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Theorem mpaaeu 30704
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 30699 . . . 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 11189 . . . . . . . 8  |-  QQ  C_  CC
4 eldifi 3626 . . . . . . . . . . . 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 11185 . . . . . . . . . . . 12  |-  ZZ  C_  QQ
7 0z 10871 . . . . . . . . . . . 12  |-  0  e.  ZZ
86, 7sselii 3501 . . . . . . . . . . 11  |-  0  e.  QQ
9 eqid 2467 . . . . . . . . . . . 12  |-  (coeff `  a )  =  (coeff `  a )
109coef2 22360 . . . . . . . . . . 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 22362 . . . . . . . . . . 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 6020 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  a ) `  (deg `  a ) )  e.  QQ )
15 eldifsni 4153 . . . . . . . . . . 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 2467 . . . . . . . . . . . . 13  |-  (deg `  a )  =  (deg
`  a )
1817, 9dgreq0 22393 . . . . . . . . . . . 12  |-  ( a  e.  (Poly `  QQ )  ->  ( a  =  0p  <->  ( (coeff `  a ) `  (deg `  a ) )  =  0 ) )
1918necon3bid 2725 . . . . . . . . . . 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 11198 . . . . . . . . 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 22335 . . . . . . . 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 11194 . . . . . . . . 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 11196 . . . . . . . . 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 22347 . . . . . . 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 22361 . . . . . . . . . . 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 6020 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( (coeff `  a ) `  (deg `  a ) )  e.  CC )
3736, 21reccld 10309 . . . . . . . 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 10310 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  =/=  0 )
39 dgrmulc 22399 . . . . . . . 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 1228 . . . . . . 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 2508 . . . . . 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 22444 . . . . . . . . 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 6307 . . . . . . . . . 10  |-  ( 1  /  ( (coeff `  a ) `  (deg `  a ) ) )  e.  _V
46 fnconstg 5771 . . . . . . . . . 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 22327 . . . . . . . . . 10  |-  ( a  e.  (Poly `  QQ )  ->  a : CC --> CC )
49 ffn 5729 . . . . . . . . . 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 9569 . . . . . . . . . 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 3707 . . . . . . . . 9  |-  ( CC 
i^i  CC )  =  CC
5445fvconst2 6114 . . . . . . . . . 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 6531 . . . . . . . 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 9774 . . . . . . 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 2508 . . . . . 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 22383 . . . . . . . . 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 5866 . . . . . . 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 30700 . . . . . . . . . 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 10848 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  a  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( (deg
`  a )  =  (degAA `  A )  /\  ( a `  A
)  =  0 ) )  ->  (degAA `  A
)  e.  NN0 )
67 fnconstg 5771 . . . . . . . . . 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 5729 . . . . . . . . . 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 10797 . . . . . . . . . 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 3707 . . . . . . . . 9  |-  ( NN0 
i^i  NN0 )  =  NN0
7445fvconst2 6114 . . . . . . . . . 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 2475 . . . . . . . . . 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 5868 . . . . . . . . 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 6531 . . . . . . . 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 10312 . . . . . . 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 2512 . . . . . 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 5864 . . . . . . . . 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 2469 . . . . . . . 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 5863 . . . . . . . . 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 2469 . . . . . . . 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 5864 . . . . . . . . . 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 5866 . . . . . . . . 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 2469 . . . . . . . 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 1299 . . . . . . 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 3214 . . . . . 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 1230 . . . . 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 2955 . . 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 997 . . . . . . . . . . 11  |-  ( ( (deg `  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 )  ->  ( p `  A )  =  0 )
97 simp2 997 . . . . . . . . . . 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 22327 . . . . . . . . . . . . . . . 16  |-  ( p  e.  (Poly `  QQ )  ->  p : CC --> CC )
100 ffn 5729 . . . . . . . . . . . . . . . 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 6531 . . . . . . . . . . . . 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 10641 . . . . . . . . . . . 12  |-  ( 0  -  0 )  =  0
111109, 110syl6eq 2524 . . . . . . . . . . 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 10890 . . . . . . . . . . . 12  |-  1  e.  ZZ
122 zq 11184 . . . . . . . . . . . 12  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
123 qnegcl 11195 . . . . . . . . . . . 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 22348 . . . . . . . . 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 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 ) ) )  -> 
(deg `  a )  =  (degAA `  A ) )
131 simprl1 1041 . . . . . . . . . . 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 2511 . . . . . . . . . 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 2555 . . . . . . . . . 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 1043 . . . . . . . . . . 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 5868 . . . . . . . . . . 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 5868 . . . . . . . . . . . 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 1046 . . . . . . . . . . . 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 2508 . . . . . . . . . . 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 2518 . . . . . . . . . 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 2467 . . . . . . . . . . 11  |-  (deg `  p )  =  (deg
`  p )
142141dgrsub2 30688 . . . . . . . . . 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 1235 . . . . . . . . 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 4471 . . . . . . . 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 30703 . . . . . . . 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 1228 . . . . . . 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 21809 . . . . . 6  |-  0p  =  ( CC  X.  { 0 } )
149147, 148syl6eq 2524 . . . . 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 10529 . . . . . . . 8  |-  ( ( CC  e.  _V  /\  p : CC --> CC  /\  a : CC --> CC )  ->  ( ( p  oF  -  a
)  =  ( CC 
X.  { 0 } )  <->  p  =  a
) )
15151, 150mp3an1 1311 . . . . . . 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 2885 . 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 5864 . . . . 5  |-  ( p  =  a  ->  (deg `  p )  =  (deg
`  a ) )
158157eqeq1d 2469 . . . 4  |-  ( p  =  a  ->  (
(deg `  p )  =  (degAA `  A )  <->  (deg `  a
)  =  (degAA `  A
) ) )
159 fveq1 5863 . . . . 5  |-  ( p  =  a  ->  (
p `  A )  =  ( a `  A ) )
160159eqeq1d 2469 . . . 4  |-  ( p  =  a  ->  (
( p `  A
)  =  0  <->  (
a `  A )  =  0 ) )
161 fveq2 5864 . . . . . 6  |-  ( p  =  a  ->  (coeff `  p )  =  (coeff `  a ) )
162161fveq1d 5866 . . . . 5  |-  ( p  =  a  ->  (
(coeff `  p ) `  (degAA `  A ) )  =  ( (coeff `  a ) `  (degAA `  A ) ) )
163162eqeq1d 2469 . . . 4  |-  ( p  =  a  ->  (
( (coeff `  p
) `  (degAA `  A
) )  =  1  <-> 
( (coeff `  a
) `  (degAA `  A
) )  =  1 ) )
164158, 160, 1633anbi123d 1299 . . 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 3297 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   E!wreu 2816   _Vcvv 3113    \ cdif 3473    C_ wss 3476   {csn 4027   class class class wbr 4447    X. cxp 4997    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282    oFcof 6520   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    < clt 9624    - cmin 9801   -ucneg 9802    / cdiv 10202   NNcn 10532   NN0cn0 10791   ZZcz 10860   QQcq 11178   0pc0p 21808  Polycply 22313  coeffccoe 22315  degcdgr 22316   AAcaa 22441  degAAcdgraa 30694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11960  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-clim 13267  df-rlim 13268  df-sum 13465  df-0p 21809  df-ply 22317  df-coe 22319  df-dgr 22320  df-aa 22442  df-dgraa 30696
This theorem is referenced by:  mpaalem  30706
  Copyright terms: Public domain W3C validator