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Theorem plydiveu 19678
Description: Lemma for plydivalg 19679. (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 21 . . . 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 19674 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  q  e.  (Poly `  S ) )  ->  R  e.  (Poly `  S ) )
122, 11mpdan 649 . . . . . . . . . . . . . . 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 19674 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  T  e.  (Poly `  S ) )
1613, 15mpdan 649 . . . . . . . . . . . . . . 15  |-  ( ph  ->  T  e.  (Poly `  S ) )
1712, 16, 3, 4, 6plysub 19601 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( R  o F  -  T )  e.  (Poly `  S )
)
18 dgrcl 19615 . . . . . . . . . . . . . 14  |-  ( ( R  o F  -  T )  e.  (Poly `  S )  ->  (deg `  ( R  o F  -  T ) )  e.  NN0 )
1917, 18syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  ( R  o F  -  T
) )  e.  NN0 )
2019nn0red 10019 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  ( R  o F  -  T
) )  e.  RR )
21 dgrcl 19615 . . . . . . . . . . . . . . 15  |-  ( T  e.  (Poly `  S
)  ->  (deg `  T
)  e.  NN0 )
2216, 21syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  (deg `  T )  e.  NN0 )
2322nn0red 10019 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  T )  e.  RR )
24 dgrcl 19615 . . . . . . . . . . . . . . 15  |-  ( R  e.  (Poly `  S
)  ->  (deg `  R
)  e.  NN0 )
2512, 24syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  (deg `  R )  e.  NN0 )
2625nn0red 10019 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  R )  e.  RR )
27 ifcl 3601 . . . . . . . . . . . . 13  |-  ( ( (deg `  T )  e.  RR  /\  (deg `  R )  e.  RR )  ->  if ( (deg
`  R )  <_ 
(deg `  T ) ,  (deg `  T ) ,  (deg `  R )
)  e.  RR )
2823, 26, 27syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  if ( (deg `  R )  <_  (deg `  T ) ,  (deg
`  T ) ,  (deg `  R )
)  e.  RR )
29 dgrcl 19615 . . . . . . . . . . . . . 14  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
308, 29syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  G )  e.  NN0 )
3130nn0red 10019 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  G )  e.  RR )
32 eqid 2283 . . . . . . . . . . . . . 14  |-  (deg `  R )  =  (deg
`  R )
33 eqid 2283 . . . . . . . . . . . . . 14  |-  (deg `  T )  =  (deg
`  T )
3432, 33dgrsub 19653 . . . . . . . . . . . . 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 642 . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . . . 17  |-  (coeff `  T )  =  (coeff `  T )
3833, 37dgrlt 19647 . . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( T  =  0 p  \/  (deg `  T )  <  (deg `  G ) )  <->  ( (deg `  T )  <_  (deg `  G )  /\  (
(coeff `  T ) `  (deg `  G )
)  =  0 ) ) )
4036, 39mpbid 201 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (deg `  T
)  <_  (deg `  G
)  /\  ( (coeff `  T ) `  (deg `  G ) )  =  0 ) )
4140simpld 445 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  T )  <_  (deg `  G )
)
42 plydiveu.qd . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) )
43 eqid 2283 . . . . . . . . . . . . . . . . 17  |-  (coeff `  R )  =  (coeff `  R )
4432, 43dgrlt 19647 . . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  <->  ( (deg `  R )  <_  (deg `  G )  /\  (
(coeff `  R ) `  (deg `  G )
)  =  0 ) ) )
4642, 45mpbid 201 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (deg `  R
)  <_  (deg `  G
