Step |
Hyp |
Ref |
Expression |
1 |
|
isassad.v |
|- ( ph -> V = ( Base ` W ) ) |
2 |
|
isassad.f |
|- ( ph -> F = ( Scalar ` W ) ) |
3 |
|
isassad.b |
|- ( ph -> B = ( Base ` F ) ) |
4 |
|
isassad.s |
|- ( ph -> .x. = ( .s ` W ) ) |
5 |
|
isassad.t |
|- ( ph -> .X. = ( .r ` W ) ) |
6 |
|
isassad.1 |
|- ( ph -> W e. LMod ) |
7 |
|
isassad.2 |
|- ( ph -> W e. Ring ) |
8 |
|
isassad.3 |
|- ( ph -> F e. CRing ) |
9 |
|
isassad.4 |
|- ( ( ph /\ ( r e. B /\ x e. V /\ y e. V ) ) -> ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) ) |
10 |
|
isassad.5 |
|- ( ( ph /\ ( r e. B /\ x e. V /\ y e. V ) ) -> ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) ) |
11 |
2 8
|
eqeltrrd |
|- ( ph -> ( Scalar ` W ) e. CRing ) |
12 |
6 7 11
|
3jca |
|- ( ph -> ( W e. LMod /\ W e. Ring /\ ( Scalar ` W ) e. CRing ) ) |
13 |
9 10
|
jca |
|- ( ( ph /\ ( r e. B /\ x e. V /\ y e. V ) ) -> ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) /\ ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) ) ) |
14 |
13
|
ralrimivvva |
|- ( ph -> A. r e. B A. x e. V A. y e. V ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) /\ ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) ) ) |
15 |
2
|
fveq2d |
|- ( ph -> ( Base ` F ) = ( Base ` ( Scalar ` W ) ) ) |
16 |
3 15
|
eqtrd |
|- ( ph -> B = ( Base ` ( Scalar ` W ) ) ) |
17 |
4
|
oveqd |
|- ( ph -> ( r .x. x ) = ( r ( .s ` W ) x ) ) |
18 |
|
eqidd |
|- ( ph -> y = y ) |
19 |
5 17 18
|
oveq123d |
|- ( ph -> ( ( r .x. x ) .X. y ) = ( ( r ( .s ` W ) x ) ( .r ` W ) y ) ) |
20 |
|
eqidd |
|- ( ph -> r = r ) |
21 |
5
|
oveqd |
|- ( ph -> ( x .X. y ) = ( x ( .r ` W ) y ) ) |
22 |
4 20 21
|
oveq123d |
|- ( ph -> ( r .x. ( x .X. y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) |
23 |
19 22
|
eqeq12d |
|- ( ph -> ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) <-> ( ( r ( .s ` W ) x ) ( .r ` W ) y ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) ) |
24 |
|
eqidd |
|- ( ph -> x = x ) |
25 |
4
|
oveqd |
|- ( ph -> ( r .x. y ) = ( r ( .s ` W ) y ) ) |
26 |
5 24 25
|
oveq123d |
|- ( ph -> ( x .X. ( r .x. y ) ) = ( x ( .r ` W ) ( r ( .s ` W ) y ) ) ) |
27 |
26 22
|
eqeq12d |
|- ( ph -> ( ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) <-> ( x ( .r ` W ) ( r ( .s ` W ) y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) ) |
28 |
23 27
|
anbi12d |
|- ( ph -> ( ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) /\ ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) ) <-> ( ( ( r ( .s ` W ) x ) ( .r ` W ) y ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) /\ ( x ( .r ` W ) ( r ( .s ` W ) y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) ) ) |
29 |
1 28
|
raleqbidv |
|- ( ph -> ( A. y e. V ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) /\ ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) ) <-> A. y e. ( Base ` W ) ( ( ( r ( .s ` W ) x ) ( .r ` W ) y ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) /\ ( x ( .r ` W ) ( r ( .s ` W ) y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) ) ) |
30 |
1 29
|
raleqbidv |
|- ( ph -> ( A. x e. V A. y e. V ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) /\ ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) ) <-> A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( ( r ( .s ` W ) x ) ( .r ` W ) y ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) /\ ( x ( .r ` W ) ( r ( .s ` W ) y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) ) ) |
31 |
16 30
|
raleqbidv |
|- ( ph -> ( A. r e. B A. x e. V A. y e. V ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) /\ ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) ) <-> A. r e. ( Base ` ( Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( ( r ( .s ` W ) x ) ( .r ` W ) y ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) /\ ( x ( .r ` W ) ( r ( .s ` W ) y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) ) ) |
32 |
14 31
|
mpbid |
|- ( ph -> A. r e. ( Base ` ( Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( ( r ( .s ` W ) x ) ( .r ` W ) y ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) /\ ( x ( .r ` W ) ( r ( .s ` W ) y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) ) |
33 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
34 |
|
eqid |
|- ( Scalar ` W ) = ( Scalar ` W ) |
35 |
|
eqid |
|- ( Base ` ( Scalar ` W ) ) = ( Base ` ( Scalar ` W ) ) |
36 |
|
eqid |
|- ( .s ` W ) = ( .s ` W ) |
37 |
|
eqid |
|- ( .r ` W ) = ( .r ` W ) |
38 |
33 34 35 36 37
|
isassa |
|- ( W e. AssAlg <-> ( ( W e. LMod /\ W e. Ring /\ ( Scalar ` W ) e. CRing ) /\ A. r e. ( Base ` ( Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( ( r ( .s ` W ) x ) ( .r ` W ) y ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) /\ ( x ( .r ` W ) ( r ( .s ` W ) y ) ) = ( r ( .s ` W ) ( x ( .r ` W ) y ) ) ) ) ) |
39 |
12 32 38
|
sylanbrc |
|- ( ph -> W e. AssAlg ) |