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Theorem mpaaeu 29645
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 29640 . . . 4  |-  ( A  e.  AA  ->  (
(degAA `
 A )  e.  NN  /\  E. a  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) ) )
21simprd 463 . . 3  |-  ( A  e.  AA  ->  E. a  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )
3 qsscn 11065 . . . . . . . 8  |-  QQ  C_  CC
4 eldifi 3576 . . . . . . . . . . . 12  |-  ( a  e.  ( (Poly `  QQ )  \  { 0p } )  -> 
a  e.  (Poly `  QQ ) )
54ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  a  e.  (Poly `  QQ ) )
6 zssq 11061 . . . . . . . . . . . 12  |-  ZZ  C_  QQ
7 0z 10758 . . . . . . . . . . . 12  |-  0  e.  ZZ
86, 7sselii 3451 . . . . . . . . . . 11  |-  0  e.  QQ
9 eqid 2451 . . . . . . . . . . . 12  |-  (coeff `  a )  =  (coeff `  a )
109coef2 21815 . . . . . . . . . . 11  |-  ( ( a  e.  (Poly `  QQ )  /\  0  e.  QQ )  ->  (coeff `  a ) : NN0 --> QQ )
115, 8, 10sylancl 662 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (coeff `  a
) : NN0 --> QQ )
12 dgrcl 21817 . . . . . . . . . . 11  |-  ( a  e.  (Poly `  QQ )  ->  (deg `  a
)  e.  NN0 )
135, 12syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (deg `  a
)  e.  NN0 )
1411, 13ffvelrnd 5943 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  a ) `  (deg `  a ) )  e.  QQ )
15 eldifsni 4099 . . . . . . . . . . 11  |-  ( a  e.  ( (Poly `  QQ )  \  { 0p } )  -> 
a  =/=  0p )
1615ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  a  =/=  0p )
17 eqid 2451 . . . . . . . . . . . . 13  |-  (deg `  a )  =  (deg
`  a )
1817, 9dgreq0 21848 . . . . . . . . . . . 12  |-  ( a  e.  (Poly `  QQ )  ->  ( a  =  0p  <->  ( (coeff `  a ) `  (deg `  a ) )  =  0 ) )
1918necon3bid 2706 . . . . . . . . . . 11  |-  ( a  e.  (Poly `  QQ )  ->  ( a  =/=  0p  <->  ( (coeff `  a ) `  (deg `  a ) )  =/=  0 ) )
205, 19syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( a  =/=  0p  <->  ( (coeff `  a ) `  (deg `  a ) )  =/=  0 ) )
2116, 20mpbid 210 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  a ) `  (deg `  a ) )  =/=  0 )
22 qreccl 11074 . . . . . . . . 9  |-  ( ( ( (coeff `  a
) `  (deg `  a
) )  e.  QQ  /\  ( (coeff `  a
) `  (deg `  a
) )  =/=  0
)  ->  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  QQ )
2314, 21, 22syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  QQ )
24 plyconst 21790 . . . . . . . 8  |-  ( ( QQ  C_  CC  /\  (
1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  QQ )  -> 
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  e.  (Poly `  QQ ) )
253, 23, 24sylancr 663 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  e.  (Poly `  QQ ) )
26 simpl 457 . . . . . . . 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 461 . . . . . . . 8  |-  ( ( ( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  -> 
a  e.  (Poly `  QQ ) )
28 qaddcl 11070 . . . . . . . . 9  |-  ( ( b  e.  QQ  /\  c  e.  QQ )  ->  ( b  +  c )  e.  QQ )
2928adantl 466 . . . . . . . 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 11072 . . . . . . . . 9  |-  ( ( b  e.  QQ  /\  c  e.  QQ )  ->  ( b  x.  c
)  e.  QQ )
3130adantl 466 . . . . . . . 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 21802 . . . . . . 7  |-  ( ( ( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  -> 
( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  oF  x.  a )  e.  (Poly `  QQ ) )
3325, 5, 32syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a )  e.  (Poly `  QQ )
)
349coef3 21816 . . . . . . . . . . 11  |-  ( a  e.  (Poly `  QQ )  ->  (coeff `  a
) : NN0 --> CC )
355, 34syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (coeff `  a
) : NN0 --> CC )
3635, 13ffvelrnd 5943 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  a ) `  (deg `  a ) )  e.  CC )
