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Theorem mpaaeu 27017
Description: An algebraic number has exactly one monic polynomial of the least degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
mpaaeu  |-  ( A  e.  AA  ->  E! p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 ) )
Distinct variable group:    A, p

Proof of Theorem mpaaeu
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dgraalem 27012 . . . 4  |-  ( A  e.  AA  ->  (
(degAA `
 A )  e.  NN  /\  E. a  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) ) )
21simprd 450 . . 3  |-  ( A  e.  AA  ->  E. a  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )
3 qsscn 10510 . . . . . . . 8  |-  QQ  C_  CC
4 eldifi 3405 . . . . . . . . . . . 12  |-  ( a  e.  ( (Poly `  QQ )  \  { 0 p } )  -> 
a  e.  (Poly `  QQ ) )
54ad2antlr 708 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  a  e.  (Poly `  QQ ) )
6 zssq 10506 . . . . . . . . . . . 12  |-  ZZ  C_  QQ
7 0z 10218 . . . . . . . . . . . 12  |-  0  e.  ZZ
86, 7sselii 3281 . . . . . . . . . . 11  |-  0  e.  QQ
9 eqid 2380 . . . . . . . . . . . 12  |-  (coeff `  a )  =  (coeff `  a )
109coef2 20010 . . . . . . . . . . 11  |-  ( ( a  e.  (Poly `  QQ )  /\  0  e.  QQ )  ->  (coeff `  a ) : NN0 --> QQ )
115, 8, 10sylancl 644 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (coeff `  a
) : NN0 --> QQ )
12 dgrcl 20012 . . . . . . . . . . 11  |-  ( a  e.  (Poly `  QQ )  ->  (deg `  a
)  e.  NN0 )
135, 12syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (deg `  a
)  e.  NN0 )
1411, 13ffvelrnd 5803 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  a ) `  (deg `  a ) )  e.  QQ )
15 eldifsni 3864 . . . . . . . . . . 11  |-  ( a  e.  ( (Poly `  QQ )  \  { 0 p } )  -> 
a  =/=  0 p )
1615ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  a  =/=  0 p )
17 eqid 2380 . . . . . . . . . . . . 13  |-  (deg `  a )  =  (deg
`  a )
1817, 9dgreq0 20043 . . . . . . . . . . . 12  |-  ( a  e.  (Poly `  QQ )  ->  ( a  =  0 p  <->  ( (coeff `  a ) `  (deg `  a ) )  =  0 ) )
1918necon3bid 2578 . . . . . . . . . . 11  |-  ( a  e.  (Poly `  QQ )  ->  ( a  =/=  0 p  <->  ( (coeff `  a ) `  (deg `  a ) )  =/=  0 ) )
205, 19syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( a  =/=  0 p  <->  ( (coeff `  a ) `  (deg `  a ) )  =/=  0 ) )
2116, 20mpbid 202 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  a ) `  (deg `  a ) )  =/=  0 )
22 qreccl 10519 . . . . . . . . 9  |-  ( ( ( (coeff `  a
) `  (deg `  a
) )  e.  QQ  /\  ( (coeff `  a
) `  (deg `  a
) )  =/=  0
)  ->  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  QQ )
2314, 21, 22syl2anc 643 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  QQ )
24 plyconst 19985 . . . . . . . 8  |-  ( ( QQ  C_  CC  /\  (
1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  QQ )  -> 
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  e.  (Poly `  QQ ) )
253, 23, 24sylancr 645 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  e.  (Poly `  QQ ) )
26 simpl 444 . . . . . . . 8  |-  ( ( ( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  -> 
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  e.  (Poly `  QQ ) )
27 simpr 448 . . . . . . . 8  |-  ( ( ( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  -> 
a  e.  (Poly `  QQ ) )
28 qaddcl 10515 . . . . . . . . 9  |-  ( ( b  e.  QQ  /\  c  e.  QQ )  ->  ( b  +  c )  e.  QQ )
2928adantl 453 . . . . . . . 8  |-  ( ( ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( b  e.  QQ  /\  c  e.  QQ ) )  -> 
( b  +  c )  e.  QQ )
30 qmulcl 10517 . . . . . . . . 9  |-  ( ( b  e.  QQ  /\  c  e.  QQ )  ->  ( b  x.  c
)  e.  QQ )
3130adantl 453 . . . . . . . 8  |-  ( ( ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( b  e.  QQ  /\  c  e.  QQ ) )  -> 
( b  x.  c
)  e.  QQ )
3226, 27, 29, 31plymul 19997 . . . . . . 7  |-  ( ( ( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  -> 
( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  o F  x.  a )  e.  (Poly `  QQ ) )
3325, 5, 32syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a )  e.  (Poly `  QQ )
)
349coef3 20011 . . . . . . . . . . 11  |-  ( a  e.  (Poly `  QQ )  ->  (coeff `  a
) : NN0 --> CC )
355, 34syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (coeff `  a
) : NN0 --> CC )
3635, 13ffvelrnd 5803 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  a ) `  (deg `  a ) )  e.  CC )
