Step |
Hyp |
Ref |
Expression |
1 |
|
r1pval.e |
|- E = ( rem1p ` R ) |
2 |
|
r1pval.p |
|- P = ( Poly1 ` R ) |
3 |
|
r1pval.b |
|- B = ( Base ` P ) |
4 |
|
r1pcl.c |
|- C = ( Unic1p ` R ) |
5 |
|
r1pdeglt.d |
|- D = ( deg1 ` R ) |
6 |
|
simp2 |
|- ( ( R e. Ring /\ F e. B /\ G e. C ) -> F e. B ) |
7 |
2 3 4
|
uc1pcl |
|- ( G e. C -> G e. B ) |
8 |
7
|
3ad2ant3 |
|- ( ( R e. Ring /\ F e. B /\ G e. C ) -> G e. B ) |
9 |
|
eqid |
|- ( quot1p ` R ) = ( quot1p ` R ) |
10 |
|
eqid |
|- ( .r ` P ) = ( .r ` P ) |
11 |
|
eqid |
|- ( -g ` P ) = ( -g ` P ) |
12 |
1 2 3 9 10 11
|
r1pval |
|- ( ( F e. B /\ G e. B ) -> ( F E G ) = ( F ( -g ` P ) ( ( F ( quot1p ` R ) G ) ( .r ` P ) G ) ) ) |
13 |
6 8 12
|
syl2anc |
|- ( ( R e. Ring /\ F e. B /\ G e. C ) -> ( F E G ) = ( F ( -g ` P ) ( ( F ( quot1p ` R ) G ) ( .r ` P ) G ) ) ) |
14 |
13
|
fveq2d |
|- ( ( R e. Ring /\ F e. B /\ G e. C ) -> ( D ` ( F E G ) ) = ( D ` ( F ( -g ` P ) ( ( F ( quot1p ` R ) G ) ( .r ` P ) G ) ) ) ) |
15 |
|
eqid |
|- ( F ( quot1p ` R ) G ) = ( F ( quot1p ` R ) G ) |
16 |
9 2 3 5 11 10 4
|
q1peqb |
|- ( ( R e. Ring /\ F e. B /\ G e. C ) -> ( ( ( F ( quot1p ` R ) G ) e. B /\ ( D ` ( F ( -g ` P ) ( ( F ( quot1p ` R ) G ) ( .r ` P ) G ) ) ) < ( D ` G ) ) <-> ( F ( quot1p ` R ) G ) = ( F ( quot1p ` R ) G ) ) ) |
17 |
15 16
|
mpbiri |
|- ( ( R e. Ring /\ F e. B /\ G e. C ) -> ( ( F ( quot1p ` R ) G ) e. B /\ ( D ` ( F ( -g ` P ) ( ( F ( quot1p ` R ) G ) ( .r ` P ) G ) ) ) < ( D ` G ) ) ) |
18 |
17
|
simprd |
|- ( ( R e. Ring /\ F e. B /\ G e. C ) -> ( D ` ( F ( -g ` P ) ( ( F ( quot1p ` R ) G ) ( .r ` P ) G ) ) ) < ( D ` G ) ) |
19 |
14 18
|
eqbrtrd |
|- ( ( R e. Ring /\ F e. B /\ G e. C ) -> ( D ` ( F E G ) ) < ( D ` G ) ) |