MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  plydiveu Unicode version

Theorem plydiveu 19694
Description: Lemma for plydivalg 19695. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
plydiv.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
plydiv.tm  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
plydiv.rc  |-  ( (
ph  /\  ( x  e.  S  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  S )
plydiv.m1  |-  ( ph  -> 
-u 1  e.  S
)
plydiv.f  |-  ( ph  ->  F  e.  (Poly `  S ) )
plydiv.g  |-  ( ph  ->  G  e.  (Poly `  S ) )
plydiv.z  |-  ( ph  ->  G  =/=  0 p )
plydiv.r  |-  R  =  ( F  o F  -  ( G  o F  x.  q )
)
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 19690 . . . . . . . . . . . . . . . 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 19690 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  (Poly `  S ) )  ->  T  e.  (Poly `  S ) )
1613, 15mpdan 649 . . . . . . . . . . . . . . 15  |-  ( ph  ->  T  e.  (Poly `  S ) )
1712, 16, 3, 4, 6plysub 19617 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( R  o F  -  T )  e.  (Poly `  S )
)
18 dgrcl 19631 . . . . . . . . . . . . . 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 10035 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  ( R  o F  -  T
) )  e.  RR )
21 dgrcl 19631 . . . . . . . . . . . . . . 15  |-  ( T  e.  (Poly `  S
)  ->  (deg `  T
)  e.  NN0 )
2216, 21syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  (deg `  T )  e.  NN0 )
2322nn0red 10035 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  T )  e.  RR )
24 dgrcl 19631 . . . . . . . . . . . . . . 15  |-  ( R  e.  (Poly `  S
)  ->  (deg `  R
)  e.  NN0 )
2512, 24syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  (deg `  R )  e.  NN0 )
2625nn0red 10035 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  R )  e.  RR )
27 ifcl 3614 . . . . . . . . . . . . 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 19631 . . . . . . . . . . . . . 14  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
308, 29syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  G )  e.  NN0 )
3130nn0red 10035 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  G )  e.  RR )
32 eqid 2296 . . . . . . . . . . . . . 14  |-  (deg `  R )  =  (deg
`  R )
33 eqid 2296 . . . . . . . . . . . . . 14  |-  (deg `  T )  =  (deg
`  T )
3432, 33dgrsub 19669 . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . . 17  |-  (coeff `  T )  =  (coeff `  T )
3833, 37dgrlt 19663 . . . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . . 17  |-  (coeff `  R )  =  (coeff `  R )
4432, 43dgrlt 19663 . . . . . . . . . . . . . . . 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 4042 . . . . . . . . . . . . . 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 4042 . . . . . . . . . . . . . 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 3609 . . . . . . . . . . . . 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 8989 . . . . . . . . . . 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 19617 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( p  o F  -  q )  e.  (Poly `  S )
)
55 dgrcl 19631 . . . . . . . . . . . . . 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 10026 . . . . . . . . . . . . 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 19596 . . . . . . . . . . . . . . . . . . . 20  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
617, 60syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F : CC --> CC )
62 ffvelrn 5679 . . . . . . . . . . . . . . . . . . 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 19616 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  o F  x.  q )  e.  (Poly `  S )
)
65 plyf 19596 . . . . . . . . . . . . . . . . . . . 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 5679 . . . . . . . . . . . . . . . . . . 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 19616 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G  o F  x.  p )  e.  (Poly `  S )
)
70 plyf 19596 . . . . . . . . . . . . . . . . . . . 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 5679 . . . . . . . . . . . . . . . . . . 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 9207 . . . . . . . . . . . . . . . . 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 4122 . . . . . . . . . . . . . . . 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 8834 . . . . . . . . . . . . . . . . . 18  |-  CC  e.  _V
7776a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  CC  e.  _V )
7863, 68subcld 9173 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  ( ( G  o F  x.  q
) `  z )
)  e.  CC )
7963, 73subcld 9173 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  ( ( G  o F  x.  p
) `  z )
)  e.  CC )
8061feqmptd 5591 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F  =  ( z  e.  CC  |->  ( F `
 z ) ) )
8166feqmptd 5591 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  o F  x.  q )  =  ( z  e.  CC  |->  ( ( G  o F  x.  q ) `  z ) ) )
8277, 63, 68, 80, 81offval2 6111 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( F  o F  -  ( G  o F  x.  q )
)  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  o F  x.  q
) `  z )
) ) )
8310, 82syl5eq 2340 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  o F  x.  q
) `  z )
) ) )
