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Theorem mpaaeu 35463
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 qsscn 11238 . . . . . 6  |-  QQ  C_  CC
2 eldifi 3565 . . . . . . . . . 10  |-  ( a  e.  ( (Poly `  QQ )  \  { 0p } )  -> 
a  e.  (Poly `  QQ ) )
32ad2antlr 725 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  a  e.  (Poly `  QQ ) )
4 zssq 11234 . . . . . . . . . 10  |-  ZZ  C_  QQ
5 0z 10916 . . . . . . . . . 10  |-  0  e.  ZZ
64, 5sselii 3439 . . . . . . . . 9  |-  0  e.  QQ
7 eqid 2402 . . . . . . . . . 10  |-  (coeff `  a )  =  (coeff `  a )
87coef2 22920 . . . . . . . . 9  |-  ( ( a  e.  (Poly `  QQ )  /\  0  e.  QQ )  ->  (coeff `  a ) : NN0 --> QQ )
93, 6, 8sylancl 660 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (coeff `  a
) : NN0 --> QQ )
10 dgrcl 22922 . . . . . . . . 9  |-  ( a  e.  (Poly `  QQ )  ->  (deg `  a
)  e.  NN0 )
113, 10syl 17 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (deg `  a
)  e.  NN0 )
129, 11ffvelrnd 6010 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  a ) `  (deg `  a ) )  e.  QQ )
13 eldifsni 4098 . . . . . . . . 9  |-  ( a  e.  ( (Poly `  QQ )  \  { 0p } )  -> 
a  =/=  0p )
1413ad2antlr 725 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  a  =/=  0p )
15 eqid 2402 . . . . . . . . . . 11  |-  (deg `  a )  =  (deg
`  a )
1615, 7dgreq0 22954 . . . . . . . . . 10  |-  ( a  e.  (Poly `  QQ )  ->  ( a  =  0p  <->  ( (coeff `  a ) `  (deg `  a ) )  =  0 ) )
1716necon3bid 2661 . . . . . . . . 9  |-  ( a  e.  (Poly `  QQ )  ->  ( a  =/=  0p  <->  ( (coeff `  a ) `  (deg `  a ) )  =/=  0 ) )
183, 17syl 17 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( a  =/=  0p  <->  ( (coeff `  a ) `  (deg `  a ) )  =/=  0 ) )
1914, 18mpbid 210 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  a ) `  (deg `  a ) )  =/=  0 )
20 qreccl 11247 . . . . . . 7  |-  ( ( ( (coeff `  a
) `  (deg `  a
) )  e.  QQ  /\  ( (coeff `  a
) `  (deg `  a
) )  =/=  0
)  ->  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  QQ )
2112, 19, 20syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  QQ )
22 plyconst 22895 . . . . . 6  |-  ( ( QQ  C_  CC  /\  (
1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  QQ )  -> 
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  e.  (Poly `  QQ ) )
231, 21, 22sylancr 661 . . . . 5  |-  ( ( ( 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 ) )
24 simpl 455 . . . . . 6  |-  ( ( ( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  -> 
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  e.  (Poly `  QQ ) )
25 simpr 459 . . . . . 6  |-  ( ( ( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  -> 
a  e.  (Poly `  QQ ) )
26 qaddcl 11243 . . . . . . 7  |-  ( ( b  e.  QQ  /\  c  e.  QQ )  ->  ( b  +  c )  e.  QQ )
2726adantl 464 . . . . . 6  |-  ( ( ( ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( b  e.  QQ  /\  c  e.  QQ ) )  -> 
( b  +  c )  e.  QQ )
28 qmulcl 11245 . . . . . . 7  |-  ( ( b  e.  QQ  /\  c  e.  QQ )  ->  ( b  x.  c
)  e.  QQ )
2928adantl 464 . . . . . 6  |-  ( ( ( ( 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 )
3024, 25, 27, 29plymul 22907 . . . . 5  |-  ( ( ( 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 ) )
3123, 3, 30syl2anc 659 . . . 4  |-  ( ( ( 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 )
)
327coef3 22921 . . . . . . . . 9  |-  ( a  e.  (Poly `  QQ )  ->  (coeff `  a
) : NN0 --> CC )
333, 32syl 17 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (coeff `  a
