MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  plydivex Structured version   Unicode version

Theorem plydivex 22424
Description: Lemma for plydivalg 22426. (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 22362 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
31, 2syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
43nn0red 10849 . . . 4  |-  ( ph  ->  (deg `  F )  e.  RR )
5 plydiv.g . . . . . 6  |-  ( ph  ->  G  e.  (Poly `  S ) )
6 dgrcl 22362 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
75, 6syl 16 . . . . 5  |-  ( ph  ->  (deg `  G )  e.  NN0 )
87nn0red 10849 . . . 4  |-  ( ph  ->  (deg `  G )  e.  RR )
94, 8resubcld 9983 . . 3  |-  ( ph  ->  ( (deg `  F
)  -  (deg `  G ) )  e.  RR )
10 arch 10788 . . 3  |-  ( ( (deg `  F )  -  (deg `  G )
)  e.  RR  ->  E. d  e.  NN  (
(deg `  F )  -  (deg `  G )
)  <  d )
119, 10syl 16 . 2  |-  ( ph  ->  E. d  e.  NN  ( (deg `  F )  -  (deg `  G )
)  <  d )
12 olc 384 . . . 4  |-  ( ( (deg `  F )  -  (deg `  G )
)  <  d  ->  ( F  =  0p  \/  ( (deg `  F )  -  (deg `  G ) )  < 
d ) )
131adantr 465 . . . . 5  |-  ( (
ph  /\  d  e.  NN )  ->  F  e.  (Poly `  S )
)
14 nnnn0 10798 . . . . . . 7  |-  ( d  e.  NN  ->  d  e.  NN0 )
15 breq2 4451 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  <  0 ) )
1615orbi2d 701 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  <  0 ) ) )
1716imbi1d 317 . . . . . . . . . 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 2903 . . . . . . . . 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 316 . . . . . . . 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 4451 . . . . . . . . . . . 12  |-  ( x  =  d  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  < 
d ) )
2120orbi2d 701 . . . . . . . . . . 11  |-  ( x  =  d  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
d ) ) )
2221imbi1d 317 . . . . . . . . . 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 2903 . . . . . . . . 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 316 . . . . . . . 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 4451 . . . . . . . . . . . 12  |-  ( x  =  ( d  +  1 )  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
2625orbi2d 701 . . . . . . . . . . 11  |-  ( x  =  ( d  +  1 )  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0p  \/  ( (deg `  f
)  -  (deg `  G ) )  < 
( d  +  1 ) ) ) )
2726imbi1d 317 . . . . . . . . . 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 2903 . . . . . . . . 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 316 . . . . . . . 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 714 . . . . . . . . . . 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 714 . . . . . . . . . . 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 714 . . . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  -u 1  e.  S )
38 simprl 755 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  f  e.  (Poly `  S ) )
395adantr 465 . . . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  G  =/=  0p )
42 eqid 2467 . . . . . . . . . . 11  |-  ( f  oF  -  ( G  oF  x.  q
) )  =  ( f  oF  -  ( G  oF  x.  q ) )
43 simprr 756 . . . . . . . . . . 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 22422 . . . . . . . . . 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 615 . . . . . . . . 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 2878 . . . . . . . 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 2471 . . . . . . . . . . . . . . . 16  |-  ( f  =  g  ->  (
f  =  0p  <-> 
g  =  0p ) )
48 fveq2 5864 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (deg `  f )  =  (deg
`  g ) )
4948oveq1d 6297 . . . . . . . . . . . . . . . . 17  |-  ( f  =  g  ->  (
(deg `  f )  -  (deg `  G )
)  =  ( (deg
`  g )  -  (deg `  G ) ) )
5049breq1d 4457 . . . . . . . . . . . . . . . 16  |-  ( f  =  g  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
d  <->  ( (deg `  g )  -  (deg `  G ) )  < 
d ) )
5147, 50orbi12d 709 . . . . . . . . . . . . . . 15  |-  ( f  =  g  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  d )  <->  ( g  =  0p  \/  ( (deg `  g
)  -  (deg `  G ) )  < 
d ) ) )
52 oveq1 6289 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (
f  oF  -  ( G  oF  x.  q ) )  =  ( g  oF  -  ( G  oF  x.  q )
) )
5352eqeq1d 2469 . . . . . . . . . . . . . . . . 17  |-  ( f  =  g  ->  (
( f  oF  -  ( G  oF  x.  q )
)  =  0p  <-> 
( g  oF  -  ( G  oF  x.  q )
)  =  0p ) )
5452fveq2d 5868 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  =  (deg
`  ( g  oF  -  ( G  oF  x.  q
) ) ) )
5554breq1d 4457 . . . . . . . . . . . . . . . . 17  |-  ( f  =  g  ->  (
(deg `  ( f  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )  <->  (deg
`  ( g  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )
) )
5653, 55orbi12d 709 . . . . . . . . . . . . . . . 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 2973 . . . . . . . . . . . . . . 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 320 . . . . . . . . . . . . . 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 3088 . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . . 16  |-  ( g  oF  -  ( G  oF  x.  p
) )  =  ( g  oF  -  ( G  oF  x.  p ) )
72 oveq1 6289 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  z  ->  (
w ^ d )  =  ( z ^
d ) )
7372oveq2d 6298 . . . . . . . . . . . . . . . . 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 4538 . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . . 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 6290 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( q  =  p  ->  ( G  oF  x.  q
)  =  ( G  oF  x.  p
) )
7776oveq2d 6298 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( q  =  p  ->  (
g  oF  -  ( G  oF  x.  q ) )  =  ( g  oF  -  ( G  oF  x.  p )
) )
7877eqeq1d 2469 . . . . . . . . . . . . . . . . . . . . 21  |-  ( q  =  p  ->  (
( g  oF  -  ( G  oF  x.  q )
)  =  0p  <-> 
( g  oF  -  ( G  oF  x.  p )
)  =  0p ) )
7977fveq2d 5868 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( q  =  p  ->  (deg `  ( g  oF  -  ( G  oF  x.  q )
) )  =  (deg
`  ( g  oF  -  ( G  oF  x.  p
) ) ) )
8079breq1d 4457 . . . . . . . . . . . . . . . . . . . . 21  |-  ( q  =  p  ->  (
(deg `  ( g  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )  <->  (deg
`  ( g  oF  -  ( G  oF  x.  p
) ) )  < 
(deg `  G )
) )
8178, 80orbi12d 709 . . . . . . . . . . . . . . . . . . . 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 3089 . . . . . . . . . . . . . . . . . . 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 312 . . . . . . . . . . . . . . . . . 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 2895 . . . . . . . . . . . . . . . . 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 196 . . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . . 16  |-  (coeff `  f )  =  (coeff `  f )
87 eqid 2467 . . . . . . . . . . . . . . . 16  |-  (coeff `  G )  =  (coeff `  G )
88 eqid 2467 . . . . . . . . . . . . . . . 16  |-  (deg `  f )  =  (deg
`  f )
89 eqid 2467 . . . . . . . . . . . . . . . 16  |-  (deg `  G )  =  (deg
`  G )
9061, 62, 63, 64, 65, 66, 67, 42, 68, 69, 70, 71, 74, 85, 86, 87, 88, 89plydivlem4 22423 . . . . . . . . . . . . . . 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 605 . . . . . . . . . . . . . 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 2882 . . . . . . . . . . . . 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 217 . . . . . . . . . . . 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 553 . . . . . . . . . . 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 22362 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  e.  (Poly `  S
)  ->  (deg `  f
)  e.  NN0 )
9695adantl 466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  f
)  e.  NN0 )
9796nn0zd 10960 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  f
)  e.  ZZ )
985ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  G  e.  (Poly `  S ) )
9998, 6syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  G
)  e.  NN0 )
10099nn0zd 10960 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  G
)  e.  ZZ )
10197, 100zsubcld 10967 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (deg `  f )  -  (deg `  G ) )  e.  ZZ )
102 nn0z 10883 . . . . . . . . . . . . . . . . . . . 20  |-  ( d  e.  NN0  ->  d  e.  ZZ )
103102ad2antlr 726 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  d  e.  ZZ )
104 zleltp1 10909 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( (deg `  f
)  -  (deg `  G ) )  e.  ZZ  /\  d  e.  ZZ )  ->  (
( (deg `  f
)  -  (deg `  G ) )  <_ 
d  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
105101, 103, 104syl2anc 661 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
(deg `  f )  -  (deg `  G )
)  <_  d  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
106101zred 10962 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (deg `  f )  -  (deg `  G ) )  e.  RR )
107 nn0re 10800 . . . . . . . . . . . . . . . . . . . 20  |-  ( d  e.  NN0  ->  d  e.  RR )
108107ad2antlr 726 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  d  e.  RR )
109106, 108leloed 9723 . . . . . . . . . . . . . . . . . 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 255 . . . . . . . . . . . . . . . . 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 701 . . . . . . . . . . . . . . . 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 pm5.63 922 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( f  =  0p  \/  ( -.  f  =  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )
