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Theorem plydiveu 21896
Description: Lemma for plydivalg 21897. (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 21892 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  q  e.  (Poly `  S ) )  ->  R  e.  (Poly `  S ) )
122, 11mpdan 668 . . . . . . . . . . . . . . 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 21892 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  T  e.  (Poly `  S ) )
1613, 15mpdan 668 . . . . . . . . . . . . . . 15  |-  ( ph  ->  T  e.  (Poly `  S ) )
1712, 16, 3, 4, 6plysub 21819 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( R  oF  -  T )  e.  (Poly `  S )
)
18 dgrcl 21833 . . . . . . . . . . . . . 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 10747 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  ( R  oF  -  T
) )  e.  RR )
21 dgrcl 21833 . . . . . . . . . . . . . . 15  |-  ( T  e.  (Poly `  S
)  ->  (deg `  T
)  e.  NN0 )
2216, 21syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  (deg `  T )  e.  NN0 )
2322nn0red 10747 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  T )  e.  RR )
24 dgrcl 21833 . . . . . . . . . . . . . . 15  |-  ( R  e.  (Poly `  S
)  ->  (deg `  R
)  e.  NN0 )
2512, 24syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  (deg `  R )  e.  NN0 )
2625nn0red 10747 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  R )  e.  RR )
27 ifcl 3938 . . . . . . . . . . . . 13  |-  ( ( (deg `  T )  e.  RR  /\  (deg `  R )  e.  RR )  ->  if ( (deg
`  R )  <_ 
(deg `  T ) ,  (deg `  T ) ,  (deg `  R )
)  e.  RR )
2823, 26, 27syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  if ( (deg `  R )  <_  (deg `  T ) ,  (deg
`  T ) ,  (deg `  R )
)  e.  RR )
29 dgrcl 21833 . . . . . . . . . . . . . 14  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
308, 29syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  G )  e.  NN0 )
3130nn0red 10747 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  G )  e.  RR )
32 eqid 2454 . . . . . . . . . . . . . 14  |-  (deg `  R )  =  (deg
`  R )
33 eqid 2454 . . . . . . . . . . . . . 14  |-  (deg `  T )  =  (deg
`  T )
3432, 33dgrsub 21871 . . . . . . . . . . . . 13  |-  ( ( R  e.  (Poly `  S )  /\  T  e.  (Poly `  S )
)  ->  (deg `  ( R  oF  -  T
) )  <_  if ( (deg `  R )  <_  (deg `  T ) ,  (deg `  T ) ,  (deg `  R )
) )
3512, 16, 34syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  ( R  oF  -  T
) )  <_  if ( (deg `  R )  <_  (deg `  T ) ,  (deg `  T ) ,  (deg `  R )
) )
36 plydiveu.pd . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( T  =  0p  \/  (deg `  T )  <  (deg `  G ) ) )
37 eqid 2454 . . . . . . . . . . . . . . . . 17  |-  (coeff `  T )  =  (coeff `  T )
3833, 37dgrlt 21865 . . . . . . . . . . . . . . . 16  |-  ( ( T  e.  (Poly `  S )  /\  (deg `  G )  e.  NN0 )  ->  ( ( T  =  0p  \/  (deg `  T )  <  (deg `  G )
)  <->  ( (deg `  T )  <_  (deg `  G )  /\  (
(coeff `  T ) `  (deg `  G )
)  =  0 ) ) )
3916, 30, 38syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( T  =  0p  \/  (deg `  T )  <  (deg `  G ) )  <->  ( (deg `  T )  <_  (deg `  G )  /\  (
(coeff `  T ) `  (deg `  G )
)  =  0 ) ) )
4036, 39mpbid 210 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (deg `  T
)  <_  (deg `  G
)  /\  ( (coeff `  T ) `  (deg `  G ) )  =  0 ) )
4140simpld 459 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  T )  <_  (deg `  G )
)
42 plydiveu.qd . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) )
43 eqid 2454 . . . . . . . . . . . . . . . . 17  |-  (coeff `  R )  =  (coeff `  R )
4432, 43dgrlt 21865 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  (Poly `  S )  /\  (deg `  G )  e.  NN0 )  ->  ( ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
