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Theorem plydiveu 22863
Description: Lemma for plydivalg 22864. (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 )
)
plydiveu.q  |-  ( ph  ->  q  e.  (Poly `  S ) )
plydiveu.qd  |-  ( ph  ->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) )
plydiveu.t  |-  T  =  ( F  oF  -  ( G  oF  x.  p )
)
plydiveu.p  |-  ( ph  ->  p  e.  (Poly `  S ) )
plydiveu.pd  |-  ( ph  ->  ( T  =  0p  \/  (deg `  T )  <  (deg `  G ) ) )
Assertion
Ref Expression
plydiveu  |-  ( ph  ->  p  =  q )
Distinct variable groups:    x, y    q, p, x, y, F    ph, x, y    x, T, y    G, p, q, x, y    R, p, x, y    S, p, q, x, y
Allowed substitution hints:    ph( q, p)    R( q)    T( q, p)

Proof of Theorem plydiveu
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 idd 24 . . . 4  |-  ( ph  ->  ( ( p  oF  -  q )  =  0p  -> 
( p  oF  -  q )  =  0p ) )
2 plydiveu.q . . . . . . . . . . . . . . . 16  |-  ( ph  ->  q  e.  (Poly `  S ) )
3 plydiv.pl . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
4 plydiv.tm . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
5 plydiv.rc . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( x  e.  S  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  S )
6 plydiv.m1 . . . . . . . . . . . . . . . . 17  |-  ( ph  -> 
-u 1  e.  S
)
7 plydiv.f . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  F  e.  (Poly `  S ) )
8 plydiv.g . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G  e.  (Poly `  S ) )
9 plydiv.z . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  G  =/=  0p )
10 plydiv.r . . . . . . . . . . . . . . . . 17  |-  R  =  ( F  oF  -  ( G  oF  x.  q )
)
113, 4, 5, 6, 7, 8, 9, 10plydivlem2 22859 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  q  e.  (Poly `  S ) )  ->  R  e.  (Poly `  S ) )
122, 11mpdan 666 . . . . . . . . . . . . . . 15  |-  ( ph  ->  R  e.  (Poly `  S ) )
13 plydiveu.p . . . . . . . . . . . . . . . 16  |-  ( ph  ->  p  e.  (Poly `  S ) )
14 plydiveu.t . . . . . . . . . . . . . . . . 17  |-  T  =  ( F  oF  -  ( G  oF  x.  p )
)
153, 4, 5, 6, 7, 8, 9, 14plydivlem2 22859 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  T  e.  (Poly `  S ) )
1613, 15mpdan 666 . . . . . . . . . . . . . . 15  |-  ( ph  ->  T  e.  (Poly `  S ) )
1712, 16, 3, 4, 6plysub 22785 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( R  oF  -  T )  e.  (Poly `  S )
)
18 dgrcl 22799 . . . . . . . . . . . . . 14  |-  ( ( R  oF  -  T )  e.  (Poly `  S )  ->  (deg `  ( R  oF  -  T ) )  e.  NN0 )
1917, 18syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  ( R  oF  -  T
) )  e.  NN0 )
2019nn0red 10849 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  ( R  oF  -  T
) )  e.  RR )
21 dgrcl 22799 . . . . . . . . . . . . . . 15  |-  ( T  e.  (Poly `  S
)  ->  (deg `  T
)  e.  NN0 )
2216, 21syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  (deg `  T )  e.  NN0 )
2322nn0red 10849 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  T )  e.  RR )
24 dgrcl 22799 . . . . . . . . . . . . . . 15  |-  ( R  e.  (Poly `  S
)  ->  (deg `  R
)  e.  NN0 )
2512, 24syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  (deg `  R )  e.  NN0 )
2625nn0red 10849 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  R )  e.  RR )
2723, 26ifcld 3972 . . . . . . . . . . . 12  |-  ( ph  ->  if ( (deg `  R )  <_  (deg `  T ) ,  (deg
`  T ) ,  (deg `  R )
)  e.  RR )