8471feqmptd 5591 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( G  o F  x.  p )  =  ( z  e.  CC  |->  ( ( G  o F  x.  p ) `  z ) ) )
8577, 63, 73, 80, 84offval2 6111 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( F  o F  -  ( G  o F  x.  p )
)  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  o F  x.  p
) `  z )
) ) )
8614, 85syl5eq 2340 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  T  =  ( z  e.  CC  |->  ( ( F `  z )  -  ( ( G  o F  x.  p
) `  z )
) ) )
8777, 78, 79, 83, 86offval2 6111 . . . . . . . . . . . . . . . 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 6111 . . . . . . . . . . . . . . . 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 2338 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( R  o F  -  T )  =  ( ( G  o F  x.  p )  o F  -  ( G  o F  x.  q
) ) )
90 plyf 19596 . . . . . . . . . . . . . . . . 17  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
918, 90syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G : CC --> CC )
92 plyf 19596 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  (Poly `  S
)  ->  p : CC
--> CC )
9313, 92syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  p : CC --> CC )
94 plyf 19596 . . . . . . . . . . . . . . . . 17  |-  ( q  e.  (Poly `  S
)  ->  q : CC
--> CC )
952, 94syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  q : CC --> CC )
96 subdi 9229 . . . . . . . . . . . . . . . . 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 6129 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G  o F  x.  ( p  o F  -  q ) )  =  ( ( G  o F  x.  p )  o F  -  ( G  o F  x.  q )
) )
9989, 98eqtr4d 2331 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( R  o F  -  T )  =  ( G  o F  x.  ( p  o F  -  q ) ) )
10099fveq2d 5545 . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . 14  |-  (deg `  G )  =  (deg
`  G )
107 eqid 2296 . . . . . . . . . . . . . 14  |-  (deg `  ( p  o F  -  q ) )  =  (deg `  (
p  o F  -  q ) )
108106, 107dgrmul 19667 . . . . . . . . . . . . 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 2328 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  ( R  o F  -  T
) )  =  ( (deg `  G )  +  (deg `  ( p  o F  -  q
) ) ) )
11159, 110breqtrrd 4065 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  (deg `  G
)  <_  (deg `  ( R  o F  -  T
) ) )
11220, 31letri3d 8977 . . . . . . . . . . 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 5545 . . . . . . . 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 19654 . . . . . . . . . . . . 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 5543 . . . . . . . . . . 11  |-  ( ph  ->  ( (coeff `  ( R  o F  -  T
) ) `  (deg `  G ) )  =  ( ( (coeff `  R )  o F  -  (coeff `  T
) ) `  (deg `  G ) ) )
11943coef3 19630 . . . . . . . . . . . . . 14  |-  ( R  e.  (Poly `  S
)  ->  (coeff `  R
) : NN0 --> CC )
120 ffn 5405 . . . . . . . . . . . . . 14  |-  ( (coeff `  R ) : NN0 --> CC 
->  (coeff `  R )  Fn  NN0 )
12112, 119, 1203syl 18 . . . . . . . . . . . . 13  |-  ( ph  ->  (coeff `  R )  Fn  NN0 )
12237coef3 19630 . . . . . . . . . . . . . 14  |-  ( T  e.  (Poly `  S
)  ->  (coeff `  T
) : NN0 --> CC )
123 ffn 5405 . . . . . . . . . . . . . 14  |-  ( (coeff `  T ) : NN0 --> CC 
->  (coeff `  T )  Fn  NN0 )
12416, 122, 1233syl 18 . . . . . . . . . . . . 13  |-  ( ph  ->  (coeff `  T )  Fn  NN0 )
125 nn0ex 9987 . . . . . . . . . . . . . 14  |-  NN0  e.  _V
126125a1i 10 . . . . . . . . . . . . 13  |-  ( ph  ->  NN0  e.  _V )
127 inidm 3391 . . . . . . . . . . . . 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 6103 . . . . . . . . . . . 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 2328 . . . . . . . . . 10  |-  ( ph  ->  ( (coeff `  ( R  o F  -  T
) ) `  (deg `  G ) )  =  ( 0  -  0 ) )
135 0cn 8847 . . . . . . . . . . 11  |-  0  e.  CC
136135subidi 9133 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
137134, 136syl6eq 2344 . . . . . . . . 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 2328 . . . . . . 7  |-  ( (
ph  /\  ( p  o F  -  q
)  =/=  0 p )  ->  ( (coeff `  ( R  o F  -  T ) ) `
 (deg `  ( R  o F  -  T
) ) )  =  0 )
140 eqid 2296 . . . . . . . . . 10  |-  (deg `  ( R  o F  -  T ) )  =  (deg `  ( R  o F  -  T
) )
141 eqid 2296 . . . . . . . . . 10  |-  (coeff `  ( R  o F  -  T ) )  =  (coeff `  ( R  o F  -  T
) )
142140, 141dgreq0 19662 . . . . . . . . 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 19677 . . . . . . 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 2304 . . . . . 6  |-  ( ph  ->  ( ( R  o F  -  T )  =  0 p  <->  ( G  o F  x.  (
p  o F  -  q ) )  =  0 p ) )
1509neneqd 2475 . . . . . . 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 2536 . . 3  |-  ( ph  ->  ( p  o F  -  q )  =  0 p )
156 df-0p 19041 . . 3  |-  0 p  =  ( CC  X.  { 0 } )
157155, 156syl6eq 2344 . 2  |-  ( ph  ->  ( p  o F  -  q )  =  ( CC  X.  {
0 } ) )
158 ofsubeq0 9759 . . 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 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   ifcif 3578   {csn 3653   class class class wbr 4039    e. cmpt 4093    X. cxp 4703    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   NN0cn0 9981   0 pc0p 19040  Polycply 19582  coeffccoe 19584  degcdgr 19585
This theorem is referenced by:  plydivalg  19695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588  df-dgr 19589
  Copyright terms: Public domain W3C validator