)  /\  ( (coeff `  R ) `  (deg `  G ) )  =  0 ) )
4746simpld 445 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  R )  <_  (deg `  G )
)
48 breq1 4026 . . . . . . . . . . . . . 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 4026 . . . . . . . . . . . . . 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 3596 . . . . . . . . . . . . 13  |-  ( ( (deg `  T )  <_  (deg `  G )  /\  (deg `  R )  <_  (deg `  G )
)  ->  if (
(deg `  R )  <_  (deg `  T ) ,  (deg `  T ) ,  (deg `  R )
)  <_  (deg `  G
) )
5141, 47, 50syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  if ( (deg `  R )  <_  (deg `  T ) ,  (deg
`  T ) ,  (deg `  R )
)  <_  (deg `  G
) )
5220, 28, 31, 35, 51letrd 8973 . . . . . . . . . . 11  |-  ( ph  ->  (deg `  ( R  o F  -  T
) )  <_  (deg `  G ) )
5352adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  ( R  o F  -  T
) )  <_  (deg `  G ) )
5413, 2, 3, 4, 6plysub 19601 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( p  o F  -  q )  e.  (Poly `  S )
)
55 dgrcl 19615 . . . . . . . . . . . . . 14  |-  ( ( p  o F  -  q )  e.  (Poly `  S )  ->  (deg `  ( p  o F  -  q ) )  e.  NN0 )
5654, 55syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  ( p  o F  -  q
) )  e.  NN0 )
57 nn0addge1 10010 . . . . . . . . . . . . 13  |-  ( ( (deg `  G )  e.  RR  /\  (deg `  ( p  o F  -  q ) )  e.  NN0 )  -> 
(deg `  G )  <_  ( (deg `  G
)  +  (deg `  ( p  o F  -  q ) ) ) )
5831, 56, 57syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  G )  <_  ( (deg `  G
)  +  (deg `  ( p  o F  -  q ) ) ) )
5958adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  G
)  <_  ( (deg `  G )  +  (deg
`  ( p  o F  -  q ) ) ) )
60 plyf 19580 . . . . . . . . . . . . . . . . . . . 20  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
617, 60syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F : CC --> CC )
62 ffvelrn 5663 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F : CC --> CC  /\  z  e.  CC )  ->  ( F `  z
)  e.  CC )
6361, 62sylan 457 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e.  CC )
648, 2, 3, 4plymul 19600 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  o F  x.  q )  e.  (Poly `  S )
)
65 plyf 19580 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( G  o F  x.  q )  e.  (Poly `  S )  ->  ( G  o F  x.  q
) : CC --> CC )
6664, 65syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  o F  x.  q ) : CC --> CC )
67 ffvelrn 5663 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( G  o F  x.  q ) : CC --> CC  /\  z  e.  CC )  ->  (
( G  o F  x.  q ) `  z )  e.  CC )
6866, 67sylan 457 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  o F  x.  q ) `  z
)  e.  CC )
698, 13, 3, 4plymul 19600 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  o F  x.  p )  e.  (Poly `  S )
)
70 plyf 19580 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( G  o F  x.  p )  e.  (Poly `  S )  ->  ( G  o F  x.  p
) : CC --> CC )
7169, 70syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  o F  x.  p ) : CC --> CC )
72 ffvelrn 5663 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( G  o F  x.  p ) : CC --> CC  /\  z  e.  CC )  ->  (
( G  o F  x.  p ) `  z )  e.  CC )
7371, 72sylan 457 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  o F  x.  p ) `  z
)  e.  CC )
7463, 68, 73nnncan1d 9191 . . . . . . . . . . . . . . . . 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 )
) )
