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Theorem plydivex 23243
Description: Lemma for plydivalg 23245. (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 23180 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
31, 2syl 17 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
43nn0red 10923 . . . 4  |-  ( ph  ->  (deg `  F )  e.  RR )
5 plydiv.g . . . . . 6  |-  ( ph  ->  G  e.  (Poly `  S ) )
6 dgrcl 23180 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
75, 6syl 17 . . . . 5  |-  ( ph  ->  (deg `  G )  e.  NN0 )
87nn0red 10923 . . . 4  |-  ( ph  ->  (deg `  G )  e.  RR )
94, 8resubcld 10044 . . 3  |-  ( ph  ->  ( (deg `  F
)  -  (deg `  G ) )  e.  RR )
10 arch 10863 . . 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 386 . . . 4  |-  ( ( (deg `  F )  -  (deg `  G )
)  <  d  ->  ( F  =  0p  \/  ( (deg `  F )  -  (deg `  G ) )  < 
d ) )
131adantr 467 . . . . 5  |-  ( (
ph  /\  d  e.  NN )  ->  F  e.  (Poly `  S )
)
14 nnnn0 10873 . . . . . . 7  |-  ( d  e.  NN  ->  d  e.  NN0 )
15 breq2 4405 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  <  0 ) )
1615orbi2d 707 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  <  0 ) ) )
1716imbi1d 319 . . . . . . . . . 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 2826 . . . . . . . . 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 318 . . . . . . . 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 4405 . . . . . . . . . . . 12  |-  ( x  =  d  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  < 
d ) )
2120orbi2d 707 . . . . . . . . . . 11  |-  ( x  =  d  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
d ) ) )
2221imbi1d 319 . . . . . . . . . 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 2826 . . . . . . . . 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 318 . . . . . . . 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 4405 . . . . . . . . . . . 12  |-  ( x  =  ( d  +  1 )  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
2625orbi2d 707 . . . . . . . . . . 11  |-  ( x  =  ( d  +  1 )  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
( d  +  1 ) ) ) )
2726imbi1d 319 . . . . . . . . . 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 2826 . . . . . . . . 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 318 . . . . . . . 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 720 . . . . . . . . . . 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 720 . . . . . . . . . . 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 720 . . . . . . . . . . 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 467 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  -u 1  e.  S )
38 simprl 763 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  f  e.  (Poly `  S ) )
395adantr 467 . . . . . . . . . . 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 467 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  G  =/=  0p )
42 eqid 2450 . . . . . . . . . . 11  |-  ( f  oF  -  ( G  oF  x.  q
) )  =  ( f  oF  -  ( G  oF  x.  q ) )
43 simprr 765 . . . . . . . . . . 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 23241 . . . . . . . . . 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 619 . . . . . . . . 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 2801 . . . . . . . 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 2454 . . . . . . . . . . . . . . . 16  |-  ( f  =  g  ->  (
f  =  0p  <-> 
g  =  0p ) )
48 fveq2 5863 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (deg `  f )  =  (deg
`  g ) )
4948oveq1d 6303 . . . . . . . . . . . . . . . . 17  |-  ( f  =  g  ->  (
(deg `  f )  -  (deg `  G )
)  =  ( (deg
`  g )  -  (deg `  G ) ) )
5049breq1d 4411 . . . . . . . . . . . . . . . 16  |-  ( f  =  g  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
d  <->  ( (deg `  g )  -  (deg `  G ) )  < 
d ) )
5147, 50orbi12d 715 . . . . . . . . . . . . . . 15  |-  ( f  =  g  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  d )  <->  ( g  =  0p  \/  ( (deg `  g
)  -  (deg `  G ) )  < 
d ) ) )
52 oveq1 6295 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (
f  oF  -  ( G  oF  x.  q ) )  =  ( g  oF  -  ( G  oF  x.  q )
) )
5352eqeq1d 2452 . . . . . . . . . . . . . . . . 17  |-  ( f  =  g  ->  (
( f  oF  -  ( G  oF  x.  q )
)  =  0p  <-> 
( g  oF  -  ( G  oF  x.  q )
)  =  0p ) )
5452fveq2d 5867 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  =  (deg
`  ( g  oF  -  ( G  oF  x.  q
) ) ) )
5554breq1d 4411 . . . . . . . . . . . . . . . . 17  |-  ( f  =  g  ->  (
(deg `  ( f  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )  <->  (deg
