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Theorem plydivex 20215
Description: Lemma for plydivalg 20217. (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  =/=  0 p )
plydiv.r  |-  R  =  ( F  o F  -  ( G  o F  x.  q )
)
Assertion
Ref Expression
plydivex  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (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 20153 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
31, 2syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
43nn0red 10276 . . . 4  |-  ( ph  ->  (deg `  F )  e.  RR )
5 plydiv.g . . . . . 6  |-  ( ph  ->  G  e.  (Poly `  S ) )
6 dgrcl 20153 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
75, 6syl 16 . . . . 5  |-  ( ph  ->  (deg `  G )  e.  NN0 )
87nn0red 10276 . . . 4  |-  ( ph  ->  (deg `  G )  e.  RR )
94, 8resubcld 9466 . . 3  |-  ( ph  ->  ( (deg `  F
)  -  (deg `  G ) )  e.  RR )
10 arch 10219 . . 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 375 . . . 4  |-  ( ( (deg `  F )  -  (deg `  G )
)  <  d  ->  ( F  =  0 p  \/  ( (deg `  F )  -  (deg `  G ) )  < 
d ) )
131adantr 453 . . . . 5  |-  ( (
ph  /\  d  e.  NN )  ->  F  e.  (Poly `  S )
)
14 nnnn0 10229 . . . . . . 7  |-  ( d  e.  NN  ->  d  e.  NN0 )
15 breq2 4217 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  <  0 ) )
1615orbi2d 684 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  0 ) ) )
1716imbi1d 310 . . . . . . . . . 10  |-  ( x  =  0  ->  (
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  x )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  0 )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
1817ralbidv 2726 . . . . . . . . 9  |-  ( x  =  0  ->  ( A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
x )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  <  0 )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
1918imbi2d 309 . . . . . . . 8  |-  ( x  =  0  ->  (
( ph  ->  A. f  e.  (Poly `  S )
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  x )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )  <->  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  <  0 )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
20 breq2 4217 . . . . . . . . . . . 12  |-  ( x  =  d  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  < 
d ) )
2120orbi2d 684 . . . . . . . . . . 11  |-  ( x  =  d  ->  (
( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  d )
) )
2221imbi1d 310 . . . . . . . . . 10  |-  ( x  =  d  ->  (
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  x )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
2322ralbidv 2726 . . . . . . . . 9  |-  ( x  =  d  ->  ( A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
x )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
2423imbi2d 309 . . . . . . . 8  |-  ( x  =  d  ->  (
( ph  ->  A. f  e.  (Poly `  S )
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  x )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )  <->  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
25 breq2 4217 . . . . . . . . . . . 12  |-  ( x  =  ( d  +  1 )  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
2625orbi2d 684 . . . . . . . . . . 11  |-  ( x  =  ( d  +  1 )  ->  (
( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  ( d  +  1 ) ) ) )
2726imbi1d 310 . . . . . . . . . 10  |-  ( x  =  ( d  +  1 )  ->  (
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  x )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  ( d  +  1 ) )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
2827ralbidv 2726 . . . . . . . . 9  |-  ( x  =  ( d  +  1 )  ->  ( A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
x )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
2928imbi2d 309 . . . . . . . 8  |-  ( x  =  ( d  +  1 )  ->  (
( ph  ->  A. f  e.  (Poly `  S )
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  x )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )  <->  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
30 plydiv.pl . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
3130adantlr 697 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  (Poly `  S )  /\  (
f  =  0 p  \/  ( (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 697 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  (Poly `  S )  /\  (
f  =  0 p  \/  ( (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 697 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  (Poly `  S )  /\  (
f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  <  0 ) ) )  /\  ( x  e.  S  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  S )
36 plydiv.m1 . . . . . . . . . . . 12  |-  ( ph  -> 
-u 1  e.  S
)
3736adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  -u 1  e.  S )
38 simprl 734 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  f  e.  (Poly `  S ) )
395adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  G  e.  (Poly `  S ) )
40 plydiv.z . . . . . . . . . . . 12  |-  ( ph  ->  G  =/=  0 p )
