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Theorem plydiveu 20168
Description: Lemma for plydivalg 20169. (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 )
)
plydiveu.q  |-  ( ph  ->  q  e.  (Poly `  S ) )
plydiveu.qd  |-  ( ph  ->  ( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) )
plydiveu.t  |-  T  =  ( F  o F  -  ( G  o F  x.  p )
)
plydiveu.p  |-  ( ph  ->  p  e.  (Poly `  S ) )
plydiveu.pd  |-  ( ph  ->  ( T  =  0 p  \/  (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 22 . . . 4  |-  ( ph  ->  ( ( p  o F  -  q )  =  0 p  -> 
( p  o F  -  q )  =  0 p ) )
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  =/=  0 p )
10 plydiv.r . . . . . . . . . . . . . . . . 17  |-  R  =  ( F  o F  -  ( G  o F  x.  q )
)
113, 4, 5, 6, 7, 8, 9, 10plydivlem2 20164 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  q  e.  (Poly `  S ) )  ->  R  e.  (Poly `  S ) )
122, 11mpdan 650 . . . . . . . . . . . . . . 15  |-  ( ph  ->  R  e.  (Poly `  S ) )
13 plydiveu.p . . . . . . . . . . . . . . . 16  |-  ( ph  ->  p  e.  (Poly `  S ) )
14 plydiveu.t . . . . . . . . . . . . . . . . 17  |-  T  =  ( F  o F  -  ( G  o F  x.  p )
)
153, 4, 5, 6, 7, 8, 9, 14plydivlem2 20164 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  T  e.  (Poly `  S ) )
1613, 15mpdan 650 . . . . . . . . . . . . . . 15  |-  ( ph  ->  T  e.  (Poly `  S ) )
1712, 16, 3, 4, 6plysub 20091 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( R  o F  -  T )  e.  (Poly `  S )
)
18 dgrcl 20105 . . . . . . . . . . . . . 14  |-  ( ( R  o F  -  T )  e.  (Poly `  S )  ->  (deg `  ( R  o F  -  T ) )  e.  NN0 )
1917, 18syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  ( R  o F  -  T
) )  e.  NN0 )
2019nn0red 10231 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  ( R  o F  -  T
) )  e.  RR )
21 dgrcl 20105 . . . . . . . . . . . . . . 15  |-  ( T  e.  (Poly `  S
)  ->  (deg `  T
)  e.  NN0 )
2216, 21syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  (deg `  T )  e.  NN0 )
2322nn0red 10231 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  T )  e.  RR )
24 dgrcl 20105 . . . . . . . . . . . . . . 15  |-  ( R  e.  (Poly `  S
)  ->  (deg `  R
)  e.  NN0 )
2512, 24syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  (deg `  R )  e.  NN0 )
2625nn0red 10231 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  R )  e.  RR )
27 ifcl 3735 . . . . . . . . . . . . 13  |-  ( ( (deg `  T )  e.  RR  /\  (deg `  R )  e.  RR )  ->  if ( (deg
`  R )  <_ 
(deg `  T ) ,  (deg `  T ) ,  (deg `  R )
)  e.  RR )
2823, 26, 27syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  if ( (deg `  R )  <_  (deg `  T ) ,  (deg
`  T ) ,  (deg `  R )
)  e.  RR )
29 dgrcl 20105 . . . . . . . . . . . . . 14  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
308, 29syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  G )  e.  NN0 )
3130nn0red 10231 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  G )  e.  RR )
32 eqid 2404 . . . . . . . . . . . . . 14  |-  (deg `  R )  =  (deg
`  R )
33 eqid 2404 . . . . . . . . . . . . . 14  |-  (deg `  T )  =  (deg
`  T )
3432, 33dgrsub 20143 . . . . . . . . . . . . 13  |-  ( ( R  e.  (Poly `  S )  /\  T  e.  (Poly `  S )
)  ->  (deg `  ( R  o F  -  T