`  ( g  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )
) )
5653, 55orbi12d 715 . . . . . . . . . . . . . . . 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 2900 . . . . . . . . . . . . . . 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 322 . . . . . . . . . . . . . 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 3018 . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . 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 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 ) ) )  ->  f  =/=  0p )
71 eqid 2450 . . . . . . . . . . . . . . . 16  |-  ( g  oF  -  ( G  oF  x.  p
) )  =  ( g  oF  -  ( G  oF  x.  p ) )
72 oveq1 6295 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  z  ->  (
w ^ d )  =  ( z ^
d ) )
7372oveq2d 6304 . . . . . . . . . . . . . . . . 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 4494 . . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . 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 6296 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( q  =  p  ->  ( G  oF  x.  q
)  =  ( G  oF  x.  p
) )
7776oveq2d 6304 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( q  =  p  ->  (
g  oF  -  ( G  oF  x.  q ) )  =  ( g  oF  -  ( G  oF  x.  p )
) )
7877eqeq1d 2452 . . . . . . . . . . . . . . . . . . . . 21  |-  ( q  =  p  ->  (
( g  oF  -  ( G  oF  x.  q )
)  =  0p  <-> 
( g  oF  -  ( G  oF  x.  p )
)  =  0p ) )
7977fveq2d 5867 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( q  =  p  ->  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  =  (deg
`  ( g  oF  -  ( G  oF  x.  p
) ) ) )
8079breq1d 4411 . . . . . . . . . . . . . . . . . . . . 21  |-  ( q  =  p  ->  (
(deg `  ( g  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )  <->  (deg
`  ( g  oF  -  ( G  oF  x.  p
) ) )  < 
(deg `  G )
) )
8178, 80orbi12d 715 . . . . . . . . . . . . . . . . . . . 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 3019 . . . . . . . . . . . . . . . . . . 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 314 . . . . . . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . . . 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 200 . . . . . . . . . . . . . . . 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 2450 . . . . . . . . . . . . . . . 16  |-  (coeff `  f )  =  (coeff `  f )
87 eqid 2450 . . . . . . . . . . . . . . . 16  |-  (coeff `  G )  =  (coeff `  G )
88 eqid 2450 . . . . . . . . . . . . . . . 16  |-  (deg `  f )  =  (deg
`  f )
89 eqid 2450 . . . . . . . . . . . . . . . 16  |-  (deg `  G )  =  (deg
`  G )
9061, 62, 63, 64, 65, 66, 67, 42, 68, 69, 70, 71, 74, 85, 86, 87, 88, 89plydivlem4 23242 . . . . . . . . . . . . . . 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 609 . . . . . . . . . . . . . 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 2805 . . . . . . . . . . . . 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 221 . . . . . . . . . . . 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 556 . . . . . . . . . . 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 23180 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  e.  (Poly `  S
)  ->  (deg `  f
)  e.  NN0 )
9695adantl 468 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  f
)  e.  NN0 )
9796nn0zd 11035 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  f
)  e.  ZZ )
985ad2antrr 731 . . . . . . . . . . . . . . . . . . . . . 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 11035 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  G
)  e.  ZZ )
10197, 100zsubcld 11042 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (deg `  f )  -  (deg `  G ) )  e.  ZZ )
102 nn0z 10957 . . . . . . . . . . . . . . . . . . . 20  |-  ( d  e.  NN0  ->  d  e.  ZZ )
103102ad2antlr 732 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  d  e.  ZZ )
104 zleltp1 10984 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( (deg `  f
)  -  (deg `  G ) )  e.  ZZ  /\  d  e.  ZZ )  ->  (
( (deg `  f
)  -  (deg `  G ) )  <_ 
d  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
105101, 103, 104syl2anc 666 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
(deg `  f )  -  (deg `  G )
)  <_  d  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
106101zred 11037 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (deg `  f )  -  (deg `  G ) )  e.  RR )
107 nn0re 10875 . . . . . . . . . . . . . . . . . . . 20  |-  ( d  e.  NN0  ->  d  e.  RR )
108107ad2antlr 732 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  d  e.  RR )