7574mpteq2dva 4106 . . . . . . . . . . . . . . . 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 ) ) ) )
76 cnex 8818 . . . . . . . . . . . . . . . . . 18  |-  CC  e.  _V
7776a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  CC  e.  _V )
7863, 68subcld 9157 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  ( ( G  o F  x.  q
) `  z )
)  e.  CC )
7963, 73subcld 9157 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  ( ( G  o F  x.  p
) `  z )
)  e.  CC )
8061feqmptd 5575 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F  =  ( z  e.  CC  |->  ( F `
 z ) ) )
8166feqmptd 5575 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  o F  x.  q )  =  ( z  e.  CC  |->  ( ( G  o F  x.  q ) `  z ) ) )
8277, 63, 68, 80, 81offval2 6095 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( F  o F  -  ( G  o F  x.  q )
)  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  o F  x.  q
) `  z )
) ) )
8310, 82syl5eq 2327 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  o F  x.  q
) `  z )
) ) )
8471feqmptd 5575 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  o F  x.  p )  =  ( z  e.  CC  |->  ( ( G  o F  x.  p ) `  z ) ) )
8577, 63, 73, 80, 84offval2 6095 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( F  o F  -  ( G  o F  x.  p )
)  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  o F  x.  p
) `  z )
) ) )
8614, 85syl5eq 2327 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  T  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  o F  x.  p
) `  z )
) ) )
8777, 78, 79, 83, 86offval2 6095 . . . . . . . . . . . . . . . 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 ) ) ) ) )
8877, 73, 68, 84, 81offval2 6095 . . . . . . . . . . . . . . . 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 ) ) ) )
8975, 87, 883eqtr4d 2325 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( R  o F  -  T )  =  ( ( G  o F  x.  p )  o F  -  ( G  o F  x.  q
) ) )
90 plyf 19580 . . . . . . . . . . . . . . . . 17  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
918, 90syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G : CC --> CC )
92 plyf 19580 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  (Poly `  S
)  ->  p : CC
--> CC )
9313, 92syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  p : CC --> CC )
94 plyf 19580 . . . . . . . . . . . . . . . . 17  |-  ( q  e.  (Poly `  S
)  ->  q : CC
--> CC )
952, 94syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  q : CC --> CC )
96 subdi 9213 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z
) ) )
9796adantl 452 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( x  x.  (
y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z ) ) )
9877, 91, 93, 95, 97caofdi 6113 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G  o F  x.  ( p  o F  -  q ) )  =  ( ( G  o F  x.  p )  o F  -  ( G  o F  x.  q )
) )
9989, 98eqtr4d 2318 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( R  o F  -  T )  =  ( G  o F  x.  ( p  o F  -  q ) ) )
10099fveq2d 5529 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  ( R  o F  -  T
) )  =  (deg
`  ( G  o F  x.  ( p  o F  -  q
) ) ) )
101100adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  ( R  o F  -  T
) )  =  (deg
`  ( G  o F  x.  ( p  o F  -  q
) ) ) )
1028adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  G  e.  (Poly `  S ) )
1039adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  G  =/=  0 p )
10454adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  ( p  o F  -  q