28 dgrcl 22799 . . . . . . . . . . . . . 14  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
298, 28syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  G )  e.  NN0 )
3029nn0red 10849 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  G )  e.  RR )
31 eqid 2454 . . . . . . . . . . . . . 14  |-  (deg `  R )  =  (deg
`  R )
32 eqid 2454 . . . . . . . . . . . . . 14  |-  (deg `  T )  =  (deg
`  T )
3331, 32dgrsub 22838 . . . . . . . . . . . . 13  |-  ( ( R  e.  (Poly `  S )  /\  T  e.  (Poly `  S )
)  ->  (deg `  ( R  oF  -  T
) )  <_  if ( (deg `  R )  <_  (deg `  T ) ,  (deg `  T ) ,  (deg `  R )
) )
3412, 16, 33syl2anc 659 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  ( R  oF  -  T
) )  <_  if ( (deg `  R )  <_  (deg `  T ) ,  (deg `  T ) ,  (deg `  R )
) )
35 plydiveu.pd . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( T  =  0p  \/  (deg `  T )  <  (deg `  G ) ) )
36 eqid 2454 . . . . . . . . . . . . . . . . 17  |-  (coeff `  T )  =  (coeff `  T )
3732, 36dgrlt 22832 . . . . . . . . . . . . . . . 16  |-  ( ( T  e.  (Poly `  S )  /\  (deg `  G )  e.  NN0 )  ->  ( ( T  =  0p  \/  (deg `  T )  <  (deg `  G )
)  <->  ( (deg `  T )  <_  (deg `  G )  /\  (
(coeff `  T ) `  (deg `  G )
)  =  0 ) ) )
3816, 29, 37syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( T  =  0p  \/  (deg `  T )  <  (deg `  G ) )  <->  ( (deg `  T )  <_  (deg `  G )  /\  (
(coeff `  T ) `  (deg `  G )
)  =  0 ) ) )
3935, 38mpbid 210 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (deg `  T
)  <_  (deg `  G
)  /\  ( (coeff `  T ) `  (deg `  G ) )  =  0 ) )
4039simpld 457 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  T )  <_  (deg `  G )
)
41 plydiveu.qd . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) )
42 eqid 2454 . . . . . . . . . . . . . . . . 17  |-  (coeff `  R )  =  (coeff `  R )
4331, 42dgrlt 22832 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  (Poly `  S )  /\  (deg `  G )  e.  NN0 )  ->  ( ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
)  <->  ( (deg `  R )  <_  (deg `  G )  /\  (
(coeff `  R ) `  (deg `  G )
)  =  0 ) ) )
4412, 29, 43syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  <->  ( (deg `  R )  <_  (deg `  G )  /\  (
(coeff `  R ) `  (deg `  G )
)  =  0 ) ) )
4541, 44mpbid 210 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (deg `  R
)  <_  (deg `  G
)  /\  ( (coeff `  R ) `  (deg `  G ) )  =  0 ) )
4645simpld 457 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  R )  <_  (deg `  G )
)
47 breq1 4442 . . . . . . . . . . . . . 14  |-  ( (deg
`  T )  =  if ( (deg `  R )  <_  (deg `  T ) ,  (deg
`  T ) ,  (deg `  R )
)  ->  ( (deg `  T )  <_  (deg `  G )  <->  if (
(deg `  R )  <_  (deg `  T ) ,  (deg `  T ) ,  (deg `  R )
)  <_  (deg `  G
) ) )
48 breq1 4442 . . . . . . . . . . . . . 14  |-  ( (deg
`  R )  =  if ( (deg `  R )  <_  (deg `  T ) ,  (deg
`  T ) ,  (deg `  R )
)  ->  ( (deg `  R )  <_  (deg `  G )  <->  if (
(deg `  R )  <_  (deg `  T ) ,  (deg `  T ) ,  (deg `  R )
)  <_  (deg `  G
) ) )
4947, 48ifboth 3965 . . . . . . . . . . . . 13  |-  ( ( (deg `  T )  <_  (deg `  G )  /\  (deg `  R )  <_  (deg `  G )
)  ->  if (
(deg `  R )  <_  (deg `  T ) ,  (deg `  T ) ,  (deg `  R )
)  <_  (deg `  G
) )
5040, 46, 49syl2anc 659 . . . . . . . . . . . 12  |-  ( ph  ->  if ( (deg `  R )  <_  (deg `  T ) ,  (deg
`  T ) ,  (deg `  R )
)  <_  (deg `  G
) )
5120, 27, 30, 34, 50letrd 9728 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  ( R  oF  -  T