)  <->  ( (deg `  R )  <_  (deg `  G )  /\  (
(coeff `  R ) `  (deg `  G )
)  =  0 ) ) )
4512, 30, 44syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  <->  ( (deg `  R )  <_  (deg `  G )  /\  (
(coeff `  R ) `  (deg `  G )
)  =  0 ) ) )
4642, 45mpbid 210 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (deg `  R
)  <_  (deg `  G
)  /\  ( (coeff `  R ) `  (deg `  G ) )  =  0 ) )
4746simpld 459 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  R )  <_  (deg `  G )
)
48 breq1 4402 . . . . . . . . . . . . . 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 4402 . . . . . . . . . . . . . 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 3932 . . . . . . . . . . . . 13  |-  ( ( (deg `  T )  <_  (deg `  G )  /\  (deg `  R )  <_  (deg `  G )
)  ->  if (
(deg `  R )  <_  (deg `  T ) ,  (deg `  T ) ,  (deg `  R )
)  <_  (deg `  G
) )
5141, 47, 50syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  if ( (deg `  R )  <_  (deg `  T ) ,  (deg
`  T ) ,  (deg `  R )
)  <_  (deg `  G
) )
5220, 28, 31, 35, 51letrd 9638 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  ( R  oF  -  T
) )  <_  (deg `  G ) )
5352adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  (deg `  ( R  oF  -  T
) )  <_  (deg `  G ) )
5413, 2, 3, 4, 6plysub 21819 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( p  oF  -  q )  e.  (Poly `  S )
)
55 dgrcl 21833 . . . . . . . . . . . . . 14  |-  ( ( p  oF  -  q )  e.  (Poly `  S )  ->  (deg `  ( p  oF  -  q ) )  e.  NN0 )
5654, 55syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  ( p  oF  -  q
) )  e.  NN0 )
57 nn0addge1 10736 . . . . . . . . . . . . 13  |-  ( ( (deg `  G )  e.  RR  /\  (deg `  ( p  oF  -  q ) )  e.  NN0 )  -> 
(deg `  G )  <_  ( (deg `  G
)  +  (deg `  ( p  oF  -  q ) ) ) )
5831, 56, 57syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  G )  <_  ( (deg `  G
)  +  (deg `  ( p  oF  -  q ) ) ) )
5958adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  (deg `  G
)  <_  ( (deg `  G )  +  (deg
`  ( p  oF  -  q ) ) ) )
60 plyf 21798 . . . . . . . . . . . . . . . . . . . 20  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
617, 60syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F : CC --> CC )
6261ffvelrnda 5951 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e.  CC )
638, 2, 3, 4plymul 21818 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  oF  x.  q )  e.  (Poly `  S )
)
64 plyf 21798 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( G  oF  x.  q )  e.  (Poly `  S )  ->  ( G  oF  x.  q
) : CC --> CC )
6563, 64syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  oF  x.  q ) : CC --> CC )
6665ffvelrnda 5951 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  oF  x.  q ) `  z
)  e.  CC )
678, 13, 3, 4plymul 21818 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  oF  x.  p )  e.  (Poly `  S )
)
68 plyf 21798 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( G  oF  x.  p )  e.  (Poly `  S )  ->  ( G  oF  x.  p
) : CC --> CC )
6967, 68syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  oF  x.  p ) : CC --> CC )
7069ffvelrnda 5951 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  oF  x.  p ) `  z
)  e.  CC )
7162, 66, 70nnncan1d 9863 . . . . . . . . . . . . . . . . 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 )
) )
7271mpteq2dva 4485 . . . . . . . . . . . . . . . 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 ) ) ) )
73 cnex 9473 . . . . . . . . . . . . . . . . . 18  |-  CC  e.  _V
7473a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  CC  e.  _V )
