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Theorem plydivex 23144
Description: Lemma for plydivalg 23146. (Contributed by Mario Carneiro, 24-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 )
)
Assertion
Ref Expression
plydivex  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) )
Distinct variable groups:    x, y,
q, F    ph, x, y    G, q, x, y    x, R, y    S, q, x, y
Allowed substitution hints:    ph( q)    R( q)

Proof of Theorem plydivex
Dummy variables  z 
f  d  p  g  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plydiv.f . . . . . 6  |-  ( ph  ->  F  e.  (Poly `  S ) )
2 dgrcl 23081 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
31, 2syl 17 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
43nn0red 10915 . . . 4  |-  ( ph  ->  (deg `  F )  e.  RR )
5 plydiv.g . . . . . 6  |-  ( ph  ->  G  e.  (Poly `  S ) )
6 dgrcl 23081 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
75, 6syl 17 . . . . 5  |-  ( ph  ->  (deg `  G )  e.  NN0 )
87nn0red 10915 . . . 4  |-  ( ph  ->  (deg `  G )  e.  RR )
94, 8resubcld 10036 . . 3  |-  ( ph  ->  ( (deg `  F
)  -  (deg `  G ) )  e.  RR )
10 arch 10855 . . 3  |-  ( ( (deg `  F )  -  (deg `  G )
)  e.  RR  ->  E. d  e.  NN  (
(deg `  F )  -  (deg `  G )
)  <  d )
119, 10syl 17 . 2  |-  ( ph  ->  E. d  e.  NN  ( (deg `  F )  -  (deg `  G )
)  <  d )
12 olc 385 . . . 4  |-  ( ( (deg `  F )  -  (deg `  G )
)  <  d  ->  ( F  =  0p  \/  ( (deg `  F )  -  (deg `  G ) )  < 
d ) )
131adantr 466 . . . . 5  |-  ( (
ph  /\  d  e.  NN )  ->  F  e.  (Poly `  S )
)
14 nnnn0 10865 . . . . . . 7  |-  ( d  e.  NN  ->  d  e.  NN0 )
15 breq2 4421 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  <  0 ) )
1615orbi2d 706 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  <  0 ) ) )
1716imbi1d 318 . . . . . . . . . 10  |-  ( x  =  0  ->  (
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  x )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  0 )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
1817ralbidv 2862 . . . . . . . . 9  |-  ( x  =  0  ->  ( A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
x )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  <  0 )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
1918imbi2d 317 . . . . . . . 8  |-  ( x  =  0  ->  (
( ph  ->  A. f  e.  (Poly `  S )
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  x )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )  <->  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  <  0 )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
20 breq2 4421 . . . . . . . . . . . 12  |-  ( x  =  d  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  < 
d ) )
2120orbi2d 706 . . . . . . . . . . 11  |-  ( x  =  d  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
d ) ) )
2221imbi1d 318 . . . . . . . . . 10  |-  ( x  =  d  ->  (
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  x )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
2322ralbidv 2862 . . . . . . . . 9  |-  ( x  =  d  ->  ( A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
x )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
2423imbi2d 317 . . . . . . . 8  |-  ( x  =  d  ->  (
( ph  ->  A. f  e.  (Poly `  S )
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  x )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )  <->  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
25 breq2 4421 . . . . . . . . . . . 12  |-  ( x  =  ( d  +  1 )  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
2625orbi2d 706 . . . . . . . . . . 11  |-  ( x  =  ( d  +  1 )  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
( d  +  1 ) ) ) )
2726imbi1d 318 . . . . . . . . . 10  |-  ( x  =  ( d  +  1 )  ->  (
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  x )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  ( d  +  1 ) )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
2827ralbidv 2862 . . . . . . . . 9  |-  ( x  =  ( d  +  1 )  ->  ( A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
x )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
2928imbi2d 317 . . . . . . . 8  |-  ( x  =  ( d  +  1 )  ->  (
( ph  ->  A. f  e.  (Poly `  S )
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  x )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )  <->  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