) )  <_  (deg `  G ) )
5251adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  (deg `  ( R  oF  -  T
) )  <_  (deg `  G ) )
5313, 2, 3, 4, 6plysub 22785 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( p  oF  -  q )  e.  (Poly `  S )
)
54 dgrcl 22799 . . . . . . . . . . . . . 14  |-  ( ( p  oF  -  q )  e.  (Poly `  S )  ->  (deg `  ( p  oF  -  q ) )  e.  NN0 )
5553, 54syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  ( p  oF  -  q
) )  e.  NN0 )
56 nn0addge1 10838 . . . . . . . . . . . . 13  |-  ( ( (deg `  G )  e.  RR  /\  (deg `  ( p  oF  -  q ) )  e.  NN0 )  -> 
(deg `  G )  <_  ( (deg `  G
)  +  (deg `  ( p  oF  -  q ) ) ) )
5730, 55, 56syl2anc 659 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  G )  <_  ( (deg `  G
)  +  (deg `  ( p  oF  -  q ) ) ) )
5857adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  (deg `  G
)  <_  ( (deg `  G )  +  (deg
`  ( p  oF  -  q ) ) ) )
59 plyf 22764 . . . . . . . . . . . . . . . . . . . 20  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
607, 59syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F : CC --> CC )
6160ffvelrnda 6007 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e.  CC )
628, 2, 3, 4plymul 22784 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  oF  x.  q )  e.  (Poly `  S )
)
63 plyf 22764 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( G  oF  x.  q )  e.  (Poly `  S )  ->  ( G  oF  x.  q
) : CC --> CC )
6462, 63syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  oF  x.  q ) : CC --> CC )
6564ffvelrnda 6007 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  oF  x.  q ) `  z
)  e.  CC )
668, 13, 3, 4plymul 22784 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  oF  x.  p )  e.  (Poly `  S )
)
67 plyf 22764 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( G  oF  x.  p )  e.  (Poly `  S )  ->  ( G  oF  x.  p
) : CC --> CC )
6866, 67syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  oF  x.  p ) : CC --> CC )
6968ffvelrnda 6007 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  oF  x.  p ) `  z
)  e.  CC )
7061, 65, 69nnncan1d 9956 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( ( F `  z
)  -  ( ( G  oF  x.  q ) `  z
) )  -  (
( F `  z
)  -  ( ( G  oF  x.  p ) `  z
) ) )  =  ( ( ( G  oF  x.  p
) `  z )  -  ( ( G  oF  x.  q
) `  z )
) )
7170mpteq2dva 4525 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( z  e.  CC  |->  ( ( ( F `
 z )  -  ( ( G  oF  x.  q ) `  z ) )  -  ( ( F `  z )  -  (
( G  oF  x.  p ) `  z ) ) ) )  =  ( z  e.  CC  |->  ( ( ( G  oF  x.  p ) `  z )  -  (
( G  oF  x.  q ) `  z ) ) ) )
72 cnex 9562 . . . . . . . . . . . . . . . . . 18  |-  CC  e.  _V
7372a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  CC  e.  _V )
7461, 65subcld 9922 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  ( ( G  oF  x.  q
) `  z )
)  e.  CC )
7561, 69subcld 9922 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  ( ( G  oF  x.  p
) `  z )
)  e.  CC )
7660feqmptd 5901 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F  =  ( z  e.  CC  |->  ( F `
 z ) ) )
7764feqmptd 5901 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  oF  x.  q )  =  ( z  e.  CC  |->  ( ( G  oF  x.  q ) `  z ) ) )
7873, 61, 65, 76, 77offval2 6529 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( F  oF  -  ( G  oF  x.  q )
)  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  oF  x.  q
) `  z )
) ) )