) )  <_  if ( (deg `  R )  <_  (deg `  T ) ,  (deg `  T ) ,  (deg `  R )
) )
3512, 16, 34syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  ( R  o F  -  T
) )  <_  if ( (deg `  R )  <_  (deg `  T ) ,  (deg `  T ) ,  (deg `  R )
) )
36 plydiveu.pd . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( T  =  0 p  \/  (deg `  T )  <  (deg `  G ) ) )
37 eqid 2404 . . . . . . . . . . . . . . . . 17  |-  (coeff `  T )  =  (coeff `  T )
3833, 37dgrlt 20137 . . . . . . . . . . . . . . . 16  |-  ( ( T  e.  (Poly `  S )  /\  (deg `  G )  e.  NN0 )  ->  ( ( T  =  0 p  \/  (deg `  T )  < 
(deg `  G )
)  <->  ( (deg `  T )  <_  (deg `  G )  /\  (
(coeff `  T ) `  (deg `  G )
)  =  0 ) ) )
3916, 30, 38syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( T  =  0 p  \/  (deg `  T )  <  (deg `  G ) )  <->  ( (deg `  T )  <_  (deg `  G )  /\  (
(coeff `  T ) `  (deg `  G )
)  =  0 ) ) )
4036, 39mpbid 202 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (deg `  T
)  <_  (deg `  G
)  /\  ( (coeff `  T ) `  (deg `  G ) )  =  0 ) )
4140simpld 446 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  T )  <_  (deg `  G )
)
42 plydiveu.qd . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) )
43 eqid 2404 . . . . . . . . . . . . . . . . 17  |-  (coeff `  R )  =  (coeff `  R )
4432, 43dgrlt 20137 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  (Poly `  S )  /\  (deg `  G )  e.  NN0 )  ->  ( ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
)  <->  ( (deg `  R )  <_  (deg `  G )  /\  (
(coeff `  R ) `  (deg `  G )
)  =  0 ) ) )
4512, 30, 44syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  <->  ( (deg `  R )  <_  (deg `  G )  /\  (
(coeff `  R ) `  (deg `  G )
)  =  0 ) ) )
4642, 45mpbid 202 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (deg `  R
)  <_  (deg `  G
)  /\  ( (coeff `  R ) `  (deg `  G ) )  =  0 ) )
4746simpld 446 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  R )  <_  (deg `  G )
)
48 breq1 4175 . . . . . . . . . . . . . 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
) ) )
49 breq1 4175 . . . . . . . . . . . . . 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
) ) )
5048, 49ifboth 3730 . . . . . . . . . . . . 13  |-  ( ( (deg `  T )  <_  (deg `  G )  /\  (deg `  R )  <_  (deg `  G )
)  ->  if (
(deg `  R )  <_  (deg `  T ) ,  (deg `  T ) ,  (deg `  R )
)  <_  (deg `  G
) )
5141, 47, 50syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  if ( (deg `  R )  <_  (deg `  T ) ,  (deg
`  T ) ,  (deg `  R )
)  <_  (deg `  G
) )
5220, 28, 31, 35, 51letrd 9183 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  ( R  o F  -  T
) )  <_  (deg `  G ) )
5352adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  ( R  o F  -  T
) )  <_  (deg `  G ) )
5413, 2, 3, 4, 6plysub 20091 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( p  o F  -  q )  e.  (Poly `  S )
)
55 dgrcl 20105 . . . . . . . . . . . . . 14  |-  ( ( p  o F  -  q )  e.  (Poly `  S )  ->  (deg `  ( p  o F  -  q ) )  e.  NN0 )
5654, 55syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  ( p  o F  -  q
) )  e.  NN0 )
57 nn0addge1 10222 . . . . . . . . . . . . 13  |-  ( ( (deg `  G )  e.  RR  /\  (deg `  ( p  o F  -  q ) )  e.  NN0 )  -> 
(deg `  G )  <_  ( (deg `  G
)  +  (deg `  ( p  o F  -  q ) ) ) )
