Step |
Hyp |
Ref |
Expression |
1 |
|
elaa |
|- ( A e. AA <-> ( A e. CC /\ E. f e. ( ( Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) ) |
2 |
|
zssq |
|- ZZ C_ QQ |
3 |
|
qsscn |
|- QQ C_ CC |
4 |
|
plyss |
|- ( ( ZZ C_ QQ /\ QQ C_ CC ) -> ( Poly ` ZZ ) C_ ( Poly ` QQ ) ) |
5 |
2 3 4
|
mp2an |
|- ( Poly ` ZZ ) C_ ( Poly ` QQ ) |
6 |
|
ssdif |
|- ( ( Poly ` ZZ ) C_ ( Poly ` QQ ) -> ( ( Poly ` ZZ ) \ { 0p } ) C_ ( ( Poly ` QQ ) \ { 0p } ) ) |
7 |
|
ssrexv |
|- ( ( ( Poly ` ZZ ) \ { 0p } ) C_ ( ( Poly ` QQ ) \ { 0p } ) -> ( E. f e. ( ( Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 -> E. f e. ( ( Poly ` QQ ) \ { 0p } ) ( f ` A ) = 0 ) ) |
8 |
5 6 7
|
mp2b |
|- ( E. f e. ( ( Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 -> E. f e. ( ( Poly ` QQ ) \ { 0p } ) ( f ` A ) = 0 ) |
9 |
8
|
anim2i |
|- ( ( A e. CC /\ E. f e. ( ( Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) -> ( A e. CC /\ E. f e. ( ( Poly ` QQ ) \ { 0p } ) ( f ` A ) = 0 ) ) |
10 |
1 9
|
sylbi |
|- ( A e. AA -> ( A e. CC /\ E. f e. ( ( Poly ` QQ ) \ { 0p } ) ( f ` A ) = 0 ) ) |
11 |
|
simpll |
|- ( ( ( A e. CC /\ f e. ( ( Poly ` QQ ) \ { 0p } ) ) /\ ( f ` A ) = 0 ) -> A e. CC ) |
12 |
|
simplr |
|- ( ( ( A e. CC /\ f e. ( ( Poly ` QQ ) \ { 0p } ) ) /\ ( f ` A ) = 0 ) -> f e. ( ( Poly ` QQ ) \ { 0p } ) ) |
13 |
|
simpr |
|- ( ( ( A e. CC /\ f e. ( ( Poly ` QQ ) \ { 0p } ) ) /\ ( f ` A ) = 0 ) -> ( f ` A ) = 0 ) |
14 |
|
eqid |
|- ( coeff ` f ) = ( coeff ` f ) |
15 |
|
fveq2 |
|- ( m = k -> ( ( coeff ` f ) ` m ) = ( ( coeff ` f ) ` k ) ) |
16 |
15
|
oveq1d |
|- ( m = k -> ( ( ( coeff ` f ) ` m ) x. j ) = ( ( ( coeff ` f ) ` k ) x. j ) ) |
17 |
16
|
eleq1d |
|- ( m = k -> ( ( ( ( coeff ` f ) ` m ) x. j ) e. ZZ <-> ( ( ( coeff ` f ) ` k ) x. j ) e. ZZ ) ) |
18 |
17
|
rabbidv |
|- ( m = k -> { j e. NN | ( ( ( coeff ` f ) ` m ) x. j ) e. ZZ } = { j e. NN | ( ( ( coeff ` f ) ` k ) x. j ) e. ZZ } ) |
19 |
|
oveq2 |
|- ( j = n -> ( ( ( coeff ` f ) ` k ) x. j ) = ( ( ( coeff ` f ) ` k ) x. n ) ) |
20 |
19
|
eleq1d |
|- ( j = n -> ( ( ( ( coeff ` f ) ` k ) x. j ) e. ZZ <-> ( ( ( coeff ` f ) ` k ) x. n ) e. ZZ ) ) |
21 |
20
|
cbvrabv |
|- { j e. NN | ( ( ( coeff ` f ) ` k ) x. j ) e. ZZ } = { n e. NN | ( ( ( coeff ` f ) ` k ) x. n ) e. ZZ } |
22 |
18 21
|
eqtrdi |
|- ( m = k -> { j e. NN | ( ( ( coeff ` f ) ` m ) x. j ) e. ZZ } = { n e. NN | ( ( ( coeff ` f ) ` k ) x. n ) e. ZZ } ) |
23 |
22
|
infeq1d |
|- ( m = k -> inf ( { j e. NN | ( ( ( coeff ` f ) ` m ) x. j ) e. ZZ } , RR , < ) = inf ( { n e. NN | ( ( ( coeff ` f ) ` k ) x. n ) e. ZZ } , RR , < ) ) |
24 |
23
|
cbvmptv |
|- ( m e. NN0 |-> inf ( { j e. NN | ( ( ( coeff ` f ) ` m ) x. j ) e. ZZ } , RR , < ) ) = ( k e. NN0 |-> inf ( { n e. NN | ( ( ( coeff ` f ) ` k ) x. n ) e. ZZ } , RR , < ) ) |
25 |
|
eqid |
|- ( seq 0 ( x. , ( m e. NN0 |-> inf ( { j e. NN | ( ( ( coeff ` f ) ` m ) x. j ) e. ZZ } , RR , < ) ) ) ` ( deg ` f ) ) = ( seq 0 ( x. , ( m e. NN0 |-> inf ( { j e. NN | ( ( ( coeff ` f ) ` m ) x. j ) e. ZZ } , RR , < ) ) ) ` ( deg ` f ) ) |
26 |
11 12 13 14 24 25
|
elqaalem3 |
|- ( ( ( A e. CC /\ f e. ( ( Poly ` QQ ) \ { 0p } ) ) /\ ( f ` A ) = 0 ) -> A e. AA ) |
27 |
26
|
r19.29an |
|- ( ( A e. CC /\ E. f e. ( ( Poly ` QQ ) \ { 0p } ) ( f ` A ) = 0 ) -> A e. AA ) |
28 |
10 27
|
impbii |
|- ( A e. AA <-> ( A e. CC /\ E. f e. ( ( Poly ` QQ ) \ { 0p } ) ( f ` A ) = 0 ) ) |