7910, 78syl5eq 2507 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  oF  x.  q
) `  z )
) ) )
8068feqmptd 5901 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  oF  x.  p )  =  ( z  e.  CC  |->  ( ( G  oF  x.  p ) `  z ) ) )
8173, 61, 69, 76, 80offval2 6529 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( F  oF  -  ( G  oF  x.  p )
)  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  oF  x.  p
) `  z )
) ) )
8214, 81syl5eq 2507 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  T  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  oF  x.  p
) `  z )
) ) )
8373, 74, 75, 79, 82offval2 6529 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( R  oF  -  T )  =  ( z  e.  CC  |->  ( ( ( F `
 z )  -  ( ( G  oF  x.  q ) `  z ) )  -  ( ( F `  z )  -  (
( G  oF  x.  p ) `  z ) ) ) ) )
8473, 69, 65, 80, 77offval2 6529 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( G  oF  x.  p )  oF  -  ( G  oF  x.  q
) )  =  ( z  e.  CC  |->  ( ( ( G  oF  x.  p ) `  z )  -  (
( G  oF  x.  q ) `  z ) ) ) )
8571, 83, 843eqtr4d 2505 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( R  oF  -  T )  =  ( ( G  oF  x.  p )  oF  -  ( G  oF  x.  q
) ) )
86 plyf 22764 . . . . . . . . . . . . . . . . 17  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
878, 86syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G : CC --> CC )
88 plyf 22764 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  (Poly `  S
)  ->  p : CC
--> CC )
8913, 88syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  p : CC --> CC )
90 plyf 22764 . . . . . . . . . . . . . . . . 17  |-  ( q  e.  (Poly `  S
)  ->  q : CC
--> CC )
912, 90syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  q : CC --> CC )
92 subdi 9986 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z
) ) )
9392adantl 464 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( x  x.  (
y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z ) ) )
9473, 87, 89, 91, 93caofdi 6549 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G  oF  x.  ( p  oF  -  q ) )  =  ( ( G  oF  x.  p )  oF  -  ( G  oF  x.  q )
) )
9585, 94eqtr4d 2498 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( R  oF  -  T )  =  ( G  oF  x.  ( p  oF  -  q ) ) )
9695fveq2d 5852 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  ( R  oF  -  T
) )  =  (deg
`  ( G  oF  x.  ( p  oF  -  q
) ) ) )
9796adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  (deg `  ( R  oF  -  T
) )  =  (deg
`  ( G  oF  x.  ( p  oF  -  q
) ) ) )
988adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  G  e.  (Poly `  S ) )
999adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  G  =/=  0p )
10053adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  ( p  oF  -  q
)  e.  (Poly `  S ) )
101 simpr 459 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  ( p  oF  -  q
)  =/=  0p )
102 eqid 2454 . . . . . . . . . . . . . 14  |-  (deg `  G )  =  (deg
`  G )
103 eqid 2454 . . . . . . . . . . . . . 14  |-  (deg `  ( p  oF  -  q ) )  =  (deg `  (
p  oF  -  q ) )
104102, 103dgrmul 22836 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  (Poly `  S )  /\  G  =/=  0p )  /\  ( ( p  oF  -  q )  e.  (Poly `  S
)  /\  ( p  oF  -  q
)  =/=  0p ) )  ->  (deg `  ( G  oF  x.  ( p  oF  -  q ) ) )  =  ( (deg `  G )  +  (deg `  ( p  oF  -  q
) ) ) )
10598, 99, 100, 101, 104syl22anc 1227 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  (deg `  ( G  oF  x.  (
p  oF  -  q ) ) )  =  ( (deg `  G )  +  (deg
