Step |
Hyp |
Ref |
Expression |
1 |
|
ply1scl.p |
|- P = ( Poly1 ` R ) |
2 |
|
ply1scl.a |
|- A = ( algSc ` P ) |
3 |
|
ply1scl1.o |
|- .1. = ( 1r ` R ) |
4 |
|
ply1scl1.n |
|- N = ( 1r ` P ) |
5 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
6 |
5 3
|
ringidcl |
|- ( R e. Ring -> .1. e. ( Base ` R ) ) |
7 |
1
|
ply1sca2 |
|- ( _I ` R ) = ( Scalar ` P ) |
8 |
|
baseid |
|- Base = Slot ( Base ` ndx ) |
9 |
8 5
|
strfvi |
|- ( Base ` R ) = ( Base ` ( _I ` R ) ) |
10 |
|
eqid |
|- ( .s ` P ) = ( .s ` P ) |
11 |
2 7 9 10 4
|
asclval |
|- ( .1. e. ( Base ` R ) -> ( A ` .1. ) = ( .1. ( .s ` P ) N ) ) |
12 |
6 11
|
syl |
|- ( R e. Ring -> ( A ` .1. ) = ( .1. ( .s ` P ) N ) ) |
13 |
|
fvi |
|- ( R e. Ring -> ( _I ` R ) = R ) |
14 |
13
|
fveq2d |
|- ( R e. Ring -> ( 1r ` ( _I ` R ) ) = ( 1r ` R ) ) |
15 |
14 3
|
eqtr4di |
|- ( R e. Ring -> ( 1r ` ( _I ` R ) ) = .1. ) |
16 |
15
|
oveq1d |
|- ( R e. Ring -> ( ( 1r ` ( _I ` R ) ) ( .s ` P ) N ) = ( .1. ( .s ` P ) N ) ) |
17 |
1
|
ply1lmod |
|- ( R e. Ring -> P e. LMod ) |
18 |
1
|
ply1ring |
|- ( R e. Ring -> P e. Ring ) |
19 |
|
eqid |
|- ( Base ` P ) = ( Base ` P ) |
20 |
19 4
|
ringidcl |
|- ( P e. Ring -> N e. ( Base ` P ) ) |
21 |
18 20
|
syl |
|- ( R e. Ring -> N e. ( Base ` P ) ) |
22 |
|
eqid |
|- ( 1r ` ( _I ` R ) ) = ( 1r ` ( _I ` R ) ) |
23 |
19 7 10 22
|
lmodvs1 |
|- ( ( P e. LMod /\ N e. ( Base ` P ) ) -> ( ( 1r ` ( _I ` R ) ) ( .s ` P ) N ) = N ) |
24 |
17 21 23
|
syl2anc |
|- ( R e. Ring -> ( ( 1r ` ( _I ` R ) ) ( .s ` P ) N ) = N ) |
25 |
12 16 24
|
3eqtr2d |
|- ( R e. Ring -> ( A ` .1. ) = N ) |