7562, 66subcld 9829 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  ( ( G  oF  x.  q
) `  z )
)  e.  CC )
7662, 70subcld 9829 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  ( ( G  oF  x.  p
) `  z )
)  e.  CC )
7761feqmptd 5852 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F  =  ( z  e.  CC  |->  ( F `
 z ) ) )
7865feqmptd 5852 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  oF  x.  q )  =  ( z  e.  CC  |->  ( ( G  oF  x.  q ) `  z ) ) )
7974, 62, 66, 77, 78offval2 6445 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( F  oF  -  ( G  oF  x.  q )
)  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  oF  x.  q
) `  z )
) ) )
8010, 79syl5eq 2507 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  oF  x.  q
) `  z )
) ) )
8169feqmptd 5852 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  oF  x.  p )  =  ( z  e.  CC  |->  ( ( G  oF  x.  p ) `  z ) ) )
8274, 62, 70, 77, 81offval2 6445 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( F  oF  -  ( G  oF  x.  p )
)  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  oF  x.  p
) `  z )
) ) )
8314, 82syl5eq 2507 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  T  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  oF  x.  p
) `  z )
) ) )
8474, 75, 76, 80, 83offval2 6445 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( R  oF  -  T )  =  ( z  e.  CC  |->  ( ( ( F `
 z )  -  ( ( G  oF  x.  q ) `  z ) )  -  ( ( F `  z )  -  (
( G  oF  x.  p ) `  z ) ) ) ) )
8574, 70, 66, 81, 78offval2 6445 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( G  oF  x.  p )  oF  -  ( G  oF  x.  q
) )  =  ( z  e.  CC  |->  ( ( ( G  oF  x.  p ) `  z )  -  (
( G  oF  x.  q ) `  z ) ) ) )
8672, 84, 853eqtr4d 2505 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( R  oF  -  T )  =  ( ( G  oF  x.  p )  oF  -  ( G  oF  x.  q
) ) )
87 plyf 21798 . . . . . . . . . . . . . . . . 17  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
888, 87syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G : CC --> CC )
89 plyf 21798 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  (Poly `  S
)  ->  p : CC
--> CC )
9013, 89syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  p : CC --> CC )
91 plyf 21798 . . . . . . . . . . . . . . . . 17  |-  ( q  e.  (Poly `  S
)  ->  q : CC
--> CC )
922, 91syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  q : CC --> CC )
93 subdi 9888 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z
) ) )
9493adantl 466 . . . . . . . . . . . . . . . 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 6465 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G  oF  x.  ( p  oF  -  q ) )  =  ( ( G  oF  x.  p )  oF  -  ( G  oF  x.  q )
) )
9686, 95eqtr4d 2498 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( R  oF  -  T )  =  ( G  oF  x.  ( p  oF  -  q ) ) )
9796fveq2d 5802 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  ( R  oF  -  T
) )  =  (deg
`  ( G  oF  x.  ( p  oF  -  q
) ) ) )
9897adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  (deg `  ( R  oF  -  T
) )  =  (deg
`  ( G  oF  x.  ( p  oF  -  q
) ) ) )
998adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  G  e.  (Poly `  S ) )
1009adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  G  =/=  0p )
10154adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  ( p  oF  -  q
)  e.  (Poly `  S ) )
102 simpr 461 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  ( p  oF  -  q
)  =/=  0p )
103 eqid 2454 . . . . . . . . . . . . . 14  |-  (deg `  G )  =  (deg
`  G )
104 eqid 2454 . . . . . . . . . . . . . 14  |-  (deg `  ( p  oF  -  q ) )  =  (deg `  (
p  oF  -  q ) )
105103, 104dgrmul 21869 . . . . . . . . . . . . 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