3736, 21reccld 10201 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  CC )
3836, 21recne0d 10202 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  =/=  0 )
39 dgrmulc 21854 . . . . . . . 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 ) ) ) } )  oF  x.  a ) )  =  (deg `  a
) )
4037, 38, 5, 39syl3anc 1219 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (deg `  (
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  oF  x.  a
) )  =  (deg
`  a ) )
41 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (deg `  a
)  =  (degAA `  A
) )
4240, 41eqtrd 2492 . . . . . 6  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (deg `  (
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  oF  x.  a
) )  =  (degAA `  A ) )
43 aacn 21899 . . . . . . . . 9  |-  ( A  e.  AA  ->  A  e.  CC )
4443ad2antrr 725 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  A  e.  CC )
45 ovex 6215 . . . . . . . . . 10  |-  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  _V
46 fnconstg 5696 . . . . . . . . . 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 )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  Fn  CC )
48 plyf 21782 . . . . . . . . . 10  |-  ( a  e.  (Poly `  QQ )  ->  a : CC --> CC )
49 ffn 5657 . . . . . . . . . 10  |-  ( a : CC --> CC  ->  a  Fn  CC )
505, 48, 493syl 20 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  a  Fn  CC )
51 cnex 9464 . . . . . . . . . 10  |-  CC  e.  _V
5251a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  CC  e.  _V )
53 inidm 3657 . . . . . . . . 9  |-  ( CC 
i^i  CC )  =  CC
5445fvconst2 6032 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } ) `
 A )  =  ( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) )
5554adantl 466 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0p } ) )  /\  ( (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 760 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  A  e.  CC )  ->  (
a `  A )  =  0 )
5747, 50, 52, 52, 53, 55, 56ofval 6429 . . . . . . . 8  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  A  e.  CC )  ->  (
( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  oF  x.  a ) `  A
)  =  ( ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  x.  0 ) )
5844, 57mpdan 668 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  oF  x.  a
) `  A )  =  ( ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  x.  0 ) )
5937mul01d 9669 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (
1  /  ( (coeff `  a ) `  (deg `  a ) ) )  x.  0 )  =  0 )
6058, 59eqtrd 2492 . . . . . 6  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  oF  x.  a
) `  A )  =  0 )
61 coemulc 21838 . . . . . . . . 9  |-  ( ( ( 1  /  (
(coeff `  a ) `  (deg `  a )
) )  e.  CC  /\  a  e.  (Poly `  QQ ) )  ->  (coeff `  ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  oF  x.  a ) )  =  ( ( NN0  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  oF  x.  (coeff `  a )
) )
6237, 5, 61syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (coeff `  (
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  oF  x.  a
) )  =  ( ( NN0  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  oF  x.  (coeff `  a ) ) )
6362fveq1d 5791 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  oF  x.  a ) ) `  (degAA `  A ) )  =  ( ( ( NN0 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  (coeff `  a
) ) `  (degAA `  A ) ) )
64 dgraacl 29641 . . . . . . . . . 10  |-  ( A  e.  AA  ->  (degAA `  A )  e.  NN )
6564ad2antrr 725 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (degAA `  A
)  e.  NN )
6665nnnn0d 10737 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (degAA `  A
)  e.  NN0 )
67 fnconstg 5696 . . . . . . . . . 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 )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( NN0  X. 
{ ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  Fn  NN0 )
69 ffn 5657 . . . . . . . . . 10  |-  ( (coeff `  a ) : NN0 --> CC 
->  (coeff `  a )  Fn  NN0 )
7035, 69syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (coeff `  a
)  Fn  NN0 )
71 nn0ex 10686 . . . . . . . . . 10  |-  NN0  e.  _V
7271a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  NN0  e.  _V )
73 inidm 3657 . . . . . . . . 9  |-  ( NN0 
i^i  NN0 )  =  NN0
7445fvconst2 6032 . . . . . . . . . 10  |-  ( (degAA `  A )  e.  NN0  ->  ( ( NN0  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } ) `  (degAA `  A
) )  =  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) )
7574adantl 466 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0p } ) )  /\  ( (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 759 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  (degAA `  A
)  e.  NN0 )  ->  (deg `  a )  =  (degAA `  A ) )
7776eqcomd 2459 . . . . . . . . . 10  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  (degAA `  A
)  e.  NN0 )  ->  (degAA `  A )  =  (deg `  a )
)
7877fveq2d 5793 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0p } ) )  /\  ( (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 6429 . . . . . . . 8  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  (degAA `  A
)  e.  NN0 )  ->  ( ( ( NN0 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  (coeff `  a
) ) `  (degAA `  A ) )  =  ( ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) )  x.  ( (coeff `  a
) `  (deg `  a
) ) ) )
8066, 79mpdan 668 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (
( NN0  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  (coeff `  a
) ) `  (degAA `  A ) )  =  ( ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) )  x.  ( (coeff `  a
) `  (deg `  a
) ) ) )
8136, 21recid2d 10204 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (
1  /  ( (coeff `  a ) `  (deg `  a ) ) )  x.  ( (coeff `  a ) `  (deg `  a ) ) )  =  1 )
8263, 80, 813eqtrd 2496 . . . . . 6  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  oF  x.  a ) ) `  (degAA `  A ) )  =  1 )
83 fveq2 5789 . . . . . . . . 9  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a )  -> 
(deg `  p )  =  (deg `  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a ) ) )
8483eqeq1d 2453 . . . . . . . 8  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a )  -> 
( (deg `  p
)  =  (degAA `  A
)  <->  (deg `  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a ) )  =  (degAA `  A ) ) )
85 fveq1 5788 . . . . . . . . 9  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a )  -> 
( p `  A
)  =  ( ( ( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  oF  x.  a
) `  A )
)
8685eqeq1d 2453 . . . . . . . 8  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a )  -> 
( ( p `  A )  =  0  <-> 
( ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a ) `  A )  =  0 ) )
87 fveq2 5789 . . . . . . . . . 10  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a )  -> 
(coeff `  p )  =  (coeff `  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a ) ) )
8887fveq1d 5791 . . . . . . . . 9  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a )  -> 
( (coeff `  p
) `  (degAA `  A
) )  =  ( (coeff `  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a ) ) `
 (degAA `  A ) ) )
8988eqeq1d 2453 . . . . . . . 8  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a )  -> 
( ( (coeff `  p ) `  (degAA `  A ) )  =  1  <->  ( (coeff `  ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  oF  x.  a ) ) `  (degAA `  A ) )  =  1 ) )
9084, 86, 893anbi123d 1290 . . . . . . 7  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a )  -> 
( ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 )  <->  ( (deg `  ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  oF  x.  a ) )  =  (degAA `  A )  /\  ( ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a ) `  A )  =  0  /\  ( (coeff `  ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  oF  x.  a ) ) `  (degAA `  A ) )  =  1 ) ) )
9190rspcev 3169 . . . . . 6  |-  ( ( ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  oF  x.  a )  e.  (Poly `  QQ )  /\  (
(deg `  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a ) )  =  (degAA `  A )  /\  ( ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a ) `  A )  =  0  /\  ( (coeff `  ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  oF  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 1221 . . . . 5  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (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 434 . . . 4  |-  ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  ->  ( (
