Step |
Hyp |
Ref |
Expression |
1 |
|
aasscn |
|- AA C_ CC |
2 |
|
eldifi |
|- ( A e. ( AA \ { 0 } ) -> A e. AA ) |
3 |
1 2
|
sseldi |
|- ( A e. ( AA \ { 0 } ) -> A e. CC ) |
4 |
|
elaa |
|- ( A e. AA <-> ( A e. CC /\ E. g e. ( ( Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 ) ) |
5 |
2 4
|
sylib |
|- ( A e. ( AA \ { 0 } ) -> ( A e. CC /\ E. g e. ( ( Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 ) ) |
6 |
5
|
simprd |
|- ( A e. ( AA \ { 0 } ) -> E. g e. ( ( Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 ) |
7 |
2
|
3ad2ant1 |
|- ( ( A e. ( AA \ { 0 } ) /\ g e. ( ( Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> A e. AA ) |
8 |
|
eldifsni |
|- ( A e. ( AA \ { 0 } ) -> A =/= 0 ) |
9 |
8
|
3ad2ant1 |
|- ( ( A e. ( AA \ { 0 } ) /\ g e. ( ( Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> A =/= 0 ) |
10 |
|
eldifi |
|- ( g e. ( ( Poly ` ZZ ) \ { 0p } ) -> g e. ( Poly ` ZZ ) ) |
11 |
10
|
3ad2ant2 |
|- ( ( A e. ( AA \ { 0 } ) /\ g e. ( ( Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> g e. ( Poly ` ZZ ) ) |
12 |
|
eldifsni |
|- ( g e. ( ( Poly ` ZZ ) \ { 0p } ) -> g =/= 0p ) |
13 |
12
|
3ad2ant2 |
|- ( ( A e. ( AA \ { 0 } ) /\ g e. ( ( Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> g =/= 0p ) |
14 |
|
simp3 |
|- ( ( A e. ( AA \ { 0 } ) /\ g e. ( ( Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> ( g ` A ) = 0 ) |
15 |
|
fveq2 |
|- ( m = n -> ( ( coeff ` g ) ` m ) = ( ( coeff ` g ) ` n ) ) |
16 |
15
|
neeq1d |
|- ( m = n -> ( ( ( coeff ` g ) ` m ) =/= 0 <-> ( ( coeff ` g ) ` n ) =/= 0 ) ) |
17 |
16
|
cbvrabv |
|- { m e. NN0 | ( ( coeff ` g ) ` m ) =/= 0 } = { n e. NN0 | ( ( coeff ` g ) ` n ) =/= 0 } |
18 |
17
|
infeq1i |
|- inf ( { m e. NN0 | ( ( coeff ` g ) ` m ) =/= 0 } , RR , < ) = inf ( { n e. NN0 | ( ( coeff ` g ) ` n ) =/= 0 } , RR , < ) |
19 |
|
fvoveq1 |
|- ( j = k -> ( ( coeff ` g ) ` ( j + inf ( { m e. NN0 | ( ( coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) = ( ( coeff ` g ) ` ( k + inf ( { m e. NN0 | ( ( coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ) |
20 |
19
|
cbvmptv |
|- ( j e. NN0 |-> ( ( coeff ` g ) ` ( j + inf ( { m e. NN0 | ( ( coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ) = ( k e. NN0 |-> ( ( coeff ` g ) ` ( k + inf ( { m e. NN0 | ( ( coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ) |
21 |
|
eqid |
|- ( z e. CC |-> sum_ k e. ( 0 ... ( ( deg ` g ) - inf ( { m e. NN0 | ( ( coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ( ( ( j e. NN0 |-> ( ( coeff ` g ) ` ( j + inf ( { m e. NN0 | ( ( coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ) ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... ( ( deg ` g ) - inf ( { m e. NN0 | ( ( coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ( ( ( j e. NN0 |-> ( ( coeff ` g ) ` ( j + inf ( { m e. NN0 | ( ( coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ) ` k ) x. ( z ^ k ) ) ) |
22 |