30 plydiv.pl . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
3130adantlr 719 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  (Poly `  S )  /\  (
f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  <  0 ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
32 plydiv.tm . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
3332adantlr 719 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  (Poly `  S )  /\  (
f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  <  0 ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
34 plydiv.rc . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  S  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  S )
3534adantlr 719 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  (Poly `  S )  /\  (
f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  <  0 ) ) )  /\  ( x  e.  S  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  S )
36 plydiv.m1 . . . . . . . . . . . 12  |-  ( ph  -> 
-u 1  e.  S
)
3736adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  -u 1  e.  S )
38 simprl 762 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  f  e.  (Poly `  S ) )
395adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  G  e.  (Poly `  S ) )
40 plydiv.z . . . . . . . . . . . 12  |-  ( ph  ->  G  =/=  0p )
4140adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  G  =/=  0p )
42 eqid 2420 . . . . . . . . . . 11  |-  ( f  oF  -  ( G  oF  x.  q
) )  =  ( f  oF  -  ( G  oF  x.  q ) )
43 simprr 764 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  <  0 ) )
4431, 33, 35, 37, 38, 39, 41, 42, 43plydivlem3 23142 . . . . . . . . . 10  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )
4544expr 618 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  (Poly `  S ) )  ->  ( ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  <  0 )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )
4645ralrimiva 2837 . . . . . . . 8  |-  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  <  0 )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )
47 eqeq1 2424 . . . . . . . . . . . . . . . 16  |-  ( f  =  g  ->  (
f  =  0p  <-> 
g  =  0p ) )
48 fveq2 5872 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (deg `  f )  =  (deg
`  g ) )
4948oveq1d 6311 . . . . . . . . . . . . . . . . 17  |-  ( f  =  g  ->  (
(deg `  f )  -  (deg `  G )
)  =  ( (deg
`  g )  -  (deg `  G ) ) )
5049breq1d 4427 . . . . . . . . . . . . . . . 16  |-  ( f  =  g  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
d  <->  ( (deg `  g )  -  (deg `  G ) )  < 
d ) )
5147, 50orbi12d 714 . . . . . . . . . . . . . . 15  |-  ( f  =  g  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  d )  <->  ( g  =  0p  \/  ( (deg `  g
)  -  (deg `  G ) )  < 
d ) ) )
52 oveq1 6303 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (
f  oF  -  ( G  oF  x.  q ) )  =  ( g  oF  -  ( G  oF  x.  q )
) )
5352eqeq1d 2422 . . . . . . . . . . . . . . . . 17  |-  ( f  =  g  ->  (
( f  oF  -  ( G  oF  x.  q )
)  =  0p  <-> 
( g  oF  -  ( G  oF  x.  q )
)  =  0p ) )
5452fveq2d 5876 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  =  (deg
`  ( g  oF  -  ( G  oF  x.  q
) ) ) )
5554breq1d 4427 . . . . . . . . . . . . . . . . 17  |-  ( f  =  g  ->  (
(deg `  ( f  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )  <->  (deg
`  ( g  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )
) )
5653, 55orbi12d 714 . . . . . . . . . . . . . . . 16  |-  ( f  =  g  ->  (
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) )  <->  ( (
g  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )
5756rexbidv 2937 . . . . . . . . . . . . . . 15  |-  ( f  =  g  ->  ( E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) )  <->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )
5851, 57imbi12d 321 . . . . . . . . . . . . . 14  |-  ( f  =  g  ->  (
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( g  =  0p  \/  (
(deg `  g )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( g  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
5958cbvralv 3053 . . . . . . . . . . . . 13  |-  ( A. f  e.  (Poly `  S
) ( ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <->  A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )
60 simplll 766 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  ph )
