Step |
Hyp |
Ref |
Expression |
1 |
|
slmdvsass.v |
|- V = ( Base ` W ) |
2 |
|
slmdvsass.f |
|- F = ( Scalar ` W ) |
3 |
|
slmdvsass.s |
|- .x. = ( .s ` W ) |
4 |
|
slmdvsass.k |
|- K = ( Base ` F ) |
5 |
|
slmdvsass.t |
|- .X. = ( .r ` F ) |
6 |
|
eqid |
|- ( +g ` W ) = ( +g ` W ) |
7 |
|
eqid |
|- ( 0g ` W ) = ( 0g ` W ) |
8 |
|
eqid |
|- ( +g ` F ) = ( +g ` F ) |
9 |
|
eqid |
|- ( 1r ` F ) = ( 1r ` F ) |
10 |
|
eqid |
|- ( 0g ` F ) = ( 0g ` F ) |
11 |
1 6 3 7 2 4 8 5 9 10
|
slmdlema |
|- ( ( W e. SLMod /\ ( Q e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( ( ( R .x. X ) e. V /\ ( R .x. ( X ( +g ` W ) X ) ) = ( ( R .x. X ) ( +g ` W ) ( R .x. X ) ) /\ ( ( Q ( +g ` F ) R ) .x. X ) = ( ( Q .x. X ) ( +g ` W ) ( R .x. X ) ) ) /\ ( ( ( Q .X. R ) .x. X ) = ( Q .x. ( R .x. X ) ) /\ ( ( 1r ` F ) .x. X ) = X /\ ( ( 0g ` F ) .x. X ) = ( 0g ` W ) ) ) ) |
12 |
11
|
simprd |
|- ( ( W e. SLMod /\ ( Q e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( ( ( Q .X. R ) .x. X ) = ( Q .x. ( R .x. X ) ) /\ ( ( 1r ` F ) .x. X ) = X /\ ( ( 0g ` F ) .x. X ) = ( 0g ` W ) ) ) |
13 |
12
|
simp1d |
|- ( ( W e. SLMod /\ ( Q e. K /\ R e. K ) /\ ( X e. V /\ X e. V ) ) -> ( ( Q .X. R ) .x. X ) = ( Q .x. ( R .x. X ) ) ) |
14 |
13
|
3expa |
|- ( ( ( W e. SLMod /\ ( Q e. K /\ R e. K ) ) /\ ( X e. V /\ X e. V ) ) -> ( ( Q .X. R ) .x. X ) = ( Q .x. ( R .x. X ) ) ) |
15 |
14
|
anabsan2 |
|- ( ( ( W e. SLMod /\ ( Q e. K /\ R e. K ) ) /\ X e. V ) -> ( ( Q .X. R ) .x. X ) = ( Q .x. ( R .x. X ) ) ) |
16 |
15
|
exp42 |
|- ( W e. SLMod -> ( Q e. K -> ( R e. K -> ( X e. V -> ( ( Q .X. R ) .x. X ) = ( Q .x. ( R .x. X ) ) ) ) ) ) |
17 |
16
|
3imp2 |
|- ( ( W e. SLMod /\ ( Q e. K /\ R e. K /\ X e. V ) ) -> ( ( Q .X. R ) .x. X ) = ( Q .x. ( R .x. X ) ) ) |