) : NN0 --> CC )
3433, 11ffvelrnd 6010 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  a ) `  (deg `  a ) )  e.  CC )
3534, 19reccld 10354 . . . . . 6  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  CC )
3634, 19recne0d 10355 . . . . . 6  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  =/=  0 )
37 dgrmulc 22960 . . . . . 6  |-  ( ( ( 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
) )
3835, 36, 3, 37syl3anc 1230 . . . . 5  |-  ( ( ( 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 ) )
39 simprl 756 . . . . 5  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (deg `  a
)  =  (degAA `  A
) )
4038, 39eqtrd 2443 . . . 4  |-  ( ( ( 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 ) )
41 aacn 23005 . . . . . . 7  |-  ( A  e.  AA  ->  A  e.  CC )
4241ad2antrr 724 . . . . . 6  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  A  e.  CC )
43 ovex 6306 . . . . . . . 8  |-  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  _V
44 fnconstg 5756 . . . . . . . 8  |-  ( ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  _V  ->  ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  Fn  CC )
4543, 44mp1i 13 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( CC  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  Fn  CC )
46 plyf 22887 . . . . . . . 8  |-  ( a  e.  (Poly `  QQ )  ->  a : CC --> CC )
47 ffn 5714 . . . . . . . 8  |-  ( a : CC --> CC  ->  a  Fn  CC )
483, 46, 473syl 18 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  a  Fn  CC )
49 cnex 9603 . . . . . . . 8  |-  CC  e.  _V
5049a1i 11 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  CC  e.  _V )
51 inidm 3648 . . . . . . 7  |-  ( CC 
i^i  CC )  =  CC
5243fvconst2 6107 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } ) `
 A )  =  ( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) )
5352adantl 464 . . . . . . 7  |-  ( ( ( ( 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 )
) ) )
54 simplrr 763 . . . . . . 7  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  A  e.  CC )  ->  (
a `  A )  =  0 )
5545, 48, 50, 50, 51, 53, 54ofval 6530 . . . . . 6  |-  ( ( ( ( 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 ) )
5642, 55mpdan 666 . . . . 5  |-  ( ( ( 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 ) )
5735mul01d 9813 . . . . 5  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (
1  /  ( (coeff `  a ) `  (deg `  a ) ) )  x.  0 )  =  0 )
5856, 57eqtrd 2443 . . . 4  |-  ( ( ( 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 )
59 coemulc 22944 . . . . . . 7  |-  ( ( ( 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 )
) )
6035, 3, 59syl2anc 659 . . . . . 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
) )  =  ( ( NN0  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  oF  x.  (coeff `  a ) ) )
6160fveq1d 5851 . . . . 5  |-  ( ( ( 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 ) ) )
62 dgraacl 35459 . . . . . . . 8  |-  ( A  e.  AA  ->  (degAA `  A )  e.  NN )
6362ad2antrr 724 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (degAA `  A
)  e.  NN )
6463nnnn0d 10893 . . . . . 6  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (degAA `  A
)  e.  NN0 )
65 fnconstg 5756 . . . . . . . 8  |-  ( ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  _V  ->  ( NN0  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  Fn  NN0 )
6643, 65mp1i 13 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( NN0  X. 
{ ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } )  Fn  NN0 )
67 ffn 5714 . . . . . . . 8  |-  ( (coeff `  a ) : NN0 --> CC 
->  (coeff `  a )  Fn  NN0 )
6833, 67syl 17 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (coeff `  a
)  Fn  NN0 )
69 nn0ex 10842 . . . . . . . 8  |-  NN0  e.  _V
7069a1i 11 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  NN0  e.  _V )
71 inidm 3648 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
7243fvconst2 6107 . . . . . . . 8  |-  ( (degAA `  A )  e.  NN0  ->  ( ( NN0  X.  { ( 1  / 
( (coeff `  a
) `  (deg `  a
) ) ) } ) `  (degAA `  A
) )  =  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) )
7372adantl 464 . . . . . . 7  |-  ( ( ( ( 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 ) ) ) )
74 simplrl 762 . . . . . . . . 9  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  (degAA `  A
)  e.  NN0 )  ->  (deg `  a )  =  (degAA `  A ) )
7574eqcomd 2410 . . . . . . . 8  |-  ( ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  { 0p } ) )  /\  ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  /\  (degAA `  A
)  e.  NN0 )  ->  (degAA `  A )  =  (deg `  a )
)
7675fveq2d 5853 . . . . . . 7  |-  ( ( ( ( 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 )
) )
7766, 68, 70, 70, 71, 73, 76ofval 6530 . . . . . 6  |-  ( ( ( ( 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
) ) ) )
7864, 77mpdan 666 . . . . 5  |-  ( ( ( 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
) ) ) )
7934, 19recid2d 10357 . . . . 5  |-  ( ( ( 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 )
8061, 78, 793eqtrd 2447 . . . 4  |-  ( ( ( 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 )
81 fveq2 5849 . . . . . . 7  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a )  -> 
(deg `  p )  =  (deg `  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a ) ) )
8281eqeq1d 2404 . . . . . 6  |-  ( 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 ) ) )
83 fveq1 5848 . . . . . . 7  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a )  -> 
( p `  A
)  =  ( ( ( CC  X.  {
( 1  /  (
(coeff `  a ) `  (deg `  a )
) ) } )  oF  x.  a
) `  A )
)
8483eqeq1d 2404 . . . . . 6  |-  ( 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 ) )
85 fveq2 5849 . . . . . . . 8  |-  ( p  =  ( ( CC 
X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a )  -> 
(coeff `  p )  =  (coeff `  ( ( CC  X.  { ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) ) } )  oF  x.  a ) ) )
8685fveq1d 5851 . . . . . . 7  |-  ( 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 ) ) )
8786eqeq1d 2404 . . . . . 6  |-  ( 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 ) )
8882, 84, 873anbi123d 1301 . . . . 5  |-  ( 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 ) ) )
8988rspcev 3160 . . . 4  |-  ( ( ( ( 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 ) )
9031, 40, 58, 80, 89syl13anc 1232 . . 3  |-  ( ( ( 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 ) )
91 dgraalem 35458 . . . 4  |-  ( A  e.  AA  ->  (
(degAA `
 A )  e.  NN  /\  E. a  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) ) )
9291simprd 461 . . 3  |-  ( A  e.  AA  ->  E. a  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )
9390, 92r19.29a 2949 . 2  |-  ( A  e.  AA  ->  E. p  e.  (Poly `  QQ )
( (deg `  p
)  =  (degAA `  A
)  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 ) )
94 simp2 998 . . . . . . . . . . 11  |-  ( ( (deg `  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 )  ->  ( p `  A )  =  0 )
95 simp2 998 . . . . . . . . . . 11  |-  ( ( (deg `  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0  /\  ( (coeff `  a
) `  (degAA `  A
) )  =  1 )  ->  ( a `  A )  =  0 )
9694, 95anim12i 564 . . . . . . . . . 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 ) )
97 plyf 22887 . . . . . . . . . . . . . . . 16  |-  ( p  e.  (Poly `  QQ )  ->  p : CC --> CC )
98 ffn 5714 . . . . . . . . . . . . . . . 16  |-  ( p : CC --> CC  ->  p  Fn  CC )
9997, 98syl 17 . . . . . . . . . . . . . . 15  |-  ( p  e.  (Poly `  QQ )  ->  p  Fn  CC )
