCcc 9538   0cc0 9540   1c1 9541    + caddc 9543    x. cmul 9545    < clt 9676    <_ cle 9677    - cmin 9861   -ucneg 9862    / cdiv 10270   NN0cn0 10870   ^cexp 12272   0pc0p 22614  Polycply 23125  coeffccoe 23127  degcdgr 23128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-rp 11304  df-fz 11786  df-fzo 11917  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-rlim 13541  df-sum 13741  df-0p 22615  df-ply 23129  df-coe 23131  df-dgr 23132
This theorem is referenced by:  plydivex  23237
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