(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 2937 . . 3  |-  ( A  e.  AA  ->  ( E. a  e.  (
(Poly `  QQ )  \  { 0p }
) ( (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 989 . . . . . . . . . . 11  |-  ( ( (deg `  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 )  ->  ( p `  A )  =  0 )
97 simp2 989 . . . . . . . . . . 11  |-  ( ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 )  ->  ( a `  A )  =  0 )
9896, 97anim12i 566 . . . . . . . . . 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 21782 . . . . . . . . . . . . . . . 16  |-  ( p  e.  (Poly `  QQ )  ->  p : CC --> CC )
100 ffn 5657 . . . . . . . . . . . . . . . 16  |-  ( p : CC --> CC  ->  p  Fn  CC )
10199, 100syl 16 . . . . . . . . . . . . . . 15  |-  ( p  e.  (Poly `  QQ )  ->  p  Fn  CC )
102101ad2antrr 725 . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . 14  |-  ( ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( p `  A )  =  0  /\  ( a `  A )  =  0 ) )  /\  A  e.  CC )  ->  (
p `  A )  =  0 )
107 simplrr 760 . . . . . . . . . . . . . 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 6429 . . . . . . . . . . . . 13  |-  ( ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( p `  A )  =  0  /\  ( a `  A )  =  0 ) )  /\  A  e.  CC )  ->  (
( p  oF  -  a ) `  A )  =  ( 0  -  0 ) )
10943, 108sylan2 474 . . . . . . . . . . . 12  |-  ( ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( p `  A )  =  0  /\  ( a `  A )  =  0 ) )  /\  A  e.  AA )  ->  (
( p  oF  -  a ) `  A )  =  ( 0  -  0 ) )
110 0m0e0 10532 . . . . . . . . . . . 12  |-  ( 0  -  0 )  =  0
111109, 110syl6eq 2508 . . . . . . . . . . 11  |-  ( ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( p `  A )  =  0  /\  ( a `  A )  =  0 ) )  /\  A  e.  AA )  ->  (
( p  oF  -  a ) `  A )  =  0 )
112111ex 434 . . . . . . . . . 10  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( (
p `  A )  =  0  /\  (
a `  A )  =  0 ) )  ->  ( A  e.  AA  ->  ( (
p  oF  -  a ) `  A
)  =  0 ) )
11398, 112sylan2 474 . . . . . . . . 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  oF  -  a ) `
 A )  =  0 ) )
114113com12 31 . . . . . . . 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  oF  -  a ) `
 A )  =  0 ) )
115114impl 620 . . . . . . 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  oF  -  a ) `
 A )  =  0 )
116 simpll 753 . . . . . . . 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 )
117 simpl 457 . . . . . . . . . 10  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  p  e.  (Poly `  QQ ) )
118 simpr 461 . . . . . . . . . 10  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  a  e.  (Poly `  QQ ) )
11928adantl 466 . . . . . . . . . 10  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( b  e.  QQ  /\  c  e.  QQ ) )  -> 
( b  +  c )  e.  QQ )
12030adantl 466 . . . . . . . . . 10  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( b  e.  QQ  /\  c  e.  QQ ) )  -> 
( b  x.  c
)  e.  QQ )
121 1z 10777 . . . . . . . . . . . 12  |-  1  e.  ZZ
122 zq 11060 . . . . . . . . . . . 12  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
123 qnegcl 11071 . . . . . . . . . . . 12  |-  ( 1  e.  QQ  ->  -u 1  e.  QQ )
124121, 122, 123mp2b 10 . . . . . . . . . . 11  |-  -u 1  e.  QQ
125124a1i 11 . . . . . . . . . 10  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  -u 1  e.  QQ )
126117, 118, 119, 120, 125plysub 21803 . . . . . . . . 9  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  ( p  oF  -  a
)  e.  (Poly `  QQ ) )
127126ad2antlr 726 . . . . . . . 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  oF  -  a )  e.  (Poly `  QQ )
)
128 simplrl 759 . . . . . . . . . 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 ) )
129 simplrr 760 . . . . . . . . . 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 ) )
130 simprr1 1036 . . . . . . . . . . 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 ) )
131 simprl1 1033 . . . . . . . . . . 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 ) )
132130, 131eqtr4d 2495 . . . . . . . . . 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 )
)
13364ad2antrr 725 . . . . . . . . . . 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 )
