Metamath Proof Explorer


Theorem ply1scl1

Description: The one scalar is the unit polynomial. (Contributed by Stefan O'Rear, 1-Apr-2015)

Ref Expression
Hypotheses ply1scl.p
|- P = ( Poly1 ` R )
ply1scl.a
|- A = ( algSc ` P )
ply1scl1.o
|- .1. = ( 1r ` R )
ply1scl1.n
|- N = ( 1r ` P )
Assertion ply1scl1
|- ( R e. Ring -> ( A ` .1. ) = N )

Proof

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 df-base
 |-  Base = Slot 1
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 )