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Theorem plydivex 19693
Description: Lemma for plydivalg 19695. (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 19631 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
31, 2syl 15 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
43nn0red 10035 . . . 4  |-  ( ph  ->  (deg `  F )  e.  RR )
5 plydiv.g . . . . . 6  |-  ( ph  ->  G  e.  (Poly `  S ) )
6 dgrcl 19631 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
75, 6syl 15 . . . . 5  |-  ( ph  ->  (deg `  G )  e.  NN0 )
87nn0red 10035 . . . 4  |-  ( ph  ->  (deg `  G )  e.  RR )
94, 8resubcld 9227 . . 3  |-  ( ph  ->  ( (deg `  F
)  -  (deg `  G ) )  e.  RR )
10 arch 9978 . . 3  |-  ( ( (deg `  F )  -  (deg `  G )
)  e.  RR  ->  E. d  e.  NN  (
(deg `  F )  -  (deg `  G )
)  <  d )
119, 10syl 15 . 2  |-  ( ph  ->  E. d  e.  NN  ( (deg `  F )  -  (deg `  G )
)  <  d )
12 olc 373 . . . 4  |-  ( ( (deg `  F )  -  (deg `  G )
)  <  d  ->  ( F  =  0 p  \/  ( (deg `  F )  -  (deg `  G ) )  < 
d ) )
131adantr 451 . . . . 5  |-  ( (
ph  /\  d  e.  NN )  ->  F  e.  (Poly `  S )
)
14 nnnn0 9988 . . . . . . 7  |-  ( d  e.  NN  ->  d  e.  NN0 )
15 breq2 4043 . . . . . . . . . . . 12  |-  ( x  =  0  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  <  0 ) )
1615orbi2d 682 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  0 ) ) )
1716imbi1d 308 . . . . . . . . . 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 2576 . . . . . . . . 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 307 . . . . . . . 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 4043 . . . . . . . . . . . 12  |-  ( x  =  d  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  < 
d ) )
2120orbi2d 682 . . . . . . . . . . 11  |-  ( x  =  d  ->  (
( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  d )
) )
2221imbi1d 308 . . . . . . . . . 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 2576 . . . . . . . . 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 307 . . . . . . . 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 4043 . . . . . . . . . . . 12  |-  ( x  =  ( d  +  1 )  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
x  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
2625orbi2d 682 . . . . . . . . . . 11  |-  ( x  =  ( d  +  1 )  ->  (
( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  x )  <->  ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G )
)  <  ( d  +  1 ) ) ) )
2726imbi1d 308 . . . . . . . . . 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 2576 . . . . . . . . 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 307 . . . . . . . 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 695 . . . . . . . . . . 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 695 . . . . . . . . . . 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 695 . . . . . . . . . . 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 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  -u 1  e.  S )
38 simprl 732 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  f  e.  (Poly `  S ) )
395adantr 451 . . . . . . . . . . 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 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  (Poly `  S )  /\  ( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  0 ) ) )  ->  G  =/=  0 p )
42 eqid 2296 . . . . . . . . . . 11  |-  ( f  o F  -  ( G  o F  x.  q
) )  =  ( f  o F  -  ( G  o F  x.  q ) )
43 simprr 733 . . . . . . . . . . 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 19691 . . . . . . . . . 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 598 . . . . . . . . 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 2639 . . . . . . . 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 2302 . . . . . . . . . . . . . . . 16  |-  ( f  =  g  ->  (
f  =  0 p  <-> 
g  =  0 p ) )
48 fveq2 5541 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (deg `  f )  =  (deg
`  g ) )
4948oveq1d 5889 . . . . . . . . . . . . . . . . 17  |-  ( f  =  g  ->  (
(deg `  f )  -  (deg `  G )
)  =  ( (deg
`  g )  -  (deg `  G ) ) )
5049breq1d 4049 . . . . . . . . . . . . . . . 16  |-  ( f  =  g  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
d  <->  ( (deg `  g )  -  (deg `  G ) )  < 
d ) )
5147, 50orbi12d 690 . . . . . . . . . . . . . . 15  |-  ( f  =  g  ->  (
( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  d )  <->  ( g  =  0 p  \/  ( (deg `  g )  -  (deg `  G )
)  <  d )
) )
52 oveq1 5881 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (
f  o F  -  ( G  o F  x.  q ) )  =  ( g  o F  -  ( G  o F  x.  q )