10099ad2antrr 724 . . . . . . . . . . . . . 14  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( (
p `  A )  =  0  /\  (
a `  A )  =  0 ) )  ->  p  Fn  CC )
10146, 47syl 17 . . . . . . . . . . . . . . 15  |-  ( a  e.  (Poly `  QQ )  ->  a  Fn  CC )
102101ad2antlr 725 . . . . . . . . . . . . . 14  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( (
p `  A )  =  0  /\  (
a `  A )  =  0 ) )  ->  a  Fn  CC )
10349a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( (
p `  A )  =  0  /\  (
a `  A )  =  0 ) )  ->  CC  e.  _V )
104 simplrl 762 . . . . . . . . . . . . . 14  |-  ( ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( p `  A )  =  0  /\  ( a `  A )  =  0 ) )  /\  A  e.  CC )  ->  (
p `  A )  =  0 )
105 simplrr 763 . . . . . . . . . . . . . 14  |-  ( ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( p `  A )  =  0  /\  ( a `  A )  =  0 ) )  /\  A  e.  CC )  ->  (
a `  A )  =  0 )
106100, 102, 103, 103, 51, 104, 105ofval 6530 . . . . . . . . . . . . 13  |-  ( ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( p `  A )  =  0  /\  ( a `  A )  =  0 ) )  /\  A  e.  CC )  ->  (
( p  oF  -  a ) `  A )  =  ( 0  -  0 ) )
10741, 106sylan2 472 . . . . . . . . . . . 12  |-  ( ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( p `  A )  =  0  /\  ( a `  A )  =  0 ) )  /\  A  e.  AA )  ->  (
( p  oF  -  a ) `  A )  =  ( 0  -  0 ) )
108 0m0e0 10686 . . . . . . . . . . . 12  |-  ( 0  -  0 )  =  0
109107, 108syl6eq 2459 . . . . . . . . . . 11  |-  ( ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ ) )  /\  ( ( p `  A )  =  0  /\  ( a `  A )  =  0 ) )  /\  A  e.  AA )  ->  (
( p  oF  -  a ) `  A )  =  0 )
110109ex 432 . . . . . . . . . 10  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( (
p `  A )  =  0  /\  (
a `  A )  =  0 ) )  ->  ( A  e.  AA  ->  ( (
p  oF  -  a ) `  A
)  =  0 ) )
11196, 110sylan2 472 . . . . . . . . 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 ) )
112111com12 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  oF  -  a ) `
 A )  =  0 ) )
113112impl 618 . . . . . . 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 )
114 simpll 752 . . . . . . . 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 )
115 simpl 455 . . . . . . . . . 10  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  p  e.  (Poly `  QQ ) )
116 simpr 459 . . . . . . . . . 10  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  a  e.  (Poly `  QQ ) )
11726adantl 464 . . . . . . . . . 10  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( b  e.  QQ  /\  c  e.  QQ ) )  -> 
( b  +  c )  e.  QQ )
11828adantl 464 . . . . . . . . . 10  |-  ( ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  /\  ( b  e.  QQ  /\  c  e.  QQ ) )  -> 
( b  x.  c
)  e.  QQ )
119 1z 10935 . . . . . . . . . . . 12  |-  1  e.  ZZ
120 zq 11233 . . . . . . . . . . . 12  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
121 qnegcl 11244 . . . . . . . . . . . 12  |-  ( 1  e.  QQ  ->  -u 1  e.  QQ )
122119, 120, 121mp2b 10 . . . . . . . . . . 11  |-  -u 1  e.  QQ
123122a1i 11 . . . . . . . . . 10  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  -u 1  e.  QQ )
124115, 116, 117, 118, 123plysub 22908 . . . . . . . . 9  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  ( p  oF  -  a
)  e.  (Poly `  QQ ) )
125124ad2antlr 725 . . . . . . . 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 )
)
126 simplrl 762 . . . . . . . . . 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 ) )
127 simplrr 763 . . . . . . . . . 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 ) )
128 simprr1 1045 . . . . . . . . . . 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 ) )
129 simprl1 1042 . . . . . . . . . . 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 ) )
130128, 129eqtr4d 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 ) ) )  -> 
(deg `  a )  =  (deg `  p )
)
13162ad2antrr 724 . . . . . . . . . . 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 )
