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Theorem mpaaeu 27223
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 27218 . . . 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 10541 . . . . . . . 8  |-  QQ  C_  CC
4 eldifi 3429 . . . . . . . . . . . 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 10537 . . . . . . . . . . . 12  |-  ZZ  C_  QQ
7 0z 10249 . . . . . . . . . . . 12  |-  0  e.  ZZ
86, 7sselii 3305 . . . . . . . . . . 11  |-  0  e.  QQ
9 eqid 2404 . . . . . . . . . . . 12  |-  (coeff `  a )  =  (coeff `  a )
109coef2 20103 . . . . . . . . . . 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 20105 . . . . . . . . . . 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 5830 . . . . . . . . 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 3888 . . . . . . . . . . 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 2404 . . . . . . . . . . . . 13  |-  (deg `  a )  =  (deg
`  a )
1817, 9dgreq0 20136 . . . . . . . . . . . 12  |-  ( a  e.  (Poly `  QQ )  ->  ( a  =  0 p  <->  ( (coeff `  a ) `  (deg `  a ) )  =  0 ) )
1918necon3bid 2602 . . . . . . . . . . 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 10550 . . . . . . . . 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 20078 . . . . . . . 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 10546 . . . . . . . . 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 10548 . . . . . . . . 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 20090 . . . . . . 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 20104 . . . . . . . . . . 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 5830 . . . . . . . . 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 9739 . . . . . . . 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 9740 . . . . . . . 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 20142 . . . . . . . 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 2436 . . . . . 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 20187 . . . . . . . . 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 6065 . . . . . . . . . 10  |-  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  _V
46 fnconstg 5590 . . . . . . . . . 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 20070 . . . . . . . . . 10  |-  ( a  e.  (Poly `  QQ )  ->  a : CC --> CC )
49 ffn 5550 . . . . . . . . . 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 9027 . . . . . . . . . 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 3510 . . . . . . . . 9  |-  ( CC 
i^i  CC )  =  CC
5445fvconst2 5906 . . . . . . . . . 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 6273 . . . . . . . 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 9221 . . . . . . 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 2436 . . . . . 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 20126 . . . . . . . . 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 5689 . . . . . . 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 27219 . . . . . . . . . 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 10230 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (degAA `  A
)  e.  NN0 )
67 fnconstg 5590 . . . . . . . . . 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 5550 . . . . . . . . . 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 10183 . . . . . . . . . 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 3510 . . . . . . . . 9  |-  ( NN0 
i^i  NN0 )  =  NN0
7445fvconst2 5906 . . . . . . . . . 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 2409 . . . . . . . . . 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 5691 . . . . . . . . 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 6273 . . . . . . . 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 9742 . . . . . . 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 2440 . . . . . 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 5687 . . . . . . . . 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 2412 . . . . . . . 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 5686 . . . . . . . . 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 2412 . . . . . . . 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 5687 . . . . . . . . . 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 5689 . . . . . . . . 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 2412 . . . . . . . 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 3012 . . . . . 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 2790 . . 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 20070 . . . . . . . . . . . . . . . 16  |-  ( p  e.  (Poly `  QQ )  ->  p : CC --> CC )
100 ffn 5550 . . . . . . . . . . . . . . . 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 6273 . . . . . . . . . . . . 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 9040 . . . . . . . . . . . . 13  |-  0  e.  CC
111110subid1i 9328 . . . . . . . . . . . 12  |-  ( 0  -  0 )  =  0
112109, 111syl6eq 2452 . . . . . . . . . . 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 10267 . . . . . . . . . . . 12  |-  1  e.  ZZ
123 zq 10536 . . . . . . . . . . . 12  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
124 qnegcl 10547 . . . . . . . . . . . 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 20091 . . . . . . . . 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 2439 . . . . . . . . . 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 2478 . . . . . . . . . 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 5691 . . . . . . . . . . 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 5691 . . . . . . . . . . . 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 2436 . . . . . . . . . . 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 2446 . . . . . . . . . 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 2404 . . . . . . . . . . 11  |-  (deg `  p )  =  (deg
`  p )
143142dgrsub2 27207 . . . . . . . . . 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 4196 . . . . . . . 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 27222 . . . . . . . 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 19515 . . . . . 6  |-  0 p  =  ( CC  X.  { 0 } )
150148, 149syl6eq 2452 . . . . 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 9953 . . . . . . . 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 2758 . 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 5687 . . . . 5  |-  ( p  =  a  ->  (deg `  p )  =  (deg
`  a ) )
159158eqeq1d 2412 . . . 4  |-  ( p  =  a  ->  (
(deg `  p )  =  (degAA `  A )  <->  (deg `  a
)  =  (degAA `  A
) ) )
160 fveq1 5686 . . . . 5  |-  ( p  =  a  ->  (
p `  A )  =  ( a `  A ) )
161160eqeq1d 2412 . . . 4  |-  ( p  =  a  ->  (
( p `  A
)  =  0  <->  (
a `  A )  =  0 ) )
162 fveq2 5687 . . . . . 6  |-  ( p  =  a  ->  (coeff `  p )  =  (coeff `  a ) )
163162fveq1d 5689 . . . . 5  |-  ( p  =  a  ->  (
(coeff `  p ) `  (degAA `  A ) )  =  ( (coeff `  a ) `  (degAA `  A ) ) )
164163eqeq1d 2412 . . . 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 3088 . 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 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   E!wreu 2668   _Vcvv 2916    \ cdif 3277    C_ wss 3280   {csn 3774   class class class wbr 4172    X. cxp 4835    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZcz 10238   QQcq 10530   0 pc0p 19514  Polycply 20056  coeffccoe 20058  degcdgr 20059   AAcaa 20184  degAAcdgraa 27213
This theorem is referenced by:  mpaalem  27225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-coe 20062  df-dgr 20063  df-aa 20185  df-dgraa 27215
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