134131, 133eqeltrd 2539 . . . . . . . . . 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 )
135 simprl3 1035 . . . . . . . . . . 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 )
136131fveq2d 5793 . . . . . . . . . . 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 ) ) )
137131fveq2d 5793 . . . . . . . . . . . 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 ) ) )
138 simprr3 1038 . . . . . . . . . . . 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 )
139137, 138eqtrd 2492 . . . . . . . . . . 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 )
140135, 136, 1393eqtr4d 2502 . . . . . . . . . 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 )
) )
141 eqid 2451 . . . . . . . . . . 11  |-  (deg `  p )  =  (deg
`  p )
142141dgrsub2 29629 . . . . . . . . . 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  oF  -  a ) )  < 
(deg `  p )
)
143128, 129, 132, 134, 140, 142syl23anc 1226 . . . . . . . . 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  oF  -  a
) )  <  (deg `  p ) )
144143, 131breqtrd 4414 . . . . . . . 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  oF  -  a
) )  <  (degAA `  A ) )
145 dgraa0p 29644 . . . . . . . 8  |-  ( ( A  e.  AA  /\  ( p  oF  -  a )  e.  (Poly `  QQ )  /\  (deg `  ( p  oF  -  a
) )  <  (degAA `  A ) )  -> 
( ( ( p  oF  -  a
) `  A )  =  0  <->  ( p  oF  -  a
)  =  0p ) )
146116, 127, 144, 145syl3anc 1219 . . . . . . 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  oF  -  a
) `  A )  =  0  <->  ( p  oF  -  a
)  =  0p ) )
147115, 146mpbid 210 . . . . . 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  oF  -  a )  =  0p )
148 df-0p 21264 . . . . . 6  |-  0p  =  ( CC  X.  { 0 } )
149147, 148syl6eq 2508 . . . . 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  oF  -  a )  =  ( CC  X.  {
0 } ) )
150 ofsubeq0 10420 . . . . . . . 8  |-  ( ( CC  e.  _V  /\  p : CC --> CC  /\  a : CC --> CC )  ->  ( ( p  oF  -  a
)  =  ( CC 
X.  { 0 } )  <->  p  =  a
) )
15151, 150mp3an1 1302 . . . . . . 7  |-  ( ( p : CC --> CC  /\  a : CC --> CC )  ->  ( ( p  oF  -  a
)  =  ( CC 
X.  { 0 } )  <->  p  =  a
) )
15299, 48, 151syl2an 477 . . . . . 6  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  ( (
p  oF  -  a )  =  ( CC  X.  { 0 } )  <->  p  =  a ) )
153152ad2antlr 726 . . . . 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  oF  -  a )  =  ( CC  X.  { 0 } )  <-> 
p  =  a ) )
154149, 153mpbid 210 . . . 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 )
155154ex 434 . . 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 ) )
156155ralrimivva 2904 . 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 ) )
157 fveq2 5789 . . . . 5  |-  ( p  =  a  ->  (deg `  p )  =  (deg
`  a ) )
158157eqeq1d 2453 . . . 4  |-  ( p  =  a  ->  (
(deg `  p )  =  (degAA `  A )  <->  (deg `  a
)  =  (degAA `  A
) ) )
159 fveq1 5788 . . . . 5  |-  ( p  =  a  ->  (
p `  A )  =  ( a `  A ) )
160159eqeq1d 2453 . . . 4  |-  ( p  =  a  ->  (
( p `  A
)  =  0  <->  (
a `  A )  =  0 ) )
161 fveq2 5789 . . . . . 6  |-  ( p  =  a  ->  (coeff `  p )  =  (coeff `  a ) )
162161fveq1d 5791 . . . . 5  |-  ( p  =  a  ->  (
(coeff `  p ) `  (degAA `  A ) )  =  ( (coeff `  a ) `  (degAA `  A ) ) )
163162eqeq1d 2453 . . . 4  |-  ( p  =  a  ->  (
( (coeff `  p
) `  (degAA `  A
) )  =  1  <-> 
( (coeff `  a
) `  (degAA `  A
) )  =  1 ) )
164158, 160, 1633anbi123d 1290 . . 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 ) ) )
165164reu4 3250 . 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 ) ) )
16695, 156, 165sylanbrc 664 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796   E!wreu 2797   _Vcvv 3068    \ cdif 3423    C_ wss 3426   {csn 3975   class class class wbr 4390    X. cxp 4936    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190    oFcof 6418   CCcc 9381   0cc0 9383   1c1 9384    + caddc 9386    x. cmul 9388    < clt 9519    - cmin 9696   -ucneg 9697    / cdiv 10094   NNcn 10423   NN0cn0 10680   ZZcz 10747   QQcq 11054   0pc0p 21263  Polycply 21768  coeffccoe 21770  degcdgr 21771   AAcaa 21896  degAAcdgraa 29635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-q 11055  df-rp 11093  df-fz 11539  df-fzo 11650  df-fl 11743  df-mod 11810  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-clim 13068  df-rlim 13069  df-sum 13266  df-0p 21264  df-ply 21772  df-coe 21774  df-dgr 21775  df-aa 21897  df-dgraa 29637
This theorem is referenced by:  mpaalem  29647
  Copyright terms: Public domain W3C validator