113 df-ne 2664 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =/=  0p  <->  -.  f  =  0p )
114113anbi1i 695 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( -.  f  =  0p  /\  ( (deg `  f
)  -  (deg `  G ) )  =  d ) )
115114orbi2i 519 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  =  0p  \/  ( f  =/=  0p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) )  <->  ( f  =  0p  \/  ( -.  f  =  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )
116112, 115bitr4i 252 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  =  0p  \/  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( f  =  0p  \/  ( f  =/=  0p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )
117116orbi2i 519 . . . . . . . . . . . . . . . . . 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 523 . . . . . . . . . . . . . . . . . 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 523 . . . . . . . . . . . . . . . . . 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 277 . . . . . . . . . . . . . . . . 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 524 . . . . . . . . . . . . . . . . 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 252 . . . . . . . . . . . . . . . 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 261 . . . . . . . . . . . . . . 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 317 . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . 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 261 . . . . . . . . . . . . 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 2900 . . . . . . . . . . . 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 2989 . . . . . . . . . . . 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 261 . . . . . . . . . . 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 234 . . . . . . . . . 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 435 . . . . . . . . 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 26 . . . . . . . 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 10953 . . . . . . 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 16 . . . . . 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 430 . . . . 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 2471 . . . . . . . 8  |-  ( f  =  F  ->  (
f  =  0p  <-> 
F  =  0p ) )
137 fveq2 5864 . . . . . . . . . 10  |-  ( f  =  F  ->  (deg `  f )  =  (deg
`  F ) )
138137oveq1d 6297 . . . . . . . . 9  |-  ( f  =  F  ->  (
(deg `  f )  -  (deg `  G )
)  =  ( (deg
`  F )  -  (deg `  G ) ) )
139138breq1d 4457 . . . . . . . 8  |-  ( f  =  F  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
d  <->  ( (deg `  F )  -  (deg `  G ) )  < 
d ) )
140136, 139orbi12d 709 . . . . . . 7  |-  ( f  =  F  ->  (
( f  =  0p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  d )  <->  ( F  =  0p  \/  ( (deg `  F
)  -  (deg `  G ) )  < 
d ) ) )
141 oveq1 6289 . . . . . . . . . . 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 2526 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f  oF  -  ( G  oF  x.  q ) )  =  R )
144143eqeq1d 2469 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f  oF  -  ( G  oF  x.  q )
)  =  0p  <-> 
R  =  0p ) )
145143fveq2d 5868 . . . . . . . . . 10  |-  ( f  =  F  ->  (deg `  ( f  oF  -  ( G  oF  x.  q )
) )  =  (deg
`  R ) )
146145breq1d 4457 . . . . . . . . 9  |-  ( f  =  F  ->  (
(deg `  ( f  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )  <->  (deg
`  R )  < 
(deg `  G )
) )
147144, 146orbi12d 709 . . . . . . . 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 2973 . . . . . . 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 320 . . . . . 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 3210 . . . . 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 60 . . . 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 32 . . 3  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( (deg `  F )  -  (deg `  G )
)  <  d  ->  E. q  e.  (Poly `  S ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
153152rexlimdva 2955 . 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 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   class class class wbr 4447    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282    oFcof 6520   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    < clt 9624    <_ cle 9625    - cmin 9801   -ucneg 9802    / cdiv 10202   NNcn 10532   NN0cn0 10791   ZZcz 10860   ^cexp 12129   0pc0p 21808  Polycply 22313  coeffccoe 22315  degcdgr 22316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-clim 13267  df-rlim 13268  df-sum 13465  df-0p 21809  df-ply 22317  df-coe 22319  df-dgr 22320
This theorem is referenced by:  plydivalg  22426
  Copyright terms: Public domain W3C validator