) )
5352eqeq1d 2304 . . . . . . . . . . . . . . . . 17  |-  ( f  =  g  ->  (
( f  o F  -  ( G  o F  x.  q )
)  =  0 p  <-> 
( g  o F  -  ( G  o F  x.  q )
)  =  0 p ) )
5452fveq2d 5545 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  g  ->  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  =  (deg
`  ( g  o F  -  ( G  o F  x.  q
) ) ) )
5554breq1d 4049 . . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . . 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 2577 . . . . . . . . . . . . . . 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 311 . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . 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 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 ) ) )  ->  ph )
6160, 30sylan 457 . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . . . 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 731 . . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . . 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 741 . . . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . 16  |-  ( g  o F  -  ( G  o F  x.  p
) )  =  ( g  o F  -  ( G  o F  x.  p ) )
72 oveq1 5881 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  z  ->  (
w ^ d )  =  ( z ^
d ) )
7372oveq2d 5890 . . . . . . . . . . . . . . . . 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 4127 . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . 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 5882 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( q  =  p  ->  ( G  o F  x.  q
)  =  ( G  o F  x.  p
) )
7776oveq2d 5890 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( q  =  p  ->  (
g  o F  -  ( G  o F  x.  q ) )  =  ( g  o F  -  ( G  o F  x.  p )
) )
7877eqeq1d 2304 . . . . . . . . . . . . . . . . . . . . 21  |-  ( q  =  p  ->  (
( g  o F  -  ( G  o F  x.  q )
)  =  0 p  <-> 
( g  o F  -  ( G  o F  x.  p )
)  =  0 p ) )
7977fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( q  =  p  ->  (deg `  ( g  o F  -  ( G  o F  x.  q )
) )  =  (deg
`  ( g  o F  -  ( G  o F  x.  p
) ) ) )
8079breq1d 4049 . . . . . . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . . . . . . 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 303 . . . . . . . . . . . . . . . . . 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 2580 . . . . . . . . . . . . . . . . 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 188 . . . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . 16  |-  (coeff `  f )  =  (coeff `  f )
87 eqid 2296 . . . . . . . . . . . . . . . 16  |-  (coeff `  G )  =  (coeff `  G )
88 eqid 2296 . . . . . . . . . . . . . . . 16  |-  (deg `  f )  =  (deg
`  f )
89 eqid 2296 . . . . . . . . . . . . . . . 16  |-  (deg `  G )  =  (deg
`  G )
9061, 62, 63, 64, 65, 66, 67, 42, 68, 69, 70, 71, 74, 85, 86, 87, 88, 89plydivlem4 19692 . . . . . . . . . . . . . . 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 588 . . . . . . . . . . . . . 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 2646 . . . . . . . . . . . . 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 208 . . . . . . . . . . . 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 536 . . . . . . . . . . 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 19631 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  e.  (Poly `  S
)  ->  (deg `  f
)  e.  NN0 )
9695adantl 452 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  f
)  e.  NN0 )
9796nn0zd 10131 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  f
)  e.  ZZ )
985ad2antrr 706 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  G  e.  (Poly `  S ) )
9998, 6syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  G
)  e.  NN0 )
10099nn0zd 10131 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  (deg `  G
)  e.  ZZ )
10197, 100zsubcld 10138 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (deg `  f )  -  (deg `  G ) )  e.  ZZ )
102 nn0z 10062 . . . . . . . . . . . . . . . . . . . 20  |-  ( d  e.  NN0  ->  d  e.  ZZ )
103102ad2antlr 707 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  d  e.  ZZ )
104 zleltp1 10084 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( (deg `  f
)  -  (deg `  G ) )  e.  ZZ  /\  d  e.  ZZ )  ->  (
( (deg `  f
)  -  (deg `  G ) )  <_ 
d  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
105101, 103, 104syl2anc 642 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (
(deg `  f )  -  (deg `  G )
)  <_  d  <->  ( (deg `  f )  -  (deg `  G ) )  < 
( d  +  1 ) ) )
106101zred 10133 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  ( (deg `  f )  -  (deg `  G ) )  e.  RR )