109106, 108leloed 9775 . . . . . . . . . . . . . . . . . 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 259 . . . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . 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 pm5.63 934 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( f  =  0p  \/  ( -.  f  =  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )
113 df-ne 2623 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =/=  0p  <->  -.  f  =  0p )
114113anbi1i 700 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( -.  f  =  0p  /\  ( (deg `  f
)  -  (deg `  G ) )  =  d ) )
115114orbi2i 522 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  =  0p  \/  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) )  <->  ( f  =  0p  \/  ( -.  f  =  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )
116112, 115bitr4i 256 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( f  =  0p  \/  ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )
117116orbi2i 522 . . . . . . . . . . . . . . . . . 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 526 . . . . . . . . . . . . . . . . . 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 526 . . . . . . . . . . . . . . . . . 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 281 . . . . . . . . . . . . . . . . 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 527 . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . 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 265 . . . . . . . . . . . . . . 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 319 . . . . . . . . . . . . . 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 791 . . . . . . . . . . . . . 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 265 . . . . . . . . . . . . 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 2823 . . . . . . . . . . . 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 2916 . . . . . . . . . . . 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 265 . . . . . . . . . . 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 238 . . . . . . . . . 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 437 . . . . . . . . 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 11027 . . . . . . 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 432 . . . . 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 2454 . . . . . . . 8  |-  ( f  =  F  ->  (
f  =  0p  <-> 
F  =  0p ) )
137 fveq2 5863 . . . . . . . . . 10  |-  ( f  =  F  ->  (deg `  f )  =  (deg
`  F ) )
138137oveq1d 6303 . . . . . . . . 9  |-  ( f  =  F  ->  (
(deg `  f )  -  (deg `  G )
)  =  ( (deg
`  F )  -  (deg `  G ) ) )
139138breq1d 4411 . . . . . . . 8  |-  ( f  =  F  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
d  <->  ( (deg `  F )  -  (deg `  G ) )  < 
d ) )
140136, 139orbi12d 715 . . . . . . 7  |-  ( f  =  F  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  d )  <->  ( F  =  0p  \/  ( (deg `  F
)  -  (deg `  G ) )  < 
d ) ) )
141 oveq1 6295 . . . . . . . . . . 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 2502 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f  oF  -  ( G  oF  x.  q ) )  =  R )
144143eqeq1d 2452 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f  oF  -  ( G  oF  x.  q )
)  =  0p  <-> 
R  =  0p ) )
145143fveq2d 5867 . . . . . . . . . 10  |-  ( f  =  F  ->  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  =  (deg
`  R ) )
146145breq1d 4411 . . . . . . . . 9  |-  ( f  =  F  ->  (
(deg `  ( f  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )  <->  (deg
`  R )  < 
(deg `  G )
) )
147144, 146orbi12d 715 . . . . . . . 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 2900 . . . . . . 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 322 . . . . . 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 3145 . . . . 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 2878 . 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 188    \/ wo 370    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736   E.wrex 2737   class class class wbr 4401    |-> cmpt 4460   ` cfv 5581  (class class class)co 6288    oFcof 6526   CCcc 9534   RRcr 9535   0cc0 9536   1c1 9537    + caddc 9539    x. cmul 9541    < clt 9672    <_ cle 9673    - cmin 9857   -ucneg 9858    / cdiv 10266   NNcn 10606   NN0cn0 10866   ZZcz 10934   ^cexp 12269   0pc0p 22620  Polycply 23131  coeffccoe 23133  degcdgr 23134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-rp 11300  df-fz 11782  df-fzo 11913  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-rlim 13546  df-sum 13746  df-0p 22621  df-ply 23135  df-coe 23137  df-dgr 23138
This theorem is referenced by:  plydivalg  23245
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