132129, 131eqeltrd 2490 . . . . . . . . . 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 )
133 simprl3 1044 . . . . . . . . . . 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 )
134129fveq2d 5853 . . . . . . . . . . 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 ) ) )
135129fveq2d 5853 . . . . . . . . . . . 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 ) ) )
136 simprr3 1047 . . . . . . . . . . . 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 )
137135, 136eqtrd 2443 . . . . . . . . . . 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 )
138133, 134, 1373eqtr4d 2453 . . . . . . . . . 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 )
) )
139 eqid 2402 . . . . . . . . . . 11  |-  (deg `  p )  =  (deg
`  p )
140139dgrsub2 35448 . . . . . . . . . 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 )
)
141126, 127, 130, 132, 138, 140syl23anc 1237 . . . . . . . . 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 ) )
142141, 129breqtrd 4419 . . . . . . . 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 ) )
143 dgraa0p 35462 . . . . . . . 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 ) )
144114, 125, 142, 143syl3anc 1230 . . . . . . 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 ) )
145113, 144mpbid 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 )
146 df-0p 22369 . . . . . 6  |-  0p  =  ( CC  X.  { 0 } )
147145, 146syl6eq 2459 . . . . 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 } ) )
148 ofsubeq0 10573 . . . . . . . 8  |-  ( ( CC  e.  _V  /\  p : CC --> CC  /\  a : CC --> CC )  ->  ( ( p  oF  -  a
)  =  ( CC 
X.  { 0 } )  <->  p  =  a
) )
14949, 148mp3an1 1313 . . . . . . 7  |-  ( ( p : CC --> CC  /\  a : CC --> CC )  ->  ( ( p  oF  -  a
)  =  ( CC 
X.  { 0 } )  <->  p  =  a
) )
15097, 46, 149syl2an 475 . . . . . 6  |-  ( ( p  e.  (Poly `  QQ )  /\  a  e.  (Poly `  QQ )
)  ->  ( (
p  oF  -  a )  =  ( CC  X.  { 0 } )  <->  p  =  a ) )
151150ad2antlr 725 . . . . 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 ) )
152147, 151mpbid 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 )
153152ex 432 . . 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 ) )
154153ralrimivva 2825 . 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 ) )
155 fveq2 5849 . . . . 5  |-  ( p  =  a  ->  (deg `  p )  =  (deg
`  a ) )
156155eqeq1d 2404 . . . 4  |-  ( p  =  a  ->  (
(deg `  p )  =  (degAA `  A )  <->  (deg `  a
)  =  (degAA `  A
) ) )
157 fveq1 5848 . . . . 5  |-  ( p  =  a  ->  (
p `  A )  =  ( a `  A ) )
158157eqeq1d 2404 . . . 4  |-  ( p  =  a  ->  (
( p `  A
)  =  0  <->  (
a `  A )  =  0 ) )
159 fveq2 5849 . . . . . 6  |-  ( p  =  a  ->  (coeff `  p )  =  (coeff `  a ) )
160159fveq1d 5851 . . . . 5  |-  ( p  =  a  ->  (
(coeff `  p ) `  (degAA `  A ) )  =  ( (coeff `  a ) `  (degAA `  A ) ) )
161160eqeq1d 2404 . . . 4  |-  ( p  =  a  ->  (
( (coeff `  p
) `  (degAA `  A
) )  =  1  <-> 
( (coeff `  a
) `  (degAA `  A
) )  =  1 ) )
162156, 158, 1613anbi123d 1301 . . 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 ) ) )
163162reu4 3243 . 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 ) ) )
16493, 154, 163sylanbrc 662 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755   E!wreu 2756   _Vcvv 3059    \ cdif 3411    C_ wss 3414   {csn 3972   class class class wbr 4395    X. cxp 4821    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278    oFcof 6519   CCcc 9520   0cc0 9522   1c1 9523    + caddc 9525    x. cmul 9527    < clt 9658    - cmin 9841   -ucneg 9842    / cdiv 10247   NNcn 10576   NN0cn0 10836   ZZcz 10905   QQcq 11227   0pc0p 22368  Polycply 22873  coeffccoe 22875  degcdgr 22876   AAcaa 23002  degAAcdgraa 35453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-q 11228  df-rp 11266  df-fz 11727  df-fzo 11855  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-rlim 13461  df-sum 13658  df-0p 22369  df-ply 22877  df-coe 22879  df-dgr 22880  df-aa 23003  df-dgraa 35455
This theorem is referenced by:  mpaalem  35465
  Copyright terms: Public domain W3C validator