5831, 56, 57syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  G )  <_  ( (deg `  G
)  +  (deg `  ( p  o F  -  q ) ) ) )
5958adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  G
)  <_  ( (deg `  G )  +  (deg
`  ( p  o F  -  q ) ) ) )
60 plyf 20070 . . . . . . . . . . . . . . . . . . . 20  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
617, 60syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F : CC --> CC )
6261ffvelrnda 5829 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e.  CC )
638, 2, 3, 4plymul 20090 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  o F  x.  q )  e.  (Poly `  S )
)
64 plyf 20070 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( G  o F  x.  q )  e.  (Poly `  S )  ->  ( G  o F  x.  q
) : CC --> CC )
6563, 64syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  o F  x.  q ) : CC --> CC )
6665ffvelrnda 5829 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  o F  x.  q ) `  z
)  e.  CC )
678, 13, 3, 4plymul 20090 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  o F  x.  p )  e.  (Poly `  S )
)
68 plyf 20070 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( G  o F  x.  p )  e.  (Poly `  S )  ->  ( G  o F  x.  p
) : CC --> CC )
6967, 68syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  o F  x.  p ) : CC --> CC )
7069ffvelrnda 5829 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  o F  x.  p ) `  z
)  e.  CC )
7162, 66, 70nnncan1d 9401 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( ( F `  z
)  -  ( ( G  o F  x.  q ) `  z
) )  -  (
( F `  z
)  -  ( ( G  o F  x.  p ) `  z
) ) )  =  ( ( ( G  o F  x.  p
) `  z )  -  ( ( G  o F  x.  q
) `  z )
) )
7271mpteq2dva 4255 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( z  e.  CC  |->  ( ( ( F `
 z )  -  ( ( G  o F  x.  q ) `  z ) )  -  ( ( F `  z )  -  (
( G  o F  x.  p ) `  z ) ) ) )  =  ( z  e.  CC  |->  ( ( ( G  o F  x.  p ) `  z )  -  (
( G  o F  x.  q ) `  z ) ) ) )
73 cnex 9027 . . . . . . . . . . . . . . . . . 18  |-  CC  e.  _V
7473a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  CC  e.  _V )
7562, 66subcld 9367 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  ( ( G  o F  x.  q
) `  z )
)  e.  CC )
7662, 70subcld 9367 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  ( ( G  o F  x.  p
) `  z )
)  e.  CC )
7761feqmptd 5738 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F  =  ( z  e.  CC  |->  ( F `
 z ) ) )
7865feqmptd 5738 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  o F  x.  q )  =  ( z  e.  CC  |->  ( ( G  o F  x.  q ) `  z ) ) )
7974, 62, 66, 77, 78offval2 6281 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( F  o F  -  ( G  o F  x.  q )
)  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  o F  x.  q
) `  z )
) ) )
8010, 79syl5eq 2448 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  o F  x.  q
) `  z )
) ) )
8169feqmptd 5738 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  o F  x.  p )  =  ( z  e.  CC  |->  ( ( G  o F  x.  p ) `  z ) ) )
8274, 62, 70, 77, 81offval2 6281 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( F  o F  -  ( G  o F  x.  p )
)  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  o F  x.  p
) `  z )
) ) )
8314, 82syl5eq 2448 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  T  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  o F  x.  p