3736, 21reccld 9708 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  CC )
3836, 21recne0d 9709 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  =/=  0 )
39 dgrmulc 20049 . . . . . . . 8  |-  ( ( ( 1  /  (
(coeff `  a ) `  (deg `  a )
) )  e.  CC  /\  ( 1  /  (
(coeff `  a ) `  (deg `  a )
) )  =/=  0  /\  a  e.  (Poly `  QQ ) )  -> 
(deg `  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a ) )  =  (deg `  a
) )
4037, 38, 5, 39syl3anc 1184 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (deg `  (
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  o F  x.  a
) )  =  (deg
`  a ) )
41 simprl 733 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (deg `  a
)  =  (degAA `  A
) )
4240, 41eqtrd 2412 . . . . . 6  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (deg `  (
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  o F  x.  a
) )  =  (degAA `  A ) )
43 aacn 20094 . . . . . . . . 9  |-  ( A  e.  AA  ->  A  e.  CC )
4443ad2antrr 707 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  A  e.  CC )
45 ovex 6038 . . . . . . . . . 10  |-  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  _V
46 fnconstg 5564 . . . . . . . . . 10  |-  ( ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  _V  ->  ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  Fn  CC )
4745, 46mp1i 12 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  Fn  CC )
48 plyf 19977 . . . . . . . . . 10  |-  ( a  e.  (Poly `  QQ )  ->  a : CC --> CC )
49 ffn 5524 . . . . . . . . . 10  |-  ( a : CC --> CC  ->  a  Fn  CC )
505, 48, 493syl 19 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  a  Fn  CC )
51 cnex 8997 . . . . . . . . . 10  |-  CC  e.  _V
5251a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  CC  e.  _V )
53 inidm 3486 . . . . . . . . 9  |-  ( CC 
i^i  CC )  =  CC
5445fvconst2 5879 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } ) `
 A )  =  ( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) )
5554adantl 453 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0 p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  A  e.  CC )  ->  (
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } ) `
 A )  =  ( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) )
56 simplrr 738 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0 p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  A  e.  CC )  ->  (
a `  A )  =  0 )
5747, 50, 52, 52, 53, 55, 56ofval 6246 . . . . . . . 8  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0 p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  A  e.  CC )  ->  (
( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  o F  x.  a ) `  A
)  =  ( ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  x.  0 ) )
5844, 57mpdan 650 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  o F  x.  a
) `  A )  =  ( ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  x.  0 ) )
5937mul01d 9190 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (
1  /  ( (coeff `  a ) `  (deg `  a ) ) )  x.  0 )  =  0 )
6058, 59eqtrd 2412 . . . . . 6  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  o F  x.  a
) `  A )  =  0 )
61 coemulc 20033 . . . . . . . . 9  |-  ( ( ( 1  /  (
(coeff `  a ) `  (deg `  a )
) )  e.  CC  /\  a  e.  (Poly `  QQ ) )  ->  (coeff `  ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  o F  x.  a ) )  =  ( ( NN0  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  o F  x.  (coeff `  a ) ) )
6237, 5, 61syl2anc 643 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (coeff `  (
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  o F  x.  a
) )  =  ( ( NN0  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  o F  x.  (coeff `  a ) ) )
6362fveq1d 5663 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  o F  x.  a ) ) `  (degAA `  A ) )  =  ( ( ( NN0 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  (coeff `  a
) ) `  (degAA `  A ) ) )
64 dgraacl 27013 . . . . . . . . . 10  |-  ( A  e.  AA  ->  (degAA `  A )  e.  NN )
6564ad2antrr 707 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (degAA `  A
)  e.  NN )
6665nnnn0d 10199 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (degAA `  A
)  e.  NN0 )
67 fnconstg 5564 . . . . . . . . . 10  |-  ( ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  _V  ->  ( NN0  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  Fn  NN0 )
6845, 67mp1i 12 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( NN0  X. 
{ ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  Fn  NN0 )
69 ffn 5524 . . . . . . . . . 10  |-  ( (coeff `  a ) : NN0 --> CC 
->  (coeff `  a )  Fn  NN0 )
7035, 69syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (coeff `  a
)  Fn  NN0 )
71 nn0ex 10152 . . . . . . . . . 10  |-  NN0  e.  _V
7271a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  NN0  e.  _V )
73 inidm 3486 . . . . . . . . 9  |-  ( NN0 
i^i  NN0 )  =  NN0
7445fvconst2 5879 . . . . . . . . . 10  |-  ( (degAA `  A )  e.  NN0  ->  ( ( NN0  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } ) `  (degAA `  A
) )  =  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) )
7574adantl 453 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0 p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  (degAA `  A
)  e.  NN0 )  ->  ( ( NN0  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } ) `  (degAA `  A
) )  =  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) )
76 simplrl 737 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0 p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  (degAA `  A
)  e.  NN0 )  ->  (deg `  a )  =  (degAA `  A ) )
7776eqcomd 2385 . . . . . . . . . 10  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0 p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  (degAA `  A
)  e.  NN0 )  ->  (degAA `  A )  =  (deg `  a )
)
7877fveq2d 5665 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0 p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  (degAA `  A
)  e.  NN0 )  ->  ( (coeff `  a
) `  (degAA `  A
) )  =  ( (coeff `  a ) `  (deg `  a )
) )
7968, 70, 72, 72, 73, 75, 78ofval 6246 . . . . . . . 8  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0 p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  (degAA `  A
)  e.  NN0 )  ->  ( ( ( NN0 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  (coeff `  a
) ) `  (degAA `  A ) )  =  ( ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) )  x.  ( (coeff `  a
) `  (deg `  a
) ) ) )
8066, 79mpdan 650 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (
( NN0  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  (coeff `  a
) ) `  (degAA `  A ) )  =  ( ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) )  x.  ( (coeff `  a
) `  (deg `  a
) ) ) )
8136, 21recid2d 9711 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (
1  /  ( (coeff `  a ) `  (deg `  a ) ) )  x.  ( (coeff `  a ) `  (deg `  a ) ) )  =  1 )
8263, 80, 813eqtrd 2416 . . . . . 6  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  o F  x.  a ) ) `  (degAA `  A ) )  =  1 )
83 fveq2 5661 . . . . . . . . 9  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a )  -> 
(deg `  p )  =  (deg `  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a ) ) )
8483eqeq1d 2388 . . . . . . . 8  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a )  -> 
( (deg `  p
)  =  (degAA `  A
)  <->  (deg `  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a ) )  =  (degAA `  A ) ) )
85 fveq1 5660 . . . . . . . . 9  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a )  -> 
( p `  A
)  =  ( ( ( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  o F  x.  a
) `  A )
)
8685eqeq1d 2388 . . . . . . . 8  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a )  -> 
( ( p `  A )  =  0  <-> 
( ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a ) `  A )  =  0 ) )
87 fveq2 5661 . . . . . . . . . 10  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a )  -> 
(coeff `  p )  =  (coeff `  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a ) ) )
8887fveq1d 5663 . . . . . . . . 9  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a )  -> 
( (coeff `  p
) `  (degAA `  A
) )  =  ( (coeff `  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a ) ) `
 (degAA `  A ) ) )
8988eqeq1d 2388 . . . . . . . 8  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a )  -> 
( ( (coeff `  p ) `  (degAA `  A ) )  =  1  <->  ( (coeff `  ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  o F  x.  a ) ) `  (degAA `  A ) )  =  1 ) )
9084, 86, 893anbi123d 1254 . . . . . . 7  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a )  -> 
( ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 )  <->  ( (deg `  ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  o F  x.  a ) )  =  (degAA `  A )  /\  ( ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a ) `  A )  =  0  /\  ( (coeff `  ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  o F  x.  a ) ) `  (degAA `  A ) )  =  1 ) ) )
9190rspcev 2988 . . . . . 6  |-  ( ( ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  o F  x.  a )  e.  (Poly `  QQ )  /\  (
(deg `  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a ) )  =  (degAA `  A )  /\  ( ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  o F  x.  a ) `  A )  =  0  /\  ( (coeff `  ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  o F  x.  a ) ) `  (degAA `  A ) )  =  1 ) )  ->  E. p  e.  (Poly `  QQ ) ( (deg
`  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 ) )
9233, 42, 60, 82, 91syl13anc 1186 . . . . 5  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  E. p  e.  (Poly `  QQ )
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 ) )
9392ex 424 . . . 4  |-  ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  ->  ( (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 )  ->  E. p  e.  (Poly `  QQ ) ( (deg
`  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 ) ) )
9493rexlimdva 2766 . . 3  |-  ( A  e.  AA  ->  ( E. a  e.  (
(Poly `  QQ )  \  { 0 p }
) ( (deg `  a )  =  (degAA `  A )  /\  (
a `  A )  =  0 )  ->  E. p  e.  (Poly `  QQ ) ( (deg
`  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 ) ) )
952, 94mpd 15 . 2  |-  ( A  e.  AA  ->  E. p  e.  (Poly `  QQ )
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 ) )
96 simp2 958 . . . . . . . . . . 11  |-  ( ( (deg `  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 )  ->  ( p `  A )  =  0 )
97 simp2 958 . . . . . . . . . . 11  |-  ( ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 )  ->  ( a `  A )  =  0 )
9896, 97anim12i 550 . . . . . . . . . 10  |-  ( ( ( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) )  ->  (
( p `  A
)  =  0  /\  ( a `  A
)  =  0 ) )
99 plyf 19977 . . . . . . . . . . . . . . . 16  |-  ( p  e.  (Poly `  QQ )  ->  p : CC --> CC )
100 ffn 5524 . . . . . . . . . . . . . . . 16  |-  ( p : CC --> CC  ->  p  Fn  CC )
10199, 100syl 16 . . . . . . . . . . . . . . 15  |-  ( p  e.  (Poly `  QQ )  ->  p  Fn  CC )
102101ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( (
p `  A )  =  0  /\  (
a `  A )  =  0 ) )  ->  p  Fn  CC )
10348, 49syl 16 . . . . . . . . . . . . . . 15  |-  ( a  e.  (Poly `  QQ )  ->  a  Fn  CC )
104103ad2antlr 708 . . . . . . . . . . . . . 14  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( (
p `  A )  =  0  /\  (
a `  A )  =  0 ) )  ->  a  Fn  CC )
10551a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( (
p `  A )  =  0  /\  (
a `  A )  =  0 ) )  ->  CC  e.  _V )
106 simplrl 737 . . . . . . . . . . . . . 14  |-  ( ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( p `  A )  =  0  /\  ( a `  A )  =  0 ) )  /\  A  e.  CC )  ->  (
p `  A )  =  0 )
107 simplrr 738 . . . . . . . . . . . . . 14  |-  ( ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( p `  A )  =  0  /\  ( a `  A )  =  0 ) )  /\  A  e.  CC )  ->  (
a `  A )  =  0 )
108102, 104, 105, 105, 53, 106, 107ofval 6246 . . . . . . . . . . . . 13  |-  ( ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( p `  A )  =  0  /\  ( a `  A )  =  0 ) )  /\  A  e.  CC )  ->  (
( p  o F  -  a ) `  A )  =  ( 0  -  0 ) )
10943, 108sylan2 461 . . . . . . . . . . . 12  |-  ( ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( p `  A )  =  0  /\  ( a `  A )  =  0 ) )  /\  A  e.  AA )  ->  (
( p  o F  -  a ) `  A )  =  ( 0  -  0 ) )
110 0cn 9010 . . . . . . . . . . . . 13  |-  0  e.  CC
111110subid1i 9297 . . . . . . . . . . . 12  |-  ( 0  -  0 )  =  0
112109, 111syl6eq 2428 . . . . . . . . . . 11  |-  ( ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( p `  A )  =  0  /\  ( a `  A )  =  0 ) )  /\  A  e.  AA )  ->  (
( p  o F  -  a ) `  A )  =  0 )