4140adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  G  =/=  0 p )
42 eqid 2437 . . . . . . . . . . 11  |-  ( f  o F  -  ( G  o F  x.  q
) )  =  ( f  o F  -  ( G  o F  x.  q ) )
43 simprr 735 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  0 ) )
4431, 33, 35, 37, 38, 39, 41, 42, 43plydivlem3 20213 . . . . . . . . . 10  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )
4544expr 600 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  (Poly `  S ) )  ->  ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  0 )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )
4645ralrimiva 2790 . . . . . . . 8  |-  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  <  0 )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )
47 eqeq1 2443 . . . . . . . . . . . . . . . 16  |-  ( f  =  g  ->  (
f  =  0 p  <-> 
g  =  0 p ) )
48 fveq2 5729 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (deg `  f )  =  (deg
`  g ) )
4948oveq1d 6097 . . . . . . . . . . . . . . . . 17  |-  ( f  =  g  ->  (
(deg `  f )  -  (deg `  G )
)  =  ( (deg
`  g )  -  (deg `  G ) ) )
5049breq1d 4223 . . . . . . . . . . . . . . . 16  |-  ( f  =  g  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
d  <->  ( (deg `  g )  -  (deg `  G ) )  < 
d ) )
5147, 50orbi12d 692 . . . . . . . . . . . . . . 15  |-  ( f  =  g  ->  (
( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  d )  <->  ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G )
)  <  d )
) )
52 oveq1 6089 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (
f  o F  -  ( G  o F  x.  q ) )  =  ( g  o F  -  ( G  o F  x.  q )
) )
5352eqeq1d 2445 . . . . . . . . . . . . . . . . 17  |-  ( f  =  g  ->  (
( f  o F  -  ( G  o F  x.  q )
)  =  0 p  <-> 
( g  o F  -  ( G  o F  x.  q )
)  =  0 p ) )
5452fveq2d 5733 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  =  (deg
`  ( g  o F  -  ( G  o F  x.  q
) ) ) )
5554breq1d 4223 . . . . . . . . . . . . . . . . 17  |-  ( f  =  g  ->  (
(deg `  ( f  o F  -  ( G  o F  x.  q
) ) )  < 
(deg `  G )  <->  (deg
`  ( g  o F  -  ( G  o F  x.  q
) ) )  < 
(deg `  G )
) )
5653, 55orbi12d 692 . . . . . . . . . . . . . . . 16  |-  ( f  =  g  ->  (
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) )  <->  ( (
g  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )
5756rexbidv 2727 . . . . . . . . . . . . . . 15  |-  ( f  =  g  ->  ( E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) )  <->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )
5851, 57imbi12d 313 . . . . . . . . . . . . . 14  |-  ( f  =  g  ->  (
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( g  =  0 p  \/  (
(deg `  g )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( g  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
5958cbvralv 2933 . . . . . . . . . . . . 13  |-  ( A. f  e.  (Poly `  S
) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <->  A. g  e.  (Poly `  S ) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )
60 simplll 736 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  ph )
6160, 30sylan 459 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S ) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  +  y )  e.  S )
6260, 32sylan 459 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S ) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  x.  y )  e.  S
)
6360, 34sylan 459 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S ) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0 p  /\  (
(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  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  -u 1  e.  S )
65 simplr 733 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0 p  /\  (
(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  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0 p  /\  (
(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  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  G  =/=  0 p )
68 simpllr 737 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  d  e.  NN0 )
69 simprrr 743 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  (
(deg `  f )  -  (deg `  G )
)  =  d )
70 simprrl 742 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  f  =/=  0 p )
71 eqid 2437 . . . . . . . . . . . . . . . 16  |-  ( g  o F  -  ( G  o F  x.  p
) )  =  ( g  o F  -  ( G  o F  x.  p ) )
72 oveq1 6089 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  z  ->  (
w ^ d )  =  ( z ^
d ) )
7372oveq2d 6098 . . . . . . . . . . . . . . . . 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 4301 . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  A. g  e.  (Poly `  S )
( ( g  =  0 p  \/  (
(deg `  g )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( g  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )
76 oveq2 6090 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( q  =  p  ->  ( G  o F  x.  q
)  =  ( G  o F  x.  p
) )
7776oveq2d 6098 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( q  =  p  ->  (
g  o F  -  ( G  o F  x.  q ) )  =  ( g  o F  -  ( G  o F  x.  p )
