Step |
Hyp |
Ref |
Expression |
1 |
|
ascldimul.a |
|- A = ( algSc ` W ) |
2 |
|
ascldimul.f |
|- F = ( Scalar ` W ) |
3 |
|
ascldimul.k |
|- K = ( Base ` F ) |
4 |
|
ascldimul.t |
|- .X. = ( .r ` W ) |
5 |
|
ascldimul.s |
|- .x. = ( .r ` F ) |
6 |
|
assaring |
|- ( W e. AssAlg -> W e. Ring ) |
7 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
8 |
|
eqid |
|- ( 1r ` W ) = ( 1r ` W ) |
9 |
7 8
|
ringidcl |
|- ( W e. Ring -> ( 1r ` W ) e. ( Base ` W ) ) |
10 |
6 9
|
syl |
|- ( W e. AssAlg -> ( 1r ` W ) e. ( Base ` W ) ) |
11 |
7 4 8
|
ringlidm |
|- ( ( W e. Ring /\ ( 1r ` W ) e. ( Base ` W ) ) -> ( ( 1r ` W ) .X. ( 1r ` W ) ) = ( 1r ` W ) ) |
12 |
6 10 11
|
syl2anc |
|- ( W e. AssAlg -> ( ( 1r ` W ) .X. ( 1r ` W ) ) = ( 1r ` W ) ) |
13 |
12
|
oveq2d |
|- ( W e. AssAlg -> ( S ( .s ` W ) ( ( 1r ` W ) .X. ( 1r ` W ) ) ) = ( S ( .s ` W ) ( 1r ` W ) ) ) |
14 |
13
|
oveq2d |
|- ( W e. AssAlg -> ( R ( .s ` W ) ( S ( .s ` W ) ( ( 1r ` W ) .X. ( 1r ` W ) ) ) ) = ( R ( .s ` W ) ( S ( .s ` W ) ( 1r ` W ) ) ) ) |
15 |
14
|
3ad2ant1 |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( R ( .s ` W ) ( S ( .s ` W ) ( ( 1r ` W ) .X. ( 1r ` W ) ) ) ) = ( R ( .s ` W ) ( S ( .s ` W ) ( 1r ` W ) ) ) ) |
16 |
|
simp1 |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> W e. AssAlg ) |
17 |
|
simp2 |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> R e. K ) |
18 |
10
|
3ad2ant1 |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( 1r ` W ) e. ( Base ` W ) ) |
19 |
|
assalmod |
|- ( W e. AssAlg -> W e. LMod ) |
20 |
19
|
3ad2ant1 |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> W e. LMod ) |
21 |
|
simp3 |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> S e. K ) |
22 |
|
eqid |
|- ( .s ` W ) = ( .s ` W ) |
23 |
7 2 22 3
|
lmodvscl |
|- ( ( W e. LMod /\ S e. K /\ ( 1r ` W ) e. ( Base ` W ) ) -> ( S ( .s ` W ) ( 1r ` W ) ) e. ( Base ` W ) ) |
24 |
20 21 18 23
|
syl3anc |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( S ( .s ` W ) ( 1r ` W ) ) e. ( Base ` W ) ) |
25 |
7 2 3 22 4
|
assaass |
|- ( ( W e. AssAlg /\ ( R e. K /\ ( 1r ` W ) e. ( Base ` W ) /\ ( S ( .s ` W ) ( 1r ` W ) ) e. ( Base ` W ) ) ) -> ( ( R ( .s ` W ) ( 1r ` W ) ) .X. ( S ( .s ` W ) ( 1r ` W ) ) ) = ( R ( .s ` W ) ( ( 1r ` W ) .X. ( S ( .s ` W ) ( 1r ` W ) ) ) ) ) |
26 |
16 17 18 24 25
|
syl13anc |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( ( R ( .s ` W ) ( 1r ` W ) ) .X. ( S ( .s ` W ) ( 1r ` W ) ) ) = ( R ( .s ` W ) ( ( 1r ` W ) .X. ( S ( .s ` W ) ( 1r ` W ) ) ) ) ) |
27 |
7 2 3 22 4
|
assaassr |