`  ( p  oF  -  q ) ) ) )
10697, 105eqtrd 2495 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  (deg `  ( R  oF  -  T
) )  =  ( (deg `  G )  +  (deg `  ( p  oF  -  q
) ) ) )
10758, 106breqtrrd 4465 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  (deg `  G
)  <_  (deg `  ( R  oF  -  T
) ) )
10820, 30letri3d 9716 . . . . . . . . . . 11  |-  ( ph  ->  ( (deg `  ( R  oF  -  T
) )  =  (deg
`  G )  <->  ( (deg `  ( R  oF  -  T ) )  <_  (deg `  G
)  /\  (deg `  G
)  <_  (deg `  ( R  oF  -  T
) ) ) ) )
109108adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  ( (deg `  ( R  oF  -  T ) )  =  (deg `  G
)  <->  ( (deg `  ( R  oF  -  T ) )  <_ 
(deg `  G )  /\  (deg `  G )  <_  (deg `  ( R  oF  -  T
) ) ) ) )
11052, 107, 109mpbir2and 920 . . . . . . . . 9  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  (deg `  ( R  oF  -  T
) )  =  (deg
`  G ) )
111110fveq2d 5852 . . . . . . . 8  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  ( (coeff `  ( R  oF  -  T ) ) `
 (deg `  ( R  oF  -  T
) ) )  =  ( (coeff `  ( R  oF  -  T
) ) `  (deg `  G ) ) )
11242, 36coesub 22823 . . . . . . . . . . . . 13  |-  ( ( R  e.  (Poly `  S )  /\  T  e.  (Poly `  S )
)  ->  (coeff `  ( R  oF  -  T
) )  =  ( (coeff `  R )  oF  -  (coeff `  T ) ) )
11312, 16, 112syl2anc 659 . . . . . . . . . . . 12  |-  ( ph  ->  (coeff `  ( R  oF  -  T
) )  =  ( (coeff `  R )  oF  -  (coeff `  T ) ) )
114113fveq1d 5850 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  ( R  oF  -  T
) ) `  (deg `  G ) )  =  ( ( (coeff `  R )  oF  -  (coeff `  T
) ) `  (deg `  G ) ) )
11542coef3 22798 . . . . . . . . . . . . . 14  |-  ( R  e.  (Poly `  S
)  ->  (coeff `  R
) : NN0 --> CC )
116 ffn 5713 . . . . . . . . . . . . . 14  |-  ( (coeff `  R ) : NN0 --> CC 
->  (coeff `  R )  Fn  NN0 )
11712, 115, 1163syl 20 . . . . . . . . . . . . 13  |-  ( ph  ->  (coeff `  R )  Fn  NN0 )
11836coef3 22798 . . . . . . . . . . . . . 14  |-  ( T  e.  (Poly `  S
)  ->  (coeff `  T
) : NN0 --> CC )
119 ffn 5713 . . . . . . . . . . . . . 14  |-  ( (coeff `  T ) : NN0 --> CC 
->  (coeff `  T )  Fn  NN0 )
12016, 118, 1193syl 20 . . . . . . . . . . . . 13  |-  ( ph  ->  (coeff `  T )  Fn  NN0 )
121 nn0ex 10797 . . . . . . . . . . . . . 14  |-  NN0  e.  _V
122121a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  NN0  e.  _V )
123 inidm 3693 . . . . . . . . . . . . 13  |-  ( NN0 
i^i  NN0 )  =  NN0
12445simprd 461 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (coeff `  R
) `  (deg `  G
) )  =  0 )
125124adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  (deg `  G
)  e.  NN0 )  ->  ( (coeff `  R
) `  (deg `  G
) )  =  0 )
12639simprd 461 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (coeff `  T
) `  (deg `  G
) )  =  0 )
127126adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  (deg `  G
)  e.  NN0 )  ->  ( (coeff `  T
) `  (deg `  G
) )  =  0 )
128117, 120, 122, 122, 123, 125, 127ofval 6522 . . . . . . . . . . . 12  |-  ( (
ph  /\  (deg `  G
)  e.  NN0 )  ->  ( ( (coeff `  R )  oF  -  (coeff `  T
) ) `  (deg `  G ) )  =  ( 0  -  0 ) )
12929, 128mpdan 666 . . . . . . . . . . 11  |-  ( ph  ->  ( ( (coeff `  R )  oF  -  (coeff `  T
) ) `  (deg `  G ) )  =  ( 0  -  0 ) )
130114, 129eqtrd 2495 . . . . . . . . . 10  |-  ( ph  ->  ( (coeff `  ( R  oF  -  T
) ) `  (deg `  G ) )  =  ( 0  -  0 ) )
131 0m0e0 10641 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
132130, 131syl6eq 2511 . . . . . . . . 9  |-  ( ph  ->  ( (coeff `  ( R  oF  -  T