) )
7877eqeq1d 2445 . . . . . . . . . . . . . . . . . . . . 21  |-  ( q  =  p  ->  (
( g  o F  -  ( G  o F  x.  q )
)  =  0 p  <-> 
( g  o F  -  ( G  o F  x.  p )
)  =  0 p ) )
7977fveq2d 5733 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( q  =  p  ->  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  =  (deg
`  ( g  o F  -  ( G  o F  x.  p
) ) ) )
8079breq1d 4223 . . . . . . . . . . . . . . . . . . . . 21  |-  ( q  =  p  ->  (
(deg `  ( g  o F  -  ( G  o F  x.  q
) ) )  < 
(deg `  G )  <->  (deg
`  ( g  o F  -  ( G  o F  x.  p
) ) )  < 
(deg `  G )
) )
8178, 80orbi12d 692 . . . . . . . . . . . . . . . . . . . 20  |-  ( q  =  p  ->  (
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) )  <->  ( (
g  o F  -  ( G  o F  x.  p ) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )
8281cbvrexv 2934 . . . . . . . . . . . . . . . . . . 19  |-  ( E. q  e.  (Poly `  S ) ( ( g  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) )  <->  E. p  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) )
8382imbi2i 305 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( g  =  0 p  \/  ( (deg
`  g )  -  (deg `  G ) )  <  d )  ->  E. q  e.  (Poly `  S ) ( ( g  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( g  =  0 p  \/  (
(deg `  g )  -  (deg `  G )
)  <  d )  ->  E. p  e.  (Poly `  S ) ( ( g  o F  -  ( G  o F  x.  p ) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )
8483ralbii 2730 . . . . . . . . . . . . . . . . 17  |-  ( A. g  e.  (Poly `  S
) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( g  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <->  A. g  e.  (Poly `  S ) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. p  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )
8575, 84sylib 190 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  A. g  e.  (Poly `  S )
( ( g  =  0 p  \/  (
(deg `  g )  -  (deg `  G )
)  <  d )  ->  E. p  e.  (Poly `  S ) ( ( g  o F  -  ( G  o F  x.  p ) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )
86 eqid 2437 . . . . . . . . . . . . . . . 16  |-  (coeff `  f )  =  (coeff `  f )
87 eqid 2437 . . . . . . . . . . . . . . . 16  |-  (coeff `  G )  =  (coeff `  G )
88 eqid 2437 . . . . . . . . . . . . . . . 16  |-  (deg `  f )  =  (deg
`  f )
89 eqid 2437 . . . . . . . . . . . . . . . 16  |-  (deg `  G )  =  (deg
`  G )
9061, 62, 63, 64, 65, 66, 67, 42, 68, 69, 70, 71, 74, 85, 86, 87, 88, 89plydivlem4 20214 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S
) )  /\  ( A. g  e.  (Poly `  S ) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( g  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )
9190exp32 590 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( A. g  e.  (Poly `  S
) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( g  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  ->  ( ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
9291ralrimdva 2797 . . . . . . . . . . . . 13  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( A. g  e.  (Poly `  S
) ( ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( g  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  ->  A. f  e.  (Poly `  S ) ( ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
9359, 92syl5bi 210 . . . . . . . . . . . 12  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( A. f  e.  (Poly `  S
) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  ->  A. f  e.  (Poly `  S ) ( ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
9493ancld 538 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( A. f  e.  (Poly `  S
) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  ->  ( A. f  e.  (Poly `  S )
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  A. f  e.  (Poly `  S )
( ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
95 dgrcl 20153 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  e.  (Poly `  S
)  ->  (deg `  f
)  e.  NN0 )
9695adantl 454 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  f
)  e.  NN0 )
9796nn0zd 10374 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  f
)  e.  ZZ )
985ad2antrr 708 . . . . . . . . . . . . . . . . . . . . . 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 10374 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  G
)  e.  ZZ )
10197, 100zsubcld 10381 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (deg `  f )  -  (deg `  G ) )  e.  ZZ )
102 nn0z 10305 . . . . . . . . . . . . . . . . . . . 20  |-  ( d  e.  NN0  ->  d  e.  ZZ )
103102ad2antlr 709 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  d  e.  ZZ )
104 zleltp1 10327 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( (deg `  f
)  -  (deg `  G ) )  e.  ZZ  /\  d  e.  ZZ )  ->  (
( (deg `  f
)  -  (deg `  G ) )  <_ 
d  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
105101, 103, 104syl2anc 644 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
(deg `  f )  -  (deg `  G )
)  <_  d  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