) `  z )
) ) )
8474, 75, 76, 80, 83offval2 6281 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( R  o F  -  T )  =  ( z  e.  CC  |->  ( ( ( F `
 z )  -  ( ( G  o F  x.  q ) `  z ) )  -  ( ( F `  z )  -  (
( G  o F  x.  p ) `  z ) ) ) ) )
8574, 70, 66, 81, 78offval2 6281 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( G  o F  x.  p )  o F  -  ( G  o F  x.  q
) )  =  ( z  e.  CC  |->  ( ( ( G  o F  x.  p ) `  z )  -  (
( G  o F  x.  q ) `  z ) ) ) )
8672, 84, 853eqtr4d 2446 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( R  o F  -  T )  =  ( ( G  o F  x.  p )  o F  -  ( G  o F  x.  q
) ) )
87 plyf 20070 . . . . . . . . . . . . . . . . 17  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
888, 87syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G : CC --> CC )
89 plyf 20070 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  (Poly `  S
)  ->  p : CC
--> CC )
9013, 89syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  p : CC --> CC )
91 plyf 20070 . . . . . . . . . . . . . . . . 17  |-  ( q  e.  (Poly `  S
)  ->  q : CC
--> CC )
922, 91syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  q : CC --> CC )
93 subdi 9423 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z
) ) )
9493adantl 453 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( x  x.  (
y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z ) ) )
9574, 88, 90, 92, 94caofdi 6299 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G  o F  x.  ( p  o F  -  q ) )  =  ( ( G  o F  x.  p )  o F  -  ( G  o F  x.  q )
) )
9686, 95eqtr4d 2439 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( R  o F  -  T )  =  ( G  o F  x.  ( p  o F  -  q ) ) )
9796fveq2d 5691 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  ( R  o F  -  T
) )  =  (deg
`  ( G  o F  x.  ( p  o F  -  q
) ) ) )
9897adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  ( R  o F  -  T
) )  =  (deg
`  ( G  o F  x.  ( p  o F  -  q
) ) ) )
998adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  G  e.  (Poly `  S ) )
1009adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  G  =/=  0 p )
10154adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  ( p  o F  -  q
)  e.  (Poly `  S ) )
102 simpr 448 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  ( p  o F  -  q
)  =/=  0 p )
103 eqid 2404 . . . . . . . . . . . . . 14  |-  (deg `  G )  =  (deg
`  G )
104 eqid 2404 . . . . . . . . . . . . . 14  |-  (deg `  ( p  o F  -  q ) )  =  (deg `  (
p  o F  -  q ) )
105103, 104dgrmul 20141 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  (Poly `  S )  /\  G  =/=  0 p )  /\  ( ( p  o F  -  q )  e.  (Poly `  S
)  /\  ( p  o F  -  q
)  =/=  0 p ) )  ->  (deg `  ( G  o F  x.  ( p  o F  -  q ) ) )  =  ( (deg `  G )  +  (deg `  ( p  o F  -  q
) ) ) )
10699, 100, 101, 102, 105syl22anc 1185 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  ( G  o F  x.  (
p  o F  -  q ) ) )  =  ( (deg `  G )  +  (deg
`  ( p  o F  -  q ) ) ) )
10798, 106eqtrd 2436 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  ( R  o F  -  T
) )  =  ( (deg `  G )  +  (deg `  ( p  o F  -  q
) ) ) )
10859, 107breqtrrd 4198 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  G
)  <_  (deg `  ( R  o F  -  T