6160, 30sylan 473 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S ) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  +  y )  e.  S )
6260, 32sylan 473 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S ) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  x.  y )  e.  S
)
6360, 34sylan 473 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S ) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  /\  (
x  e.  S  /\  x  =/=  0 ) )  ->  ( 1  /  x )  e.  S
)
6460, 36syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  -u 1  e.  S )
65 simplr 760 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  f  e.  (Poly `  S )
)
6660, 5syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  G  e.  (Poly `  S )
)
6760, 40syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  G  =/=  0p )
68 simpllr 767 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  d  e.  NN0 )
69 simprrr 773 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  (
(deg `  f )  -  (deg `  G )
)  =  d )
70 simprrl 772 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  f  =/=  0p )
71 eqid 2420 . . . . . . . . . . . . . . . 16  |-  ( g  oF  -  ( G  oF  x.  p
) )  =  ( g  oF  -  ( G  oF  x.  p ) )
72 oveq1 6303 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  z  ->  (
w ^ d )  =  ( z ^
d ) )
7372oveq2d 6312 . . . . . . . . . . . . . . . . 17  |-  ( w  =  z  ->  (
( ( (coeff `  f ) `  (deg `  f ) )  / 
( (coeff `  G
) `  (deg `  G
) ) )  x.  ( w ^ d
) )  =  ( ( ( (coeff `  f ) `  (deg `  f ) )  / 
( (coeff `  G
) `  (deg `  G
) ) )  x.  ( z ^ d
) ) )
7473cbvmptv 4509 . . . . . . . . . . . . . . . 16  |-  ( w  e.  CC  |->  ( ( ( (coeff `  f
) `  (deg `  f
) )  /  (
(coeff `  G ) `  (deg `  G )
) )  x.  (
w ^ d ) ) )  =  ( z  e.  CC  |->  ( ( ( (coeff `  f ) `  (deg `  f ) )  / 
( (coeff `  G
) `  (deg `  G
) ) )  x.  ( z ^ d
) ) )
75 simprl 762 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  A. g  e.  (Poly `  S )
( ( g  =  0p  \/  (
(deg `  g )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( g  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )
76 oveq2 6304 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( q  =  p  ->  ( G  oF  x.  q
)  =  ( G  oF  x.  p
) )
7776oveq2d 6312 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( q  =  p  ->  (
g  oF  -  ( G  oF  x.  q ) )  =  ( g  oF  -  ( G  oF  x.  p )
) )
7877eqeq1d 2422 . . . . . . . . . . . . . . . . . . . . 21  |-  ( q  =  p  ->  (
( g  oF  -  ( G  oF  x.  q )
)  =  0p  <-> 
( g  oF  -  ( G  oF  x.  p )
)  =  0p ) )
7977fveq2d 5876 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( q  =  p  ->  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  =  (deg
`  ( g  oF  -  ( G  oF  x.  p
) ) ) )
8079breq1d 4427 . . . . . . . . . . . . . . . . . . . . 21  |-  ( q  =  p  ->  (
(deg `  ( g  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )  <->  (deg
`  ( g  oF  -  ( G  oF  x.  p
) ) )  < 
(deg `  G )
) )
8178, 80orbi12d 714 . . . . . . . . . . . . . . . . . . . 20  |-  ( q  =  p  ->  (
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) )  <->  ( (
g  oF  -  ( G  oF  x.  p ) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )
8281cbvrexv 3054 . . . . . . . . . . . . . . . . . . 19  |-  ( E. q  e.  (Poly `  S ) ( ( g  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) )  <->  E. p  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) )
8382imbi2i 313 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( g  =  0p  \/  ( (deg
`  g )  -  (deg `  G ) )  <  d )  ->  E. q  e.  (Poly `  S ) ( ( g  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( g  =  0p  \/  (
(deg `  g )  -  (deg `  G )
)  <  d )  ->  E. p  e.  (Poly `  S ) ( ( g  oF  -  ( G  oF  x.  p ) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )
8483ralbii 2854 . . . . . . . . . . . . . . . . 17  |-  ( A. g  e.  (Poly `  S
) ( ( g  =  0p  \/  ( (deg `  g
)  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <->  A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. p  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )
8575, 84sylib 199 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  A. g  e.  (Poly `  S )
( ( g  =  0p  \/  (
(deg `  g )  -  (deg `  G )
)  <  d )  ->  E. p  e.  (Poly `  S ) ( ( g  oF  -  ( G  oF  x.  p ) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )
86 eqid 2420 . . . . . . . . . . . . . . . 16  |-  (coeff `  f )  =  (coeff `  f )
87 eqid 2420 . . . . . . . . . . . . . . . 16  |-  (coeff `  G )  =  (coeff `  G )
88 eqid 2420 . . . . . . . . . . . . . . . 16  |-  (deg `  f )  =  (deg
`  f )
89 eqid 2420 . . . . . . . . . . . . . . . 16  |-  (deg `  G )  =  (deg
`  G )
9061, 62, 63, 64, 65, 66, 67, 42, 68, 69, 70, 71, 74, 85, 86, 87, 88, 89plydivlem4 23143 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )
9190exp32 608 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( A. g  e.  (Poly `  S
) ( ( g  =  0p  \/  ( (deg `  g
)  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  ->  ( ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
9291ralrimdva 2841 . . . . . . . . . . . . 13  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( A. g  e.  (Poly `  S
) ( ( g  =  0p  \/  ( (deg `  g
)  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  ->  A. f  e.  (Poly `  S ) ( ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
9359, 92syl5bi 220 . . . . . . . . . . . 12  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( A. f  e.  (Poly `  S
) ( ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  ->  A. f  e.  (Poly `  S ) ( ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
9493ancld 555 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( A. f  e.  (Poly `  S
) ( ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  ->  ( A. f  e.  (Poly `  S )
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  A. f  e.  (Poly `  S )
( ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
95 dgrcl 23081 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  e.  (Poly `  S
)  ->  (deg `  f
)  e.  NN0 )
9695adantl 467 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  f
)  e.  NN0 )
9796nn0zd 11027 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  f
)  e.  ZZ )
985ad2antrr 730 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  G  e.  (Poly `  S ) )
9998, 6syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  G
)  e.  NN0 )
10099nn0zd 11027 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  G
)  e.  ZZ )
10197, 100zsubcld 11034 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (deg `  f )  -  (deg `  G ) )  e.  ZZ )
102 nn0z 10949 . . . . . . . . . . . . . . . . . . . 20  |-  ( d  e.  NN0  ->  d  e.  ZZ )
103102ad2antlr 731 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  d  e.  ZZ )
104 zleltp1 10976 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( (deg `  f
)  -  (deg `  G ) )  e.  ZZ  /\  d  e.  ZZ )  ->  (
( (deg `  f
)  -  (deg `  G ) )  <_ 
d  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
105101, 103, 104syl2anc 665 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
(deg `  f )  -  (deg `  G )
)  <_  d  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
106101zred 11029 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (deg `  f )  -  (deg `  G ) )  e.  RR )
107 nn0re 10867 . . . . . . . . . . . . . . . . . . . 20  |-  ( d  e.  NN0  ->  d  e.  RR )
108107ad2antlr 731 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  d  e.  RR )
109106, 108leloed 9767 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
(deg `  f )  -  (deg `  G )
)  <_  d  <->  ( (
(deg `  f )  -  (deg `  G )
)  <  d  \/  ( (deg `  f )  -  (deg `  G )
)  =  d ) ) )
110105, 109bitr3d 258 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
(deg `  f )  -  (deg `  G )
)  <  ( d  +  1 )  <->  ( (
(deg `  f )  -  (deg `  G )
)  <  d  \/  ( (deg `  f )  -  (deg `  G )
)  =  d ) ) )
111110orbi2d 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) )  <->  ( f  =  0p  \/  ( ( (deg `  f )  -  (deg `  G ) )  < 
d  \/  ( (deg
`  f )  -  (deg `  G ) )  =  d ) ) ) )
112 pm5.63 932 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( f  =  0p  \/  ( -.  f  =  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )
113 df-ne 2618 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =/=  0p  <->  -.  f  =  0p )
114113anbi1i 699 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( -.  f  =  0p  /\  ( (deg `  f
)  -  (deg `  G ) )  =  d ) )
115114orbi2i 521 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  =  0p  \/  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) )  <->  ( f  =  0p  \/  ( -.  f  =  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )
116112, 115bitr4i 255 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( f  =  0p  \/  ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )
117116orbi2i 521 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( (deg `  f
)  -  (deg `  G ) )  < 
d  \/  ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  =  d ) )  <->  ( (
(deg `  f )  -  (deg `  G )
)  <  d  \/  ( f  =  0p  \/  ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d ) ) ) )
118 or12 525 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  =  0p  \/  ( ( (deg
`  f )  -  (deg `  G ) )  <  d  \/  (
(deg `  f )  -  (deg `  G )
)  =  d ) )  <->  ( ( (deg
`  f )  -  (deg `  G ) )  <  d  \/  (
f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )
119 or12 525 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  =  0p  \/  ( ( (deg
`  f )  -  (deg `  G ) )  <  d  \/  (
f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )  <-> 
( ( (deg `  f )  -  (deg `  G ) )  < 
d  \/  ( f  =  0p  \/  ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) ) )
120117, 118, 1193bitr4i 280 . . . . . . . . . . . . . . . . 17  |-  ( ( f  =  0p  \/  ( ( (deg
`  f )  -  (deg `  G ) )  <  d  \/  (
(deg `  f )  -  (deg `  G )
)  =  d ) )  <->  ( f  =  0p  \/  (
( (deg `  f
)  -  (deg `  G ) )  < 
d  \/  ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d ) ) ) )
121 orass 526 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  d )  \/  ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) )  <->  ( f  =  0p  \/  ( ( (deg `  f )  -  (deg `  G ) )  < 
d  \/  ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d ) ) ) )
122120, 121bitr4i 255 . . . . . . . . . . . . . . . 16  |-  ( ( f  =  0p  \/  ( ( (deg
`  f )  -  (deg `  G ) )  <  d  \/  (
(deg `  f )  -  (deg `  G )
)  =  d ) )  <->  ( ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
d )  \/  (
f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )
123111, 122syl6bb 264 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) )  <->  ( (
f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
d )  \/  (
f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) ) )
124123imbi1d 318 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  ( d  +  1 ) )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
d )  \/  (
f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
125 jaob 790 . . . . . . . . . . . . . 14  |-  ( ( ( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  \/  ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
126124, 125syl6bb 264 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  ( d  +  1 ) )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
127126ralbidva 2859 . . . . . . . . . . . 12  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( A. f  e.  (Poly `  S
) ( ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
( d  +  1 ) )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <->  A. f  e.  (Poly `  S ) ( ( ( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
128 r19.26 2953 . . . . . . . . . . . 12  |-  ( A. f  e.  (Poly `  S
) ( ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  ( ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )  <->  ( A. f  e.  (Poly `  S )
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  A. f  e.  (Poly `  S )
( ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
129127, 128syl6bb 264 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( A. f  e.  (Poly `  S
) ( ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
( d  +  1 ) )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( A. f  e.  (Poly `  S )
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  /\  A. f  e.  (Poly `  S )
( ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
13094, 129sylibrd 237 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( A. f  e.  (Poly `  S
) ( ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  ->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
131130expcom 436 . . . . . . . . 9  |-  ( d  e.  NN0  ->  ( ph  ->  ( A. f  e.  (Poly `  S )
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  ->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
132131a2d 29 . . . . . . . 8  |-  ( d  e.  NN0  ->  ( (
ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )  ->  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