) ) ) )
10699, 100, 101, 102, 105syl22anc 1220 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  (deg `  ( G  oF  x.  (
p  oF  -  q ) ) )  =  ( (deg `  G )  +  (deg
`  ( p  oF  -  q ) ) ) )
10798, 106eqtrd 2495 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  (deg `  ( R  oF  -  T
) )  =  ( (deg `  G )  +  (deg `  ( p  oF  -  q
) ) ) )
10859, 107breqtrrd 4425 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  (deg `  G
)  <_  (deg `  ( R  oF  -  T
) ) )
10920, 31letri3d 9626 . . . . . . . . . . 11  |-  ( ph  ->  ( (deg `  ( R  oF  -  T
) )  =  (deg
`  G )  <->  ( (deg `  ( R  oF  -  T ) )  <_  (deg `  G
)  /\  (deg `  G
)  <_  (deg `  ( R  oF  -  T
) ) ) ) )
110109adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  ( (deg `  ( R  oF  -  T ) )  =  (deg `  G
)  <->  ( (deg `  ( R  oF  -  T ) )  <_ 
(deg `  G )  /\  (deg `  G )  <_  (deg `  ( R  oF  -  T
) ) ) ) )
11153, 108, 110mpbir2and 913 . . . . . . . . 9  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  (deg `  ( R  oF  -  T
) )  =  (deg
`  G ) )
112111fveq2d 5802 . . . . . . . 8  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  ( (coeff `  ( R  oF  -  T ) ) `
 (deg `  ( R  oF  -  T
) ) )  =  ( (coeff `  ( R  oF  -  T
) ) `  (deg `  G ) ) )
11343, 37coesub 21856 . . . . . . . . . . . . 13  |-  ( ( R  e.  (Poly `  S )  /\  T  e.  (Poly `  S )
)  ->  (coeff `  ( R  oF  -  T
) )  =  ( (coeff `  R )  oF  -  (coeff `  T ) ) )
11412, 16, 113syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  (coeff `  ( R  oF  -  T
) )  =  ( (coeff `  R )  oF  -  (coeff `  T ) ) )
115114fveq1d 5800 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  ( R  oF  -  T
) ) `  (deg `  G ) )  =  ( ( (coeff `  R )  oF  -  (coeff `  T
) ) `  (deg `  G ) ) )
11643coef3 21832 . . . . . . . . . . . . . 14  |-  ( R  e.  (Poly `  S
)  ->  (coeff `  R
) : NN0 --> CC )
117 ffn 5666 . . . . . . . . . . . . . 14  |-  ( (coeff `  R ) : NN0 --> CC 
->  (coeff `  R )  Fn  NN0 )
11812, 116, 1173syl 20 . . . . . . . . . . . . 13  |-  ( ph  ->  (coeff `  R )  Fn  NN0 )
11937coef3 21832 . . . . . . . . . . . . . 14  |-  ( T  e.  (Poly `  S
)  ->  (coeff `  T
) : NN0 --> CC )
120 ffn 5666 . . . . . . . . . . . . . 14  |-  ( (coeff `  T ) : NN0 --> CC 
->  (coeff `  T )  Fn  NN0 )
12116, 119, 1203syl 20 . . . . . . . . . . . . 13  |-  ( ph  ->  (coeff `  T )  Fn  NN0 )
122 nn0ex 10695 . . . . . . . . . . . . . 14  |-  NN0  e.  _V
123122a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  NN0  e.  _V )
124 inidm 3666 . . . . . . . . . . . . 13  |-  ( NN0 
i^i  NN0 )  =  NN0
12546simprd 463 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (coeff `  R
) `  (deg `  G
) )  =  0 )
126125adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  (deg `  G
)  e.  NN0 )  ->  ( (coeff `  R
) `  (deg `  G
) )  =  0 )
12740simprd 463 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (coeff `  T
) `  (deg `  G
) )  =  0 )
128127adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  (deg `  G
)  e.  NN0 )  ->  ( (coeff `  T
) `  (deg `  G
) )  =  0 )
129118, 121, 123, 123, 124, 126, 128ofval 6438 . . . . . . . . . . . 12  |-  ( (
ph  /\  (deg `  G
)  e.  NN0 )  ->  ( ( (coeff `  R )  oF  -  (coeff `  T
) ) `  (deg `  G ) )  =  ( 0  -  0 ) )
13030, 129mpdan 668 . . . . . . . . . . 11  |-  ( ph  ->  ( ( (coeff `  R )  oF  -  (coeff `  T
) ) `  (deg `  G ) )  =  ( 0  -  0 ) )
131115, 130eqtrd 2495 . . . . . . . . . 10  |-  ( ph  ->  ( (coeff `  ( R  oF  -  T
) ) `  (deg `  G ) )  =  ( 0  -  0 ) )
132 0m0e0 10541 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
133131, 132syl6eq 2511 . . . . . . . . 9  |-  ( ph  ->  ( (coeff `  ( R  oF  -  T