)  e.  (Poly `  S ) )
105 simpr 447 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  ( p  o F  -  q
)  =/=  0 p )
106 eqid 2283 . . . . . . . . . . . . . 14  |-  (deg `  G )  =  (deg
`  G )
107 eqid 2283 . . . . . . . . . . . . . 14  |-  (deg `  ( p  o F  -  q ) )  =  (deg `  (
p  o F  -  q ) )
108106, 107dgrmul 19651 . . . . . . . . . . . . 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
) ) ) )
109102, 103, 104, 105, 108syl22anc 1183 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  ( G  o F  x.  (
p  o F  -  q ) ) )  =  ( (deg `  G )  +  (deg
`  ( p  o F  -  q ) ) ) )
110101, 109eqtrd 2315 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  ( R  o F  -  T
) )  =  ( (deg `  G )  +  (deg `  ( p  o F  -  q
) ) ) )
11159, 110breqtrrd 4049 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  G
)  <_  (deg `  ( R  o F  -  T
) ) )
11220, 31letri3d 8961 . . . . . . . . . . 11  |-  ( ph  ->  ( (deg `  ( R  o F  -  T
) )  =  (deg
`  G )  <->  ( (deg `  ( R  o F  -  T ) )  <_  (deg `  G
)  /\  (deg `  G
)  <_  (deg `  ( R  o F  -  T
) ) ) ) )
113112adantr 451 . . . . . . . . . 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
) ) ) ) )
11453, 111, 113mpbir2and 888 . . . . . . . . 9  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  ( R  o F  -  T
) )  =  (deg
`  G ) )
115114fveq2d 5529 . . . . . . . 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 ) ) )
11643, 37coesub 19638 . . . . . . . . . . . . 13  |-  ( ( R  e.  (Poly `  S )  /\  T  e.  (Poly `  S )
)  ->  (coeff `  ( R  o F  -  T
) )  =  ( (coeff `  R )  o F  -  (coeff `  T ) ) )
11712, 16, 116syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  (coeff `  ( R  o F  -  T
) )  =  ( (coeff `  R )  o F  -  (coeff `  T ) ) )
118117fveq1d 5527 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  ( R  o F  -  T
) ) `  (deg `  G ) )  =  ( ( (coeff `  R )  o F  -  (coeff `  T
) ) `  (deg `  G ) ) )
11943coef3 19614 . . . . . . . . . . . . . 14  |-  ( R  e.  (Poly `  S
)  ->  (coeff `  R
) : NN0 --> CC )
120 ffn 5389 . . . . . . . . . . . . . 14  |-  ( (coeff `  R ) : NN0 --> CC 
->  (coeff `  R )  Fn  NN0 )
12112, 119, 1203syl 18 . . . . . . . . . . . . 13  |-  ( ph  ->  (coeff `  R )  Fn  NN0 )
12237coef3 19614 . . . . . . . . . . . . . 14  |-  ( T  e.  (Poly `  S
)  ->  (coeff `  T
) : NN0 --> CC )
123 ffn 5389 . . . . . . . . . . . . . 14  |-  ( (coeff `  T ) : NN0 --> CC 
->  (coeff `  T )  Fn  NN0 )
12416, 122, 1233syl 18 . . . . . . . . . . . . 13  |-  ( ph  ->  (coeff `  T )  Fn  NN0 )
125 nn0ex 9971 . . . . . . . . . . . . . 14  |-  NN0  e.  _V
126125a1i 10 . . . . . . . . . . . . 13  |-  ( ph  ->  NN0  e.  _V )
127 inidm 3378 . . . . . . . . . . . . 13  |-  ( NN0 
i^i  NN0 )  =  NN0
12846simprd 449 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (coeff `  R
) `  (deg `  G
) )  =  0 )
129128adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  (deg `  G
)  e.  NN0 )  ->  ( (coeff `  R
) `  (deg `  G
) )  =  0 )
13040simprd 449 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (coeff `  T
) `  (deg `  G
) )  =  0 )
131130adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  (deg `  G
)  e.  NN0 )  ->  ( (coeff `  T
) `  (deg `  G
) )  =  0 )
132121, 124, 126, 126, 127, 129, 131ofval 6087 . . . . . . . . . . . 12  |-  ( (
ph  /\  (deg `  G
)  e.  NN0 )  ->  ( ( (coeff `  R )  o F  -  (coeff `  T
) ) `  (deg `  G ) )  =  ( 0  -  0 ) )