107 nn0re 9990 . . . . . . . . . . . . . . . . . . . 20  |-  ( d  e.  NN0  ->  d  e.  RR )
108107ad2antlr 707 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  d  e.  NN0 )  /\  f  e.  (Poly `  S )
)  ->  d  e.  RR )
109106, 108leloed 8978 . . . . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . . . . . 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 682 . . . . . . . . . . . . . . . 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 pm5.63 890 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( f  =  0 p  \/  ( -.  f  = 
0 p  /\  (
(deg `  f )  -  (deg `  G )
)  =  d ) ) )
113 df-ne 2461 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =/=  0 p  <->  -.  f  =  0 p )
114113anbi1i 676 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( -.  f  =  0 p  /\  ( (deg `  f
)  -  (deg `  G ) )  =  d ) )
115114orbi2i 505 . . . . . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  =  0 p  \/  ( (deg `  f )  -  (deg `  G ) )  =  d )  <->  ( f  =  0 p  \/  ( f  =/=  0 p  /\  ( (deg `  f )  -  (deg `  G ) )  =  d ) ) )
117116orbi2i 505 . . . . . . . . . . . . . . . . . 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 509 . . . . . . . . . . . . . . . . . 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 509 . . . . . . . . . . . . . . . . . 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 268 . . . . . . . . . . . . . . . . 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 510 . . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . . 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 252 . . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . 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 252 . . . . . . . . . . . . 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 2572 . . . . . . . . . . . 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 2688 . . . . . . . . . . . 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 252 . . . . . . . . . . 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 225 . . . . . . . . . 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 424 . . . . . . . . 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 23 . . . . . . . 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 10124 . . . . . . 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 15 . . . . . 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 419 . . . . 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 2302 . . . . . . . 8  |-  ( f  =  F  ->  (
f  =  0 p  <-> 
F  =  0 p ) )
137 fveq2 5541 . . . . . . . . . 10  |-  ( f  =  F  ->  (deg `  f )  =  (deg
`  F ) )
138137oveq1d 5889 . . . . . . . . 9  |-  ( f  =  F  ->  (
(deg `  f )  -  (deg `  G )
)  =  ( (deg
`  F )  -  (deg `  G ) ) )
139138breq1d 4049 . . . . . . . 8  |-  ( f  =  F  ->  (
( (deg `  f
)  -  (deg `  G ) )  < 
d  <->  ( (deg `  F )  -  (deg `  G ) )  < 
d ) )
140136, 139orbi12d 690 . . . . . . 7  |-  ( f  =  F  ->  (
( f  =  0 p  \/  ( (deg
`  f )  -  (deg `  G ) )  <  d )  <->  ( F  =  0 p  \/  ( (deg `  F )  -  (deg `  G )
)  <  d )
) )
141 oveq1 5881 . . . . . . . . . . 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 2346 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f  o F  -  ( G  o F  x.  q ) )  =  R )
144143eqeq1d 2304 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f  o F  -  ( G  o F  x.  q )
)  =  0 p  <-> 
R  =  0 p ) )
145143fveq2d 5545 . . . . . . . . . 10  |-  ( f  =  F  ->  (deg `  ( f  o F  -  ( G  o F  x.  q )
) )  =  (deg
`  R ) )
146145breq1d 4049 . . . . . . . . 9  |-  ( f  =  F  ->  (
(deg `  ( f  o F  -  ( G  o F  x.  q
) ) )  < 
(deg `  G )  <->  (deg
`  R )  < 
(deg `  G )
) )
147144, 146orbi12d 690 . . . . . . . 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 2577 . . . . . . 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 311 . . . . . 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 2893 . . . . 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 56 . . . 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 28 . . 3  |-  ( (
ph  /\  d  e.  NN )  ->  ( ( (deg `  F )  -  (deg `  G )
)  <  d  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
153152rexlimdva 2680 . 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 14 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 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ^cexp 11120   0 pc0p 19040  Polycply 19582  coeffccoe 19584  degcdgr 19585
This theorem is referenced by:  plydivalg  19695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588  df-dgr 19589
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