Metamath Proof Explorer

Theorem asclmul1

Description: Left multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015)

Ref Expression
Hypotheses asclmul1.a 
|- A = ( algSc ` W )
asclmul1.f 
|- F = ( Scalar ` W )
asclmul1.k 
|- K = ( Base ` F )
asclmul1.v 
|- V = ( Base ` W )
asclmul1.t 
|- .X. = ( .r ` W )
asclmul1.s 
|- .x. = ( .s ` W )
Assertion asclmul1 
|- ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> ( ( A ` R ) .X. X ) = ( R .x. X ) )

Proof

Step Hyp Ref Expression
1 asclmul1.a 
 |-  A = ( algSc ` W )
2 asclmul1.f 
 |-  F = ( Scalar ` W )
3 asclmul1.k 
 |-  K = ( Base ` F )
4 asclmul1.v 
 |-  V = ( Base ` W )
5 asclmul1.t 
 |-  .X. = ( .r ` W )
6 asclmul1.s 
 |-  .x. = ( .s ` W )
7 eqid 
 |-  ( 1r ` W ) = ( 1r ` W )
8 1 2 3 6 7 asclval 
 |-  ( R e. K -> ( A ` R ) = ( R .x. ( 1r ` W ) ) )
9 8 3ad2ant2 
 |-  ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> ( A ` R ) = ( R .x. ( 1r ` W ) ) )
10 9 oveq1d 
 |-  ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> ( ( A ` R ) .X. X ) = ( ( R .x. ( 1r ` W ) ) .X. X ) )
11 simp1 
 |-  ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> W e. AssAlg )
12 simp2 
 |-  ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> R e. K )
13 assaring 
 |-  ( W e. AssAlg -> W e. Ring )
14 13 3ad2ant1 
 |-  ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> W e. Ring )
15 4 7 ringidcl 
 |-  ( W e. Ring -> ( 1r ` W ) e. V )
16 14 15 syl 
 |-  ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> ( 1r ` W ) e. V )
17 simp3 
 |-  ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> X e. V )
18 4 2 3 6 5 assaass 
 |-  ( ( W e. AssAlg /\ ( R e. K /\ ( 1r ` W ) e. V /\ X e. V ) ) -> ( ( R .x. ( 1r ` W ) ) .X. X ) = ( R .x. ( ( 1r ` W ) .X. X ) ) )
19 11 12 16 17 18 syl13anc 
 |-  ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> ( ( R .x. ( 1r ` W ) ) .X. X ) = ( R .x. ( ( 1r ` W ) .X. X ) ) )
20 4 5 7 ringlidm 
 |-  ( ( W e. Ring /\ X e. V ) -> ( ( 1r ` W ) .X. X ) = X )
21 14 17 20 syl2anc 
 |-  ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> ( ( 1r ` W ) .X. X ) = X )
22 21 oveq2d 
 |-  ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> ( R .x. ( ( 1r ` W ) .X. X ) ) = ( R .x. X ) )
23 10 19 22 3eqtrd 
 |-  ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> ( ( A ` R ) .X. X ) = ( R .x. X ) )