13319, 24, 29, 24, 46, 132nn0ind 11019 . . . . . . 7  |-  ( d  e.  NN0  ->  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
13414, 133syl 17 . . . . . 6  |-  ( d  e.  NN  ->  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) ) )
135134impcom 431 . . . . 5  |-  ( (
ph  /\  d  e.  NN )  ->  A. f  e.  (Poly `  S )
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )
136 eqeq1 2424 . . . . . . . 8  |-  ( f  =  F  ->  (
f  =  0p  <-> 
F  =  0p ) )
137 fveq2 5872 . . . . . . . . . 10  |-  ( f  =  F  ->  (deg `  f )  =  (deg
`  F ) )
138137oveq1d 6311 . . . . . . . . 9  |-  ( f  =  F  ->  (
(deg `  f )  -  (deg `  G )
)  =  ( (deg
`  F )  -  (deg `  G ) ) )
139138breq1d 4427 . . . . . . . 8  |-  ( f  =  F  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
d  <->  ( (deg `  F )  -  (deg `  G ) )  < 
d ) )
140136, 139orbi12d 714 . . . . . . 7  |-  ( f  =  F  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  d )  <->  ( F  =  0p  \/  ( (deg `  F
)  -  (deg `  G ) )  < 
d ) ) )
141 oveq1 6303 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f  oF  -  ( G  oF  x.  q ) )  =  ( F  oF  -  ( G  oF  x.  q )
) )
142 plydiv.r . . . . . . . . . . 11  |-  R  =  ( F  oF  -  ( G  oF  x.  q )
)
143141, 142syl6eqr 2479 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f  oF  -  ( G  oF  x.  q ) )  =  R )
144143eqeq1d 2422 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f  oF  -  ( G  oF  x.  q )
)  =  0p  <-> 
R  =  0p ) )
145143fveq2d 5876 . . . . . . . . . 10  |-  ( f  =  F  ->  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  =  (deg
`  R ) )
146145breq1d 4427 . . . . . . . . 9  |-  ( f  =  F  ->  (
(deg `  ( f  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )  <->  (deg
`  R )  < 
(deg `  G )
) )
147144, 146orbi12d 714 . . . . . . . 8  |-  ( f  =  F  ->  (
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) )  <->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
148147rexbidv 2937 . . . . . . 7  |-  ( f  =  F  ->  ( E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) )  <->  E. q  e.  (Poly `  S )
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) ) )
149140, 148imbi12d 321 . . . . . 6  |-  ( f  =  F  ->  (
( ( f  =  0p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( F  =  0p  \/  (
(deg `  F )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) ) )
150149rspcv 3175 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  ( A. f  e.  (Poly `  S
) ( ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  ->  ( ( F  =  0p  \/  ( (deg `  F
)  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) ) ) )
15113, 135, 150sylc 62 . . . 4  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( F  =  0p  \/  ( (deg `  F )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) ) )
15212, 151syl5 33 . . 3  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( (deg `  F )  -  (deg `  G )
)  <  d  ->  E. q  e.  (Poly `  S ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
153152rexlimdva 2915 . 2  |-  ( ph  ->  ( E. d  e.  NN  ( (deg `  F )  -  (deg `  G ) )  < 
d  ->  E. q  e.  (Poly `  S )
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) ) )
15411, 153mpd 15 1  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   A.wral 2773   E.wrex 2774   class class class wbr 4417    |-> cmpt 4475   ` cfv 5592  (class class class)co 6296    oFcof 6534   CCcc 9526   RRcr 9527   0cc0 9528   1c1 9529    + caddc 9531    x. cmul 9533    < clt 9664    <_ cle 9665    - cmin 9849   -ucneg 9850    / cdiv 10258   NNcn 10598   NN0cn0 10858   ZZcz 10926   ^cexp 12258   0pc0p 22521  Polycply 23032  coeffccoe 23034  degcdgr 23035
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 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606  ax-addf 9607
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 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-inf 7954  df-oi 8016  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-fz 11772  df-fzo 11903  df-fl 12014  df-seq 12200  df-exp 12259  df-hash 12502  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-clim 13519  df-rlim 13520  df-sum 13720  df-0p 22522  df-ply 23036  df-coe 23038  df-dgr 23039
This theorem is referenced by:  plydivalg  23146
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