) ) )
10920, 31letri3d 9171 . . . . . . . . . . 11  |-  ( ph  ->  ( (deg `  ( R  o F  -  T
) )  =  (deg
`  G )  <->  ( (deg `  ( R  o F  -  T ) )  <_  (deg `  G
)  /\  (deg `  G
)  <_  (deg `  ( R  o F  -  T
) ) ) ) )
110109adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  ( (deg `  ( R  o F  -  T ) )  =  (deg `  G
)  <->  ( (deg `  ( R  o F  -  T ) )  <_ 
(deg `  G )  /\  (deg `  G )  <_  (deg `  ( R  o F  -  T
) ) ) ) )
11153, 108, 110mpbir2and 889 . . . . . . . . 9  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  ( R  o F  -  T
) )  =  (deg
`  G ) )
112111fveq2d 5691 . . . . . . . 8  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  ( (coeff `  ( R  o F  -  T ) ) `
 (deg `  ( R  o F  -  T
) ) )  =  ( (coeff `  ( R  o F  -  T
) ) `  (deg `  G ) ) )
11343, 37coesub 20128 . . . . . . . . . . . . 13  |-  ( ( R  e.  (Poly `  S )  /\  T  e.  (Poly `  S )
)  ->  (coeff `  ( R  o F  -  T
) )  =  ( (coeff `  R )  o F  -  (coeff `  T ) ) )
11412, 16, 113syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  (coeff `  ( R  o F  -  T
) )  =  ( (coeff `  R )  o F  -  (coeff `  T ) ) )
115114fveq1d 5689 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  ( R  o F  -  T
) ) `  (deg `  G ) )  =  ( ( (coeff `  R )  o F  -  (coeff `  T
) ) `  (deg `  G ) ) )
11643coef3 20104 . . . . . . . . . . . . . 14  |-  ( R  e.  (Poly `  S
)  ->  (coeff `  R
) : NN0 --> CC )
117 ffn 5550 . . . . . . . . . . . . . 14  |-  ( (coeff `  R ) : NN0 --> CC 
->  (coeff `  R )  Fn  NN0 )
11812, 116, 1173syl 19 . . . . . . . . . . . . 13  |-  ( ph  ->  (coeff `  R )  Fn  NN0 )
11937coef3 20104 . . . . . . . . . . . . . 14  |-  ( T  e.  (Poly `  S
)  ->  (coeff `  T
) : NN0 --> CC )
120 ffn 5550 . . . . . . . . . . . . . 14  |-  ( (coeff `  T ) : NN0 --> CC 
->  (coeff `  T )  Fn  NN0 )
12116, 119, 1203syl 19 . . . . . . . . . . . . 13  |-  ( ph  ->  (coeff `  T )  Fn  NN0 )
122 nn0ex 10183 . . . . . . . . . . . . . 14  |-  NN0  e.  _V
123122a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  NN0  e.  _V )
124 inidm 3510 . . . . . . . . . . . . 13  |-  ( NN0 
i^i  NN0 )  =  NN0
12546simprd 450 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (coeff `  R
) `  (deg `  G
) )  =  0 )
126125adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  (deg `  G
)  e.  NN0 )  ->  ( (coeff `  R
) `  (deg `  G
) )  =  0 )
12740simprd 450 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (coeff `  T
) `  (deg `  G
) )  =  0 )
128127adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  (deg `  G
)  e.  NN0 )  ->  ( (coeff `  T
) `  (deg `  G
) )  =  0 )
129118, 121, 123, 123, 124, 126, 128ofval 6273 . . . . . . . . . . . 12  |-  ( (
ph  /\  (deg `  G
)  e.  NN0 )  ->  ( ( (coeff `  R )  o F  -  (coeff `  T
) ) `  (deg `  G ) )  =  ( 0  -  0 ) )
13030, 129mpdan 650 . . . . . . . . . . 11  |-  ( ph  ->  ( ( (coeff `  R )  o F  -  (coeff `  T
) ) `  (deg `  G ) )  =  ( 0  -  0 ) )
131115, 130eqtrd 2436 . . . . . . . . . 10  |-  ( ph  ->  ( (coeff `  ( R  o F  -  T
) ) `  (deg `  G ) )  =  ( 0  -  0 ) )
132 0cn 9040 . . . . . . . . . . 11  |-  0  e.  CC
133132subidi 9327 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
134131, 133syl6eq 2452 . . . . . . . . 9  |-  ( ph  ->  ( (coeff `  ( R  o F  -  T
) ) `  (deg `  G ) )  =  0 )