7 9 11 13 14 18 20 21
|
elaa2lem |
|- ( ( A e. ( AA \ { 0 } ) /\ g e. ( ( Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) |
23 |
22
|
rexlimdv3a |
|- ( A e. ( AA \ { 0 } ) -> ( E. g e. ( ( Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 -> E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) ) |
24 |
6 23
|
mpd |
|- ( A e. ( AA \ { 0 } ) -> E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) |
25 |
3 24
|
jca |
|- ( A e. ( AA \ { 0 } ) -> ( A e. CC /\ E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) ) |
26 |
|
simpl |
|- ( ( f e. ( Poly ` ZZ ) /\ ( ( coeff ` f ) ` 0 ) =/= 0 ) -> f e. ( Poly ` ZZ ) ) |
27 |
|
fveq2 |
|- ( f = 0p -> ( coeff ` f ) = ( coeff ` 0p ) ) |
28 |
|
coe0 |
|- ( coeff ` 0p ) = ( NN0 X. { 0 } ) |
29 |
27 28
|
eqtrdi |
|- ( f = 0p -> ( coeff ` f ) = ( NN0 X. { 0 } ) ) |
30 |
29
|
fveq1d |
|- ( f = 0p -> ( ( coeff ` f ) ` 0 ) = ( ( NN0 X. { 0 } ) ` 0 ) ) |
31 |
|
0nn0 |
|- 0 e. NN0 |
32 |
|
fvconst2g |
|- ( ( 0 e. NN0 /\ 0 e. NN0 ) -> ( ( NN0 X. { 0 } ) ` 0 ) = 0 ) |
33 |
31 31 32
|
mp2an |
|- ( ( NN0 X. { 0 } ) ` 0 ) = 0 |
34 |
30 33
|
eqtrdi |
|- ( f = 0p -> ( ( coeff ` f ) ` 0 ) = 0 ) |
35 |
34
|
adantl |
|- ( ( ( f e. ( Poly ` ZZ ) /\ ( ( coeff ` f ) ` 0 ) =/= 0 ) /\ f = 0p ) -> ( ( coeff ` f ) ` 0 ) = 0 ) |
36 |
|
neneq |
|- ( ( ( coeff ` f ) ` 0 ) =/= 0 -> -. ( ( coeff ` f ) ` 0 ) = 0 ) |
37 |
36
|
ad2antlr |
|- ( ( ( f e. ( Poly ` ZZ ) /\ ( ( coeff ` f ) ` 0 ) =/= 0 ) /\ f = 0p ) -> -. ( ( coeff ` f ) ` 0 ) = 0 ) |
38 |
35 37
|
pm2.65da |
|- ( ( f e. ( Poly ` ZZ ) /\ ( ( coeff ` f ) ` 0 ) =/= 0 ) -> -. f = 0p ) |
39 |
|
velsn |
|- ( f e. { 0p } <-> f = 0p ) |
40 |
38 39
|
sylnibr |
|- ( ( f e. ( Poly ` ZZ ) /\ ( ( coeff ` f ) ` 0 ) =/= 0 ) -> -. f e. { 0p } ) |
41 |
26 40
|
eldifd |
|- ( ( f e. ( Poly ` ZZ ) /\ ( ( coeff ` f ) ` 0 ) =/= 0 ) -> f e. ( ( Poly ` ZZ ) \ { 0p } ) ) |
42 |
41
|
adantrr |
|- ( ( f e. ( Poly ` ZZ ) /\ ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> f e. ( ( Poly ` ZZ ) \ { 0p } ) ) |
43 |
|
simprr |
|- ( ( f e. ( Poly ` ZZ ) /\ ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( f ` A ) = 0 ) |
44 |
42 43
|
jca |
|- ( ( f e. ( Poly ` ZZ ) /\ ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( f e. ( ( Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) |
45 |
44
|
reximi2 |
|- ( E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) -> E. f e. ( ( Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) |
46 |
45
|
anim2i |
|- ( ( A e. CC /\ E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( A e. CC /\ E. f e. ( ( Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) ) |
47 |
|
elaa |
|- ( A e. AA <-> ( A e. CC /\ E. f e. ( ( Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) ) |
48 |
46 47
|
sylibr |
|- ( ( A e. CC /\ E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> A e. AA ) |
49 |
|
simpr |
|- ( ( A e. CC /\ E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) |
