Step |
Hyp |
Ref |
Expression |
1 |
|
dgrub.1 |
|- A = ( coeff ` F ) |
2 |
|
dgrub.2 |
|- N = ( deg ` F ) |
3 |
|
simp2 |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> M e. NN0 ) |
4 |
3
|
nn0red |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> M e. RR ) |
5 |
|
simp1 |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> F e. ( Poly ` S ) ) |
6 |
|
dgrcl |
|- ( F e. ( Poly ` S ) -> ( deg ` F ) e. NN0 ) |
7 |
2 6
|
eqeltrid |
|- ( F e. ( Poly ` S ) -> N e. NN0 ) |
8 |
5 7
|
syl |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> N e. NN0 ) |
9 |
8
|
nn0red |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> N e. RR ) |
10 |
1
|
dgrval |
|- ( F e. ( Poly ` S ) -> ( deg ` F ) = sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) ) |
11 |
2 10
|
syl5eq |
|- ( F e. ( Poly ` S ) -> N = sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) ) |
12 |
5 11
|
syl |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> N = sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) ) |
13 |
1
|
coef3 |
|- ( F e. ( Poly ` S ) -> A : NN0 --> CC ) |
14 |
5 13
|
syl |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> A : NN0 --> CC ) |
15 |
14 3
|
ffvelrnd |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> ( A ` M ) e. CC ) |
16 |
|
simp3 |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> ( A ` M ) =/= 0 ) |
17 |
|
eldifsn |
|- ( ( A ` M ) e. ( CC \ { 0 } ) <-> ( ( A ` M ) e. CC /\ ( A ` M ) =/= 0 ) ) |
18 |
15 16 17
|
sylanbrc |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> ( A ` M ) e. ( CC \ { 0 } ) ) |
19 |
1
|
coef |
|- ( F e. ( Poly ` S ) -> A : NN0 --> ( S u. { 0 } ) ) |
20 |
|
ffn |
|- ( A : NN0 --> ( S u. { 0 } ) -> A Fn NN0 ) |
21 |
|
elpreima |
|- ( A Fn NN0 -> ( M e. ( `' A " ( CC \ { 0 } ) ) <-> ( M e. NN0 /\ ( A ` M ) e. ( CC \ { 0 } ) ) ) ) |
22 |
5 19 20 21
|
4syl |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> ( M e. ( `' A " ( CC \ { 0 } ) ) <-> ( M e. NN0 /\ ( A ` M ) e. ( CC \ { 0 } ) ) ) ) |
23 |
3 18 22
|
mpbir2and |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> M e. ( `' A " ( CC \ { 0 } ) ) ) |
24 |
|
nn0ssre |
|- NN0 C_ RR |
25 |
|
ltso |
|- < Or RR |
26 |
|
soss |
|- ( NN0 C_ RR -> ( < Or RR -> < Or NN0 ) ) |
27 |
24 25 26
|
mp2 |
|- < Or NN0 |
28 |
27
|
a1i |
|- ( F e. ( Poly ` S ) -> < Or NN0 ) |
29 |
|
0zd |
|- ( F e. ( Poly ` S ) -> 0 e. ZZ ) |
30 |
|
cnvimass |
|- ( `' A " ( CC \ { 0 } ) ) C_ dom A |
31 |
30 19
|
fssdm |
|- ( F e. ( Poly ` S ) -> ( `' A " ( CC \ { 0 } ) ) C_ NN0 ) |
32 |
1
|
dgrlem |
|- ( F e. ( Poly ` S ) -> ( A : NN0 --> ( S u. { 0 } ) /\ E. n e. ZZ A. x e. ( `' A " ( CC \ { 0 } ) ) x <_ n ) ) |
33 |
32
|
simprd |
|- ( F e. ( Poly ` S ) -> E. n e. ZZ A. x e. ( `' A " ( CC \ { 0 } ) ) x <_ n ) |
34 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
35 |
34
|
uzsupss |
|- ( ( 0 e. ZZ /\ ( `' A " ( CC \ { 0 } ) ) C_ NN0 /\ E. n e. ZZ A. x e. ( `' A " ( CC \ { 0 } ) ) x <_ n ) -> E. n e. NN0 ( A. x e. ( `' A " ( CC \ { 0 } ) ) -. n < x /\ A. x e. NN0 ( x < n -> E. y e. ( `' A " ( CC \ { 0 } ) ) x < y ) ) ) |
36 |
29 31 33 35
|
syl3anc |
|- ( F e. ( Poly ` S ) -> E. n e. NN0 ( A. x e. ( `' A " ( CC \ { 0 } ) ) -. n < x /\ A. x e. NN0 ( x < n -> E. y e. ( `' A " ( CC \ { 0 } ) ) x < y ) ) ) |
37 |
28 36
|
supub |
|- ( F e. ( Poly ` S ) -> ( M e. ( `' A " ( CC \ { 0 } ) ) -> -. sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) < M ) ) |
38 |
5 23 37
|
sylc |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> -. sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) < M ) |
39 |
12 38
|
eqnbrtrd |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> -. N < M ) |
40 |
4 9 39
|
nltled |
|- ( ( F e. ( Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> M <_ N ) |