135134adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  ( (coeff `  ( R  o F  -  T ) ) `
 (deg `  G
) )  =  0 )
136112, 135eqtrd 2436 . . . . . . 7  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  ( (coeff `  ( R  o F  -  T ) ) `
 (deg `  ( R  o F  -  T
) ) )  =  0 )
137 eqid 2404 . . . . . . . . . 10  |-  (deg `  ( R  o F  -  T ) )  =  (deg `  ( R  o F  -  T
) )
138 eqid 2404 . . . . . . . . . 10  |-  (coeff `  ( R  o F  -  T ) )  =  (coeff `  ( R  o F  -  T
) )
139137, 138dgreq0 20136 . . . . . . . . 9  |-  ( ( R  o F  -  T )  e.  (Poly `  S )  ->  (
( R  o F  -  T )  =  0 p  <->  ( (coeff `  ( R  o F  -  T ) ) `
 (deg `  ( R  o F  -  T
) ) )  =  0 ) )
14017, 139syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( R  o F  -  T )  =  0 p  <->  ( (coeff `  ( R  o F  -  T ) ) `
 (deg `  ( R  o F  -  T
) ) )  =  0 ) )
141140biimpar 472 . . . . . . 7  |-  ( (
ph  /\  ( (coeff `  ( R  o F  -  T ) ) `
 (deg `  ( R  o F  -  T
) ) )  =  0 )  ->  ( R  o F  -  T
)  =  0 p )
142136, 141syldan 457 . . . . . 6  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  ( R  o F  -  T
)  =  0 p )
143142ex 424 . . . . 5  |-  ( ph  ->  ( ( p  o F  -  q )  =/=  0 p  -> 
( R  o F  -  T )  =  0 p ) )
144 plymul0or 20151 . . . . . . 7  |-  ( ( G  e.  (Poly `  S )  /\  (
p  o F  -  q )  e.  (Poly `  S ) )  -> 
( ( G  o F  x.  ( p  o F  -  q
) )  =  0 p  <->  ( G  =  0 p  \/  (
p  o F  -  q )  =  0 p ) ) )
1458, 54, 144syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( G  o F  x.  ( p  o F  -  q
) )  =  0 p  <->  ( G  =  0 p  \/  (
p  o F  -  q )  =  0 p ) ) )
14696eqeq1d 2412 . . . . . 6  |-  ( ph  ->  ( ( R  o F  -  T )  =  0 p  <->  ( G  o F  x.  (
p  o F  -  q ) )  =  0 p ) )
1479neneqd 2583 . . . . . . 7  |-  ( ph  ->  -.  G  =  0 p )
148 biorf 395 . . . . . . 7  |-  ( -.  G  =  0 p  ->  ( ( p  o F  -  q
)  =  0 p  <-> 
( G  =  0 p  \/  ( p  o F  -  q
)  =  0 p ) ) )
149147, 148syl 16 . . . . . 6  |-  ( ph  ->  ( ( p  o F  -  q )  =  0 p  <->  ( G  =  0 p  \/  ( p  o F  -  q )  =  0 p ) ) )
150145, 146, 1493bitr4d 277 . . . . 5  |-  ( ph  ->  ( ( R  o F  -  T )  =  0 p  <->  ( p  o F  -  q
)  =  0 p ) )
151143, 150sylibd 206 . . . 4  |-  ( ph  ->  ( ( p  o F  -  q )  =/=  0 p  -> 
( p  o F  -  q )  =  0 p ) )
1521, 151pm2.61dne 2644 . . 3  |-  ( ph  ->  ( p  o F  -  q )  =  0 p )
153 df-0p 19515 . . 3  |-  0 p  =  ( CC  X.  { 0 } )
154152, 153syl6eq 2452 . 2  |-  ( ph  ->  ( p  o F  -  q )  =  ( CC  X.  {
0 } ) )
155 ofsubeq0 9953 . . 3  |-  ( ( CC  e.  _V  /\  p : CC --> CC  /\  q : CC --> CC )  ->  ( ( p  o F  -  q
)  =  ( CC 
X.  { 0 } )  <->  p  =  q
) )
15674, 90, 92, 155syl3anc 1184 . 2  |-  ( ph  ->  ( ( p  o F  -  q )  =  ( CC  X.  { 0 } )  <-> 
p  =  q ) )
157154, 156mpbid 202 1  |-  ( ph  ->  p  =  q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916   ifcif 3699   {csn 3774   class class class wbr 4172    e. cmpt 4226    X. cxp 4835    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247   -ucneg 9248    / cdiv 9633   NN0cn0 10177   0 pc0p 19514  Polycply 20056  coeffccoe 20058  degcdgr 20059
This theorem is referenced by:  plydivalg  20169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-coe 20062  df-dgr 20063
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