113112ex 424 . . . . . . . . . 10  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( (
p `  A )  =  0  /\  (
a `  A )  =  0 ) )  ->  ( A  e.  AA  ->  ( (
p  o F  -  a ) `  A
)  =  0 ) )
11498, 113sylan2 461 . . . . . . . . 9  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( (
(deg `  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
( A  e.  AA  ->  ( ( p  o F  -  a ) `
 A )  =  0 ) )
115114com12 29 . . . . . . . 8  |-  ( A  e.  AA  ->  (
( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
( ( p  o F  -  a ) `
 A )  =  0 ) )
116115impl 604 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
( ( p  o F  -  a ) `
 A )  =  0 )
117 simpll 731 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  ->  A  e.  AA )
118 simpl 444 . . . . . . . . . 10  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  p  e.  (Poly `  QQ ) )
119 simpr 448 . . . . . . . . . 10  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  a  e.  (Poly `  QQ ) )
12028adantl 453 . . . . . . . . . 10  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( b  e.  QQ  /\  c  e.  QQ ) )  -> 
( b  +  c )  e.  QQ )
12130adantl 453 . . . . . . . . . 10  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( b  e.  QQ  /\  c  e.  QQ ) )  -> 
( b  x.  c
)  e.  QQ )
122 1z 10236 . . . . . . . . . . . 12  |-  1  e.  ZZ
123 zq 10505 . . . . . . . . . . . 12  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
124 qnegcl 10516 . . . . . . . . . . . 12  |-  ( 1  e.  QQ  ->  -u 1  e.  QQ )
125122, 123, 124mp2b 10 . . . . . . . . . . 11  |-  -u 1  e.  QQ
126125a1i 11 . . . . . . . . . 10  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  -u 1  e.  QQ )
127118, 119, 120, 121, 126plysub 19998 . . . . . . . . 9  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  ( p  o F  -  a
)  e.  (Poly `  QQ ) )
128127ad2antlr 708 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
( p  o F  -  a )  e.  (Poly `  QQ )
)
129 simplrl 737 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  ->  p  e.  (Poly `  QQ ) )
130 simplrr 738 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
a  e.  (Poly `  QQ ) )
131 simprr1 1005 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
(deg `  a )  =  (degAA `  A ) )
132 simprl1 1002 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
(deg `  p )  =  (degAA `  A ) )
133131, 132eqtr4d 2415 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
(deg `  a )  =  (deg `  p )
)
13464ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
(degAA `
 A )  e.  NN )
135132, 134eqeltrd 2454 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
(deg `  p )  e.  NN )
136 simprl3 1004 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
( (coeff `  p
) `  (degAA `  A
) )  =  1 )
137132fveq2d 5665 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
( (coeff `  p
) `  (deg `  p
) )  =  ( (coeff `  p ) `  (degAA `  A ) ) )
138132fveq2d 5665 . . . . . . . . . . . 12  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
( (coeff `  a
) `  (deg `  p
) )  =  ( (coeff `  a ) `  (degAA `  A ) ) )
139 simprr3 1007 . . . . . . . . . . . 12  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
( (coeff `  a
) `  (degAA `  A
) )  =  1 )
140138, 139eqtrd 2412 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
( (coeff `  a
) `  (deg `  p
) )  =  1 )
141136, 137, 1403eqtr4d 2422 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
( (coeff `  p
) `  (deg `  p
) )  =  ( (coeff `  a ) `  (deg `  p )
) )
142 eqid 2380 . . . . . . . . . . 11  |-  (deg `  p )  =  (deg
`  p )
143142dgrsub2 27001 . . . . . . . . . 10  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( (deg `  a )  =  (deg
`  p )  /\  (deg `  p )  e.  NN  /\  ( (coeff `  p ) `  (deg `  p ) )  =  ( (coeff `  a
) `  (deg `  p
) ) ) )  ->  (deg `  (
p  o F  -  a ) )  < 
(deg `  p )
)
144129, 130, 133, 135, 141, 143syl23anc 1191 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
(deg `  ( p  o F  -  a
) )  <  (deg `  p ) )
145144, 132breqtrd 4170 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
(deg `  ( p  o F  -  a
) )  <  (degAA `  A ) )
146 dgraa0p 27016 . . . . . . . 8  |-  ( ( A  e.  AA  /\  ( p  o F  -  a )  e.  (Poly `  QQ )  /\  (deg `  ( p  o F  -  a