) ) `  (deg `  G ) )  =  0 )
133132adantr 463 . . . . . . . 8  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  ( (coeff `  ( R  oF  -  T ) ) `
 (deg `  G
) )  =  0 )
134111, 133eqtrd 2495 . . . . . . 7  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  ( (coeff `  ( R  oF  -  T ) ) `
 (deg `  ( R  oF  -  T
) ) )  =  0 )
135 eqid 2454 . . . . . . . . . 10  |-  (deg `  ( R  oF  -  T ) )  =  (deg `  ( R  oF  -  T
) )
136 eqid 2454 . . . . . . . . . 10  |-  (coeff `  ( R  oF  -  T ) )  =  (coeff `  ( R  oF  -  T
) )
137135, 136dgreq0 22831 . . . . . . . . 9  |-  ( ( R  oF  -  T )  e.  (Poly `  S )  ->  (
( R  oF  -  T )  =  0p  <->  ( (coeff `  ( R  oF  -  T ) ) `
 (deg `  ( R  oF  -  T
) ) )  =  0 ) )
13817, 137syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( R  oF  -  T )  =  0p  <->  ( (coeff `  ( R  oF  -  T ) ) `
 (deg `  ( R  oF  -  T
) ) )  =  0 ) )
139138biimpar 483 . . . . . . 7  |-  ( (
ph  /\  ( (coeff `  ( R  oF  -  T ) ) `
 (deg `  ( R  oF  -  T
) ) )  =  0 )  ->  ( R  oF  -  T
)  =  0p )
140134, 139syldan 468 . . . . . 6  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  ( R  oF  -  T
)  =  0p )
141140ex 432 . . . . 5  |-  ( ph  ->  ( ( p  oF  -  q )  =/=  0p  -> 
( R  oF  -  T )  =  0p ) )
142 plymul0or 22846 . . . . . . 7  |-  ( ( G  e.  (Poly `  S )  /\  (
p  oF  -  q )  e.  (Poly `  S ) )  -> 
( ( G  oF  x.  ( p  oF  -  q
) )  =  0p  <->  ( G  =  0p  \/  (
p  oF  -  q )  =  0p ) ) )
1438, 53, 142syl2anc 659 . . . . . 6  |-  ( ph  ->  ( ( G  oF  x.  ( p  oF  -  q
) )  =  0p  <->  ( G  =  0p  \/  (
p  oF  -  q )  =  0p ) ) )
14495eqeq1d 2456 . . . . . 6  |-  ( ph  ->  ( ( R  oF  -  T )  =  0p  <->  ( G  oF  x.  (
p  oF  -  q ) )  =  0p ) )
1459neneqd 2656 . . . . . . 7  |-  ( ph  ->  -.  G  =  0p )
146 biorf 403 . . . . . . 7  |-  ( -.  G  =  0p  ->  ( ( p  oF  -  q
)  =  0p  <-> 
( G  =  0p  \/  ( p  oF  -  q
)  =  0p ) ) )
147145, 146syl 16 . . . . . 6  |-  ( ph  ->  ( ( p  oF  -  q )  =  0p  <->  ( G  =  0p  \/  ( p  oF  -  q )  =  0p ) ) )
148143, 144, 1473bitr4d 285 . . . . 5  |-  ( ph  ->  ( ( R  oF  -  T )  =  0p  <->  ( p  oF  -  q
)  =  0p ) )
149141, 148sylibd 214 . . . 4  |-  ( ph  ->  ( ( p  oF  -  q )  =/=  0p  -> 
( p  oF  -  q )  =  0p ) )
1501, 149pm2.61dne 2771 . . 3  |-  ( ph  ->  ( p  oF  -  q )  =  0p )
151 df-0p 22246 . . 3  |-  0p  =  ( CC  X.  { 0 } )
152150, 151syl6eq 2511 . 2  |-  ( ph  ->  ( p  oF  -  q )  =  ( CC  X.  {
0 } ) )
153 ofsubeq0 10528 . . 3  |-  ( ( CC  e.  _V  /\  p : CC --> CC  /\  q : CC --> CC )  ->  ( ( p  oF  -  q
)  =  ( CC 
X.  { 0 } )  <->  p  =  q
) )
15473, 89, 91, 153syl3anc 1226 . 2  |-  ( ph  ->  ( ( p  oF  -  q )  =  ( CC  X.  { 0 } )  <-> 
p  =  q ) )
155152, 154mpbid 210 1  |-  ( ph  ->  p  =  q )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106   ifcif 3929   {csn 4016   class class class wbr 4439    |-> cmpt 4497    X. cxp 4986    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    oFcof 6511   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    < clt 9617    <_ cle 9618    - cmin 9796   -ucneg 9797    / cdiv 10202   NN0cn0 10791   0pc0p 22245  Polycply 22750  coeffccoe 22752  degcdgr 22753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-clim 13396  df-rlim 13397  df-sum 13594  df-0p 22246  df-ply 22754  df-coe 22756  df-dgr 22757
This theorem is referenced by:  plydivalg  22864
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