13330, 132mpdan 649 . . . . . . . . . . 11  |-  ( ph  ->  ( ( (coeff `  R )  o F  -  (coeff `  T
) ) `  (deg `  G ) )  =  ( 0  -  0 ) )
134118, 133eqtrd 2315 . . . . . . . . . 10  |-  ( ph  ->  ( (coeff `  ( R  o F  -  T
) ) `  (deg `  G ) )  =  ( 0  -  0 ) )
135 0cn 8831 . . . . . . . . . . 11  |-  0  e.  CC
136135subidi 9117 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
137134, 136syl6eq 2331 . . . . . . . . 9  |-  ( ph  ->  ( (coeff `  ( R  o F  -  T
) ) `  (deg `  G ) )  =  0 )
138137adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  ( (coeff `  ( R  o F  -  T ) ) `
 (deg `  G
) )  =  0 )
139115, 138eqtrd 2315 . . . . . . 7  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  ( (coeff `  ( R  o F  -  T ) ) `
 (deg `  ( R  o F  -  T
) ) )  =  0 )
140 eqid 2283 . . . . . . . . . 10  |-  (deg `  ( R  o F  -  T ) )  =  (deg `  ( R  o F  -  T
) )
141 eqid 2283 . . . . . . . . . 10  |-  (coeff `  ( R  o F  -  T ) )  =  (coeff `  ( R  o F  -  T
) )
142140, 141dgreq0 19646 . . . . . . . . 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 ) )
14317, 142syl 15 . . . . . . . 8  |-  ( ph  ->  ( ( R  o F  -  T )  =  0 p  <->  ( (coeff `  ( R  o F  -  T ) ) `
 (deg `  ( R  o F  -  T
) ) )  =  0 ) )
144143biimpar 471 . . . . . . 7  |-  ( (
ph  /\  ( (coeff `  ( R  o F  -  T ) ) `
 (deg `  ( R  o F  -  T
) ) )  =  0 )  ->  ( R  o F  -  T
)  =  0 p )
145139, 144syldan 456 . . . . . 6  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  ( R  o F  -  T
)  =  0 p )
146145ex 423 . . . . 5  |-  ( ph  ->  ( ( p  o F  -  q )  =/=  0 p  -> 
( R  o F  -  T )  =  0 p ) )
147 plymul0or 19661 . . . . . . 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 ) ) )
1488, 54, 147syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( G  o F  x.  ( p  o F  -  q
) )  =  0 p  <->  ( G  =  0 p  \/  (
p  o F  -  q )  =  0 p ) ) )
14999eqeq1d 2291 . . . . . 6  |-  ( ph  ->  ( ( R  o F  -  T )  =  0 p  <->  ( G  o F  x.  (
p  o F  -  q ) )  =  0 p ) )
1509neneqd 2462 . . . . . . 7  |-  ( ph  ->  -.  G  =  0 p )
151 biorf 394 . . . . . . 7  |-  ( -.  G  =  0 p  ->  ( ( p  o F  -  q
)  =  0 p  <-> 
( G  =  0 p  \/  ( p  o F  -  q
)  =  0 p ) ) )
152150, 151syl 15 . . . . . 6  |-  ( ph  ->  ( ( p  o F  -  q )  =  0 p  <->  ( G  =  0 p  \/  ( p  o F  -  q )  =  0 p ) ) )
153148, 149, 1523bitr4d 276 . . . . 5  |-  ( ph  ->  ( ( R  o F  -  T )  =  0 p  <->  ( p  o F  -  q
)  =  0 p ) )
154146, 153sylibd 205 . . . 4  |-  ( ph  ->  ( ( p  o F  -  q )  =/=  0 p  -> 
( p  o F  -  q )  =  0 p ) )
1551, 154pm2.61dne 2523 . . 3  |-  ( ph  ->  ( p  o F  -  q )  =  0 p )
156 df-0p 19025 . . 3  |-  0 p  =  ( CC  X.  { 0 } )
157155, 156syl6eq 2331 . 2  |-  ( ph  ->  ( p  o F  -  q )  =  ( CC  X.  {
0 } ) )
158 ofsubeq0 9743 . . 3  |-  ( ( CC  e.  _V  /\  p : CC --> CC  /\  q : CC --> CC )  ->  ( ( p  o F  -  q
)  =  ( CC 
X.  { 0 } )  <->  p  =  q
) )
15977, 93, 95, 158syl3anc 1182 . 2  |-  ( ph  ->  ( ( p  o F  -  q )  =  ( CC  X.  { 0 } )  <-> 
p  =  q ) )
160157, 159mpbid 201 1  |-  ( ph  ->  p  =  q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   ifcif 3565   {csn 3640   class class class wbr 4023    e. cmpt 4077    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038    / cdiv 9423   NN0cn0 9965   0 pc0p 19024  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  plydivalg  19679
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573
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