) ) `  (deg `  G ) )  =  0 )
134133adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  ( (coeff `  ( R  oF  -  T ) ) `
 (deg `  G
) )  =  0 )
135112, 134eqtrd 2495 . . . . . . 7  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  ( (coeff `  ( R  oF  -  T ) ) `
 (deg `  ( R  oF  -  T
) ) )  =  0 )
136 eqid 2454 . . . . . . . . . 10  |-  (deg `  ( R  oF  -  T ) )  =  (deg `  ( R  oF  -  T
) )
137 eqid 2454 . . . . . . . . . 10  |-  (coeff `  ( R  oF  -  T ) )  =  (coeff `  ( R  oF  -  T
) )
138136, 137dgreq0 21864 . . . . . . . . 9  |-  ( ( R  oF  -  T )  e.  (Poly `  S )  ->  (
( R  oF  -  T )  =  0p  <->  ( (coeff `  ( R  oF  -  T ) ) `
 (deg `  ( R  oF  -  T
) ) )  =  0 ) )
13917, 138syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( R  oF  -  T )  =  0p  <->  ( (coeff `  ( R  oF  -  T ) ) `
 (deg `  ( R  oF  -  T
) ) )  =  0 ) )
140139biimpar 485 . . . . . . 7  |-  ( (
ph  /\  ( (coeff `  ( R  oF  -  T ) ) `
 (deg `  ( R  oF  -  T
) ) )  =  0 )  ->  ( R  oF  -  T
)  =  0p )
141135, 140syldan 470 . . . . . 6  |-  ( (
ph  /\  ( p  oF  -  q
)  =/=  0p )  ->  ( R  oF  -  T
)  =  0p )
142141ex 434 . . . . 5  |-  ( ph  ->  ( ( p  oF  -  q )  =/=  0p  -> 
( R  oF  -  T )  =  0p ) )
143 plymul0or 21879 . . . . . . 7  |-  ( ( G  e.  (Poly `  S )  /\  (
p  oF  -  q )  e.  (Poly `  S ) )  -> 
( ( G  oF  x.  ( p  oF  -  q
) )  =  0p  <->  ( G  =  0p  \/  (
p  oF  -  q )  =  0p ) ) )
1448, 54, 143syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( G  oF  x.  ( p  oF  -  q
) )  =  0p  <->  ( G  =  0p  \/  (
p  oF  -  q )  =  0p ) ) )
14596eqeq1d 2456 . . . . . 6  |-  ( ph  ->  ( ( R  oF  -  T )  =  0p  <->  ( G  oF  x.  (
p  oF  -  q ) )  =  0p ) )
1469neneqd 2654 . . . . . . 7  |-  ( ph  ->  -.  G  =  0p )
147 biorf 405 . . . . . . 7  |-  ( -.  G  =  0p  ->  ( ( p  oF  -  q
)  =  0p  <-> 
( G  =  0p  \/  ( p  oF  -  q
)  =  0p ) ) )
148146, 147syl 16 . . . . . 6  |-  ( ph  ->  ( ( p  oF  -  q )  =  0p  <->  ( G  =  0p  \/  ( p  oF  -  q )  =  0p ) ) )
149144, 145, 1483bitr4d 285 . . . . 5  |-  ( ph  ->  ( ( R  oF  -  T )  =  0p  <->  ( p  oF  -  q
)  =  0p ) )
150142, 149sylibd 214 . . . 4  |-  ( ph  ->  ( ( p  oF  -  q )  =/=  0p  -> 
( p  oF  -  q )  =  0p ) )
1511, 150pm2.61dne 2768 . . 3  |-  ( ph  ->  ( p  oF  -  q )  =  0p )
152 df-0p 21280 . . 3  |-  0p  =  ( CC  X.  { 0 } )
153151, 152syl6eq 2511 . 2  |-  ( ph  ->  ( p  oF  -  q )  =  ( CC  X.  {
0 } ) )
154 ofsubeq0 10429 . . 3  |-  ( ( CC  e.  _V  /\  p : CC --> CC  /\  q : CC --> CC )  ->  ( ( p  oF  -  q
)  =  ( CC 
X.  { 0 } )  <->  p  =  q
) )
15574, 90, 92, 154syl3anc 1219 . 2  |-  ( ph  ->  ( ( p  oF  -  q )  =  ( CC  X.  { 0 } )  <-> 
p  =  q ) )
156153, 155mpbid 210 1  |-  ( ph  ->  p  =  q )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   _Vcvv 3076   ifcif 3898   {csn 3984   class class class wbr 4399    |-> cmpt 4457    X. cxp 4945    Fn wfn 5520   -->wf 5521   ` cfv 5525  (class class class)co 6199    oFcof 6427   CCcc 9390   RRcr 9391   0cc0 9392   1c1 9393    + caddc 9395    x. cmul 9397    < clt 9528    <_ cle 9529    - cmin 9705   -ucneg 9706    / cdiv 10103   NN0cn0 10689   0pc0p 21279  Polycply 21784  coeffccoe 21786  degcdgr 21787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-sup 7801  df-oi 7834  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-n0 10690  df-z 10757  df-uz 10972  df-rp 11102  df-fz 11554  df-fzo 11665  df-fl 11758  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-clim 13083  df-rlim 13084  df-sum 13281  df-0p 21280  df-ply 21788  df-coe 21790  df-dgr 21791
This theorem is referenced by:  plydivalg  21897
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