|- ( ( W e. AssAlg /\ ( S e. K /\ ( 1r ` W ) e. ( Base ` W ) /\ ( 1r ` W ) e. ( Base ` W ) ) ) -> ( ( 1r ` W ) .X. ( S ( .s ` W ) ( 1r ` W ) ) ) = ( S ( .s ` W ) ( ( 1r ` W ) .X. ( 1r ` W ) ) ) ) |
28 |
16 21 18 18 27
|
syl13anc |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( ( 1r ` W ) .X. ( S ( .s ` W ) ( 1r ` W ) ) ) = ( S ( .s ` W ) ( ( 1r ` W ) .X. ( 1r ` W ) ) ) ) |
29 |
28
|
oveq2d |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( R ( .s ` W ) ( ( 1r ` W ) .X. ( S ( .s ` W ) ( 1r ` W ) ) ) ) = ( R ( .s ` W ) ( S ( .s ` W ) ( ( 1r ` W ) .X. ( 1r ` W ) ) ) ) ) |
30 |
26 29
|
eqtrd |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( ( R ( .s ` W ) ( 1r ` W ) ) .X. ( S ( .s ` W ) ( 1r ` W ) ) ) = ( R ( .s ` W ) ( S ( .s ` W ) ( ( 1r ` W ) .X. ( 1r ` W ) ) ) ) ) |
31 |
7 2 22 3 5
|
lmodvsass |
|- ( ( W e. LMod /\ ( R e. K /\ S e. K /\ ( 1r ` W ) e. ( Base ` W ) ) ) -> ( ( R .x. S ) ( .s ` W ) ( 1r ` W ) ) = ( R ( .s ` W ) ( S ( .s ` W ) ( 1r ` W ) ) ) ) |
32 |
20 17 21 18 31
|
syl13anc |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( ( R .x. S ) ( .s ` W ) ( 1r ` W ) ) = ( R ( .s ` W ) ( S ( .s ` W ) ( 1r ` W ) ) ) ) |
33 |
15 30 32
|
3eqtr4rd |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( ( R .x. S ) ( .s ` W ) ( 1r ` W ) ) = ( ( R ( .s ` W ) ( 1r ` W ) ) .X. ( S ( .s ` W ) ( 1r ` W ) ) ) ) |
34 |
2
|
assasca |
|- ( W e. AssAlg -> F e. CRing ) |
35 |
|
crngring |
|- ( F e. CRing -> F e. Ring ) |
36 |
34 35
|
syl |
|- ( W e. AssAlg -> F e. Ring ) |
37 |
3 5
|
ringcl |
|- ( ( F e. Ring /\ R e. K /\ S e. K ) -> ( R .x. S ) e. K ) |
38 |
36 37
|
syl3an1 |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( R .x. S ) e. K ) |
39 |
1 2 3 22 8
|
asclval |
|- ( ( R .x. S ) e. K -> ( A ` ( R .x. S ) ) = ( ( R .x. S ) ( .s ` W ) ( 1r ` W ) ) ) |
40 |
38 39
|
syl |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( A ` ( R .x. S ) ) = ( ( R .x. S ) ( .s ` W ) ( 1r ` W ) ) ) |
41 |
1 2 3 22 8
|
asclval |
|- ( R e. K -> ( A ` R ) = ( R ( .s ` W ) ( 1r ` W ) ) ) |
42 |
17 41
|
syl |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( A ` R ) = ( R ( .s ` W ) ( 1r ` W ) ) ) |
43 |
1 2 3 22 8
|
asclval |
|- ( S e. K -> ( A ` S ) = ( S ( .s ` W ) ( 1r ` W ) ) ) |
44 |
21 43
|
syl |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( A ` S ) = ( S ( .s ` W ) ( 1r ` W ) ) ) |
45 |
42 44
|
oveq12d |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( ( A ` R ) .X. ( A ` S ) ) = ( ( R ( .s ` W ) ( 1r ` W ) ) .X. ( S ( .s ` W ) ( 1r ` W ) ) ) ) |
46 |
33 40 45
|
3eqtr4d |
|- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( A ` ( R .x. S ) ) = ( ( A ` R ) .X. ( A ` S ) ) ) |