50 |
|
nfv |
|- F/ f A e. CC |
51 |
|
nfre1 |
|- F/ f E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) |
52 |
50 51
|
nfan |
|- F/ f ( A e. CC /\ E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) |
53 |
|
nfv |
|- F/ f -. A e. { 0 } |
54 |
|
simpl3r |
|- ( ( ( A e. CC /\ f e. ( Poly ` ZZ ) /\ ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A = 0 ) -> ( f ` A ) = 0 ) |
55 |
|
fveq2 |
|- ( A = 0 -> ( f ` A ) = ( f ` 0 ) ) |
56 |
|
eqid |
|- ( coeff ` f ) = ( coeff ` f ) |
57 |
56
|
coefv0 |
|- ( f e. ( Poly ` ZZ ) -> ( f ` 0 ) = ( ( coeff ` f ) ` 0 ) ) |
58 |
55 57
|
sylan9eqr |
|- ( ( f e. ( Poly ` ZZ ) /\ A = 0 ) -> ( f ` A ) = ( ( coeff ` f ) ` 0 ) ) |
59 |
58
|
adantlr |
|- ( ( ( f e. ( Poly ` ZZ ) /\ ( ( coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( f ` A ) = ( ( coeff ` f ) ` 0 ) ) |
60 |
|
simplr |
|- ( ( ( f e. ( Poly ` ZZ ) /\ ( ( coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( ( coeff ` f ) ` 0 ) =/= 0 ) |
61 |
59 60
|
eqnetrd |
|- ( ( ( f e. ( Poly ` ZZ ) /\ ( ( coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( f ` A ) =/= 0 ) |
62 |
61
|
neneqd |
|- ( ( ( f e. ( Poly ` ZZ ) /\ ( ( coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> -. ( f ` A ) = 0 ) |
63 |
62
|
adantlrr |
|- ( ( ( f e. ( Poly ` ZZ ) /\ ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A = 0 ) -> -. ( f ` A ) = 0 ) |
64 |
63
|
3adantl1 |
|- ( ( ( A e. CC /\ f e. ( Poly ` ZZ ) /\ ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A = 0 ) -> -. ( f ` A ) = 0 ) |
65 |
54 64
|
pm2.65da |
|- ( ( A e. CC /\ f e. ( Poly ` ZZ ) /\ ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> -. A = 0 ) |
66 |
|
elsng |
|- ( A e. CC -> ( A e. { 0 } <-> A = 0 ) ) |
67 |
66
|
biimpa |
|- ( ( A e. CC /\ A e. { 0 } ) -> A = 0 ) |
68 |
67
|
3ad2antl1 |
|- ( ( ( A e. CC /\ f e. ( Poly ` ZZ ) /\ ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A e. { 0 } ) -> A = 0 ) |
69 |
65 68
|
mtand |
|- ( ( A e. CC /\ f e. ( Poly ` ZZ ) /\ ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> -. A e. { 0 } ) |
70 |
69
|
3exp |
|- ( A e. CC -> ( f e. ( Poly ` ZZ ) -> ( ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) -> -. A e. { 0 } ) ) ) |
71 |
70
|
adantr |
|- ( ( A e. CC /\ E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( f e. ( Poly ` ZZ ) -> ( ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) -> -. A e. { 0 } ) ) ) |
72 |
52 53 71
|
rexlimd |
|- ( ( A e. CC /\ E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) -> -. A e. { 0 } ) ) |
73 |
49 72
|
mpd |
|- ( ( A e. CC /\ E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> -. A e. { 0 } ) |
74 |
48 73
|
eldifd |
|- ( ( A e. CC /\ E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> A e. ( AA \ { 0 } ) ) |
75 |
25 74
|
impbii |
|- ( A e. ( AA \ { 0 } ) <-> ( A e. CC /\ E. f e. ( Poly ` ZZ ) ( ( ( coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) ) |