Step |
Hyp |
Ref |
Expression |
1 |
|
mdeg0.d |
|- D = ( I mDeg R ) |
2 |
|
mdeg0.p |
|- P = ( I mPoly R ) |
3 |
|
mdeg0.z |
|- .0. = ( 0g ` P ) |
4 |
|
ringgrp |
|- ( R e. Ring -> R e. Grp ) |
5 |
2
|
mplgrp |
|- ( ( I e. V /\ R e. Grp ) -> P e. Grp ) |
6 |
4 5
|
sylan2 |
|- ( ( I e. V /\ R e. Ring ) -> P e. Grp ) |
7 |
|
eqid |
|- ( Base ` P ) = ( Base ` P ) |
8 |
7 3
|
grpidcl |
|- ( P e. Grp -> .0. e. ( Base ` P ) ) |
9 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
10 |
|
eqid |
|- { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } = { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } |
11 |
|
eqid |
|- ( y e. { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } |-> ( CCfld gsum y ) ) = ( y e. { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } |-> ( CCfld gsum y ) ) |
12 |
1 2 7 9 10 11
|
mdegval |
|- ( .0. e. ( Base ` P ) -> ( D ` .0. ) = sup ( ( ( y e. { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } |-> ( CCfld gsum y ) ) " ( .0. supp ( 0g ` R ) ) ) , RR* , < ) ) |
13 |
6 8 12
|
3syl |
|- ( ( I e. V /\ R e. Ring ) -> ( D ` .0. ) = sup ( ( ( y e. { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } |-> ( CCfld gsum y ) ) " ( .0. supp ( 0g ` R ) ) ) , RR* , < ) ) |
14 |
|
simpl |
|- ( ( I e. V /\ R e. Ring ) -> I e. V ) |
15 |
4
|
adantl |
|- ( ( I e. V /\ R e. Ring ) -> R e. Grp ) |
16 |
2 10 9 3 14 15
|
mpl0 |
|- ( ( I e. V /\ R e. Ring ) -> .0. = ( { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } X. { ( 0g ` R ) } ) ) |
17 |
|
fvex |
|- ( 0g ` R ) e. _V |
18 |
|
fnconstg |
|- ( ( 0g ` R ) e. _V -> ( { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } X. { ( 0g ` R ) } ) Fn { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } ) |
19 |
17 18
|
mp1i |
|- ( ( I e. V /\ R e. Ring ) -> ( { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } X. { ( 0g ` R ) } ) Fn { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } ) |
20 |
16
|
fneq1d |
|- ( ( I e. V /\ R e. Ring ) -> ( .0. Fn { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } <-> ( { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } X. { ( 0g ` R ) } ) Fn { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } ) ) |
21 |
19 20
|
mpbird |
|- ( ( I e. V /\ R e. Ring ) -> .0. Fn { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } ) |
22 |
|
ovex |
|- ( NN0 ^m I ) e. _V |
23 |
22
|
rabex |
|- { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } e. _V |
24 |
23
|
a1i |
|- ( ( I e. V /\ R e. Ring ) -> { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } e. _V ) |
25 |
17
|
a1i |
|- ( ( I e. V /\ R e. Ring ) -> ( 0g ` R ) e. _V ) |
26 |
|
fnsuppeq0 |
|- ( ( .0. Fn { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } /\ { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } e. _V /\ ( 0g ` R ) e. _V ) -> ( ( .0. supp ( 0g ` R ) ) = (/) <-> .0. = ( { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } X. { ( 0g ` R ) } ) ) ) |
27 |
21 24 25 26
|
syl3anc |
|- ( ( I e. V /\ R e. Ring ) -> ( ( .0. supp ( 0g ` R ) ) = (/) <-> .0. = ( { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } X. { ( 0g ` R ) } ) ) ) |
28 |
16 27
|
mpbird |
|- ( ( I e. V /\ R e. Ring ) -> ( .0. supp ( 0g ` R ) ) = (/) ) |
29 |
28
|
imaeq2d |
|- ( ( I e. V /\ R e. Ring ) -> ( ( y e. { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } |-> ( CCfld gsum y ) ) " ( .0. supp ( 0g ` R ) ) ) = ( ( y e. { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } |-> ( CCfld gsum y ) ) " (/) ) ) |
30 |
|
ima0 |
|- ( ( y e. { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } |-> ( CCfld gsum y ) ) " (/) ) = (/) |
31 |
29 30
|
eqtrdi |
|- ( ( I e. V /\ R e. Ring ) -> ( ( y e. { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } |-> ( CCfld gsum y ) ) " ( .0. supp ( 0g ` R ) ) ) = (/) ) |
32 |
31
|
supeq1d |
|- ( ( I e. V /\ R e. Ring ) -> sup ( ( ( y e. { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } |-> ( CCfld gsum y ) ) " ( .0. supp ( 0g ` R ) ) ) , RR* , < ) = sup ( (/) , RR* , < ) ) |
33 |
|
xrsup0 |
|- sup ( (/) , RR* , < ) = -oo |
34 |
32 33
|
eqtrdi |
|- ( ( I e. V /\ R e. Ring ) -> sup ( ( ( y e. { x e. ( NN0 ^m I ) | ( `' x " NN ) e. Fin } |-> ( CCfld gsum y ) ) " ( .0. supp ( 0g ` R ) ) ) , RR* , < ) = -oo ) |
35 |
13 34
|
eqtrd |
|- ( ( I e. V /\ R e. Ring ) -> ( D ` .0. ) = -oo ) |