106101zred 10376 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (deg `  f )  -  (deg `  G ) )  e.  RR )
107 nn0re 10231 . . . . . . . . . . . . . . . . . . . 20  |-  ( d  e.  NN0  ->  d  e.  RR )
108107ad2antlr 709 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  d  e.  RR )
109106, 108leloed 9217 . . . . . . . . . . . . . . . . . 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 248 . . . . . . . . . . . . . . . . 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 684 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) )  <->  ( f  =  0 p  \/  ( ( (deg `  f )  -  (deg `  G ) )  < 
d  \/  ( (deg
`  f )  -  (deg `  G ) )  =  d ) ) ) )
112 pm5.63 892 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( f  =  0 p  \/  ( -.  f  = 
0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )
113 df-ne 2602 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =/=  0 p  <->  -.  f  =  0 p )
114113anbi1i 678 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( -.  f  =  0 p  /\  ( (deg `  f
)  -  (deg `  G ) )  =  d ) )
115114orbi2i 507 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  =  0 p  \/  ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) )  <->  ( f  =  0 p  \/  ( -.  f  =  0 p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )
116112, 115bitr4i 245 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( f  =  0 p  \/  ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )
117116orbi2i 507 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( (deg `  f
)  -  (deg `  G ) )  < 
d  \/  ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  =  d ) )  <->  ( ( (deg
`  f )  -  (deg `  G ) )  <  d  \/  (
f  =  0 p  \/  ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) ) )
118 or12 511 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  =  0 p  \/  ( ( (deg
`  f )  -  (deg `  G ) )  <  d  \/  (
(deg `  f )  -  (deg `  G )
)  =  d ) )  <->  ( ( (deg
`  f )  -  (deg `  G ) )  <  d  \/  (
f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )
119 or12 511 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  =  0 p  \/  ( ( (deg
`  f )  -  (deg `  G ) )  <  d  \/  (
f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )  <-> 
( ( (deg `  f )  -  (deg `  G ) )  < 
d  \/  ( f  =  0 p  \/  ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) ) )
120117, 118, 1193bitr4i 270 . . . . . . . . . . . . . . . . 17  |-  ( ( f  =  0 p  \/  ( ( (deg
`  f )  -  (deg `  G ) )  <  d  \/  (
(deg `  f )  -  (deg `  G )
)  =  d ) )  <->  ( f  =  0 p  \/  (
( (deg `  f
)  -  (deg `  G ) )  < 
d  \/  ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d ) ) ) )
121 orass 512 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  d )  \/  ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) )  <->  ( f  =  0 p  \/  ( ( (deg `  f )  -  (deg `  G ) )  < 
d  \/  ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d ) ) ) )
122120, 121bitr4i 245 . . . . . . . . . . . . . . . 16  |-  ( ( f  =  0 p  \/  ( ( (deg
`  f )  -  (deg `  G ) )  <  d  \/  (
(deg `  f )  -  (deg `  G )
)  =  d ) )  <->  ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  d )  \/  ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )
123111, 122syl6bb 254 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) )  <->  ( (
f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
d )  \/  (
f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) ) )
124123imbi1d 310 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  ( d  +  1 ) )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  d )  \/  ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
125 jaob 760 . . . . . . . . . . . . . 14  |-  ( ( ( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  \/  ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
126124, 125syl6bb 254 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  ( d  +  1 ) )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
127126ralbidva 2722 . . . . . . . . . . . 12  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( A. f  e.  (Poly `  S
) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  ( d  +  1 ) )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <->  A. f  e.  (Poly `  S ) ( ( ( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
128 r19.26 2839 . . . . . . . . . . . 12  |-  ( A. f  e.  (Poly `  S
) ( ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  ( ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )  <->  ( A. f  e.  (Poly `  S )
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  A. f  e.  (Poly `  S )
( ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
129127, 128syl6bb 254 . . . . . . . . . . 11  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( A. f  e.  (Poly `  S
) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  ( d  +  1 ) )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( A. f  e.  (Poly `  S )
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  /\  A. f  e.  (Poly `  S )
( ( f  =/=  0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