) )  <  (degAA `  A ) )  -> 
( ( ( p  o F  -  a
) `  A )  =  0  <->  ( p  o F  -  a
)  =  0 p ) )
147117, 128, 145, 146syl3anc 1184 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
( ( ( p  o F  -  a
) `  A )  =  0  <->  ( p  o F  -  a
)  =  0 p ) )
148116, 147mpbid 202 . . . . . 6  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
( p  o F  -  a )  =  0 p )
149 df-0p 19422 . . . . . 6  |-  0 p  =  ( CC  X.  { 0 } )
150148, 149syl6eq 2428 . . . . 5  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
( p  o F  -  a )  =  ( CC  X.  {
0 } ) )
151 ofsubeq0 9922 . . . . . . . 8  |-  ( ( CC  e.  _V  /\  p : CC --> CC  /\  a : CC --> CC )  ->  ( ( p  o F  -  a
)  =  ( CC 
X.  { 0 } )  <->  p  =  a
) )
15251, 151mp3an1 1266 . . . . . . 7  |-  ( ( p : CC --> CC  /\  a : CC --> CC )  ->  ( ( p  o F  -  a
)  =  ( CC 
X.  { 0 } )  <->  p  =  a
) )
15399, 48, 152syl2an 464 . . . . . 6  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  ( (
p  o F  -  a )  =  ( CC  X.  { 0 } )  <->  p  =  a ) )
154153ad2antlr 708 . . . . 5  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  -> 
( ( p  o F  -  a )  =  ( CC  X.  { 0 } )  <-> 
p  =  a ) )
155150, 154mpbid 202 . . . 4  |-  ( ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  /\  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) ) )  ->  p  =  a )
156155ex 424 . . 3  |-  ( ( A  e.  AA  /\  ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
) )  ->  (
( ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) )  ->  p  =  a ) )
157156ralrimivva 2734 . 2  |-  ( A  e.  AA  ->  A. p  e.  (Poly `  QQ ) A. a  e.  (Poly `  QQ ) ( ( ( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  (
(deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) )  ->  p  =  a ) )
158 fveq2 5661 . . . . 5  |-  ( p  =  a  ->  (deg `  p )  =  (deg
`  a ) )
159158eqeq1d 2388 . . . 4  |-  ( p  =  a  ->  (
(deg `  p )  =  (degAA `  A )  <->  (deg `  a
)  =  (degAA `  A
) ) )
160 fveq1 5660 . . . . 5  |-  ( p  =  a  ->  (
p `  A )  =  ( a `  A ) )
161160eqeq1d 2388 . . . 4  |-  ( p  =  a  ->  (
( p `  A
)  =  0  <->  (
a `  A )  =  0 ) )
162 fveq2 5661 . . . . . 6  |-  ( p  =  a  ->  (coeff `  p )  =  (coeff `  a ) )
163162fveq1d 5663 . . . . 5  |-  ( p  =  a  ->  (
(coeff `  p ) `  (degAA `  A ) )  =  ( (coeff `  a ) `  (degAA `  A ) ) )
164163eqeq1d 2388 . . . 4  |-  ( p  =  a  ->  (
( (coeff `  p
) `  (degAA `  A
) )  =  1  <-> 
( (coeff `  a
) `  (degAA `  A
) )  =  1 ) )
165159, 161, 1643anbi123d 1254 . . 3  |-  ( p  =  a  ->  (
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 )  <->  ( (deg `  a )  =  (degAA `  A )  /\  (
a `  A )  =  0  /\  (
(coeff `  a ) `  (degAA `  A ) )  =  1 ) ) )
166165reu4 3064 . 2  |-  ( E! p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 )  <->  ( E. p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 )  /\  A. p  e.  (Poly `  QQ ) A. a  e.  (Poly `  QQ )
( ( ( (deg
`  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 ) )  ->  p  =  a ) ) )
16795, 157, 166sylanbrc 646 1  |-  ( A  e.  AA  ->  E! p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   A.wral 2642   E.wrex 2643   E!wreu 2644   _Vcvv 2892    \ cdif 3253    C_ wss 3256   {csn 3750   class class class wbr 4146    X. cxp 4809    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013    o Fcof 6235   CCcc 8914   0cc0 8916   1c1 8917    + caddc 8919    x. cmul 8921    < clt 9046    - cmin 9216   -ucneg 9217    / cdiv 9602   NNcn 9925   NN0cn0 10146   ZZcz 10207   QQcq 10499   0 pc0p 19421  Polycply 19963  coeffccoe 19965  degcdgr 19966   AAcaa 20091  degAAcdgraa 27007
This theorem is referenced by:  mpaalem  27019
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-addf 8995
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-q 10500  df-rp 10538  df-fz 10969  df-fzo 11059  df-fl 11122  df-mod 11171  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-rlim 12203  df-sum 12400  df-0p 19422  df-ply 19967  df-coe 19969  df-dgr 19970  df-aa 20092  df-dgraa 27009
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