13094, 129sylibrd 227 . . . . . . . . . 10  |-  ( (
ph  /\  d  e.  NN0 )  ->  ( A. f  e.  (Poly `  S
) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  ->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
131130expcom 426 . . . . . . . . 9  |-  ( d  e.  NN0  ->  ( ph  ->  ( A. f  e.  (Poly `  S )
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  ->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
132131a2d 25 . . . . . . . 8  |-  ( d  e.  NN0  ->  ( (
ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )  ->  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) ) )
13319, 24, 29, 24, 46, 132nn0ind 10367 . . . . . . 7  |-  ( d  e.  NN0  ->  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
13414, 133syl 16 . . . . . 6  |-  ( d  e.  NN  ->  ( ph  ->  A. f  e.  (Poly `  S ) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) ) )
135134impcom 421 . . . . 5  |-  ( (
ph  /\  d  e.  NN )  ->  A. f  e.  (Poly `  S )
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )
136 eqeq1 2443 . . . . . . . 8  |-  ( f  =  F  ->  (
f  =  0 p  <-> 
F  =  0 p ) )
137 fveq2 5729 . . . . . . . . . 10  |-  ( f  =  F  ->  (deg `  f )  =  (deg
`  F ) )
138137oveq1d 6097 . . . . . . . . 9  |-  ( f  =  F  ->  (
(deg `  f )  -  (deg `  G )
)  =  ( (deg
`  F )  -  (deg `  G ) ) )
139138breq1d 4223 . . . . . . . 8  |-  ( f  =  F  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
d  <->  ( (deg `  F )  -  (deg `  G ) )  < 
d ) )
140136, 139orbi12d 692 . . . . . . 7  |-  ( f  =  F  ->  (
( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  d )  <->  ( F  =  0 p  \/  ( (deg `  F )  -  (deg `  G )
)  <  d )
) )
141 oveq1 6089 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f  o F  -  ( G  o F  x.  q ) )  =  ( F  o F  -  ( G  o F  x.  q )
) )
142 plydiv.r . . . . . . . . . . 11  |-  R  =  ( F  o F  -  ( G  o F  x.  q )
)
143141, 142syl6eqr 2487 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f  o F  -  ( G  o F  x.  q ) )  =  R )
144143eqeq1d 2445 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f  o F  -  ( G  o F  x.  q )
)  =  0 p  <-> 
R  =  0 p ) )
145143fveq2d 5733 . . . . . . . . . 10  |-  ( f  =  F  ->  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  =  (deg
`  R ) )
146145breq1d 4223 . . . . . . . . 9  |-  ( f  =  F  ->  (
(deg `  ( f  o F  -  ( G  o F  x.  q
) ) )  < 
(deg `  G )  <->  (deg
`  R )  < 
(deg `  G )
) )
147144, 146orbi12d 692 . . . . . . . 8  |-  ( f  =  F  ->  (
( ( f  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) )  <->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
148147rexbidv 2727 . . . . . . 7  |-  ( f  =  F  ->  ( E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) )  <->  E. q  e.  (Poly `  S )
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) ) )
149140, 148imbi12d 313 . . . . . 6  |-  ( f  =  F  ->  (
( ( f  =  0 p  \/  (
(deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  <-> 
( ( F  =  0 p  \/  (
(deg `  F )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) ) )
150149rspcv 3049 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  ( A. f  e.  (Poly `  S
) ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( ( f  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  ->  ( ( F  =  0 p  \/  ( (deg `  F )  -  (deg `  G )
)  <  d )  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) ) )
15113, 135, 150sylc 59 . . . 4  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( F  =  0 p  \/  ( (deg `  F )  -  (deg `  G ) )  < 
d )  ->  E. q  e.  (Poly `  S )
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) ) )
15212, 151syl5 31 . . 3  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( (deg `  F )  -  (deg `  G )
)  <  d  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
153152rexlimdva 2831 . 2  |-  ( ph  ->  ( E. d  e.  NN  ( (deg `  F )  -  (deg `  G ) )  < 
d  ->  E. q  e.  (Poly `  S )
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) ) )
15411, 153mpd 15 1  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706   E.wrex 2707   class class class wbr 4213    e. cmpt 4267   ` cfv 5455  (class class class)co 6082    o Fcof 6304   CCcc 8989   RRcr 8990   0cc0 8991   1c1 8992    + caddc 8994    x. cmul 8996    < clt 9121    <_ cle 9122    - cmin 9292   -ucneg 9293    / cdiv 9678   NNcn 10001   NN0cn0 10222   ZZcz 10283   ^cexp 11383   0 pc0p 19562  Polycply 20104  coeffccoe 20106  degcdgr 20107
This theorem is referenced by:  plydivalg  20217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069  ax-addf 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-pm 7022  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-oi 7480  df-card 7827  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-fz 11045  df-fzo 11137  df-fl 11203  df-seq 11325  df-exp 11384  df-hash 11620  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-clim 12283  df-rlim 12284  df-sum 12481  df-0p 19563  df-ply 20108  df-coe 20110  df-dgr 20111
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